TPTP Problem File: SLH0424^1.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    : Safe_Range_RC/0024_Restrict_Frees/prob_00289_012251__17908010_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1830 ( 513 unt; 557 typ;   0 def)
%            Number of atoms       : 5190 (2250 equ;   0 cnn)
%            Maximal formula atoms :   23 (   4 avg)
%            Number of connectives : 20459 (1062   ~; 101   |; 673   &;15960   @)
%                                         (   0 <=>;2663  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   9 avg)
%            Number of types       :   90 (  89 usr)
%            Number of type conns  : 2194 (2194   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  471 ( 468 usr;  28 con; 0-4 aty)
%            Number of variables   : 5350 ( 434   ^;4581   !; 335   ?;5350   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:27:48.386
%------------------------------------------------------------------------------
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thf(ty_n_t__HashMap__Ohashmap_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Ounit_J,type,
    hashma7010742005768827389t_unit: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_Pr5488025237498180813et_nat: $tType ).

thf(ty_n_t__RBT__Orbt_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Ounit_J,type,
    rbt_Pr1701002678351477513t_unit: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc8373899037510109440at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc7248412053542808358at_nat: $tType ).

thf(ty_n_t__List__Olist_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    list_P9162950289778280392at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_Pr4048851178543822343list_a: $tType ).

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

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    list_l3264859301627795341at_nat: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_li5450038453877631591at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc7819656566062154093et_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    produc9164743771328383783list_a: $tType ).

thf(ty_n_t__Dlist__Odlist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    dlist_2221377526271503905at_nat: $tType ).

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

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

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

thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc7491599851749785783at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    produc2400336064389900727et_nat: $tType ).

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

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

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

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

thf(ty_n_t__Relational____Calculus__Ofmla_Itf__a_Mtf__b_J,type,
    relational_fmla_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
    product_prod_b_nat: $tType ).

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

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

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

thf(ty_n_t__Relational____Calculus__Oterm_Itf__a_J,type,
    relational_term_a: $tType ).

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

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

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

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

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

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

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

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

% Explicit typings (468)
thf(sy_c_AList_Odelete__aux_001t__Nat__Onat_001t__Nat__Onat,type,
    delete_aux_nat_nat: nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_AList_Odelete__aux_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    delete5918753289527851494at_nat: product_prod_nat_nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_AList_Odelete__aux__rel_001t__Nat__Onat_001t__Nat__Onat,type,
    delete1397635992828230261at_nat: produc8472197452120411308at_nat > produc8472197452120411308at_nat > $o ).

thf(sy_c_AList_Odelete__aux__rel_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    delete4570897403666713875at_nat: produc5065989759464263123at_nat > produc5065989759464263123at_nat > $o ).

thf(sy_c_AList_Omap__default_001t__Nat__Onat_001t__Nat__Onat,type,
    map_default_nat_nat: nat > nat > ( nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_AList_Omap__default_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    map_de6203744875203570968at_nat: product_prod_nat_nat > product_prod_nat_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_AList_Omap__ran_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    map_ran_nat_nat_nat: ( nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_AList_Omap__ran_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    map_ra4011674865287502026at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_AList_Oupdate__with__aux_001t__Nat__Onat_001t__Nat__Onat,type,
    update528237659335440164at_nat: nat > nat > ( nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_AList_Oupdate__with__aux_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    update8621587714000398146at_nat: product_prod_nat_nat > product_prod_nat_nat > ( product_prod_nat_nat > product_prod_nat_nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

thf(sy_c_Autoref__Bindings__HOL_Oop__list__append__elem_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    autore7828122933726872089at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

thf(sy_c_Autoref__Bindings__HOL_Oop__list__singleton_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    autore447854781811194899at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat ).

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

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr3130287167067265568at_nat: set_li5450038453877631591at_nat > product_prod_nat_nat > set_li5450038453877631591at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_Gr9155129491085760262at_nat: set_li3197816953174176717at_nat > produc859450856879609959at_nat > set_li3197816953174176717at_nat ).

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

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

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

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bNF_Gr5363859321595349404at_nat: set_li5450038453877631591at_nat > list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bNF_Gr8603829661560577154at_nat: set_li3197816953174176717at_nat > list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).

thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
    finite_card_list_a: set_list_a > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite711546835091564841at_nat: set_Pr1261947904930325089at_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite1207074278014112911at_nat: set_Pr8693737435421807431at_nat > nat ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    finite1660835950917165235list_a: set_list_list_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
    finite8100373058378681591st_nat: set_list_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o ).

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

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    finite4392333629123659920at_nat: set_Pr8693737435421807431at_nat > $o ).

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

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
    finite1152437895449049373et_nat: set_set_nat > $o ).

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

thf(sy_c_Finite__Set_Ofolding__on_OF_001t__Nat__Onat_001t__Nat__Onat,type,
    finite3337649020256630879at_nat: ( nat > nat > nat ) > nat > set_nat > nat ).

thf(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set_Pr8693737435421807431at_nat ).

thf(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set_Pr8693737435421807431at_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

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__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    if_lis9186351972506106189at_nat: $o > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    if_lis7763640049307703347at_nat: $o > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

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

thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    if_Pro969932733043319213at_nat: $o > produc6392793444374437607at_nat > produc6392793444374437607at_nat > produc6392793444374437607at_nat ).

thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    if_Pro5163289783556962317at_nat: $o > produc5956105832733135175at_nat > produc5956105832733135175at_nat > produc5956105832733135175at_nat ).

thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
    if_set_nat: $o > set_nat > set_nat > set_nat ).

thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Nat__Onat,type,
    infini8530281810654367211te_nat: set_nat > nat > nat ).

thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    infini875302759086991654at_nat: set_Pr1261947904930325089at_nat > nat > product_prod_nat_nat ).

thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    infini1315406684128855308at_nat: set_Pr8693737435421807431at_nat > nat > produc859450856879609959at_nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
    lattic5043722365632780795_a_nat: ( list_a > nat ) > set_list_a > list_a ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
    lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    lattic4984276347100956536at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > product_prod_nat_nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    lattic390166758595302878at_nat: ( produc859450856879609959at_nat > nat ) > set_Pr8693737435421807431at_nat > produc859450856879609959at_nat ).

thf(sy_c_List_OBleast_001t__Nat__Onat,type,
    bleast_nat: set_nat > ( nat > $o ) > nat ).

thf(sy_c_List_OBleast_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    bleast198245214120542575at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).

thf(sy_c_List_OBleast_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bleast3078576362778749909at_nat: set_Pr8693737435421807431at_nat > ( produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat ).

thf(sy_c_List_Oabort__Bleast_001t__Nat__Onat,type,
    abort_Bleast_nat: set_nat > ( nat > $o ) > nat ).

thf(sy_c_List_Oabort__Bleast_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    abort_5709735807243680467at_nat: set_Pr1261947904930325089at_nat > ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).

thf(sy_c_List_Oabort__Bleast_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    abort_8368135353768099641at_nat: set_Pr8693737435421807431at_nat > ( produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat ).

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

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

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    append8751754712269456642at_nat: list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat ).

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

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thf(sy_c_Misc_Omergesort__remdups_001t__Nat__Onat,type,
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thf(sy_c_Misc_Oquicksort__by__rel_001t__Nat__Onat,type,
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    same_fst_nat_nat: ( nat > $o ) > ( nat > set_Pr1261947904930325089at_nat ) > set_Pr8693737435421807431at_nat ).

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    same_f4956014544515070124at_nat: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > set_Pr8693737435421807431at_nat ) > set_Pr553994874890374343at_nat ).

thf(sy_c_fChoice_001t__Nat__Onat,type,
    fChoice_nat: ( nat > $o ) > nat ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    fChoic6978938873391328853at_nat: ( product_prod_nat_nat > $o ) > product_prod_nat_nat ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    fChoic6673916560228149179at_nat: ( produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat ).

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

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

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member3067507820990806192at_nat: list_P6011104703257516679at_nat > set_li5450038453877631591at_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
    member_list_a: list_a > set_list_a > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member5927835510201945878at_nat: produc4961808859980711423at_nat > set_Pr3493580773761353909at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member7775084264787586014at_nat: produc4471711990508489141at_nat > set_Pr4668228411283691157at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member139686735972302228at_nat: produc2687737633280426365at_nat > set_Pr3272603486844509235at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
    member7866418522119297528at_nat: produc2780847497663810273at_nat > set_Pr7760760537764498071at_nat > $o ).

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    member412078224538685225at_nat: produc2366258654402830848at_nat > set_Pr1709944248907917408at_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6693912407220327184at_nat: produc6392793444374437607at_nat > set_Pr1542805901266377927at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4574794575480667280at_nat: produc289266582803401575at_nat > set_Pr4087777274317423175at_nat > $o ).

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member3048279782668807382at_nat: produc6277219514840344877at_nat > set_Pr575275573428919693at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    member1995966531042493578at_nat: produc5224906263214031073at_nat > set_Pr7116486347545156417at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8062223511168850704at_nat: produc349518998152878311at_nat > set_Pr553994874890374343at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member2637305593812847696list_a: produc3865747550613471015list_a > set_Pr5391735021705396743list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member5855424355840516880at_nat: produc6487378988399798503at_nat > set_Pr5297940549829899463at_nat > $o ).

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

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

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

thf(sy_v_I,type,
    i: product_prod_b_nat > set_list_a ).

thf(sy_v_Qa____,type,
    qa: relational_fmla_a_b ).

thf(sy_v_x____,type,
    x: nat ).

thf(sy_v_xysa____,type,
    xysa: list_P6011104703257516679at_nat ).

thf(sy_v_y____,type,
    y: nat ).

% Relevant facts (1254)
thf(fact_0__C2_Oprems_C_I1_J,axiom,
    ~ ( finite_finite_list_a @ ( relational_eval_a_b @ qa @ i ) ) ).

% "2.prems"(1)
thf(fact_1__C2_Oprems_C_I2_J,axiom,
    restri7641029068292571945sh_a_b @ qa @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ x @ y ) @ xysa ) ).

% "2.prems"(2)
thf(fact_2__C2_Ohyps_C,axiom,
    ( ~ ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Conj_a_b @ qa @ ( relational_Eq_a_b @ x @ ( relational_Var_a @ y ) ) ) @ i ) )
   => ( ( restri7641029068292571945sh_a_b @ ( relational_Conj_a_b @ qa @ ( relational_Eq_a_b @ x @ ( relational_Var_a @ y ) ) ) @ xysa )
     => ~ ( finite_finite_list_a @ ( relational_eval_a_b @ ( restrict_Conjs_a_b @ ( relational_Conj_a_b @ qa @ ( relational_Eq_a_b @ x @ ( relational_Var_a @ y ) ) ) @ xysa ) @ i ) ) ) ) ).

% "2.hyps"
thf(fact_3_finite__Collect__conjI,axiom,
    ! [P: list_P6011104703257516679at_nat > $o,Q: list_P6011104703257516679at_nat > $o] :
      ( ( ( finite500796754983035824at_nat @ ( collec3343600615725829874at_nat @ P ) )
        | ( finite500796754983035824at_nat @ ( collec3343600615725829874at_nat @ Q ) ) )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [X: list_P6011104703257516679at_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_4_finite__Collect__conjI,axiom,
    ! [P: list_list_a > $o,Q: list_list_a > $o] :
      ( ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P ) )
        | ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q ) ) )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [X: list_list_a] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_5_finite__Collect__conjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6_finite__Collect__conjI,axiom,
    ! [P: set_list_a > $o,Q: set_list_a > $o] :
      ( ( ( finite5282473924520328461list_a @ ( collect_set_list_a @ P ) )
        | ( finite5282473924520328461list_a @ ( collect_set_list_a @ Q ) ) )
     => ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [X: set_list_a] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_7_finite__Collect__conjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_8_finite__Collect__conjI,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( ( finite_finite_a @ ( collect_a @ P ) )
        | ( finite_finite_a @ ( collect_a @ Q ) ) )
     => ( finite_finite_a
        @ ( collect_a
          @ ^ [X: a] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_9_finite__Collect__conjI,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
        | ( finite_finite_list_a @ ( collect_list_a @ Q ) ) )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X: list_a] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_10_finite__Collect__conjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
        | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_11_finite__Collect__disjI,axiom,
    ! [P: list_P6011104703257516679at_nat > $o,Q: list_P6011104703257516679at_nat > $o] :
      ( ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [X: list_P6011104703257516679at_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite500796754983035824at_nat @ ( collec3343600615725829874at_nat @ P ) )
        & ( finite500796754983035824at_nat @ ( collec3343600615725829874at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_12_finite__Collect__disjI,axiom,
    ! [P: list_list_a > $o,Q: list_list_a > $o] :
      ( ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [X: list_list_a] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P ) )
        & ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_13_finite__Collect__disjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_14_finite__Collect__disjI,axiom,
    ! [P: set_list_a > $o,Q: set_list_a > $o] :
      ( ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [X: set_list_a] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite5282473924520328461list_a @ ( collect_set_list_a @ P ) )
        & ( finite5282473924520328461list_a @ ( collect_set_list_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_15_finite__Collect__disjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_16_finite__Collect__disjI,axiom,
    ! [P: a > $o,Q: a > $o] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [X: a] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_a @ ( collect_a @ P ) )
        & ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_17_finite__Collect__disjI,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X: list_a] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_list_a @ ( collect_list_a @ P ) )
        & ( finite_finite_list_a @ ( collect_list_a @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_18_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_19_list_Oinject,axiom,
    ! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
      ( ( ( cons_list_a @ X21 @ X22 )
        = ( cons_list_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_20_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X22 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_21_list_Oinject,axiom,
    ! [X21: product_prod_nat_nat > nat,X22: list_P9162950289778280392at_nat,Y21: product_prod_nat_nat > nat,Y22: list_P9162950289778280392at_nat] :
      ( ( ( cons_P4861729644591583992at_nat @ X21 @ X22 )
        = ( cons_P4861729644591583992at_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_22_list_Oinject,axiom,
    ! [X21: nat > nat,X22: list_nat_nat,Y21: nat > nat,Y22: list_nat_nat] :
      ( ( ( cons_nat_nat @ X21 @ X22 )
        = ( cons_nat_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_23_list_Oinject,axiom,
    ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
      ( ( ( cons_a @ X21 @ X22 )
        = ( cons_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_24_list_Oinject,axiom,
    ! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,Y21: product_prod_nat_nat,Y22: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ X21 @ X22 )
        = ( cons_P6512896166579812791at_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_25_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat > product_prod_nat_nat > $o,X2: list_P6011104703257516679at_nat,Y1: product_prod_nat_nat > product_prod_nat_nat > $o,Y2: list_P6011104703257516679at_nat] :
      ( ( ( produc3352296309980913008at_nat @ X1 @ X2 )
        = ( produc3352296309980913008at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_26_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,X2: produc289266582803401575at_nat,Y1: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Y2: produc289266582803401575at_nat] :
      ( ( ( produc2486214636825388819at_nat @ X1 @ X2 )
        = ( produc2486214636825388819at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_27_prod_Oinject,axiom,
    ! [X1: nat > nat,X2: list_P6011104703257516679at_nat,Y1: nat > nat,Y2: list_P6011104703257516679at_nat] :
      ( ( ( produc1236331799044183215at_nat @ X1 @ X2 )
        = ( produc1236331799044183215at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_28_prod_Oinject,axiom,
    ! [X1: nat > nat > nat,X2: produc7248412053542808358at_nat,Y1: nat > nat > nat,Y2: produc7248412053542808358at_nat] :
      ( ( ( produc3209952032786966637at_nat @ X1 @ X2 )
        = ( produc3209952032786966637at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_29_prod_Oinject,axiom,
    ! [X1: nat > nat > nat > nat,X2: produc6392793444374437607at_nat,Y1: nat > nat > nat > nat,Y2: produc6392793444374437607at_nat] :
      ( ( ( produc8375354190041410609at_nat @ X1 @ X2 )
        = ( produc8375354190041410609at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_30_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_31_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X2 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_32_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat > product_prod_nat_nat > $o,B: list_P6011104703257516679at_nat,A2: product_prod_nat_nat > product_prod_nat_nat > $o,B2: list_P6011104703257516679at_nat] :
      ( ( ( produc3352296309980913008at_nat @ A @ B )
        = ( produc3352296309980913008at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_33_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,B: produc289266582803401575at_nat,A2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,B2: produc289266582803401575at_nat] :
      ( ( ( produc2486214636825388819at_nat @ A @ B )
        = ( produc2486214636825388819at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_34_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: list_P6011104703257516679at_nat,A2: nat > nat,B2: list_P6011104703257516679at_nat] :
      ( ( ( produc1236331799044183215at_nat @ A @ B )
        = ( produc1236331799044183215at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_35_old_Oprod_Oinject,axiom,
    ! [A: nat > nat > nat,B: produc7248412053542808358at_nat,A2: nat > nat > nat,B2: produc7248412053542808358at_nat] :
      ( ( ( produc3209952032786966637at_nat @ A @ B )
        = ( produc3209952032786966637at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_36_old_Oprod_Oinject,axiom,
    ! [A: nat > nat > nat > nat,B: produc6392793444374437607at_nat,A2: nat > nat > nat > nat,B2: produc6392793444374437607at_nat] :
      ( ( ( produc8375354190041410609at_nat @ A @ B )
        = ( produc8375354190041410609at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_37_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_38_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_39_Conjs_Osimps_I2_J,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat,Xys: list_P6011104703257516679at_nat] :
      ( ( restrict_Conjs_a_b @ Q @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ Xys ) )
      = ( restrict_Conjs_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) ) @ Xys ) ) ).

% Conjs.simps(2)
thf(fact_40_not__finite__existsD,axiom,
    ! [P: list_P6011104703257516679at_nat > $o] :
      ( ~ ( finite500796754983035824at_nat @ ( collec3343600615725829874at_nat @ P ) )
     => ? [X_1: list_P6011104703257516679at_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_41_not__finite__existsD,axiom,
    ! [P: list_list_a > $o] :
      ( ~ ( finite1660835950917165235list_a @ ( collect_list_list_a @ P ) )
     => ? [X_1: list_list_a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_42_not__finite__existsD,axiom,
    ! [P: list_nat > $o] :
      ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
     => ? [X_1: list_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_43_not__finite__existsD,axiom,
    ! [P: set_list_a > $o] :
      ( ~ ( finite5282473924520328461list_a @ ( collect_set_list_a @ P ) )
     => ? [X_1: set_list_a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_44_not__finite__existsD,axiom,
    ! [P: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
     => ? [X_1: set_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_45_not__finite__existsD,axiom,
    ! [P: a > $o] :
      ( ~ ( finite_finite_a @ ( collect_a @ P ) )
     => ? [X_1: a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_46_not__finite__existsD,axiom,
    ! [P: list_a > $o] :
      ( ~ ( finite_finite_list_a @ ( collect_list_a @ P ) )
     => ? [X_1: list_a] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_47_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_48_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A3 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_49_pigeonhole__infinite__rel,axiom,
    ! [A3: set_list_a,B3: set_nat,R: list_a > nat > $o] :
      ( ~ ( finite_finite_list_a @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X4: list_a] :
              ( ( member_list_a @ X4 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A4: list_a] :
                        ( ( member_list_a @ A4 @ A3 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_50_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_list_a,R: nat > list_a > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_list_a @ B3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A3 )
             => ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ B3 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: list_a] :
              ( ( member_list_a @ X4 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A3 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_51_pigeonhole__infinite__rel,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_nat,R: product_prod_nat_nat > nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
              & ~ ( finite6177210948735845034at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [A4: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ A4 @ A3 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_52_pigeonhole__infinite__rel,axiom,
    ! [A3: set_list_a,B3: set_list_a,R: list_a > list_a > $o] :
      ( ~ ( finite_finite_list_a @ A3 )
     => ( ( finite_finite_list_a @ B3 )
       => ( ! [X4: list_a] :
              ( ( member_list_a @ X4 @ A3 )
             => ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ B3 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: list_a] :
              ( ( member_list_a @ X4 @ B3 )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A4: list_a] :
                        ( ( member_list_a @ A4 @ A3 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_53_pigeonhole__infinite__rel,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_list_a,R: product_prod_nat_nat > list_a > $o] :
      ( ~ ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite_finite_list_a @ B3 )
       => ( ! [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A3 )
             => ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ B3 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: list_a] :
              ( ( member_list_a @ X4 @ B3 )
              & ~ ( finite6177210948735845034at_nat
                  @ ( collec3392354462482085612at_nat
                    @ ^ [A4: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ A4 @ A3 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_54_pigeonhole__infinite__rel,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_nat,R: produc859450856879609959at_nat > nat > $o] :
      ( ~ ( finite4392333629123659920at_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
              & ~ ( finite4392333629123659920at_nat
                  @ ( collec7088162979684241874at_nat
                    @ ^ [A4: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ A4 @ A3 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_55_pigeonhole__infinite__rel,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,B3: set_list_a,R: produc859450856879609959at_nat > list_a > $o] :
      ( ~ ( finite4392333629123659920at_nat @ A3 )
     => ( ( finite_finite_list_a @ B3 )
       => ( ! [X4: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X4 @ A3 )
             => ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ B3 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: list_a] :
              ( ( member_list_a @ X4 @ B3 )
              & ~ ( finite4392333629123659920at_nat
                  @ ( collec7088162979684241874at_nat
                    @ ^ [A4: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ A4 @ A3 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_56_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B3: set_a,R: nat > a > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_a @ B3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A3 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B3 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: a] :
              ( ( member_a @ X4 @ B3 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A3 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_57_pigeonhole__infinite__rel,axiom,
    ! [A3: set_a,B3: set_nat,R: a > nat > $o] :
      ( ~ ( finite_finite_a @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ! [X4: a] :
              ( ( member_a @ X4 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B3 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B3 )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A4: a] :
                        ( ( member_a @ A4 @ A3 )
                        & ( R @ A4 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_58_pred__equals__eq2,axiom,
    ! [R: set_Pr1709944248907917408at_nat,S: set_Pr1709944248907917408at_nat] :
      ( ( ( ^ [X: product_prod_nat_nat > product_prod_nat_nat > $o,Y3: list_P6011104703257516679at_nat] : ( member412078224538685225at_nat @ ( produc3352296309980913008at_nat @ X @ Y3 ) @ R ) )
        = ( ^ [X: product_prod_nat_nat > product_prod_nat_nat > $o,Y3: list_P6011104703257516679at_nat] : ( member412078224538685225at_nat @ ( produc3352296309980913008at_nat @ X @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_59_pred__equals__eq2,axiom,
    ! [R: set_Pr7760760537764498071at_nat,S: set_Pr7760760537764498071at_nat] :
      ( ( ( ^ [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Y3: produc289266582803401575at_nat] : ( member7866418522119297528at_nat @ ( produc2486214636825388819at_nat @ X @ Y3 ) @ R ) )
        = ( ^ [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Y3: produc289266582803401575at_nat] : ( member7866418522119297528at_nat @ ( produc2486214636825388819at_nat @ X @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_60_pred__equals__eq2,axiom,
    ! [R: set_Pr3272603486844509235at_nat,S: set_Pr3272603486844509235at_nat] :
      ( ( ( ^ [X: nat > nat,Y3: list_P6011104703257516679at_nat] : ( member139686735972302228at_nat @ ( produc1236331799044183215at_nat @ X @ Y3 ) @ R ) )
        = ( ^ [X: nat > nat,Y3: list_P6011104703257516679at_nat] : ( member139686735972302228at_nat @ ( produc1236331799044183215at_nat @ X @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_61_pred__equals__eq2,axiom,
    ! [R: set_Pr4668228411283691157at_nat,S: set_Pr4668228411283691157at_nat] :
      ( ( ( ^ [X: nat > nat > nat,Y3: produc7248412053542808358at_nat] : ( member7775084264787586014at_nat @ ( produc3209952032786966637at_nat @ X @ Y3 ) @ R ) )
        = ( ^ [X: nat > nat > nat,Y3: produc7248412053542808358at_nat] : ( member7775084264787586014at_nat @ ( produc3209952032786966637at_nat @ X @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_62_pred__equals__eq2,axiom,
    ! [R: set_Pr3493580773761353909at_nat,S: set_Pr3493580773761353909at_nat] :
      ( ( ( ^ [X: nat > nat > nat > nat,Y3: produc6392793444374437607at_nat] : ( member5927835510201945878at_nat @ ( produc8375354190041410609at_nat @ X @ Y3 ) @ R ) )
        = ( ^ [X: nat > nat > nat > nat,Y3: produc6392793444374437607at_nat] : ( member5927835510201945878at_nat @ ( produc8375354190041410609at_nat @ X @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_63_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
        = ( ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_64_pred__equals__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ( ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R ) )
        = ( ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_65_prod__decode__aux_Ocases,axiom,
    ! [X3: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] :
          ( X3
         != ( product_Pair_nat_nat @ K @ M ) ) ).

% prod_decode_aux.cases
thf(fact_66_prod__induct4,axiom,
    ! [P: produc4471711990508489141at_nat > $o,X3: produc4471711990508489141at_nat] :
      ( ! [A5: nat > nat > nat,B4: nat,C: nat,D: nat] : ( P @ ( produc3209952032786966637at_nat @ A5 @ ( produc487386426758144856at_nat @ B4 @ ( product_Pair_nat_nat @ C @ D ) ) ) )
     => ( P @ X3 ) ) ).

% prod_induct4
thf(fact_67_prod__induct3,axiom,
    ! [P: produc2780847497663810273at_nat > $o,X3: produc2780847497663810273at_nat] :
      ( ! [A5: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,B4: list_P8469869581646625389at_nat,C: list_P8469869581646625389at_nat] : ( P @ ( produc2486214636825388819at_nat @ A5 @ ( produc1338542795132623831at_nat @ B4 @ C ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_68_prod__induct3,axiom,
    ! [P: produc4471711990508489141at_nat > $o,X3: produc4471711990508489141at_nat] :
      ( ! [A5: nat > nat > nat,B4: nat,C: product_prod_nat_nat] : ( P @ ( produc3209952032786966637at_nat @ A5 @ ( produc487386426758144856at_nat @ B4 @ C ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_69_prod__induct3,axiom,
    ! [P: produc4961808859980711423at_nat > $o,X3: produc4961808859980711423at_nat] :
      ( ! [A5: nat > nat > nat > nat,B4: list_P6011104703257516679at_nat,C: list_P6011104703257516679at_nat] : ( P @ ( produc8375354190041410609at_nat @ A5 @ ( produc5943733680697469783at_nat @ B4 @ C ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_70_prod__induct3,axiom,
    ! [P: produc859450856879609959at_nat > $o,X3: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: nat,C: nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B4 @ C ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_71_prod__cases4,axiom,
    ! [Y: produc4471711990508489141at_nat] :
      ~ ! [A5: nat > nat > nat,B4: nat,C: nat,D: nat] :
          ( Y
         != ( produc3209952032786966637at_nat @ A5 @ ( produc487386426758144856at_nat @ B4 @ ( product_Pair_nat_nat @ C @ D ) ) ) ) ).

% prod_cases4
thf(fact_72_prod__cases3,axiom,
    ! [Y: produc2780847497663810273at_nat] :
      ~ ! [A5: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,B4: list_P8469869581646625389at_nat,C: list_P8469869581646625389at_nat] :
          ( Y
         != ( produc2486214636825388819at_nat @ A5 @ ( produc1338542795132623831at_nat @ B4 @ C ) ) ) ).

% prod_cases3
thf(fact_73_prod__cases3,axiom,
    ! [Y: produc4471711990508489141at_nat] :
      ~ ! [A5: nat > nat > nat,B4: nat,C: product_prod_nat_nat] :
          ( Y
         != ( produc3209952032786966637at_nat @ A5 @ ( produc487386426758144856at_nat @ B4 @ C ) ) ) ).

% prod_cases3
thf(fact_74_prod__cases3,axiom,
    ! [Y: produc4961808859980711423at_nat] :
      ~ ! [A5: nat > nat > nat > nat,B4: list_P6011104703257516679at_nat,C: list_P6011104703257516679at_nat] :
          ( Y
         != ( produc8375354190041410609at_nat @ A5 @ ( produc5943733680697469783at_nat @ B4 @ C ) ) ) ).

% prod_cases3
thf(fact_75_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A5: product_prod_nat_nat,B4: nat,C: nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B4 @ C ) ) ) ).

% prod_cases3
thf(fact_76_Pair__inject,axiom,
    ! [A: product_prod_nat_nat > product_prod_nat_nat > $o,B: list_P6011104703257516679at_nat,A2: product_prod_nat_nat > product_prod_nat_nat > $o,B2: list_P6011104703257516679at_nat] :
      ( ( ( produc3352296309980913008at_nat @ A @ B )
        = ( produc3352296309980913008at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_77_Pair__inject,axiom,
    ! [A: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,B: produc289266582803401575at_nat,A2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,B2: produc289266582803401575at_nat] :
      ( ( ( produc2486214636825388819at_nat @ A @ B )
        = ( produc2486214636825388819at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_78_Pair__inject,axiom,
    ! [A: nat > nat,B: list_P6011104703257516679at_nat,A2: nat > nat,B2: list_P6011104703257516679at_nat] :
      ( ( ( produc1236331799044183215at_nat @ A @ B )
        = ( produc1236331799044183215at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_79_Pair__inject,axiom,
    ! [A: nat > nat > nat,B: produc7248412053542808358at_nat,A2: nat > nat > nat,B2: produc7248412053542808358at_nat] :
      ( ( ( produc3209952032786966637at_nat @ A @ B )
        = ( produc3209952032786966637at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_80_Pair__inject,axiom,
    ! [A: nat > nat > nat > nat,B: produc6392793444374437607at_nat,A2: nat > nat > nat > nat,B2: produc6392793444374437607at_nat] :
      ( ( ( produc8375354190041410609at_nat @ A @ B )
        = ( produc8375354190041410609at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_81_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_82_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_83_prod__cases,axiom,
    ! [P: produc2366258654402830848at_nat > $o,P2: produc2366258654402830848at_nat] :
      ( ! [A5: product_prod_nat_nat > product_prod_nat_nat > $o,B4: list_P6011104703257516679at_nat] : ( P @ ( produc3352296309980913008at_nat @ A5 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_84_prod__cases,axiom,
    ! [P: produc2780847497663810273at_nat > $o,P2: produc2780847497663810273at_nat] :
      ( ! [A5: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,B4: produc289266582803401575at_nat] : ( P @ ( produc2486214636825388819at_nat @ A5 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_85_prod__cases,axiom,
    ! [P: produc2687737633280426365at_nat > $o,P2: produc2687737633280426365at_nat] :
      ( ! [A5: nat > nat,B4: list_P6011104703257516679at_nat] : ( P @ ( produc1236331799044183215at_nat @ A5 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_86_prod__cases,axiom,
    ! [P: produc4471711990508489141at_nat > $o,P2: produc4471711990508489141at_nat] :
      ( ! [A5: nat > nat > nat,B4: produc7248412053542808358at_nat] : ( P @ ( produc3209952032786966637at_nat @ A5 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_87_prod__cases,axiom,
    ! [P: produc4961808859980711423at_nat > $o,P2: produc4961808859980711423at_nat] :
      ( ! [A5: nat > nat > nat > nat,B4: produc6392793444374437607at_nat] : ( P @ ( produc8375354190041410609at_nat @ A5 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_88_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A5: nat,B4: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_89_prod__cases,axiom,
    ! [P: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ B4 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_90_surj__pair,axiom,
    ! [P2: produc2366258654402830848at_nat] :
    ? [X4: product_prod_nat_nat > product_prod_nat_nat > $o,Y4: list_P6011104703257516679at_nat] :
      ( P2
      = ( produc3352296309980913008at_nat @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_91_surj__pair,axiom,
    ! [P2: produc2780847497663810273at_nat] :
    ? [X4: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Y4: produc289266582803401575at_nat] :
      ( P2
      = ( produc2486214636825388819at_nat @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_92_surj__pair,axiom,
    ! [P2: produc2687737633280426365at_nat] :
    ? [X4: nat > nat,Y4: list_P6011104703257516679at_nat] :
      ( P2
      = ( produc1236331799044183215at_nat @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_93_surj__pair,axiom,
    ! [P2: produc4471711990508489141at_nat] :
    ? [X4: nat > nat > nat,Y4: produc7248412053542808358at_nat] :
      ( P2
      = ( produc3209952032786966637at_nat @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_94_surj__pair,axiom,
    ! [P2: produc4961808859980711423at_nat] :
    ? [X4: nat > nat > nat > nat,Y4: produc6392793444374437607at_nat] :
      ( P2
      = ( produc8375354190041410609at_nat @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_95_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X4: nat,Y4: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_96_surj__pair,axiom,
    ! [P2: produc859450856879609959at_nat] :
    ? [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
      ( P2
      = ( produc6161850002892822231at_nat @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_97_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2366258654402830848at_nat] :
      ~ ! [A5: product_prod_nat_nat > product_prod_nat_nat > $o,B4: list_P6011104703257516679at_nat] :
          ( Y
         != ( produc3352296309980913008at_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_98_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2780847497663810273at_nat] :
      ~ ! [A5: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,B4: produc289266582803401575at_nat] :
          ( Y
         != ( produc2486214636825388819at_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_99_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2687737633280426365at_nat] :
      ~ ! [A5: nat > nat,B4: list_P6011104703257516679at_nat] :
          ( Y
         != ( produc1236331799044183215at_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_100_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4471711990508489141at_nat] :
      ~ ! [A5: nat > nat > nat,B4: produc7248412053542808358at_nat] :
          ( Y
         != ( produc3209952032786966637at_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_101_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4961808859980711423at_nat] :
      ~ ! [A5: nat > nat > nat > nat,B4: produc6392793444374437607at_nat] :
          ( Y
         != ( produc8375354190041410609at_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_102_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A5: nat,B4: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_103_old_Oprod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A5 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_104_not__Cons__self2,axiom,
    ! [X3: list_a,Xs: list_list_a] :
      ( ( cons_list_a @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_105_not__Cons__self2,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( cons_nat @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_106_not__Cons__self2,axiom,
    ! [X3: product_prod_nat_nat > nat,Xs: list_P9162950289778280392at_nat] :
      ( ( cons_P4861729644591583992at_nat @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_107_not__Cons__self2,axiom,
    ! [X3: nat > nat,Xs: list_nat_nat] :
      ( ( cons_nat_nat @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_108_not__Cons__self2,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( cons_a @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_109_not__Cons__self2,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( cons_P6512896166579812791at_nat @ X3 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_110_term_Oinject_I2_J,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( relational_Var_a @ X2 )
        = ( relational_Var_a @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% term.inject(2)
thf(fact_111_fmla_Oinject_I5_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,Y51: relational_fmla_a_b,Y52: relational_fmla_a_b] :
      ( ( ( relational_Conj_a_b @ X51 @ X52 )
        = ( relational_Conj_a_b @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% fmla.inject(5)
thf(fact_112_fmla_Oinject_I3_J,axiom,
    ! [X31: nat,X32: relational_term_a,Y31: nat,Y32: relational_term_a] :
      ( ( ( relational_Eq_a_b @ X31 @ X32 )
        = ( relational_Eq_a_b @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% fmla.inject(3)
thf(fact_113_Conjs_Oelims,axiom,
    ! [X3: relational_fmla_a_b,Xa2: list_P6011104703257516679at_nat,Y: relational_fmla_a_b] :
      ( ( ( restrict_Conjs_a_b @ X3 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
         => ( Y != X3 ) )
       => ~ ! [X4: nat,Y4: nat,Xys2: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) )
             => ( Y
               != ( restrict_Conjs_a_b @ ( relational_Conj_a_b @ X3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y4 ) ) ) @ Xys2 ) ) ) ) ) ).

% Conjs.elims
thf(fact_114_fmla_Odistinct_I25_J,axiom,
    ! [X31: nat,X32: relational_term_a,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(25)
thf(fact_115_leftfresh_Osimps_I2_J,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat,Xys: list_P6011104703257516679at_nat] :
      ( ( restri7641029068292571945sh_a_b @ Q @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ Xys ) )
      = ( ~ ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
        & ( restri7641029068292571945sh_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) ) @ Xys ) ) ) ).

% leftfresh.simps(2)
thf(fact_116_leftfresh_Oelims_I3_J,axiom,
    ! [X3: relational_fmla_a_b,Xa2: list_P6011104703257516679at_nat] :
      ( ~ ( restri7641029068292571945sh_a_b @ X3 @ Xa2 )
     => ~ ! [X4: nat,Y4: nat,Xys2: list_P6011104703257516679at_nat] :
            ( ( Xa2
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) )
           => ( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ X3 ) )
              & ( restri7641029068292571945sh_a_b @ ( relational_Conj_a_b @ X3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y4 ) ) ) @ Xys2 ) ) ) ) ).

% leftfresh.elims(3)
thf(fact_117_infinite__eval__Conj,axiom,
    ! [X3: nat,Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Y: nat] :
      ( ~ ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
     => ( ~ ( finite_finite_list_a @ ( relational_eval_a_b @ Q @ I ) )
       => ~ ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) ) @ I ) ) ) ) ).

% infinite_eval_Conj
thf(fact_118_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X3: produc4471711990508489141at_nat] :
      ~ ! [F: nat > nat > nat,A5: nat,B4: nat,Acc: nat] :
          ( X3
         != ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B4 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_119_mem__Collect__eq,axiom,
    ! [A: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > $o] :
      ( ( member3067507820990806192at_nat @ A @ ( collec3343600615725829874at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_120_mem__Collect__eq,axiom,
    ! [A: list_list_a,P: list_list_a > $o] :
      ( ( member_list_list_a @ A @ ( collect_list_list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_121_mem__Collect__eq,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_122_mem__Collect__eq,axiom,
    ! [A: set_list_a,P: set_list_a > $o] :
      ( ( member_set_list_a @ A @ ( collect_set_list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_123_mem__Collect__eq,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_124_mem__Collect__eq,axiom,
    ! [A: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ A @ ( collec7088162979684241874at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
    ! [A: list_a,P: list_a > $o] :
      ( ( member_list_a @ A @ ( collect_list_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_128_Collect__mem__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] :
      ( ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_129_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_130_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_131_Collect__mem__eq,axiom,
    ! [A3: set_list_a] :
      ( ( collect_list_a
        @ ^ [X: list_a] : ( member_list_a @ X @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_132_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_133_Collect__cong,axiom,
    ! [P: list_a > $o,Q: list_a > $o] :
      ( ! [X4: list_a] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_list_a @ P )
        = ( collect_list_a @ Q ) ) ) ).

% Collect_cong
thf(fact_134_transpose_Ocases,axiom,
    ! [X3: list_l3264859301627795341at_nat] :
      ( ( X3 != nil_li8973309667444810893at_nat )
     => ( ! [Xss: list_l3264859301627795341at_nat] :
            ( X3
           != ( cons_l7612840610449961021at_nat @ nil_Pr5478986624290739719at_nat @ Xss ) )
       => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Xss: list_l3264859301627795341at_nat] :
              ( X3
             != ( cons_l7612840610449961021at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_135_subst__var_Ocases,axiom,
    ! [X3: produc5938564224262359745at_nat] :
      ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Z: product_prod_nat_nat] :
          ( X3
         != ( produc5588385650412007345at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( produc625733983951899089at_nat @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) @ Z ) ) )
     => ( ! [Uv: list_P6011104703257516679at_nat,Z: product_prod_nat_nat] :
            ( X3
           != ( produc5588385650412007345at_nat @ nil_Pr5478986624290739719at_nat @ ( produc625733983951899089at_nat @ Uv @ Z ) ) )
       => ~ ! [Uu: list_P6011104703257516679at_nat,Z: product_prod_nat_nat] :
              ( X3
             != ( produc5588385650412007345at_nat @ Uu @ ( produc625733983951899089at_nat @ nil_Pr5478986624290739719at_nat @ Z ) ) ) ) ) ).

% subst_var.cases
thf(fact_136_foldr1_Ocases,axiom,
    ! [X3: produc260040116751333809at_nat] :
      ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Z: product_prod_nat_nat] :
          ( X3
         != ( produc78076565302966945at_nat @ F @ ( produc625733983951899089at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ Z ) ) )
     => ~ ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Z: product_prod_nat_nat] :
            ( X3
           != ( produc78076565302966945at_nat @ F @ ( produc625733983951899089at_nat @ nil_Pr5478986624290739719at_nat @ Z ) ) ) ) ).

% foldr1.cases
thf(fact_137_list_Odistinct_I1_J,axiom,
    ! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] :
      ( nil_Pr5478986624290739719at_nat
     != ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_138_list_OdiscI,axiom,
    ! [List: list_P6011104703257516679at_nat,X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] :
      ( ( List
        = ( cons_P6512896166579812791at_nat @ X21 @ X22 ) )
     => ( List != nil_Pr5478986624290739719at_nat ) ) ).

% list.discI
thf(fact_139_list_Oexhaust,axiom,
    ! [Y: list_P6011104703257516679at_nat] :
      ( ( Y != nil_Pr5478986624290739719at_nat )
     => ~ ! [X212: product_prod_nat_nat,X222: list_P6011104703257516679at_nat] :
            ( Y
           != ( cons_P6512896166579812791at_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_140_min__list_Ocases,axiom,
    ! [X3: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
          ( X3
         != ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
     => ( X3 = nil_Pr5478986624290739719at_nat ) ) ).

% min_list.cases
thf(fact_141_remdups__adj_Ocases,axiom,
    ! [X3: list_P6011104703257516679at_nat] :
      ( ( X3 != nil_Pr5478986624290739719at_nat )
     => ( ! [X4: product_prod_nat_nat] :
            ( X3
           != ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( X3
             != ( cons_P6512896166579812791at_nat @ X4 @ ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_142_neq__Nil__conv,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( Xs != nil_Pr5478986624290739719at_nat )
      = ( ? [Y3: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
            ( Xs
            = ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_143_list__induct2_H,axiom,
    ! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
     => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ nil_Pr5478986624290739719at_nat )
       => ( ! [Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
         => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
                ( ( P @ Xs2 @ Ys )
               => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) )
           => ( P @ Xs @ Ys3 ) ) ) ) ) ).

% list_induct2'
thf(fact_144_list__nonempty__induct,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > $o] :
      ( ( Xs != nil_Pr5478986624290739719at_nat )
     => ( ! [X4: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) )
       => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( Xs2 != nil_Pr5478986624290739719at_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_145_successively_Ocases,axiom,
    ! [X3: produc2366258654402830848at_nat] :
      ( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o] :
          ( X3
         != ( produc3352296309980913008at_nat @ P3 @ nil_Pr5478986624290739719at_nat ) )
     => ( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat] :
            ( X3
           != ( produc3352296309980913008at_nat @ P3 @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) ) )
       => ~ ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc3352296309980913008at_nat @ P3 @ ( cons_P6512896166579812791at_nat @ X4 @ ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_146_sorted__wrt_Ocases,axiom,
    ! [X3: produc2366258654402830848at_nat] :
      ( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o] :
          ( X3
         != ( produc3352296309980913008at_nat @ P3 @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc3352296309980913008at_nat @ P3 @ ( cons_P6512896166579812791at_nat @ X4 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_147_shuffles_Ocases,axiom,
    ! [X3: produc6392793444374437607at_nat] :
      ( ! [Ys: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Ys ) )
     => ( ! [Xs2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc5943733680697469783at_nat @ Xs2 @ nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_148_Conjs_Osimps_I1_J,axiom,
    ! [Q: relational_fmla_a_b] :
      ( ( restrict_Conjs_a_b @ Q @ nil_Pr5478986624290739719at_nat )
      = Q ) ).

% Conjs.simps(1)
thf(fact_149_leftfresh_Osimps_I1_J,axiom,
    ! [Q: relational_fmla_a_b] : ( restri7641029068292571945sh_a_b @ Q @ nil_Pr5478986624290739719at_nat ) ).

% leftfresh.simps(1)
thf(fact_150_leftfresh_Oelims_I2_J,axiom,
    ! [X3: relational_fmla_a_b,Xa2: list_P6011104703257516679at_nat] :
      ( ( restri7641029068292571945sh_a_b @ X3 @ Xa2 )
     => ( ( Xa2 != nil_Pr5478986624290739719at_nat )
       => ~ ! [X4: nat,Y4: nat,Xys2: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) )
             => ~ ( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ X3 ) )
                  & ( restri7641029068292571945sh_a_b @ ( relational_Conj_a_b @ X3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y4 ) ) ) @ Xys2 ) ) ) ) ) ).

% leftfresh.elims(2)
thf(fact_151_leftfresh_Oelims_I1_J,axiom,
    ! [X3: relational_fmla_a_b,Xa2: list_P6011104703257516679at_nat,Y: $o] :
      ( ( ( restri7641029068292571945sh_a_b @ X3 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
         => ~ Y )
       => ~ ! [X4: nat,Y4: nat,Xys2: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) )
             => ( Y
                = ( ~ ( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ X3 ) )
                      & ( restri7641029068292571945sh_a_b @ ( relational_Conj_a_b @ X3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y4 ) ) ) @ Xys2 ) ) ) ) ) ) ) ).

% leftfresh.elims(1)
thf(fact_152_delete__aux_Ocases,axiom,
    ! [X3: produc8472197452120411308at_nat] :
      ( ! [K: nat] :
          ( X3
         != ( produc6109913384486294878at_nat @ K @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [K: nat,K2: nat,V: nat,Xs2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc6109913384486294878at_nat @ K @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_153_delete__aux_Ocases,axiom,
    ! [X3: produc5065989759464263123at_nat] :
      ( ! [K: product_prod_nat_nat] :
          ( X3
         != ( produc4832291263848940611at_nat @ K @ nil_Pr2582115297535392877at_nat ) )
     => ~ ! [K: product_prod_nat_nat,K2: product_prod_nat_nat,V: product_prod_nat_nat,Xs2: list_P8469869581646625389at_nat] :
            ( X3
           != ( produc4832291263848940611at_nat @ K @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_154_ord_Osunion__with_Ocases,axiom,
    ! [X3: produc4961808859980711423at_nat] :
      ( ! [F: nat > nat > nat > nat,K: nat,V: nat,As: list_P6011104703257516679at_nat,K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
     => ( ! [F: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
       => ~ ! [F: nat > nat > nat > nat,As: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ As @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% ord.sunion_with.cases
thf(fact_155_ord_Osunion__with_Ocases,axiom,
    ! [X3: produc2780847497663810273at_nat] :
      ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat,K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
          ( X3
         != ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
     => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
            ( X3
           != ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Bs ) ) )
       => ~ ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
              ( X3
             != ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ As @ nil_Pr2582115297535392877at_nat ) ) ) ) ) ).

% ord.sunion_with.cases
thf(fact_156_sunion__with_Ocases,axiom,
    ! [X3: produc4961808859980711423at_nat] :
      ( ! [F: nat > nat > nat > nat,K: nat,V: nat,As: list_P6011104703257516679at_nat,K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
     => ( ! [F: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
       => ~ ! [F: nat > nat > nat > nat,As: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ As @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% sunion_with.cases
thf(fact_157_sunion__with_Ocases,axiom,
    ! [X3: produc2780847497663810273at_nat] :
      ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat,K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
          ( X3
         != ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
     => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
            ( X3
           != ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Bs ) ) )
       => ~ ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
              ( X3
             != ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ As @ nil_Pr2582115297535392877at_nat ) ) ) ) ) ).

% sunion_with.cases
thf(fact_158_clearjunk_Ocases,axiom,
    ! [X3: list_P6011104703257516679at_nat] :
      ( ( X3 != nil_Pr5478986624290739719at_nat )
     => ~ ! [P4: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
            ( X3
           != ( cons_P6512896166579812791at_nat @ P4 @ Ps ) ) ) ).

% clearjunk.cases
thf(fact_159_map__default_Ocases,axiom,
    ! [X3: produc5405368317271509971at_nat] :
      ( ! [K: nat,V: nat,F: nat > nat] :
          ( X3
         != ( produc2291548248119593221at_nat @ K @ ( produc1709345877921393766at_nat @ V @ ( produc1236331799044183215at_nat @ F @ nil_Pr5478986624290739719at_nat ) ) ) )
     => ~ ! [K: nat,V: nat,F: nat > nat,P4: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc2291548248119593221at_nat @ K @ ( produc1709345877921393766at_nat @ V @ ( produc1236331799044183215at_nat @ F @ ( cons_P6512896166579812791at_nat @ P4 @ Ps ) ) ) ) ) ) ).

% map_default.cases
thf(fact_160_map__entry_Ocases,axiom,
    ! [X3: produc6121082497140218670at_nat] :
      ( ! [K: nat,F: nat > nat] :
          ( X3
         != ( produc1709345877921393766at_nat @ K @ ( produc1236331799044183215at_nat @ F @ nil_Pr5478986624290739719at_nat ) ) )
     => ~ ! [K: nat,F: nat > nat,P4: product_prod_nat_nat,Ps: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc1709345877921393766at_nat @ K @ ( produc1236331799044183215at_nat @ F @ ( cons_P6512896166579812791at_nat @ P4 @ Ps ) ) ) ) ) ).

% map_entry.cases
thf(fact_161_bex2I,axiom,
    ! [A: nat,B: nat,S: set_Pr1261947904930325089at_nat,P: nat > nat > $o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S )
     => ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S )
         => ( P @ A @ B ) )
       => ? [A5: nat,B4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ S )
            & ( P @ A5 @ B4 ) ) ) ) ).

% bex2I
thf(fact_162_bex2I,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,S: set_Pr8693737435421807431at_nat,P: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ S )
     => ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ S )
         => ( P @ A @ B ) )
       => ? [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ B4 ) @ S )
            & ( P @ A5 @ B4 ) ) ) ) ).

% bex2I
thf(fact_163_list__tail__coinc,axiom,
    ! [N1: product_prod_nat_nat,R1: list_P6011104703257516679at_nat,N2: product_prod_nat_nat,R2: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ N1 @ R1 )
        = ( cons_P6512896166579812791at_nat @ N2 @ R2 ) )
     => ( ( N1 = N2 )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
thf(fact_164_list__all__zip_Ocases,axiom,
    ! [X3: produc4450861684445579872at_nat] :
      ( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o] :
          ( X3
         != ( produc6665944465064540112at_nat @ P3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat ) ) )
     => ( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,A5: product_prod_nat_nat,As: list_P6011104703257516679at_nat,B4: product_prod_nat_nat,Bs: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc6665944465064540112at_nat @ P3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ A5 @ As ) @ ( cons_P6512896166579812791at_nat @ B4 @ Bs ) ) ) )
       => ( ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc6665944465064540112at_nat @ P3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) @ nil_Pr5478986624290739719at_nat ) ) )
         => ~ ! [P3: product_prod_nat_nat > product_prod_nat_nat > $o,V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
                ( X3
               != ( produc6665944465064540112at_nat @ P3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) ) ) ) ) ) ) ).

% list_all_zip.cases
thf(fact_165_partition__rev_Ocases,axiom,
    ! [X3: produc6471691067047545792at_nat] :
      ( ! [P3: product_prod_nat_nat > $o,Yes: list_P6011104703257516679at_nat,No: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc8971229750533022322at_nat @ P3 @ ( produc8267574230603542263at_nat @ ( produc5943733680697469783at_nat @ Yes @ No ) @ nil_Pr5478986624290739719at_nat ) ) )
     => ~ ! [P3: product_prod_nat_nat > $o,Yes: list_P6011104703257516679at_nat,No: list_P6011104703257516679at_nat,X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc8971229750533022322at_nat @ P3 @ ( produc8267574230603542263at_nat @ ( produc5943733680697469783at_nat @ Yes @ No ) @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ) ) ).

% partition_rev.cases
thf(fact_166_quicksort__by__rel_Ocases,axiom,
    ! [X3: produc4450861684445579872at_nat] :
      ( ! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,Sl: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc6665944465064540112at_nat @ R3 @ ( produc5943733680697469783at_nat @ Sl @ nil_Pr5478986624290739719at_nat ) ) )
     => ~ ! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,Sl: list_P6011104703257516679at_nat,X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc6665944465064540112at_nat @ R3 @ ( produc5943733680697469783at_nat @ Sl @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ) ) ).

% quicksort_by_rel.cases
thf(fact_167_mergesort__by__rel__merge_Ocases,axiom,
    ! [X3: produc4450861684445579872at_nat] :
      ( ! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc6665944465064540112at_nat @ R3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) )
     => ( ! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,Xs2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc6665944465064540112at_nat @ R3 @ ( produc5943733680697469783at_nat @ Xs2 @ nil_Pr5478986624290739719at_nat ) ) )
       => ~ ! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc6665944465064540112at_nat @ R3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) ) ) ) ) ) ).

% mergesort_by_rel_merge.cases
thf(fact_168_mergesort__by__rel__split_Ocases,axiom,
    ! [X3: produc465025108790790279at_nat] :
      ( ! [Xs1: list_P6011104703257516679at_nat,Xs22: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc8267574230603542263at_nat @ ( produc5943733680697469783at_nat @ Xs1 @ Xs22 ) @ nil_Pr5478986624290739719at_nat ) )
     => ( ! [Xs1: list_P6011104703257516679at_nat,Xs22: list_P6011104703257516679at_nat,X4: product_prod_nat_nat] :
            ( X3
           != ( produc8267574230603542263at_nat @ ( produc5943733680697469783at_nat @ Xs1 @ Xs22 ) @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) ) )
       => ~ ! [Xs1: list_P6011104703257516679at_nat,Xs22: list_P6011104703257516679at_nat,X12: product_prod_nat_nat,X23: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc8267574230603542263at_nat @ ( produc5943733680697469783at_nat @ Xs1 @ Xs22 ) @ ( cons_P6512896166579812791at_nat @ X12 @ ( cons_P6512896166579812791at_nat @ X23 @ Xs2 ) ) ) ) ) ) ).

% mergesort_by_rel_split.cases
thf(fact_169_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ! [Xs2: list_P6011104703257516679at_nat] : ( P @ Xs2 @ nil_Pr5478986624290739719at_nat )
     => ( ! [X_1: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ X_1 )
       => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
              ( ( R @ X4 @ Y4 )
             => ( ( P @ Xs2 @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
               => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) )
         => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
                ( ~ ( R @ X4 @ Y4 )
               => ( ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ Ys )
                 => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) )
           => ( P @ Xs @ Ys3 ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_170_list__induct__first2,axiom,
    ! [P: list_P6011104703257516679at_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( P @ nil_Pr5478986624290739719at_nat )
     => ( ! [X4: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) )
       => ( ! [X12: product_prod_nat_nat,X23: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( P @ Xs2 )
             => ( P @ ( cons_P6512896166579812791at_nat @ X12 @ ( cons_P6512896166579812791at_nat @ X23 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_171_list__2pre__induct,axiom,
    ! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,W1: list_P6011104703257516679at_nat,W2: list_P6011104703257516679at_nat] :
      ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
     => ( ! [E: product_prod_nat_nat,W12: list_P6011104703257516679at_nat,W22: list_P6011104703257516679at_nat] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_P6512896166579812791at_nat @ E @ W12 ) @ W22 ) )
       => ( ! [E: product_prod_nat_nat,W13: list_P6011104703257516679at_nat,W23: list_P6011104703257516679at_nat] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_P6512896166579812791at_nat @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_172_neq__NilE,axiom,
    ! [L: list_P6011104703257516679at_nat] :
      ( ( L != nil_Pr5478986624290739719at_nat )
     => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( L
           != ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_173_lookup_Ocases,axiom,
    ! [X3: produc5938564224262359745at_nat] :
      ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Z: product_prod_nat_nat] :
          ( X3
         != ( produc5588385650412007345at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( produc625733983951899089at_nat @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) @ Z ) ) )
     => ( ! [Uv: list_P6011104703257516679at_nat,Uw: product_prod_nat_nat] :
            ( X3
           != ( produc5588385650412007345at_nat @ nil_Pr5478986624290739719at_nat @ ( produc625733983951899089at_nat @ Uv @ Uw ) ) )
       => ~ ! [Uu: list_P6011104703257516679at_nat,Uw: product_prod_nat_nat] :
              ( X3
             != ( produc5588385650412007345at_nat @ Uu @ ( produc625733983951899089at_nat @ nil_Pr5478986624290739719at_nat @ Uw ) ) ) ) ) ).

% lookup.cases
thf(fact_174_restrict_Ocases,axiom,
    ! [X3: produc8722808085745974893at_nat] :
      ( ! [A6: set_Pr1261947904930325089at_nat,X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc1349757789725428829at_nat @ A6 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) )
     => ( ! [A6: set_Pr1261947904930325089at_nat,Uv: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc1349757789725428829at_nat @ A6 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Uv ) ) )
       => ~ ! [A6: set_Pr1261947904930325089at_nat,Uu: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc1349757789725428829at_nat @ A6 @ ( produc5943733680697469783at_nat @ Uu @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% restrict.cases
thf(fact_175_Autoref__Bindings__HOL_Olist__eq_Ocases,axiom,
    ! [X3: produc4450861684445579872at_nat] :
      ( ! [Eq: product_prod_nat_nat > product_prod_nat_nat > $o] :
          ( X3
         != ( produc6665944465064540112at_nat @ Eq @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat ) ) )
     => ( ! [Eq: product_prod_nat_nat > product_prod_nat_nat > $o,A5: product_prod_nat_nat,L2: list_P6011104703257516679at_nat,A7: product_prod_nat_nat,L3: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc6665944465064540112at_nat @ Eq @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ A5 @ L2 ) @ ( cons_P6512896166579812791at_nat @ A7 @ L3 ) ) ) )
       => ( ! [Uu: product_prod_nat_nat > product_prod_nat_nat > $o,V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc6665944465064540112at_nat @ Uu @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) @ nil_Pr5478986624290739719at_nat ) ) )
         => ~ ! [Uu: product_prod_nat_nat > product_prod_nat_nat > $o,V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
                ( X3
               != ( produc6665944465064540112at_nat @ Uu @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) ) ) ) ) ) ) ).

% Autoref_Bindings_HOL.list_eq.cases
thf(fact_176_subset__eq__mset__impl_Ocases,axiom,
    ! [X3: produc6392793444374437607at_nat] :
      ( ! [Ys: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Ys ) )
     => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_177_subset__sorted_Ocases,axiom,
    ! [X3: produc6392793444374437607at_nat] :
      ( ! [L22: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ L22 ) )
     => ( ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat,X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ ( cons_P6512896166579812791at_nat @ X23 @ L22 ) ) ) ) ) ).

% subset_sorted.cases
thf(fact_178_lss__basic_Og__from__list__aux_Ocases,axiom,
    ! [X3: produc6392793444374437607at_nat] :
      ( ! [Accs: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc5943733680697469783at_nat @ Accs @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [Accs: list_P6011104703257516679at_nat,X4: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc5943733680697469783at_nat @ Accs @ ( cons_P6512896166579812791at_nat @ X4 @ L2 ) ) ) ) ).

% lss_basic.g_from_list_aux.cases
thf(fact_179_lsnd__basic_Og__from__list__aux_Ocases,axiom,
    ! [X3: produc6392793444374437607at_nat] :
      ( ! [Accs: list_P6011104703257516679at_nat] :
          ( X3
         != ( produc5943733680697469783at_nat @ Accs @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [Accs: list_P6011104703257516679at_nat,X4: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc5943733680697469783at_nat @ Accs @ ( cons_P6512896166579812791at_nat @ X4 @ L2 ) ) ) ) ).

% lsnd_basic.g_from_list_aux.cases
thf(fact_180_insertion__sort_Ocases,axiom,
    ! [X3: produc7489448085829838189at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( X3
         != ( produc1593612501639298397at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc1593612501639298397at_nat @ X4 @ ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) ) ) ).

% insertion_sort.cases
thf(fact_181_memb__sorted_Ocases,axiom,
    ! [X3: produc4481152233128974049at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( X3
         != ( produc625733983951899089at_nat @ nil_Pr5478986624290739719at_nat @ X4 ) )
     => ~ ! [Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,X4: product_prod_nat_nat] :
            ( X3
           != ( produc625733983951899089at_nat @ ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) @ X4 ) ) ) ).

% memb_sorted.cases
thf(fact_182_Conjs_Opelims,axiom,
    ! [X3: relational_fmla_a_b,Xa2: list_P6011104703257516679at_nat,Y: relational_fmla_a_b] :
      ( ( ( restrict_Conjs_a_b @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P4726741148976992084at_nat @ restri7125274854013146766el_a_b @ ( produc5858919939532109903at_nat @ X3 @ Xa2 ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( ( Y = X3 )
             => ~ ( accp_P4726741148976992084at_nat @ restri7125274854013146766el_a_b @ ( produc5858919939532109903at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) )
         => ~ ! [X4: nat,Y4: nat,Xys2: list_P6011104703257516679at_nat] :
                ( ( Xa2
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) )
               => ( ( Y
                    = ( restrict_Conjs_a_b @ ( relational_Conj_a_b @ X3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y4 ) ) ) @ Xys2 ) )
                 => ~ ( accp_P4726741148976992084at_nat @ restri7125274854013146766el_a_b @ ( produc5858919939532109903at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) ) ) ) ) ) ) ) ).

% Conjs.pelims
thf(fact_183_leftfresh_Opelims_I1_J,axiom,
    ! [X3: relational_fmla_a_b,Xa2: list_P6011104703257516679at_nat,Y: $o] :
      ( ( ( restri7641029068292571945sh_a_b @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P4726741148976992084at_nat @ restri5669634700885871058el_a_b @ ( produc5858919939532109903at_nat @ X3 @ Xa2 ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( Y
             => ~ ( accp_P4726741148976992084at_nat @ restri5669634700885871058el_a_b @ ( produc5858919939532109903at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) )
         => ~ ! [X4: nat,Y4: nat,Xys2: list_P6011104703257516679at_nat] :
                ( ( Xa2
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) )
               => ( ( Y
                    = ( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ X3 ) )
                      & ( restri7641029068292571945sh_a_b @ ( relational_Conj_a_b @ X3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y4 ) ) ) @ Xys2 ) ) )
                 => ~ ( accp_P4726741148976992084at_nat @ restri5669634700885871058el_a_b @ ( produc5858919939532109903at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) ) ) ) ) ) ) ) ).

% leftfresh.pelims(1)
thf(fact_184_leftfresh_Opelims_I2_J,axiom,
    ! [X3: relational_fmla_a_b,Xa2: list_P6011104703257516679at_nat] :
      ( ( restri7641029068292571945sh_a_b @ X3 @ Xa2 )
     => ( ( accp_P4726741148976992084at_nat @ restri5669634700885871058el_a_b @ ( produc5858919939532109903at_nat @ X3 @ Xa2 ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ~ ( accp_P4726741148976992084at_nat @ restri5669634700885871058el_a_b @ ( produc5858919939532109903at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) )
         => ~ ! [X4: nat,Y4: nat,Xys2: list_P6011104703257516679at_nat] :
                ( ( Xa2
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) )
               => ( ( accp_P4726741148976992084at_nat @ restri5669634700885871058el_a_b @ ( produc5858919939532109903at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) ) )
                 => ~ ( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ X3 ) )
                      & ( restri7641029068292571945sh_a_b @ ( relational_Conj_a_b @ X3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y4 ) ) ) @ Xys2 ) ) ) ) ) ) ) ).

% leftfresh.pelims(2)
thf(fact_185_quicksort_Ocases,axiom,
    ! [X3: list_P6011104703257516679at_nat] :
      ( ( X3 != nil_Pr5478986624290739719at_nat )
     => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( X3
           != ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ).

% quicksort.cases
thf(fact_186_ahs__sbm_Obasic_Og__from__list__aux_Ocases,axiom,
    ! [X3: produc2286679691103333188at_nat] :
      ( ! [Accs: hashma5287334503967765098t_unit] :
          ( X3
         != ( produc6571620835003027764at_nat @ Accs @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [Accs: hashma5287334503967765098t_unit,X4: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc6571620835003027764at_nat @ Accs @ ( cons_P6512896166579812791at_nat @ X4 @ L2 ) ) ) ) ).

% ahs_sbm.basic.g_from_list_aux.cases
thf(fact_187_rs__sbm_Oobasic_Og__from__list__aux_Ocases,axiom,
    ! [X3: produc4873611444601972787at_nat] :
      ( ! [Accs: rbt_Pr1701002678351477513t_unit] :
          ( X3
         != ( produc1852532052082862949at_nat @ Accs @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [Accs: rbt_Pr1701002678351477513t_unit,X4: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc1852532052082862949at_nat @ Accs @ ( cons_P6512896166579812791at_nat @ X4 @ L2 ) ) ) ) ).

% rs_sbm.obasic.g_from_list_aux.cases
thf(fact_188_ls__basic_Og__from__list__aux_Ocases,axiom,
    ! [X3: produc8052360320355758925at_nat] :
      ( ! [Accs: dlist_2221377526271503905at_nat] :
          ( X3
         != ( produc5189926092923711549at_nat @ Accs @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [Accs: dlist_2221377526271503905at_nat,X4: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc5189926092923711549at_nat @ Accs @ ( cons_P6512896166579812791at_nat @ X4 @ L2 ) ) ) ) ).

% ls_basic.g_from_list_aux.cases
thf(fact_189_hs__sbm_Obasic_Og__from__list__aux_Ocases,axiom,
    ! [X3: produc5258074957289258047at_nat] :
      ( ! [Accs: hashma7010742005768827389t_unit] :
          ( X3
         != ( produc2625458324064763249at_nat @ Accs @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [Accs: hashma7010742005768827389t_unit,X4: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc2625458324064763249at_nat @ Accs @ ( cons_P6512896166579812791at_nat @ X4 @ L2 ) ) ) ) ).

% hs_sbm.basic.g_from_list_aux.cases
thf(fact_190_leftfresh_Opelims_I3_J,axiom,
    ! [X3: relational_fmla_a_b,Xa2: list_P6011104703257516679at_nat] :
      ( ~ ( restri7641029068292571945sh_a_b @ X3 @ Xa2 )
     => ( ( accp_P4726741148976992084at_nat @ restri5669634700885871058el_a_b @ ( produc5858919939532109903at_nat @ X3 @ Xa2 ) )
       => ~ ! [X4: nat,Y4: nat,Xys2: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) )
             => ( ( accp_P4726741148976992084at_nat @ restri5669634700885871058el_a_b @ ( produc5858919939532109903at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ Xys2 ) ) )
               => ( ~ ( member_nat @ X4 @ ( relational_fv_a_b @ X3 ) )
                  & ( restri7641029068292571945sh_a_b @ ( relational_Conj_a_b @ X3 @ ( relational_Eq_a_b @ X4 @ ( relational_Var_a @ Y4 ) ) ) @ Xys2 ) ) ) ) ) ) ).

% leftfresh.pelims(3)
thf(fact_191_op__list__singleton__def,axiom,
    ( autore447854781811194899at_nat
    = ( ^ [X: product_prod_nat_nat] : ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) ) ).

% op_list_singleton_def
thf(fact_192_sr__Conj__eq,axiom,
    ! [Q: relational_fmla_a_b,X3: nat,Y: nat] :
      ( ( relational_sr_a_b @ Q )
     => ( ( ( member_nat @ X3 @ ( relational_fv_a_b @ Q ) )
          | ( member_nat @ Y @ ( relational_fv_a_b @ Q ) ) )
       => ( relational_sr_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) ) ) ) ) ).

% sr_Conj_eq
thf(fact_193_delete__aux_Opelims,axiom,
    ! [X3: nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( delete_aux_nat_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P1391465874720445027at_nat @ delete1397635992828230261at_nat @ ( produc6109913384486294878at_nat @ X3 @ Xa2 ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( ( Y = nil_Pr5478986624290739719at_nat )
             => ~ ( accp_P1391465874720445027at_nat @ delete1397635992828230261at_nat @ ( produc6109913384486294878at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) )
         => ~ ! [K2: nat,V: nat,Xs2: list_P6011104703257516679at_nat] :
                ( ( Xa2
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ Xs2 ) )
               => ( ( ( ( X3 = K2 )
                     => ( Y = Xs2 ) )
                    & ( ( X3 != K2 )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ ( delete_aux_nat_nat @ X3 @ Xs2 ) ) ) ) )
                 => ~ ( accp_P1391465874720445027at_nat @ delete1397635992828230261at_nat @ ( produc6109913384486294878at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_194_delete__aux_Opelims,axiom,
    ! [X3: product_prod_nat_nat,Xa2: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( delete5918753289527851494at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P2469863912318080476at_nat @ delete4570897403666713875at_nat @ ( produc4832291263848940611at_nat @ X3 @ Xa2 ) )
       => ( ( ( Xa2 = nil_Pr2582115297535392877at_nat )
           => ( ( Y = nil_Pr2582115297535392877at_nat )
             => ~ ( accp_P2469863912318080476at_nat @ delete4570897403666713875at_nat @ ( produc4832291263848940611at_nat @ X3 @ nil_Pr2582115297535392877at_nat ) ) ) )
         => ~ ! [K2: product_prod_nat_nat,V: product_prod_nat_nat,Xs2: list_P8469869581646625389at_nat] :
                ( ( Xa2
                  = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ Xs2 ) )
               => ( ( ( ( X3 = K2 )
                     => ( Y = Xs2 ) )
                    & ( ( X3 != K2 )
                     => ( Y
                        = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ ( delete5918753289527851494at_nat @ X3 @ Xs2 ) ) ) ) )
                 => ~ ( accp_P2469863912318080476at_nat @ delete4570897403666713875at_nat @ ( produc4832291263848940611at_nat @ X3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.pelims
thf(fact_195_map__default_Osimps_I1_J,axiom,
    ! [K3: nat,V3: nat,F2: nat > nat] :
      ( ( map_default_nat_nat @ K3 @ V3 @ F2 @ nil_Pr5478986624290739719at_nat )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ nil_Pr5478986624290739719at_nat ) ) ).

% map_default.simps(1)
thf(fact_196_map__default_Osimps_I1_J,axiom,
    ! [K3: product_prod_nat_nat,V3: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( map_de6203744875203570968at_nat @ K3 @ V3 @ F2 @ nil_Pr2582115297535392877at_nat )
      = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ nil_Pr2582115297535392877at_nat ) ) ).

% map_default.simps(1)
thf(fact_197_update__with__aux_Osimps_I1_J,axiom,
    ! [V3: nat,K3: nat,F2: nat > nat] :
      ( ( update528237659335440164at_nat @ V3 @ K3 @ F2 @ nil_Pr5478986624290739719at_nat )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ ( F2 @ V3 ) ) @ nil_Pr5478986624290739719at_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_198_update__with__aux_Osimps_I1_J,axiom,
    ! [V3: product_prod_nat_nat,K3: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( update8621587714000398146at_nat @ V3 @ K3 @ F2 @ nil_Pr2582115297535392877at_nat )
      = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ ( F2 @ V3 ) ) @ nil_Pr2582115297535392877at_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_199_delete__aux__eq__Nil__conv,axiom,
    ! [K3: nat,Ts: list_P6011104703257516679at_nat] :
      ( ( ( delete_aux_nat_nat @ K3 @ Ts )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Ts = nil_Pr5478986624290739719at_nat )
        | ? [V4: nat] :
            ( Ts
            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V4 ) @ nil_Pr5478986624290739719at_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_200_delete__aux__eq__Nil__conv,axiom,
    ! [K3: product_prod_nat_nat,Ts: list_P8469869581646625389at_nat] :
      ( ( ( delete5918753289527851494at_nat @ K3 @ Ts )
        = nil_Pr2582115297535392877at_nat )
      = ( ( Ts = nil_Pr2582115297535392877at_nat )
        | ? [V4: product_prod_nat_nat] :
            ( Ts
            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V4 ) @ nil_Pr2582115297535392877at_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_201_delete__aux_Osimps_I2_J,axiom,
    ! [K3: nat,K4: nat,V3: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( K3 = K4 )
       => ( ( delete_aux_nat_nat @ K3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V3 ) @ Xs ) )
          = Xs ) )
      & ( ( K3 != K4 )
       => ( ( delete_aux_nat_nat @ K3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V3 ) @ Xs ) )
          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V3 ) @ ( delete_aux_nat_nat @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_202_delete__aux_Osimps_I2_J,axiom,
    ! [K3: product_prod_nat_nat,K4: product_prod_nat_nat,V3: product_prod_nat_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ( K3 = K4 )
       => ( ( delete5918753289527851494at_nat @ K3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V3 ) @ Xs ) )
          = Xs ) )
      & ( ( K3 != K4 )
       => ( ( delete5918753289527851494at_nat @ K3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V3 ) @ Xs ) )
          = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V3 ) @ ( delete5918753289527851494at_nat @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_203_delete__aux_Oelims,axiom,
    ! [X3: nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( delete_aux_nat_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [K2: nat,V: nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ Xs2 ) )
             => ~ ( ( ( X3 = K2 )
                   => ( Y = Xs2 ) )
                  & ( ( X3 != K2 )
                   => ( Y
                      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ ( delete_aux_nat_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_204_delete__aux_Oelims,axiom,
    ! [X3: product_prod_nat_nat,Xa2: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( delete5918753289527851494at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr2582115297535392877at_nat )
         => ( Y != nil_Pr2582115297535392877at_nat ) )
       => ~ ! [K2: product_prod_nat_nat,V: product_prod_nat_nat,Xs2: list_P8469869581646625389at_nat] :
              ( ( Xa2
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ Xs2 ) )
             => ~ ( ( ( X3 = K2 )
                   => ( Y = Xs2 ) )
                  & ( ( X3 != K2 )
                   => ( Y
                      = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V ) @ ( delete5918753289527851494at_nat @ X3 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_205_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R4 ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X4: nat,Y4: nat,Xs2: list_nat] :
              ( ( A1
                = ( cons_nat @ X4 @ Xs2 ) )
             => ! [Ys: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R4 )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R4 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_206_listrel_Ocases,axiom,
    ! [A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A1 @ A22 ) @ ( listre818007680106770737at_nat @ R4 ) )
     => ( ( ( A1 = nil_Pr5478986624290739719at_nat )
         => ( A22 != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( A1
                = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
             => ! [Ys: list_P6011104703257516679at_nat] :
                  ( ( A22
                    = ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
                 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R4 )
                   => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys ) @ ( listre818007680106770737at_nat @ R4 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_207_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R4 ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_nat ) )
        | ? [X: nat,Y3: nat,Xs3: list_nat,Ys2: list_nat] :
            ( ( A1
              = ( cons_nat @ X @ Xs3 ) )
            & ( A22
              = ( cons_nat @ Y3 @ Ys2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R4 )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( listrel_nat_nat @ R4 ) ) ) ) ) ).

% listrel.simps
thf(fact_208_listrel_Osimps,axiom,
    ! [A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ A1 @ A22 ) @ ( listre818007680106770737at_nat @ R4 ) )
      = ( ( ( A1 = nil_Pr5478986624290739719at_nat )
          & ( A22 = nil_Pr5478986624290739719at_nat ) )
        | ? [X: product_prod_nat_nat,Y3: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
            ( ( A1
              = ( cons_P6512896166579812791at_nat @ X @ Xs3 ) )
            & ( A22
              = ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R4 )
            & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs3 @ Ys2 ) @ ( listre818007680106770737at_nat @ R4 ) ) ) ) ) ).

% listrel.simps
thf(fact_209_insert__Nil,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( insert8944034826898310173at_nat @ X3 @ nil_Pr5478986624290739719at_nat )
      = ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ).

% insert_Nil
thf(fact_210_partition__rev_Oelims,axiom,
    ! [X3: product_prod_nat_nat > $o,Xa2: produc6392793444374437607at_nat,Xb: list_P6011104703257516679at_nat,Y: produc6392793444374437607at_nat] :
      ( ( ( partit1285214537004407631at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ! [Yes: list_P6011104703257516679at_nat,No: list_P6011104703257516679at_nat] :
            ( ( Xa2
              = ( produc5943733680697469783at_nat @ Yes @ No ) )
           => ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y
               != ( produc5943733680697469783at_nat @ Yes @ No ) ) ) )
       => ~ ! [Yes: list_P6011104703257516679at_nat,No: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( produc5943733680697469783at_nat @ Yes @ No ) )
             => ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
                 => ( Y
                   != ( partit1285214537004407631at_nat @ X3 @ ( if_Pro969932733043319213at_nat @ ( X3 @ X4 ) @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Yes ) @ No ) @ ( produc5943733680697469783at_nat @ Yes @ ( cons_P6512896166579812791at_nat @ X4 @ No ) ) ) @ Xs2 ) ) ) ) ) ) ).

% partition_rev.elims
thf(fact_211_lexord__Nil__left,axiom,
    ! [Y: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Y ) @ ( lexord2841853652668343668at_nat @ R4 ) )
      = ( ? [A4: product_prod_nat_nat,X: list_P6011104703257516679at_nat] :
            ( Y
            = ( cons_P6512896166579812791at_nat @ A4 @ X ) ) ) ) ).

% lexord_Nil_left
thf(fact_212_lexord__cons__cons,axiom,
    ! [A: nat,X3: list_nat,B: nat,Y: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X3 ) @ ( cons_nat @ B @ Y ) ) @ ( lexord_nat @ R4 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
        | ( ( A = B )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ ( lexord_nat @ R4 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_213_lexord__cons__cons,axiom,
    ! [A: product_prod_nat_nat,X3: list_P6011104703257516679at_nat,B: product_prod_nat_nat,Y: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ A @ X3 ) @ ( cons_P6512896166579812791at_nat @ B @ Y ) ) @ ( lexord2841853652668343668at_nat @ R4 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R4 )
        | ( ( A = B )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X3 @ Y ) @ ( lexord2841853652668343668at_nat @ R4 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_214_lexord__irreflexive,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R4 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lexord_nat @ R4 ) ) ) ).

% lexord_irreflexive
thf(fact_215_lexord__irreflexive,axiom,
    ! [R4: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R4 )
     => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Xs ) @ ( lexord2841853652668343668at_nat @ R4 ) ) ) ).

% lexord_irreflexive
thf(fact_216_lexord__linear,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,X3: list_nat,Y: list_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ R4 )
          | ( A5 = B4 )
          | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B4 @ A5 ) @ R4 ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ ( lexord_nat @ R4 ) )
        | ( X3 = Y )
        | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X3 ) @ ( lexord_nat @ R4 ) ) ) ) ).

% lexord_linear
thf(fact_217_lexord__linear,axiom,
    ! [R4: set_Pr8693737435421807431at_nat,X3: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ B4 ) @ R4 )
          | ( A5 = B4 )
          | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B4 @ A5 ) @ R4 ) )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X3 @ Y ) @ ( lexord2841853652668343668at_nat @ R4 ) )
        | ( X3 = Y )
        | ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Y @ X3 ) @ ( lexord2841853652668343668at_nat @ R4 ) ) ) ) ).

% lexord_linear
thf(fact_218_partition__rev_Osimps_I2_J,axiom,
    ! [P: product_prod_nat_nat > $o,Yes2: list_P6011104703257516679at_nat,No2: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( partit1285214537004407631at_nat @ P @ ( produc5943733680697469783at_nat @ Yes2 @ No2 ) @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( partit1285214537004407631at_nat @ P @ ( if_Pro969932733043319213at_nat @ ( P @ X3 ) @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Yes2 ) @ No2 ) @ ( produc5943733680697469783at_nat @ Yes2 @ ( cons_P6512896166579812791at_nat @ X3 @ No2 ) ) ) @ Xs ) ) ).

% partition_rev.simps(2)
thf(fact_219_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys3 ) ) @ ( listrel_nat_nat @ R4 ) )
     => ~ ! [X4: nat,Xs2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs2 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ R4 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys3 ) @ ( listrel_nat_nat @ R4 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_220_listrel__Cons2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) @ ( listre818007680106770737at_nat @ R4 ) )
     => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y ) @ R4 )
             => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys3 ) @ ( listre818007680106770737at_nat @ R4 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_221_listrel__Cons1,axiom,
    ! [Y: nat,Ys3: list_nat,Xs: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys3 ) @ Xs ) @ ( listrel_nat_nat @ R4 ) )
     => ~ ! [Y4: nat,Ys: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y4 ) @ R4 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys3 @ Ys ) @ ( listrel_nat_nat @ R4 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_222_listrel__Cons1,axiom,
    ! [Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) @ Xs ) @ ( listre818007680106770737at_nat @ R4 ) )
     => ~ ! [Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Y4 ) @ R4 )
             => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys3 @ Ys ) @ ( listre818007680106770737at_nat @ R4 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_223_listrel_OCons,axiom,
    ! [X3: nat,Y: nat,R4: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys3: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R4 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( listrel_nat_nat @ R4 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys3 ) ) @ ( listrel_nat_nat @ R4 ) ) ) ) ).

% listrel.Cons
thf(fact_224_listrel_OCons,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R4 )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys3 ) @ ( listre818007680106770737at_nat @ R4 ) )
       => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) @ ( listre818007680106770737at_nat @ R4 ) ) ) ) ).

% listrel.Cons
thf(fact_225_listrelp__listrel__eq,axiom,
    ! [R4: set_Pr1261947904930325089at_nat] :
      ( ( listrelp_nat_nat
        @ ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R4 ) )
      = ( ^ [X: list_nat,Y3: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y3 ) @ ( listrel_nat_nat @ R4 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_226_listrelp__listrel__eq,axiom,
    ! [R4: set_Pr8693737435421807431at_nat] :
      ( ( listre5588581128925307775at_nat
        @ ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R4 ) )
      = ( ^ [X: list_P6011104703257516679at_nat,Y3: list_P6011104703257516679at_nat] : ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X @ Y3 ) @ ( listre818007680106770737at_nat @ R4 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_227_enumerate__simps_I2_J,axiom,
    ! [N: nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( enumer8411992446310978662at_nat @ N @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( cons_P4943146402254145264at_nat @ ( produc487386426758144856at_nat @ N @ X3 ) @ ( enumer8411992446310978662at_nat @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_228_enumerate__simps_I2_J,axiom,
    ! [N: nat,X3: nat,Xs: list_nat] :
      ( ( enumerate_nat @ N @ ( cons_nat @ X3 @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X3 ) @ ( enumerate_nat @ ( suc @ N ) @ Xs ) ) ) ).

% enumerate_simps(2)
thf(fact_229_partition__rev_Opelims,axiom,
    ! [X3: product_prod_nat_nat > $o,Xa2: produc6392793444374437607at_nat,Xb: list_P6011104703257516679at_nat,Y: produc6392793444374437607at_nat] :
      ( ( ( partit1285214537004407631at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P5620345288600724343at_nat @ partit533615971937049976at_nat @ ( produc8971229750533022322at_nat @ X3 @ ( produc8267574230603542263at_nat @ Xa2 @ Xb ) ) )
       => ( ! [Yes: list_P6011104703257516679at_nat,No: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( produc5943733680697469783at_nat @ Yes @ No ) )
             => ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y
                    = ( produc5943733680697469783at_nat @ Yes @ No ) )
                 => ~ ( accp_P5620345288600724343at_nat @ partit533615971937049976at_nat @ ( produc8971229750533022322at_nat @ X3 @ ( produc8267574230603542263at_nat @ ( produc5943733680697469783at_nat @ Yes @ No ) @ nil_Pr5478986624290739719at_nat ) ) ) ) ) )
         => ~ ! [Yes: list_P6011104703257516679at_nat,No: list_P6011104703257516679at_nat] :
                ( ( Xa2
                  = ( produc5943733680697469783at_nat @ Yes @ No ) )
               => ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                    ( ( Xb
                      = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
                   => ( ( Y
                        = ( partit1285214537004407631at_nat @ X3 @ ( if_Pro969932733043319213at_nat @ ( X3 @ X4 ) @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Yes ) @ No ) @ ( produc5943733680697469783at_nat @ Yes @ ( cons_P6512896166579812791at_nat @ X4 @ No ) ) ) @ Xs2 ) )
                     => ~ ( accp_P5620345288600724343at_nat @ partit533615971937049976at_nat @ ( produc8971229750533022322at_nat @ X3 @ ( produc8267574230603542263at_nat @ ( produc5943733680697469783at_nat @ Yes @ No ) @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).

% partition_rev.pelims
thf(fact_230_cpropagated__simps_I3_J,axiom,
    ! [X3: nat,T: relational_term_a] :
      ( ( relati1591879772219623554ed_a_b @ ( relational_Eq_a_b @ X3 @ T ) )
      = ( T
       != ( relational_Var_a @ X3 ) ) ) ).

% cpropagated_simps(3)
thf(fact_231_map__ran__simps_I2_J,axiom,
    ! [F2: nat > nat > nat,K3: nat,V3: nat,Ps2: list_P6011104703257516679at_nat] :
      ( ( map_ran_nat_nat_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Ps2 ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ ( F2 @ K3 @ V3 ) ) @ ( map_ran_nat_nat_nat @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_232_map__ran__simps_I2_J,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K3: product_prod_nat_nat,V3: product_prod_nat_nat,Ps2: list_P8469869581646625389at_nat] :
      ( ( map_ra4011674865287502026at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ Ps2 ) )
      = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ ( F2 @ K3 @ V3 ) ) @ ( map_ra4011674865287502026at_nat @ F2 @ Ps2 ) ) ) ).

% map_ran_simps(2)
thf(fact_233_lexord__append__left__rightI,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat,U: list_nat,X3: list_nat,Y: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X3 ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y ) ) ) @ ( lexord_nat @ R4 ) ) ) ).

% lexord_append_left_rightI
thf(fact_234_lexord__append__left__rightI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat,U: list_P6011104703257516679at_nat,X3: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R4 )
     => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ U @ ( cons_P6512896166579812791at_nat @ A @ X3 ) ) @ ( append985823374593552924at_nat @ U @ ( cons_P6512896166579812791at_nat @ B @ Y ) ) ) @ ( lexord2841853652668343668at_nat @ R4 ) ) ) ).

% lexord_append_left_rightI
thf(fact_235_Cons__listrel1__Cons,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys3 ) ) @ ( listrel1_nat @ R4 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R4 )
          & ( Xs = Ys3 ) )
        | ( ( X3 = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( listrel1_nat @ R4 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_236_Cons__listrel1__Cons,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) @ ( listre4828114922151135584at_nat @ R4 ) )
      = ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R4 )
          & ( Xs = Ys3 ) )
        | ( ( X3 = Y )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys3 ) @ ( listre4828114922151135584at_nat @ R4 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_237_append1__eq__conv,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
      ( ( ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) )
        = ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) )
      = ( ( Xs = Ys3 )
        & ( X3 = Y ) ) ) ).

% append1_eq_conv
thf(fact_238_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_P6011104703257516679at_nat,E2: product_prod_nat_nat,L23: list_P6011104703257516679at_nat,E3: product_prod_nat_nat] :
      ( ( ( append985823374593552924at_nat @ L12 @ ( cons_P6512896166579812791at_nat @ E2 @ L23 ) )
        = ( cons_P6512896166579812791at_nat @ E3 @ nil_Pr5478986624290739719at_nat ) )
      = ( ( L12 = nil_Pr5478986624290739719at_nat )
        & ( E2 = E3 )
        & ( L23 = nil_Pr5478986624290739719at_nat ) ) ) ).

% list_e_eq_lel(2)
thf(fact_239_list__e__eq__lel_I1_J,axiom,
    ! [E3: product_prod_nat_nat,L12: list_P6011104703257516679at_nat,E2: product_prod_nat_nat,L23: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ E3 @ nil_Pr5478986624290739719at_nat )
        = ( append985823374593552924at_nat @ L12 @ ( cons_P6512896166579812791at_nat @ E2 @ L23 ) ) )
      = ( ( L12 = nil_Pr5478986624290739719at_nat )
        & ( E2 = E3 )
        & ( L23 = nil_Pr5478986624290739719at_nat ) ) ) ).

% list_e_eq_lel(1)
thf(fact_240_list__se__match_I4_J,axiom,
    ! [L23: list_P6011104703257516679at_nat,A: product_prod_nat_nat,L12: list_P6011104703257516679at_nat] :
      ( ( L23 != nil_Pr5478986624290739719at_nat )
     => ( ( ( cons_P6512896166579812791at_nat @ A @ nil_Pr5478986624290739719at_nat )
          = ( append985823374593552924at_nat @ L12 @ L23 ) )
        = ( ( L12 = nil_Pr5478986624290739719at_nat )
          & ( L23
            = ( cons_P6512896166579812791at_nat @ A @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% list_se_match(4)
thf(fact_241_list__se__match_I3_J,axiom,
    ! [L12: list_P6011104703257516679at_nat,A: product_prod_nat_nat,L23: list_P6011104703257516679at_nat] :
      ( ( L12 != nil_Pr5478986624290739719at_nat )
     => ( ( ( cons_P6512896166579812791at_nat @ A @ nil_Pr5478986624290739719at_nat )
          = ( append985823374593552924at_nat @ L12 @ L23 ) )
        = ( ( L12
            = ( cons_P6512896166579812791at_nat @ A @ nil_Pr5478986624290739719at_nat ) )
          & ( L23 = nil_Pr5478986624290739719at_nat ) ) ) ) ).

% list_se_match(3)
thf(fact_242_list__se__match_I2_J,axiom,
    ! [L23: list_P6011104703257516679at_nat,L12: list_P6011104703257516679at_nat,A: product_prod_nat_nat] :
      ( ( L23 != nil_Pr5478986624290739719at_nat )
     => ( ( ( append985823374593552924at_nat @ L12 @ L23 )
          = ( cons_P6512896166579812791at_nat @ A @ nil_Pr5478986624290739719at_nat ) )
        = ( ( L12 = nil_Pr5478986624290739719at_nat )
          & ( L23
            = ( cons_P6512896166579812791at_nat @ A @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% list_se_match(2)
thf(fact_243_list__se__match_I1_J,axiom,
    ! [L12: list_P6011104703257516679at_nat,L23: list_P6011104703257516679at_nat,A: product_prod_nat_nat] :
      ( ( L12 != nil_Pr5478986624290739719at_nat )
     => ( ( ( append985823374593552924at_nat @ L12 @ L23 )
          = ( cons_P6512896166579812791at_nat @ A @ nil_Pr5478986624290739719at_nat ) )
        = ( ( L12
            = ( cons_P6512896166579812791at_nat @ A @ nil_Pr5478986624290739719at_nat ) )
          & ( L23 = nil_Pr5478986624290739719at_nat ) ) ) ) ).

% list_se_match(1)
thf(fact_244_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_P6011104703257516679at_nat,E1: product_prod_nat_nat,E22: product_prod_nat_nat,L23: list_P6011104703257516679at_nat,E12: product_prod_nat_nat,E23: product_prod_nat_nat] :
      ( ( ( append985823374593552924at_nat @ L12 @ ( cons_P6512896166579812791at_nat @ E1 @ ( cons_P6512896166579812791at_nat @ E22 @ L23 ) ) )
        = ( cons_P6512896166579812791at_nat @ E12 @ ( cons_P6512896166579812791at_nat @ E23 @ nil_Pr5478986624290739719at_nat ) ) )
      = ( ( L12 = nil_Pr5478986624290739719at_nat )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_Pr5478986624290739719at_nat ) ) ) ).

% list_ee_eq_leel(2)
thf(fact_245_list__ee__eq__leel_I1_J,axiom,
    ! [E12: product_prod_nat_nat,E23: product_prod_nat_nat,L12: list_P6011104703257516679at_nat,E1: product_prod_nat_nat,E22: product_prod_nat_nat,L23: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ E12 @ ( cons_P6512896166579812791at_nat @ E23 @ nil_Pr5478986624290739719at_nat ) )
        = ( append985823374593552924at_nat @ L12 @ ( cons_P6512896166579812791at_nat @ E1 @ ( cons_P6512896166579812791at_nat @ E22 @ L23 ) ) ) )
      = ( ( L12 = nil_Pr5478986624290739719at_nat )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_Pr5478986624290739719at_nat ) ) ) ).

% list_ee_eq_leel(1)
thf(fact_246_append__Cons,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ Ys3 )
      = ( cons_P6512896166579812791at_nat @ X3 @ ( append985823374593552924at_nat @ Xs @ Ys3 ) ) ) ).

% append_Cons
thf(fact_247_Cons__eq__appendI,axiom,
    ! [X3: product_prod_nat_nat,Xs12: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ X3 @ Xs12 )
        = Ys3 )
     => ( ( Xs
          = ( append985823374593552924at_nat @ Xs12 @ Zs ) )
       => ( ( cons_P6512896166579812791at_nat @ X3 @ Xs )
          = ( append985823374593552924at_nat @ Ys3 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_248_list__match__lel__lel,axiom,
    ! [C1: list_P6011104703257516679at_nat,Qs: product_prod_nat_nat,C2: list_P6011104703257516679at_nat,C12: list_P6011104703257516679at_nat,Qs2: product_prod_nat_nat,C22: list_P6011104703257516679at_nat] :
      ( ( ( append985823374593552924at_nat @ C1 @ ( cons_P6512896166579812791at_nat @ Qs @ C2 ) )
        = ( append985823374593552924at_nat @ C12 @ ( cons_P6512896166579812791at_nat @ Qs2 @ C22 ) ) )
     => ( ! [C21: list_P6011104703257516679at_nat] :
            ( ( C1
              = ( append985823374593552924at_nat @ C12 @ ( cons_P6512896166579812791at_nat @ Qs2 @ C21 ) ) )
           => ( C22
             != ( append985823374593552924at_nat @ C21 @ ( cons_P6512896166579812791at_nat @ Qs @ C2 ) ) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C22 != C2 ) ) )
         => ~ ! [C212: list_P6011104703257516679at_nat] :
                ( ( C12
                  = ( append985823374593552924at_nat @ C1 @ ( cons_P6512896166579812791at_nat @ Qs @ C212 ) ) )
               => ( C2
                 != ( append985823374593552924at_nat @ C212 @ ( cons_P6512896166579812791at_nat @ Qs2 @ C22 ) ) ) ) ) ) ) ).

% list_match_lel_lel
thf(fact_249_listrel1E,axiom,
    ! [Xs: list_nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( listrel1_nat @ R4 ) )
     => ~ ! [X4: nat,Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R4 )
           => ! [Us: list_nat,Vs: list_nat] :
                ( ( Xs
                  = ( append_nat @ Us @ ( cons_nat @ X4 @ Vs ) ) )
               => ( Ys3
                 != ( append_nat @ Us @ ( cons_nat @ Y4 @ Vs ) ) ) ) ) ) ).

% listrel1E
thf(fact_250_listrel1E,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys3 ) @ ( listre4828114922151135584at_nat @ R4 ) )
     => ~ ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
            ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R4 )
           => ! [Us: list_P6011104703257516679at_nat,Vs: list_P6011104703257516679at_nat] :
                ( ( Xs
                  = ( append985823374593552924at_nat @ Us @ ( cons_P6512896166579812791at_nat @ X4 @ Vs ) ) )
               => ( Ys3
                 != ( append985823374593552924at_nat @ Us @ ( cons_P6512896166579812791at_nat @ Y4 @ Vs ) ) ) ) ) ) ).

% listrel1E
thf(fact_251_listrel1I,axiom,
    ! [X3: nat,Y: nat,R4: set_Pr1261947904930325089at_nat,Xs: list_nat,Us2: list_nat,Vs2: list_nat,Ys3: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R4 )
     => ( ( Xs
          = ( append_nat @ Us2 @ ( cons_nat @ X3 @ Vs2 ) ) )
       => ( ( Ys3
            = ( append_nat @ Us2 @ ( cons_nat @ Y @ Vs2 ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( listrel1_nat @ R4 ) ) ) ) ) ).

% listrel1I
thf(fact_252_listrel1I,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Us2: list_P6011104703257516679at_nat,Vs2: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R4 )
     => ( ( Xs
          = ( append985823374593552924at_nat @ Us2 @ ( cons_P6512896166579812791at_nat @ X3 @ Vs2 ) ) )
       => ( ( Ys3
            = ( append985823374593552924at_nat @ Us2 @ ( cons_P6512896166579812791at_nat @ Y @ Vs2 ) ) )
         => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys3 ) @ ( listre4828114922151135584at_nat @ R4 ) ) ) ) ) ).

% listrel1I
thf(fact_253_listrelp_OCons,axiom,
    ! [R4: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( R4 @ X3 @ Y )
     => ( ( listre5588581128925307775at_nat @ R4 @ Xs @ Ys3 )
       => ( listre5588581128925307775at_nat @ R4 @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) ) ) ).

% listrelp.Cons
thf(fact_254_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_nat,X3: nat,Ys3: list_nat,Y: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) @ ( append_nat @ Ys3 @ ( cons_nat @ Y @ nil_nat ) ) ) @ ( listrel1_nat @ R4 ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( listrel1_nat @ R4 ) )
          & ( X3 = Y ) )
        | ( ( Xs = Ys3 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R4 ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_255_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) @ ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) ) @ ( listre4828114922151135584at_nat @ R4 ) )
      = ( ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys3 ) @ ( listre4828114922151135584at_nat @ R4 ) )
          & ( X3 = Y ) )
        | ( ( Xs = Ys3 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R4 ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_256_rev__nonempty__induct,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > $o] :
      ( ( Xs != nil_Pr5478986624290739719at_nat )
     => ( ! [X4: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) )
       => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( Xs2 != nil_Pr5478986624290739719at_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_257_append__eq__Cons__conv,axiom,
    ! [Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( append985823374593552924at_nat @ Ys3 @ Zs )
        = ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( ( ( Ys3 = nil_Pr5478986624290739719at_nat )
          & ( Zs
            = ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) )
        | ? [Ys4: list_P6011104703257516679at_nat] :
            ( ( Ys3
              = ( cons_P6512896166579812791at_nat @ X3 @ Ys4 ) )
            & ( ( append985823374593552924at_nat @ Ys4 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_258_Cons__eq__append__conv,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ X3 @ Xs )
        = ( append985823374593552924at_nat @ Ys3 @ Zs ) )
      = ( ( ( Ys3 = nil_Pr5478986624290739719at_nat )
          & ( ( cons_P6512896166579812791at_nat @ X3 @ Xs )
            = Zs ) )
        | ? [Ys4: list_P6011104703257516679at_nat] :
            ( ( ( cons_P6512896166579812791at_nat @ X3 @ Ys4 )
              = Ys3 )
            & ( Xs
              = ( append985823374593552924at_nat @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_259_rev__exhaust,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( Xs != nil_Pr5478986624290739719at_nat )
     => ~ ! [Ys: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat] :
            ( Xs
           != ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) ) ) ) ).

% rev_exhaust
thf(fact_260_rev__induct,axiom,
    ! [P: list_P6011104703257516679at_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( P @ nil_Pr5478986624290739719at_nat )
     => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( ( P @ Xs2 )
           => ( P @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_261_neq__Nil__revE,axiom,
    ! [L: list_P6011104703257516679at_nat] :
      ( ( L != nil_Pr5478986624290739719at_nat )
     => ~ ! [Ll: list_P6011104703257516679at_nat,E: product_prod_nat_nat] :
            ( L
           != ( append985823374593552924at_nat @ Ll @ ( cons_P6512896166579812791at_nat @ E @ nil_Pr5478986624290739719at_nat ) ) ) ) ).

% neq_Nil_revE
thf(fact_262_rev__induct2_H,axiom,
    ! [P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
     => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] : ( P @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) ) @ nil_Pr5478986624290739719at_nat )
       => ( ! [Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) ) )
         => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
                ( ( P @ Xs2 @ Ys )
               => ( P @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) ) @ ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) ) ) )
           => ( P @ Xs @ Ys3 ) ) ) ) ) ).

% rev_induct2'
thf(fact_263_neq__Nil__rev__conv,axiom,
    ! [L: list_P6011104703257516679at_nat] :
      ( ( L != nil_Pr5478986624290739719at_nat )
      = ( ? [Xs3: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
            ( L
            = ( append985823374593552924at_nat @ Xs3 @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% neq_Nil_rev_conv
thf(fact_264_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( Xs != nil_Pr5478986624290739719at_nat )
     => ( ( Ys3 != nil_Pr5478986624290739719at_nat )
       => ( ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] : ( P @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) )
         => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat] :
                ( ( Xs2 != nil_Pr5478986624290739719at_nat )
               => ( P @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) ) @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) ) )
           => ( ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
                  ( ( Ys != nil_Pr5478986624290739719at_nat )
                 => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) @ ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) ) ) )
             => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
                    ( ( P @ Xs2 @ Ys )
                   => ( ( Xs2 != nil_Pr5478986624290739719at_nat )
                     => ( ( Ys != nil_Pr5478986624290739719at_nat )
                       => ( P @ ( append985823374593552924at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) ) @ ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) ) ) ) ) )
               => ( P @ Xs @ Ys3 ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_265_list__Cons__eq__append__cases,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ X3 @ Xs )
        = ( append985823374593552924at_nat @ Ys3 @ Zs ) )
     => ( ( ( Ys3 = nil_Pr5478986624290739719at_nat )
         => ( Zs
           != ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) )
       => ~ ! [Ys5: list_P6011104703257516679at_nat] :
              ( ( Ys3
                = ( cons_P6512896166579812791at_nat @ X3 @ Ys5 ) )
             => ( ( append985823374593552924at_nat @ Ys5 @ Zs )
               != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
thf(fact_266_list__append__eq__Cons__cases,axiom,
    ! [Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( append985823374593552924at_nat @ Ys3 @ Zs )
        = ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
     => ( ( ( Ys3 = nil_Pr5478986624290739719at_nat )
         => ( Zs
           != ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) )
       => ~ ! [Ys5: list_P6011104703257516679at_nat] :
              ( ( Ys3
                = ( cons_P6512896166579812791at_nat @ X3 @ Ys5 ) )
             => ( ( append985823374593552924at_nat @ Ys5 @ Zs )
               != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
thf(fact_267_listrel1I2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat,X3: product_prod_nat_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys3 ) @ ( listre4828114922151135584at_nat @ R4 ) )
     => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ X3 @ Ys3 ) ) @ ( listre4828114922151135584at_nat @ R4 ) ) ) ).

% listrel1I2
thf(fact_268_lexord__append__leftD,axiom,
    ! [X3: list_nat,U: list_nat,V3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X3 @ U ) @ ( append_nat @ X3 @ V3 ) ) @ ( lexord_nat @ R4 ) )
     => ( ! [A5: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ A5 ) @ R4 )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V3 ) @ ( lexord_nat @ R4 ) ) ) ) ).

% lexord_append_leftD
thf(fact_269_lexord__append__leftD,axiom,
    ! [X3: list_P6011104703257516679at_nat,U: list_P6011104703257516679at_nat,V3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ X3 @ U ) @ ( append985823374593552924at_nat @ X3 @ V3 ) ) @ ( lexord2841853652668343668at_nat @ R4 ) )
     => ( ! [A5: product_prod_nat_nat] :
            ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A5 @ A5 ) @ R4 )
       => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ U @ V3 ) @ ( lexord2841853652668343668at_nat @ R4 ) ) ) ) ).

% lexord_append_leftD
thf(fact_270_lexord__append__rightI,axiom,
    ! [Y: list_P6011104703257516679at_nat,X3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ? [B5: product_prod_nat_nat,Z2: list_P6011104703257516679at_nat] :
          ( Y
          = ( cons_P6512896166579812791at_nat @ B5 @ Z2 ) )
     => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X3 @ ( append985823374593552924at_nat @ X3 @ Y ) ) @ ( lexord2841853652668343668at_nat @ R4 ) ) ) ).

% lexord_append_rightI
thf(fact_271_listrel1I1,axiom,
    ! [X3: nat,Y: nat,R4: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R4 )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Xs ) ) @ ( listrel1_nat @ R4 ) ) ) ).

% listrel1I1
thf(fact_272_listrel1I1,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R4 )
     => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) @ ( listre4828114922151135584at_nat @ R4 ) ) ) ).

% listrel1I1
thf(fact_273_Cons__listrel1E1,axiom,
    ! [X3: nat,Xs: list_nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ Ys3 ) @ ( listrel1_nat @ R4 ) )
     => ( ! [Y4: nat] :
            ( ( Ys3
              = ( cons_nat @ Y4 @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R4 ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Ys3
                = ( cons_nat @ X3 @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( listrel1_nat @ R4 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_274_Cons__listrel1E1,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ Ys3 ) @ ( listre4828114922151135584at_nat @ R4 ) )
     => ( ! [Y4: product_prod_nat_nat] :
            ( ( Ys3
              = ( cons_P6512896166579812791at_nat @ Y4 @ Xs ) )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y4 ) @ R4 ) )
       => ~ ! [Zs2: list_P6011104703257516679at_nat] :
              ( ( Ys3
                = ( cons_P6512896166579812791at_nat @ X3 @ Zs2 ) )
             => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Zs2 ) @ ( listre4828114922151135584at_nat @ R4 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_275_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y: nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys3 ) ) @ ( listrel1_nat @ R4 ) )
     => ( ! [X4: nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Ys3 ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y ) @ R4 ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys3 ) @ ( listrel1_nat @ R4 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_276_Cons__listrel1E2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) @ ( listre4828114922151135584at_nat @ R4 ) )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ X4 @ Ys3 ) )
           => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y ) @ R4 ) )
       => ~ ! [Zs2: list_P6011104703257516679at_nat] :
              ( ( Xs
                = ( cons_P6512896166579812791at_nat @ Y @ Zs2 ) )
             => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Zs2 @ Ys3 ) @ ( listre4828114922151135584at_nat @ R4 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_277_listrelp_Osimps,axiom,
    ( listre5588581128925307775at_nat
    = ( ^ [R5: product_prod_nat_nat > product_prod_nat_nat > $o,A12: list_P6011104703257516679at_nat,A23: list_P6011104703257516679at_nat] :
          ( ( ( A12 = nil_Pr5478986624290739719at_nat )
            & ( A23 = nil_Pr5478986624290739719at_nat ) )
          | ? [X: product_prod_nat_nat,Y3: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
              ( ( A12
                = ( cons_P6512896166579812791at_nat @ X @ Xs3 ) )
              & ( A23
                = ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) )
              & ( R5 @ X @ Y3 )
              & ( listre5588581128925307775at_nat @ R5 @ Xs3 @ Ys2 ) ) ) ) ) ).

% listrelp.simps
thf(fact_278_listrelp_Ocases,axiom,
    ! [R4: product_prod_nat_nat > product_prod_nat_nat > $o,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat] :
      ( ( listre5588581128925307775at_nat @ R4 @ A1 @ A22 )
     => ( ( ( A1 = nil_Pr5478986624290739719at_nat )
         => ( A22 != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( A1
                = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
             => ! [Ys: list_P6011104703257516679at_nat] :
                  ( ( A22
                    = ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
                 => ( ( R4 @ X4 @ Y4 )
                   => ~ ( listre5588581128925307775at_nat @ R4 @ Xs2 @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_279_op__list__append__elem__def,axiom,
    ( autore7828122933726872089at_nat
    = ( ^ [S2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] : ( append985823374593552924at_nat @ S2 @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) ) ) ).

% op_list_append_elem_def
thf(fact_280_Conjs__pull__out,axiom,
    ! [Q: relational_fmla_a_b,Xys: list_P6011104703257516679at_nat,X3: nat,Y: nat,Xys3: list_P6011104703257516679at_nat] : ( relational_equiv_a_b @ ( restrict_Conjs_a_b @ Q @ ( append985823374593552924at_nat @ Xys @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ Xys3 ) ) ) @ ( restrict_Conjs_a_b @ ( relational_Conj_a_b @ Q @ ( relational_Eq_a_b @ X3 @ ( relational_Var_a @ Y ) ) ) @ ( append985823374593552924at_nat @ Xys @ Xys3 ) ) ) ).

% Conjs_pull_out
thf(fact_281_quicksort__by__rel_Osimps_I2_J,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o,Sl2: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( quicks6042341746770278769at_nat @ R @ Sl2 @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( produc390734169182146075at_nat
        @ ^ [Xs_s: list_P6011104703257516679at_nat,Xs_b: list_P6011104703257516679at_nat] : ( quicks6042341746770278769at_nat @ R @ ( cons_P6512896166579812791at_nat @ X3 @ ( quicks6042341746770278769at_nat @ R @ Sl2 @ Xs_b ) ) @ Xs_s )
        @ ( partit1285214537004407631at_nat
          @ ^ [Y3: product_prod_nat_nat] : ( R @ Y3 @ X3 )
          @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
          @ Xs ) ) ) ).

% quicksort_by_rel.simps(2)
thf(fact_282_quicksort__by__rel_Oelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > $o,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( quicks6042341746770278769at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xb = nil_Pr5478986624290739719at_nat )
         => ( Y != Xa2 ) )
       => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( Xb
                = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
             => ( Y
               != ( produc390734169182146075at_nat
                  @ ^ [Xs_s: list_P6011104703257516679at_nat,Xs_b: list_P6011104703257516679at_nat] : ( quicks6042341746770278769at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ X4 @ ( quicks6042341746770278769at_nat @ X3 @ Xa2 @ Xs_b ) ) @ Xs_s )
                  @ ( partit1285214537004407631at_nat
                    @ ^ [Y3: product_prod_nat_nat] : ( X3 @ Y3 @ X4 )
                    @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
                    @ Xs2 ) ) ) ) ) ) ).

% quicksort_by_rel.elims
thf(fact_283_gen__length__code_I2_J,axiom,
    ! [N: nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( gen_le2383899666085517716at_nat @ N @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( gen_le2383899666085517716at_nat @ ( suc @ N ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_284_equiv__Conj__cong,axiom,
    ! [Q1: relational_fmla_a_b,Q12: relational_fmla_a_b,Q2: relational_fmla_a_b,Q22: relational_fmla_a_b] :
      ( ( relational_equiv_a_b @ Q1 @ Q12 )
     => ( ( relational_equiv_a_b @ Q2 @ Q22 )
       => ( relational_equiv_a_b @ ( relational_Conj_a_b @ Q1 @ Q2 ) @ ( relational_Conj_a_b @ Q12 @ Q22 ) ) ) ) ).

% equiv_Conj_cong
thf(fact_285_equiv__Conjs__cong,axiom,
    ! [Q: relational_fmla_a_b,Q3: relational_fmla_a_b,Xys: list_P6011104703257516679at_nat] :
      ( ( relational_equiv_a_b @ Q @ Q3 )
     => ( relational_equiv_a_b @ ( restrict_Conjs_a_b @ Q @ Xys ) @ ( restrict_Conjs_a_b @ Q3 @ Xys ) ) ) ).

% equiv_Conjs_cong
thf(fact_286_param__case__prod_H_H,axiom,
    ! [P2: product_prod_nat_nat,P5: product_prod_nat_nat,F2: nat > nat > nat,F3: nat > nat > nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat,B4: nat,A7: nat,B6: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( ( P5
              = ( product_Pair_nat_nat @ A7 @ B6 ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A7 @ B6 ) ) @ R ) ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( produc6842872674320459806at_nat @ F2 @ P2 ) @ ( produc6842872674320459806at_nat @ F3 @ P5 ) ) @ R ) ) ).

% param_case_prod''
thf(fact_287_param__case__prod_H_H,axiom,
    ! [P2: product_prod_nat_nat,P5: product_prod_nat_nat,F2: nat > nat > product_prod_nat_nat,F3: nat > nat > product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ! [A5: nat,B4: nat,A7: nat,B6: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( ( P5
              = ( product_Pair_nat_nat @ A7 @ B6 ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A7 @ B6 ) ) @ R ) ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( produc2626176000494625587at_nat @ F2 @ P2 ) @ ( produc2626176000494625587at_nat @ F3 @ P5 ) ) @ R ) ) ).

% param_case_prod''
thf(fact_288_param__case__prod_H_H,axiom,
    ! [P2: product_prod_nat_nat,P5: produc859450856879609959at_nat,F2: nat > nat > nat,F3: product_prod_nat_nat > product_prod_nat_nat > nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat,B4: nat,A7: product_prod_nat_nat,B6: product_prod_nat_nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( ( P5
              = ( produc6161850002892822231at_nat @ A7 @ B6 ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A7 @ B6 ) ) @ R ) ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( produc6842872674320459806at_nat @ F2 @ P2 ) @ ( produc6237124055692578492at_nat @ F3 @ P5 ) ) @ R ) ) ).

% param_case_prod''
thf(fact_289_param__case__prod_H_H,axiom,
    ! [P2: product_prod_nat_nat,P5: produc859450856879609959at_nat,F2: nat > nat > product_prod_nat_nat,F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ! [A5: nat,B4: nat,A7: product_prod_nat_nat,B6: product_prod_nat_nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( ( P5
              = ( produc6161850002892822231at_nat @ A7 @ B6 ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A7 @ B6 ) ) @ R ) ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( produc2626176000494625587at_nat @ F2 @ P2 ) @ ( produc2982569246487150357at_nat @ F3 @ P5 ) ) @ R ) ) ).

% param_case_prod''
thf(fact_290_param__case__prod_H_H,axiom,
    ! [P2: produc859450856879609959at_nat,P5: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat > nat,F3: nat > nat > nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat,A7: nat,B6: nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( ( P5
              = ( product_Pair_nat_nat @ A7 @ B6 ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A7 @ B6 ) ) @ R ) ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( produc6237124055692578492at_nat @ F2 @ P2 ) @ ( produc6842872674320459806at_nat @ F3 @ P5 ) ) @ R ) ) ).

% param_case_prod''
thf(fact_291_param__case__prod_H_H,axiom,
    ! [P2: produc859450856879609959at_nat,P5: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,F3: nat > nat > product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat,A7: nat,B6: nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( ( P5
              = ( product_Pair_nat_nat @ A7 @ B6 ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A7 @ B6 ) ) @ R ) ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( produc2982569246487150357at_nat @ F2 @ P2 ) @ ( produc2626176000494625587at_nat @ F3 @ P5 ) ) @ R ) ) ).

% param_case_prod''
thf(fact_292_param__case__prod_H_H,axiom,
    ! [P2: produc859450856879609959at_nat,P5: produc859450856879609959at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > nat,F3: product_prod_nat_nat > product_prod_nat_nat > nat,R: set_Pr1261947904930325089at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat,A7: product_prod_nat_nat,B6: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( ( P5
              = ( produc6161850002892822231at_nat @ A7 @ B6 ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A7 @ B6 ) ) @ R ) ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( produc6237124055692578492at_nat @ F2 @ P2 ) @ ( produc6237124055692578492at_nat @ F3 @ P5 ) ) @ R ) ) ).

% param_case_prod''
thf(fact_293_param__case__prod_H_H,axiom,
    ! [P2: produc859450856879609959at_nat,P5: produc859450856879609959at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat,A7: product_prod_nat_nat,B6: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( ( P5
              = ( produc6161850002892822231at_nat @ A7 @ B6 ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A7 @ B6 ) ) @ R ) ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( produc2982569246487150357at_nat @ F2 @ P2 ) @ ( produc2982569246487150357at_nat @ F3 @ P5 ) ) @ R ) ) ).

% param_case_prod''
thf(fact_294_equiv__eval__eqI,axiom,
    ! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q3: relational_fmla_a_b] :
      ( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
     => ( ( ( relational_fv_a_b @ Q )
          = ( relational_fv_a_b @ Q3 ) )
       => ( ( relational_equiv_a_b @ Q @ Q3 )
         => ( ( relational_eval_a_b @ Q @ I )
            = ( relational_eval_a_b @ Q3 @ I ) ) ) ) ) ).

% equiv_eval_eqI
thf(fact_295_mem__case__prodI,axiom,
    ! [Z3: nat,C3: nat > nat > set_nat,A: nat,B: nat] :
      ( ( member_nat @ Z3 @ ( C3 @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc6189476227299908564et_nat @ C3 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_296_mem__case__prodI,axiom,
    ! [Z3: produc859450856879609959at_nat,C3: nat > nat > set_Pr8693737435421807431at_nat,A: nat,B: nat] :
      ( ( member8206827879206165904at_nat @ Z3 @ ( C3 @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z3 @ ( produc2423807293333670521at_nat @ C3 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_297_mem__case__prodI,axiom,
    ! [Z3: product_prod_nat_nat,C3: nat > nat > set_Pr1261947904930325089at_nat,A: nat,B: nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( C3 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc8197505143624133779at_nat @ C3 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_298_mem__case__prodI,axiom,
    ! [Z3: nat,C3: product_prod_nat_nat > product_prod_nat_nat > set_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member_nat @ Z3 @ ( C3 @ A @ B ) )
     => ( member_nat @ Z3 @ ( produc1370685125915216498et_nat @ C3 @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_299_mem__case__prodI,axiom,
    ! [Z3: produc859450856879609959at_nat,C3: product_prod_nat_nat > product_prod_nat_nat > set_Pr8693737435421807431at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z3 @ ( C3 @ A @ B ) )
     => ( member8206827879206165904at_nat @ Z3 @ ( produc773927175282099035at_nat @ C3 @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_300_mem__case__prodI,axiom,
    ! [Z3: product_prod_nat_nat,C3: product_prod_nat_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( C3 @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc3380114247675707509at_nat @ C3 @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_301_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z3: nat,C3: nat > nat > set_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member_nat @ Z3 @ ( C3 @ A5 @ B4 ) ) )
     => ( member_nat @ Z3 @ ( produc6189476227299908564et_nat @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_302_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z3: produc859450856879609959at_nat,C3: nat > nat > set_Pr8693737435421807431at_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member8206827879206165904at_nat @ Z3 @ ( C3 @ A5 @ B4 ) ) )
     => ( member8206827879206165904at_nat @ Z3 @ ( produc2423807293333670521at_nat @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_303_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,Z3: product_prod_nat_nat,C3: nat > nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( member8440522571783428010at_nat @ Z3 @ ( C3 @ A5 @ B4 ) ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc8197505143624133779at_nat @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_304_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z3: nat,C3: product_prod_nat_nat > product_prod_nat_nat > set_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( member_nat @ Z3 @ ( C3 @ A5 @ B4 ) ) )
     => ( member_nat @ Z3 @ ( produc1370685125915216498et_nat @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_305_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z3: produc859450856879609959at_nat,C3: product_prod_nat_nat > product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( member8206827879206165904at_nat @ Z3 @ ( C3 @ A5 @ B4 ) ) )
     => ( member8206827879206165904at_nat @ Z3 @ ( produc773927175282099035at_nat @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_306_mem__case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,Z3: product_prod_nat_nat,C3: product_prod_nat_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( member8440522571783428010at_nat @ Z3 @ ( C3 @ A5 @ B4 ) ) )
     => ( member8440522571783428010at_nat @ Z3 @ ( produc3380114247675707509at_nat @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_307_case__prodI,axiom,
    ! [F2: nat > nat > $o,A: nat,B: nat] :
      ( ( F2 @ A @ B )
     => ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_308_case__prodI,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( F2 @ A @ B )
     => ( produc6590410687421337004_nat_o @ F2 @ ( produc6161850002892822231at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_309_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C3: nat > nat > $o] :
      ( ! [A5: nat,B4: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( C3 @ A5 @ B4 ) )
     => ( produc6081775807080527818_nat_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_310_case__prodI2,axiom,
    ! [P2: produc859450856879609959at_nat,C3: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( C3 @ A5 @ B4 ) )
     => ( produc6590410687421337004_nat_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_311_mem__case__prodE,axiom,
    ! [Z3: nat,C3: nat > nat > set_nat,P2: product_prod_nat_nat] :
      ( ( member_nat @ Z3 @ ( produc6189476227299908564et_nat @ C3 @ P2 ) )
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y4 ) )
           => ~ ( member_nat @ Z3 @ ( C3 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_312_mem__case__prodE,axiom,
    ! [Z3: produc859450856879609959at_nat,C3: nat > nat > set_Pr8693737435421807431at_nat,P2: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ Z3 @ ( produc2423807293333670521at_nat @ C3 @ P2 ) )
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y4 ) )
           => ~ ( member8206827879206165904at_nat @ Z3 @ ( C3 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_313_mem__case__prodE,axiom,
    ! [Z3: product_prod_nat_nat,C3: nat > nat > set_Pr1261947904930325089at_nat,P2: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( produc8197505143624133779at_nat @ C3 @ P2 ) )
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y4 ) )
           => ~ ( member8440522571783428010at_nat @ Z3 @ ( C3 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_314_mem__case__prodE,axiom,
    ! [Z3: nat,C3: product_prod_nat_nat > product_prod_nat_nat > set_nat,P2: produc859450856879609959at_nat] :
      ( ( member_nat @ Z3 @ ( produc1370685125915216498et_nat @ C3 @ P2 ) )
     => ~ ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X4 @ Y4 ) )
           => ~ ( member_nat @ Z3 @ ( C3 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_315_mem__case__prodE,axiom,
    ! [Z3: produc859450856879609959at_nat,C3: product_prod_nat_nat > product_prod_nat_nat > set_Pr8693737435421807431at_nat,P2: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ Z3 @ ( produc773927175282099035at_nat @ C3 @ P2 ) )
     => ~ ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X4 @ Y4 ) )
           => ~ ( member8206827879206165904at_nat @ Z3 @ ( C3 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_316_mem__case__prodE,axiom,
    ! [Z3: product_prod_nat_nat,C3: product_prod_nat_nat > product_prod_nat_nat > set_Pr1261947904930325089at_nat,P2: produc859450856879609959at_nat] :
      ( ( member8440522571783428010at_nat @ Z3 @ ( produc3380114247675707509at_nat @ C3 @ P2 ) )
     => ~ ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X4 @ Y4 ) )
           => ~ ( member8440522571783428010at_nat @ Z3 @ ( C3 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_317_case__prodD,axiom,
    ! [F2: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F2 @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_318_case__prodD,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( produc6590410687421337004_nat_o @ F2 @ ( produc6161850002892822231at_nat @ A @ B ) )
     => ( F2 @ A @ B ) ) ).

% case_prodD
thf(fact_319_case__prodE,axiom,
    ! [C3: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C3 @ P2 )
     => ~ ! [X4: nat,Y4: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X4 @ Y4 ) )
           => ~ ( C3 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_320_case__prodE,axiom,
    ! [C3: product_prod_nat_nat > product_prod_nat_nat > $o,P2: produc859450856879609959at_nat] :
      ( ( produc6590410687421337004_nat_o @ C3 @ P2 )
     => ~ ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
            ( ( P2
              = ( produc6161850002892822231at_nat @ X4 @ Y4 ) )
           => ~ ( C3 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_321_listrel__def,axiom,
    ( listrel_nat_nat
    = ( ^ [R5: set_Pr1261947904930325089at_nat] :
          ( collec1570431334306492044st_nat
          @ ( produc6382697403170970154_nat_o
            @ ( listrelp_nat_nat
              @ ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R5 ) ) ) ) ) ) ).

% listrel_def
thf(fact_322_listrel__def,axiom,
    ( listre818007680106770737at_nat
    = ( ^ [R5: set_Pr8693737435421807431at_nat] :
          ( collec2094671920101799250at_nat
          @ ( produc2929669097083596076_nat_o
            @ ( listre5588581128925307775at_nat
              @ ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R5 ) ) ) ) ) ) ).

% listrel_def
thf(fact_323_EVAL__def,axiom,
    ( restrict_EVAL_a_b
    = ( ^ [Q4: relational_fmla_a_b,Qfin: relational_fmla_a_b,Qinf: relational_fmla_a_b] :
        ! [I2: product_prod_b_nat > set_list_a] :
          ( ( finite_finite_a @ ( relational_adom_b_a @ I2 ) )
         => ( ( ( ( relational_eval_a_b @ Qinf @ I2 )
                = bot_bot_set_list_a )
             => ( ( relational_eval_a_b @ Qfin @ I2 )
                = ( relational_eval_a_b @ Q4 @ I2 ) ) )
            & ( ( ( relational_eval_a_b @ Qinf @ I2 )
               != bot_bot_set_list_a )
             => ~ ( finite_finite_list_a @ ( relational_eval_a_b @ Q4 @ I2 ) ) ) ) ) ) ) ).

% EVAL_def
thf(fact_324_part__code_I2_J,axiom,
    ! [F2: product_prod_nat_nat > nat,Pivot: nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( linord9222846411292336904at_nat @ F2 @ Pivot @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( produc4082587324116209787at_nat
        @ ^ [Lts: list_P6011104703257516679at_nat] :
            ( produc1862374959283844315at_nat
            @ ^ [Eqs: list_P6011104703257516679at_nat,Gts: list_P6011104703257516679at_nat] : ( if_Pro5163289783556962317at_nat @ ( ord_less_nat @ ( F2 @ X3 ) @ Pivot ) @ ( produc913954598053515191at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Lts ) @ ( produc5943733680697469783at_nat @ Eqs @ Gts ) ) @ ( if_Pro5163289783556962317at_nat @ ( ord_less_nat @ Pivot @ ( F2 @ X3 ) ) @ ( produc913954598053515191at_nat @ Lts @ ( produc5943733680697469783at_nat @ Eqs @ ( cons_P6512896166579812791at_nat @ X3 @ Gts ) ) ) @ ( produc913954598053515191at_nat @ Lts @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Eqs ) @ Gts ) ) ) ) )
        @ ( linord9222846411292336904at_nat @ F2 @ Pivot @ Xs ) ) ) ).

% part_code(2)
thf(fact_325_finite__Collect__less__nat,axiom,
    ! [K3: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K3 ) ) ) ).

% finite_Collect_less_nat
thf(fact_326_memb__imp__not__empty,axiom,
    ! [X3: produc859450856879609959at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ S )
     => ( S != bot_bo5327735625951526323at_nat ) ) ).

% memb_imp_not_empty
thf(fact_327_memb__imp__not__empty,axiom,
    ! [X3: product_prod_nat_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ S )
     => ( S != bot_bo2099793752762293965at_nat ) ) ).

% memb_imp_not_empty
thf(fact_328_memb__imp__not__empty,axiom,
    ! [X3: nat,S: set_nat] :
      ( ( member_nat @ X3 @ S )
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_329_set__notEmptyE,axiom,
    ! [S: set_Pr8693737435421807431at_nat] :
      ( ( S != bot_bo5327735625951526323at_nat )
     => ~ ! [X4: produc859450856879609959at_nat] :
            ~ ( member8206827879206165904at_nat @ X4 @ S ) ) ).

% set_notEmptyE
thf(fact_330_set__notEmptyE,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ~ ! [X4: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X4 @ S ) ) ).

% set_notEmptyE
thf(fact_331_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X4: nat] :
            ~ ( member_nat @ X4 @ S ) ) ).

% set_notEmptyE
thf(fact_332_finite__psubset__induct,axiom,
    ! [A3: set_list_a,P: set_list_a > $o] :
      ( ( finite_finite_list_a @ A3 )
     => ( ! [A6: set_list_a] :
            ( ( finite_finite_list_a @ A6 )
           => ( ! [B7: set_list_a] :
                  ( ( ord_less_set_list_a @ B7 @ A6 )
                 => ( P @ B7 ) )
             => ( P @ A6 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_333_finite__psubset__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A6: set_nat] :
            ( ( finite_finite_nat @ A6 )
           => ( ! [B7: set_nat] :
                  ( ( ord_less_set_nat @ B7 @ A6 )
                 => ( P @ B7 ) )
             => ( P @ A6 ) ) )
       => ( P @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_334_infinite__imp__nonempty,axiom,
    ! [S: set_list_a] :
      ( ~ ( finite_finite_list_a @ S )
     => ( S != bot_bot_set_list_a ) ) ).

% infinite_imp_nonempty
thf(fact_335_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_336_finite_OemptyI,axiom,
    finite_finite_list_a @ bot_bot_set_list_a ).

% finite.emptyI
thf(fact_337_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_338_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ N4 )
         => ( ord_less_nat @ X4 @ N ) )
     => ( finite_finite_nat @ N4 ) ) ).

% bounded_nat_set_is_finite
thf(fact_339_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M2: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N5 )
         => ( ord_less_nat @ X @ M2 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_340_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I3: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K5: nat] :
            ( ( P @ K5 )
            & ( ord_less_nat @ K5 @ I3 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_341_case__prod__Pair__iden,axiom,
    ! [P2: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_342_case__prod__Pair__iden,axiom,
    ! [P2: produc859450856879609959at_nat] :
      ( ( produc6744312701629110395at_nat @ produc6161850002892822231at_nat @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_343_Cons__less__Cons,axiom,
    ! [A: product_prod_nat_nat,X3: list_P6011104703257516679at_nat,B: product_prod_nat_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ord_le7035401960408590643at_nat @ ( cons_P6512896166579812791at_nat @ A @ X3 ) @ ( cons_P6512896166579812791at_nat @ B @ Y ) )
      = ( ( ord_le1203424502768444845at_nat @ A @ B )
        | ( ( A = B )
          & ( ord_le7035401960408590643at_nat @ X3 @ Y ) ) ) ) ).

% Cons_less_Cons
thf(fact_344_Cons__less__Cons,axiom,
    ! [A: nat,X3: list_nat,B: nat,Y: list_nat] :
      ( ( ord_less_list_nat @ ( cons_nat @ A @ X3 ) @ ( cons_nat @ B @ Y ) )
      = ( ( ord_less_nat @ A @ B )
        | ( ( A = B )
          & ( ord_less_list_nat @ X3 @ Y ) ) ) ) ).

% Cons_less_Cons
thf(fact_345_list__less__def,axiom,
    ( ord_less_list_nat
    = ( ^ [Xs3: list_nat,Ys2: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( lexord_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ) ) ) ).

% list_less_def
thf(fact_346_EVAL_H__def,axiom,
    ( restrict_EVAL_a_b2
    = ( ^ [Q4: relational_fmla_a_b,Qfin: relational_fmla_a_b,Qinf: relational_fmla_a_b] :
        ! [I2: product_prod_b_nat > set_list_a] :
          ( ( finite_finite_a @ ( relational_adom_b_a @ I2 ) )
         => ( ( ( ( relational_eval_a_b @ Qinf @ I2 )
                = bot_bot_set_list_a )
             => ( ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q4 ) @ Qfin @ I2 )
                = ( relational_eval_a_b @ Q4 @ I2 ) ) )
            & ( ( ( relational_eval_a_b @ Qinf @ I2 )
               != bot_bot_set_list_a )
             => ~ ( finite_finite_list_a @ ( relational_eval_a_b @ Q4 @ I2 ) ) ) ) ) ) ) ).

% EVAL'_def
thf(fact_347_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ S )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S )
                  & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_348_infinite__growing,axiom,
    ! [X5: set_Pr8693737435421807431at_nat] :
      ( ( X5 != bot_bo5327735625951526323at_nat )
     => ( ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ X5 )
           => ? [Xa: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Xa @ X5 )
                & ( ord_le9033551061567896339at_nat @ X4 @ Xa ) ) )
       => ~ ( finite4392333629123659920at_nat @ X5 ) ) ) ).

% infinite_growing
thf(fact_349_infinite__growing,axiom,
    ! [X5: set_Pr1261947904930325089at_nat] :
      ( ( X5 != bot_bo2099793752762293965at_nat )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ X5 )
           => ? [Xa: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa @ X5 )
                & ( ord_le1203424502768444845at_nat @ X4 @ Xa ) ) )
       => ~ ( finite6177210948735845034at_nat @ X5 ) ) ) ).

% infinite_growing
thf(fact_350_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X5 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X5 )
                & ( ord_less_nat @ X4 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X5 ) ) ) ).

% infinite_growing
thf(fact_351_finite__transitivity__chain,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,R: produc859450856879609959at_nat > produc859450856879609959at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ A3 )
     => ( ! [X4: produc859450856879609959at_nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: produc859450856879609959at_nat,Y4: produc859450856879609959at_nat,Z: produc859450856879609959at_nat] :
              ( ( R @ X4 @ Y4 )
             => ( ( R @ Y4 @ Z )
               => ( R @ X4 @ Z ) ) )
         => ( ! [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ A3 )
               => ? [Y5: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Y5 @ A3 )
                    & ( R @ X4 @ Y5 ) ) )
           => ( A3 = bot_bo5327735625951526323at_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_352_finite__transitivity__chain,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ! [X4: product_prod_nat_nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Z: product_prod_nat_nat] :
              ( ( R @ X4 @ Y4 )
             => ( ( R @ Y4 @ Z )
               => ( R @ X4 @ Z ) ) )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A3 )
               => ? [Y5: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y5 @ A3 )
                    & ( R @ X4 @ Y5 ) ) )
           => ( A3 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_353_finite__transitivity__chain,axiom,
    ! [A3: set_list_a,R: list_a > list_a > $o] :
      ( ( finite_finite_list_a @ A3 )
     => ( ! [X4: list_a] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: list_a,Y4: list_a,Z: list_a] :
              ( ( R @ X4 @ Y4 )
             => ( ( R @ Y4 @ Z )
               => ( R @ X4 @ Z ) ) )
         => ( ! [X4: list_a] :
                ( ( member_list_a @ X4 @ A3 )
               => ? [Y5: list_a] :
                    ( ( member_list_a @ Y5 @ A3 )
                    & ( R @ X4 @ Y5 ) ) )
           => ( A3 = bot_bot_set_list_a ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_354_finite__transitivity__chain,axiom,
    ! [A3: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X4: nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y4: nat,Z: nat] :
              ( ( R @ X4 @ Y4 )
             => ( ( R @ Y4 @ Z )
               => ( R @ X4 @ Z ) ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A3 )
               => ? [Y5: nat] :
                    ( ( member_nat @ Y5 @ A3 )
                    & ( R @ X4 @ Y5 ) ) )
           => ( A3 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_355_Relational__Calculus_Oeval__def,axiom,
    ( relational_eval_a_b
    = ( ^ [Q4: relational_fmla_a_b] : ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q4 ) @ Q4 ) ) ) ).

% Relational_Calculus.eval_def
thf(fact_356_unbounded__k__infinite,axiom,
    ! [K3: nat,S: set_nat] :
      ( ! [M: nat] :
          ( ( ord_less_nat @ K3 @ M )
         => ? [N6: nat] :
              ( ( ord_less_nat @ M @ N6 )
              & ( member_nat @ N6 @ S ) ) )
     => ~ ( finite_finite_nat @ S ) ) ).

% unbounded_k_infinite
thf(fact_357_infinite__nat__iff__unbounded,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M2: nat] :
          ? [N3: nat] :
            ( ( ord_less_nat @ M2 @ N3 )
            & ( member_nat @ N3 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_358_less__list__code_I3_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( ord_le7035401960408590643at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) )
      = ( ( ord_le1203424502768444845at_nat @ X3 @ Y )
        | ( ( X3 = Y )
          & ( ord_le7035401960408590643at_nat @ Xs @ Ys3 ) ) ) ) ).

% less_list_code(3)
thf(fact_359_less__list__code_I3_J,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,Ys3: list_nat] :
      ( ( ord_less_list_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys3 ) )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ( X3 = Y )
          & ( ord_less_list_nat @ Xs @ Ys3 ) ) ) ) ).

% less_list_code(3)
thf(fact_360_less__list__code_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] : ( ord_le7035401960408590643at_nat @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) ).

% less_list_code(2)
thf(fact_361_Nil__less__Cons,axiom,
    ! [A: product_prod_nat_nat,X3: list_P6011104703257516679at_nat] : ( ord_le7035401960408590643at_nat @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ A @ X3 ) ) ).

% Nil_less_Cons
thf(fact_362_equiv__eval__on__eval__eqI,axiom,
    ! [I: product_prod_b_nat > set_list_a,Q: relational_fmla_a_b,Q3: relational_fmla_a_b] :
      ( ( finite_finite_a @ ( relational_adom_b_a @ I ) )
     => ( ( ord_less_eq_set_nat @ ( relational_fv_a_b @ Q ) @ ( relational_fv_a_b @ Q3 ) )
       => ( ( relational_equiv_a_b @ Q @ Q3 )
         => ( ( relati8814510239606734169on_a_b @ ( relational_fv_a_b @ Q3 ) @ Q @ I )
            = ( relational_eval_a_b @ Q3 @ I ) ) ) ) ) ).

% equiv_eval_on_eval_eqI
thf(fact_363_arg__min__if__finite_I2_J,axiom,
    ! [S: set_list_a,F2: list_a > nat] :
      ( ( finite_finite_list_a @ S )
     => ( ( S != bot_bot_set_list_a )
       => ~ ? [X6: list_a] :
              ( ( member_list_a @ X6 @ S )
              & ( ord_less_nat @ ( F2 @ X6 ) @ ( F2 @ ( lattic5043722365632780795_a_nat @ F2 @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_364_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X6: nat] :
              ( ( member_nat @ X6 @ S )
              & ( ord_less_nat @ ( F2 @ X6 ) @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_365_bot__prod__def,axiom,
    ( bot_bo2769642828321324397at_nat
    = ( product_Pair_nat_nat @ bot_bot_nat @ bot_bot_nat ) ) ).

% bot_prod_def
thf(fact_366_bot__prod__def,axiom,
    ( bot_bo7480491830437098195at_nat
    = ( produc6161850002892822231at_nat @ bot_bo2769642828321324397at_nat @ bot_bo2769642828321324397at_nat ) ) ).

% bot_prod_def
thf(fact_367_bot__prod__def,axiom,
    ( bot_bo3047382831089536473et_nat
    = ( produc4532415448927165861et_nat @ bot_bot_set_nat @ bot_bot_set_nat ) ) ).

% bot_prod_def
thf(fact_368_in__measures_I2_J,axiom,
    ! [X3: nat,Y: nat,F2: nat > nat,Fs: list_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X3 )
            = ( F2 @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_369_in__measures_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,F2: product_prod_nat_nat > nat,Fs: list_P9162950289778280392at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X3 )
            = ( F2 @ Y ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_370_insertion__sort_Opelims,axiom,
    ! [X3: product_prod_nat_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( sorted252080234835893926at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P909202496322092790at_nat @ sorted8891877951980152545at_nat @ ( produc1593612501639298397at_nat @ X3 @ Xa2 ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( ( Y
                = ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) )
             => ~ ( accp_P909202496322092790at_nat @ sorted8891877951980152545at_nat @ ( produc1593612501639298397at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) )
         => ~ ! [Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                ( ( Xa2
                  = ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) )
               => ( ( ( ( ord_le1203424502768444845at_nat @ Y4 @ X3 )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ Y4 @ ( sorted252080234835893926at_nat @ X3 @ Xs2 ) ) ) )
                    & ( ~ ( ord_le1203424502768444845at_nat @ Y4 @ X3 )
                     => ( ( ( X3 = Y4 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) )
                        & ( ( X3 != Y4 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) ) ) ) ) )
                 => ~ ( accp_P909202496322092790at_nat @ sorted8891877951980152545at_nat @ ( produc1593612501639298397at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ).

% insertion_sort.pelims
thf(fact_371_insertion__sort_Opelims,axiom,
    ! [X3: nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( sorted4128666710326294635rt_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P8032398731279750170st_nat @ sorted4594844048289928688el_nat @ ( produc8282810413953273033st_nat @ X3 @ Xa2 ) )
       => ( ( ( Xa2 = nil_nat )
           => ( ( Y
                = ( cons_nat @ X3 @ nil_nat ) )
             => ~ ( accp_P8032398731279750170st_nat @ sorted4594844048289928688el_nat @ ( produc8282810413953273033st_nat @ X3 @ nil_nat ) ) ) )
         => ~ ! [Y4: nat,Xs2: list_nat] :
                ( ( Xa2
                  = ( cons_nat @ Y4 @ Xs2 ) )
               => ( ( ( ( ord_less_nat @ Y4 @ X3 )
                     => ( Y
                        = ( cons_nat @ Y4 @ ( sorted4128666710326294635rt_nat @ X3 @ Xs2 ) ) ) )
                    & ( ~ ( ord_less_nat @ Y4 @ X3 )
                     => ( ( ( X3 = Y4 )
                         => ( Y
                            = ( cons_nat @ Y4 @ Xs2 ) ) )
                        & ( ( X3 != Y4 )
                         => ( Y
                            = ( cons_nat @ X3 @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ) )
                 => ~ ( accp_P8032398731279750170st_nat @ sorted4594844048289928688el_nat @ ( produc8282810413953273033st_nat @ X3 @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ).

% insertion_sort.pelims
thf(fact_372_delete__sorted_Opelims,axiom,
    ! [X3: product_prod_nat_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( sorted8108290951004058757at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P909202496322092790at_nat @ sorted6093607279699462082at_nat @ ( produc1593612501639298397at_nat @ X3 @ Xa2 ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( ( Y = nil_Pr5478986624290739719at_nat )
             => ~ ( accp_P909202496322092790at_nat @ sorted6093607279699462082at_nat @ ( produc1593612501639298397at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) )
         => ~ ! [Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                ( ( Xa2
                  = ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) )
               => ( ( ( ( ord_le1203424502768444845at_nat @ Y4 @ X3 )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ Y4 @ ( sorted8108290951004058757at_nat @ X3 @ Xs2 ) ) ) )
                    & ( ~ ( ord_le1203424502768444845at_nat @ Y4 @ X3 )
                     => ( ( ( X3 = Y4 )
                         => ( Y = Xs2 ) )
                        & ( ( X3 != Y4 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) ) ) ) )
                 => ~ ( accp_P909202496322092790at_nat @ sorted6093607279699462082at_nat @ ( produc1593612501639298397at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_sorted.pelims
thf(fact_373_delete__sorted_Opelims,axiom,
    ! [X3: nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( sorted5501677424865158476ed_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P8032398731279750170st_nat @ sorted2826361768923046479el_nat @ ( produc8282810413953273033st_nat @ X3 @ Xa2 ) )
       => ( ( ( Xa2 = nil_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_P8032398731279750170st_nat @ sorted2826361768923046479el_nat @ ( produc8282810413953273033st_nat @ X3 @ nil_nat ) ) ) )
         => ~ ! [Y4: nat,Xs2: list_nat] :
                ( ( Xa2
                  = ( cons_nat @ Y4 @ Xs2 ) )
               => ( ( ( ( ord_less_nat @ Y4 @ X3 )
                     => ( Y
                        = ( cons_nat @ Y4 @ ( sorted5501677424865158476ed_nat @ X3 @ Xs2 ) ) ) )
                    & ( ~ ( ord_less_nat @ Y4 @ X3 )
                     => ( ( ( X3 = Y4 )
                         => ( Y = Xs2 ) )
                        & ( ( X3 != Y4 )
                         => ( Y
                            = ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) )
                 => ~ ( accp_P8032398731279750170st_nat @ sorted2826361768923046479el_nat @ ( produc8282810413953273033st_nat @ X3 @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_sorted.pelims
thf(fact_374_finite__Collect__subsets,axiom,
    ! [A3: set_list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [B8: set_list_a] : ( ord_le8861187494160871172list_a @ B8 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_375_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B8: set_nat] : ( ord_less_eq_set_nat @ B8 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_376_in__measures_I1_J,axiom,
    ! [X3: nat,Y: nat] :
      ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ nil_nat_nat ) ) ).

% in_measures(1)
thf(fact_377_in__measures_I1_J,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ nil_Pr2865493887535707976at_nat ) ) ).

% in_measures(1)
thf(fact_378_less__eq__prod__simp,axiom,
    ! [X1: product_prod_nat_nat,Y1: product_prod_nat_nat,X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ord_le6722121967309221895at_nat @ ( produc6161850002892822231at_nat @ X1 @ Y1 ) @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) )
      = ( ( ord_le1203424502768444845at_nat @ X1 @ X2 )
        | ( ( ord_le8460144461188290721at_nat @ X1 @ X2 )
          & ( ord_le8460144461188290721at_nat @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_379_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ X1 @ X2 )
        | ( ( ord_less_eq_nat @ X1 @ X2 )
          & ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_380_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: set_nat,X2: nat,Y2: set_nat] :
      ( ( ord_le4284901688344473943et_nat @ ( produc4207506657711014383et_nat @ X1 @ Y1 ) @ ( produc4207506657711014383et_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ X1 @ X2 )
        | ( ( ord_less_eq_nat @ X1 @ X2 )
          & ( ord_less_eq_set_nat @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_381_less__eq__prod__simp,axiom,
    ! [X1: set_nat,Y1: nat,X2: set_nat,Y2: nat] :
      ( ( ord_le152793438849583191at_nat @ ( produc641871753055645167at_nat @ X1 @ Y1 ) @ ( produc641871753055645167at_nat @ X2 @ Y2 ) )
      = ( ( ord_less_set_nat @ X1 @ X2 )
        | ( ( ord_less_eq_set_nat @ X1 @ X2 )
          & ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_382_less__eq__prod__simp,axiom,
    ! [X1: set_nat,Y1: set_nat,X2: set_nat,Y2: set_nat] :
      ( ( ord_le2041963031926835469et_nat @ ( produc4532415448927165861et_nat @ X1 @ Y1 ) @ ( produc4532415448927165861et_nat @ X2 @ Y2 ) )
      = ( ( ord_less_set_nat @ X1 @ X2 )
        | ( ( ord_less_eq_set_nat @ X1 @ X2 )
          & ( ord_less_eq_set_nat @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_383_less__prod__simp,axiom,
    ! [X1: product_prod_nat_nat,Y1: product_prod_nat_nat,X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ord_le9033551061567896339at_nat @ ( produc6161850002892822231at_nat @ X1 @ Y1 ) @ ( produc6161850002892822231at_nat @ X2 @ Y2 ) )
      = ( ( ord_le1203424502768444845at_nat @ X1 @ X2 )
        | ( ( ord_le8460144461188290721at_nat @ X1 @ X2 )
          & ( ord_le1203424502768444845at_nat @ Y1 @ Y2 ) ) ) ) ).

% less_prod_simp
thf(fact_384_less__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
      ( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ X1 @ X2 )
        | ( ( ord_less_eq_nat @ X1 @ X2 )
          & ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).

% less_prod_simp
thf(fact_385_less__prod__simp,axiom,
    ! [X1: set_nat,Y1: nat,X2: set_nat,Y2: nat] :
      ( ( ord_le4872869340735563107at_nat @ ( produc641871753055645167at_nat @ X1 @ Y1 ) @ ( produc641871753055645167at_nat @ X2 @ Y2 ) )
      = ( ( ord_less_set_nat @ X1 @ X2 )
        | ( ( ord_less_eq_set_nat @ X1 @ X2 )
          & ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).

% less_prod_simp
thf(fact_386_subset__Collect__conv,axiom,
    ! [S: set_list_a,P: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ S @ ( collect_list_a @ P ) )
      = ( ! [X: list_a] :
            ( ( member_list_a @ X @ S )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_387_subset__Collect__conv,axiom,
    ! [S: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ S )
           => ( P @ X ) ) ) ) ).

% subset_Collect_conv
thf(fact_388_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C3: nat,D2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C3 )
       => ( ( C3 = D2 )
         => ( ord_less_eq_nat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_389_ord__eq__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C3: set_nat,D2: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C3 )
       => ( ( C3 = D2 )
         => ( ord_less_eq_set_nat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_390_bot__empty__eq,axiom,
    ( bot_bo7573314457883560170_nat_o
    = ( ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq
thf(fact_391_bot__empty__eq,axiom,
    ( bot_bo482883023278783056_nat_o
    = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq
thf(fact_392_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_393_finite__has__minimal2,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
      ( ( finite4392333629123659920at_nat @ A3 )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ? [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A3 )
            & ( ord_le6722121967309221895at_nat @ X4 @ A )
            & ! [Xa: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Xa @ A3 )
               => ( ( ord_le6722121967309221895at_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_394_finite__has__minimal2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ? [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
            & ( ord_le8460144461188290721at_nat @ X4 @ A )
            & ! [Xa: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa @ A3 )
               => ( ( ord_le8460144461188290721at_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_395_finite__has__minimal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ( ord_less_eq_nat @ X4 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_396_finite__has__minimal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A @ A3 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
            & ( ord_less_eq_set_nat @ X4 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A3 )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_397_finite__has__maximal2,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,A: produc859450856879609959at_nat] :
      ( ( finite4392333629123659920at_nat @ A3 )
     => ( ( member8206827879206165904at_nat @ A @ A3 )
       => ? [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ A3 )
            & ( ord_le6722121967309221895at_nat @ A @ X4 )
            & ! [Xa: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Xa @ A3 )
               => ( ( ord_le6722121967309221895at_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_398_finite__has__maximal2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( member8440522571783428010at_nat @ A @ A3 )
       => ? [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A3 )
            & ( ord_le8460144461188290721at_nat @ A @ X4 )
            & ! [Xa: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Xa @ A3 )
               => ( ( ord_le8460144461188290721at_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_399_finite__has__maximal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ( ord_less_eq_nat @ A @ X4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_400_finite__has__maximal2,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A @ A3 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
            & ( ord_less_eq_set_nat @ A @ X4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A3 )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_401_rev__finite__subset,axiom,
    ! [B3: set_list_a,A3: set_list_a] :
      ( ( finite_finite_list_a @ B3 )
     => ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
       => ( finite_finite_list_a @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_402_rev__finite__subset,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_403_infinite__super,axiom,
    ! [S: set_list_a,T2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ S @ T2 )
     => ( ~ ( finite_finite_list_a @ S )
       => ~ ( finite_finite_list_a @ T2 ) ) ) ).

% infinite_super
thf(fact_404_infinite__super,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T2 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_405_finite__subset,axiom,
    ! [A3: set_list_a,B3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
     => ( ( finite_finite_list_a @ B3 )
       => ( finite_finite_list_a @ A3 ) ) ) ).

% finite_subset
thf(fact_406_finite__subset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( finite_finite_nat @ B3 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% finite_subset
thf(fact_407_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_408_bot__empty__eq2,axiom,
    ( bot_bo4898103413517107610_nat_o
    = ( ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ bot_bo5327735625951526323at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_409_arg__min__least,axiom,
    ! [S: set_Pr8693737435421807431at_nat,Y: produc859450856879609959at_nat,F2: produc859450856879609959at_nat > nat] :
      ( ( finite4392333629123659920at_nat @ S )
     => ( ( S != bot_bo5327735625951526323at_nat )
       => ( ( member8206827879206165904at_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic390166758595302878at_nat @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_410_arg__min__least,axiom,
    ! [S: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,F2: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( S != bot_bo2099793752762293965at_nat )
       => ( ( member8440522571783428010at_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic4984276347100956536at_nat @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_411_arg__min__least,axiom,
    ! [S: set_list_a,Y: list_a,F2: list_a > nat] :
      ( ( finite_finite_list_a @ S )
     => ( ( S != bot_bot_set_list_a )
       => ( ( member_list_a @ Y @ S )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic5043722365632780795_a_nat @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_412_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F2: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_413_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_414_finite__has__minimal,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A3 )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_415_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_416_finite__has__maximal,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A3 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A3 )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_417_insertion__sort_Osimps_I2_J,axiom,
    ! [Y: product_prod_nat_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( ord_le1203424502768444845at_nat @ Y @ X3 )
       => ( ( sorted252080234835893926at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) )
          = ( cons_P6512896166579812791at_nat @ Y @ ( sorted252080234835893926at_nat @ X3 @ Xs ) ) ) )
      & ( ~ ( ord_le1203424502768444845at_nat @ Y @ X3 )
       => ( ( ( X3 = Y )
           => ( ( sorted252080234835893926at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) )
              = ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) )
          & ( ( X3 != Y )
           => ( ( sorted252080234835893926at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) )
              = ( cons_P6512896166579812791at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) ) ) ) ) ) ).

% insertion_sort.simps(2)
thf(fact_418_insertion__sort_Osimps_I2_J,axiom,
    ! [Y: nat,X3: nat,Xs: list_nat] :
      ( ( ( ord_less_nat @ Y @ X3 )
       => ( ( sorted4128666710326294635rt_nat @ X3 @ ( cons_nat @ Y @ Xs ) )
          = ( cons_nat @ Y @ ( sorted4128666710326294635rt_nat @ X3 @ Xs ) ) ) )
      & ( ~ ( ord_less_nat @ Y @ X3 )
       => ( ( ( X3 = Y )
           => ( ( sorted4128666710326294635rt_nat @ X3 @ ( cons_nat @ Y @ Xs ) )
              = ( cons_nat @ Y @ Xs ) ) )
          & ( ( X3 != Y )
           => ( ( sorted4128666710326294635rt_nat @ X3 @ ( cons_nat @ Y @ Xs ) )
              = ( cons_nat @ X3 @ ( cons_nat @ Y @ Xs ) ) ) ) ) ) ) ).

% insertion_sort.simps(2)
thf(fact_419_delete__sorted_Osimps_I2_J,axiom,
    ! [Y: product_prod_nat_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( ord_le1203424502768444845at_nat @ Y @ X3 )
       => ( ( sorted8108290951004058757at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) )
          = ( cons_P6512896166579812791at_nat @ Y @ ( sorted8108290951004058757at_nat @ X3 @ Xs ) ) ) )
      & ( ~ ( ord_le1203424502768444845at_nat @ Y @ X3 )
       => ( ( ( X3 = Y )
           => ( ( sorted8108290951004058757at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) )
              = Xs ) )
          & ( ( X3 != Y )
           => ( ( sorted8108290951004058757at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Xs ) )
              = ( cons_P6512896166579812791at_nat @ Y @ Xs ) ) ) ) ) ) ).

% delete_sorted.simps(2)
thf(fact_420_delete__sorted_Osimps_I2_J,axiom,
    ! [Y: nat,X3: nat,Xs: list_nat] :
      ( ( ( ord_less_nat @ Y @ X3 )
       => ( ( sorted5501677424865158476ed_nat @ X3 @ ( cons_nat @ Y @ Xs ) )
          = ( cons_nat @ Y @ ( sorted5501677424865158476ed_nat @ X3 @ Xs ) ) ) )
      & ( ~ ( ord_less_nat @ Y @ X3 )
       => ( ( ( X3 = Y )
           => ( ( sorted5501677424865158476ed_nat @ X3 @ ( cons_nat @ Y @ Xs ) )
              = Xs ) )
          & ( ( X3 != Y )
           => ( ( sorted5501677424865158476ed_nat @ X3 @ ( cons_nat @ Y @ Xs ) )
              = ( cons_nat @ Y @ Xs ) ) ) ) ) ) ).

% delete_sorted.simps(2)
thf(fact_421_insertion__sort_Osimps_I1_J,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( sorted252080234835893926at_nat @ X3 @ nil_Pr5478986624290739719at_nat )
      = ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ).

% insertion_sort.simps(1)
thf(fact_422_measures__less,axiom,
    ! [F2: nat > nat,X3: nat,Y: nat,Fs: list_nat_nat] :
      ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_423_measures__less,axiom,
    ! [F2: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
      ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_424_insertion__sort_Oelims,axiom,
    ! [X3: product_prod_nat_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( sorted252080234835893926at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
         => ( Y
           != ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) )
       => ~ ! [Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) )
             => ~ ( ( ( ord_le1203424502768444845at_nat @ Y4 @ X3 )
                   => ( Y
                      = ( cons_P6512896166579812791at_nat @ Y4 @ ( sorted252080234835893926at_nat @ X3 @ Xs2 ) ) ) )
                  & ( ~ ( ord_le1203424502768444845at_nat @ Y4 @ X3 )
                   => ( ( ( X3 = Y4 )
                       => ( Y
                          = ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) )
                      & ( ( X3 != Y4 )
                       => ( Y
                          = ( cons_P6512896166579812791at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).

% insertion_sort.elims
thf(fact_425_insertion__sort_Oelims,axiom,
    ! [X3: nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( sorted4128666710326294635rt_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_nat )
         => ( Y
           != ( cons_nat @ X3 @ nil_nat ) ) )
       => ~ ! [Y4: nat,Xs2: list_nat] :
              ( ( Xa2
                = ( cons_nat @ Y4 @ Xs2 ) )
             => ~ ( ( ( ord_less_nat @ Y4 @ X3 )
                   => ( Y
                      = ( cons_nat @ Y4 @ ( sorted4128666710326294635rt_nat @ X3 @ Xs2 ) ) ) )
                  & ( ~ ( ord_less_nat @ Y4 @ X3 )
                   => ( ( ( X3 = Y4 )
                       => ( Y
                          = ( cons_nat @ Y4 @ Xs2 ) ) )
                      & ( ( X3 != Y4 )
                       => ( Y
                          = ( cons_nat @ X3 @ ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).

% insertion_sort.elims
thf(fact_426_delete__sorted_Oelims,axiom,
    ! [X3: product_prod_nat_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( sorted8108290951004058757at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) )
             => ~ ( ( ( ord_le1203424502768444845at_nat @ Y4 @ X3 )
                   => ( Y
                      = ( cons_P6512896166579812791at_nat @ Y4 @ ( sorted8108290951004058757at_nat @ X3 @ Xs2 ) ) ) )
                  & ( ~ ( ord_le1203424502768444845at_nat @ Y4 @ X3 )
                   => ( ( ( X3 = Y4 )
                       => ( Y = Xs2 ) )
                      & ( ( X3 != Y4 )
                       => ( Y
                          = ( cons_P6512896166579812791at_nat @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ).

% delete_sorted.elims
thf(fact_427_delete__sorted_Oelims,axiom,
    ! [X3: nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( sorted5501677424865158476ed_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_nat )
         => ( Y != nil_nat ) )
       => ~ ! [Y4: nat,Xs2: list_nat] :
              ( ( Xa2
                = ( cons_nat @ Y4 @ Xs2 ) )
             => ~ ( ( ( ord_less_nat @ Y4 @ X3 )
                   => ( Y
                      = ( cons_nat @ Y4 @ ( sorted5501677424865158476ed_nat @ X3 @ Xs2 ) ) ) )
                  & ( ~ ( ord_less_nat @ Y4 @ X3 )
                   => ( ( ( X3 = Y4 )
                       => ( Y = Xs2 ) )
                      & ( ( X3 != Y4 )
                       => ( Y
                          = ( cons_nat @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ).

% delete_sorted.elims
thf(fact_428_case__prod__mono,axiom,
    ! [P2: product_prod_nat_nat,F2: nat > nat > nat,F3: nat > nat > nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( ord_less_eq_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A5 @ B4 ) ) )
     => ( ord_less_eq_nat @ ( produc6842872674320459806at_nat @ F2 @ P2 ) @ ( produc6842872674320459806at_nat @ F3 @ P2 ) ) ) ).

% case_prod_mono
thf(fact_429_case__prod__mono,axiom,
    ! [P2: produc859450856879609959at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > nat,F3: product_prod_nat_nat > product_prod_nat_nat > nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( ord_less_eq_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A5 @ B4 ) ) )
     => ( ord_less_eq_nat @ ( produc6237124055692578492at_nat @ F2 @ P2 ) @ ( produc6237124055692578492at_nat @ F3 @ P2 ) ) ) ).

% case_prod_mono
thf(fact_430_case__prod__mono,axiom,
    ! [P2: product_prod_nat_nat,F2: nat > nat > set_nat,F3: nat > nat > set_nat] :
      ( ! [A5: nat,B4: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A5 @ B4 ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A5 @ B4 ) ) )
     => ( ord_less_eq_set_nat @ ( produc6189476227299908564et_nat @ F2 @ P2 ) @ ( produc6189476227299908564et_nat @ F3 @ P2 ) ) ) ).

% case_prod_mono
thf(fact_431_case__prod__mono,axiom,
    ! [P2: produc859450856879609959at_nat,F2: product_prod_nat_nat > product_prod_nat_nat > set_nat,F3: product_prod_nat_nat > product_prod_nat_nat > set_nat] :
      ( ! [A5: product_prod_nat_nat,B4: product_prod_nat_nat] :
          ( ( P2
            = ( produc6161850002892822231at_nat @ A5 @ B4 ) )
         => ( ord_less_eq_set_nat @ ( F2 @ A5 @ B4 ) @ ( F3 @ A5 @ B4 ) ) )
     => ( ord_less_eq_set_nat @ ( produc1370685125915216498et_nat @ F2 @ P2 ) @ ( produc1370685125915216498et_nat @ F3 @ P2 ) ) ) ).

% case_prod_mono
thf(fact_432_splice_Opinduct,axiom,
    ! [A0: list_P6011104703257516679at_nat,A1: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P7052990409830227952at_nat @ splice425205989056337325at_nat @ ( produc5943733680697469783at_nat @ A0 @ A1 ) )
     => ( ! [Ys: list_P6011104703257516679at_nat] :
            ( ( accp_P7052990409830227952at_nat @ splice425205989056337325at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Ys ) )
           => ( P @ nil_Pr5478986624290739719at_nat @ Ys ) )
       => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
              ( ( accp_P7052990409830227952at_nat @ splice425205989056337325at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ Ys ) )
             => ( ( P @ Ys @ Xs2 )
               => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ Ys ) ) )
         => ( P @ A0 @ A1 ) ) ) ) ).

% splice.pinduct
thf(fact_433_pair__less__iff1,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( product_Pair_nat_nat @ X3 @ Z3 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z3 ) ) ).

% pair_less_iff1
thf(fact_434_lexordp__conv__lexord,axiom,
    ( ord_lexordp_nat
    = ( ^ [Xs3: list_nat,Ys2: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( lexord_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ) ) ) ).

% lexordp_conv_lexord
thf(fact_435_shuffles_Opinduct,axiom,
    ! [A0: list_P6011104703257516679at_nat,A1: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P7052990409830227952at_nat @ shuffl479420286952424683at_nat @ ( produc5943733680697469783at_nat @ A0 @ A1 ) )
     => ( ! [Ys: list_P6011104703257516679at_nat] :
            ( ( accp_P7052990409830227952at_nat @ shuffl479420286952424683at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Ys ) )
           => ( P @ nil_Pr5478986624290739719at_nat @ Ys ) )
       => ( ! [Xs2: list_P6011104703257516679at_nat] :
              ( ( accp_P7052990409830227952at_nat @ shuffl479420286952424683at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ nil_Pr5478986624290739719at_nat ) )
             => ( P @ Xs2 @ nil_Pr5478986624290739719at_nat ) )
         => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
                ( ( accp_P7052990409830227952at_nat @ shuffl479420286952424683at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) )
               => ( ( P @ Xs2 @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
                 => ( ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ Ys )
                   => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) ) )
           => ( P @ A0 @ A1 ) ) ) ) ) ).

% shuffles.pinduct
thf(fact_436_finite__Collect__le__nat,axiom,
    ! [K3: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K3 ) ) ) ).

% finite_Collect_le_nat
thf(fact_437_Cons__le__Cons,axiom,
    ! [A: product_prod_nat_nat,X3: list_P6011104703257516679at_nat,B: product_prod_nat_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ord_le2270365492076597799at_nat @ ( cons_P6512896166579812791at_nat @ A @ X3 ) @ ( cons_P6512896166579812791at_nat @ B @ Y ) )
      = ( ( ord_le1203424502768444845at_nat @ A @ B )
        | ( ( A = B )
          & ( ord_le2270365492076597799at_nat @ X3 @ Y ) ) ) ) ).

% Cons_le_Cons
thf(fact_438_Cons__le__Cons,axiom,
    ! [A: nat,X3: list_nat,B: nat,Y: list_nat] :
      ( ( ord_less_eq_list_nat @ ( cons_nat @ A @ X3 ) @ ( cons_nat @ B @ Y ) )
      = ( ( ord_less_nat @ A @ B )
        | ( ( A = B )
          & ( ord_less_eq_list_nat @ X3 @ Y ) ) ) ) ).

% Cons_le_Cons
thf(fact_439_lexordp__simps_I3_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( ord_le1029496871892194115at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) )
      = ( ( ord_le1203424502768444845at_nat @ X3 @ Y )
        | ( ~ ( ord_le1203424502768444845at_nat @ Y @ X3 )
          & ( ord_le1029496871892194115at_nat @ Xs @ Ys3 ) ) ) ) ).

% lexordp_simps(3)
thf(fact_440_lexordp__simps_I3_J,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,Ys3: list_nat] :
      ( ( ord_lexordp_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys3 ) )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ~ ( ord_less_nat @ Y @ X3 )
          & ( ord_lexordp_nat @ Xs @ Ys3 ) ) ) ) ).

% lexordp_simps(3)
thf(fact_441_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R )
        @ ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ S ) )
      = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_442_pred__subset__eq2,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ord_le5604493270027003598_nat_o
        @ ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R )
        @ ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ S ) )
      = ( ord_le3000389064537975527at_nat @ R @ S ) ) ).

% pred_subset_eq2
thf(fact_443_subrelI,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R4 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ S3 ) )
     => ( ord_le3146513528884898305at_nat @ R4 @ S3 ) ) ).

% subrelI
thf(fact_444_subrelI,axiom,
    ! [R4: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R4 )
         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ S3 ) )
     => ( ord_le3000389064537975527at_nat @ R4 @ S3 ) ) ).

% subrelI
thf(fact_445_exists__leI,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [N7: nat] :
            ( ( ord_less_nat @ N7 @ N )
           => ~ ( P @ N7 ) )
       => ( P @ N ) )
     => ? [N8: nat] :
          ( ( ord_less_eq_nat @ N8 @ N )
          & ( P @ N8 ) ) ) ).

% exists_leI
thf(fact_446_lexordp__irreflexive,axiom,
    ! [Xs: list_nat] :
      ( ! [X4: nat] :
          ~ ( ord_less_nat @ X4 @ X4 )
     => ~ ( ord_lexordp_nat @ Xs @ Xs ) ) ).

% lexordp_irreflexive
thf(fact_447_pair__lessI2,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_448_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set_nat] :
      ( ( ~ ( finite_finite_nat @ S ) )
      = ( ! [M2: nat] :
          ? [N3: nat] :
            ( ( ord_less_eq_nat @ M2 @ N3 )
            & ( member_nat @ N3 @ S ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_449_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M2: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N5 )
         => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_450_pred__subset__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
      ( ( ord_le7858099551454983350_nat_o
        @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ R )
        @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ S ) )
      = ( ord_le3000389064537975527at_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_451_pred__subset__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ord_le704812498762024988_nat_o
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ R )
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S ) )
      = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_452_pred__subset__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S ) )
      = ( ord_less_eq_set_nat @ R @ S ) ) ).

% pred_subset_eq
thf(fact_453_finite__less__ub,axiom,
    ! [F2: nat > nat,U: nat] :
      ( ! [N9: nat] : ( ord_less_eq_nat @ N9 @ ( F2 @ N9 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F2 @ N3 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_454_less__eq__list__code_I3_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( ord_le2270365492076597799at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) )
      = ( ( ord_le1203424502768444845at_nat @ X3 @ Y )
        | ( ( X3 = Y )
          & ( ord_le2270365492076597799at_nat @ Xs @ Ys3 ) ) ) ) ).

% less_eq_list_code(3)
thf(fact_455_less__eq__list__code_I3_J,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,Ys3: list_nat] :
      ( ( ord_less_eq_list_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys3 ) )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ( X3 = Y )
          & ( ord_less_eq_list_nat @ Xs @ Ys3 ) ) ) ) ).

% less_eq_list_code(3)
thf(fact_456_less__eq__list__code_I1_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ~ ( ord_le2270365492076597799at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ nil_Pr5478986624290739719at_nat ) ).

% less_eq_list_code(1)
thf(fact_457_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( ord_less_eq_nat @ B @ ( suc @ A ) ) )
      = ( B
        = ( suc @ A ) ) ) ).

% nat_in_between_eq(1)
thf(fact_458_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_nat @ B @ ( suc @ A ) ) )
      = ( B = A ) ) ).

% nat_in_between_eq(2)
thf(fact_459_lexordp_OCons__eq,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ~ ( ord_le1203424502768444845at_nat @ X3 @ Y )
     => ( ~ ( ord_le1203424502768444845at_nat @ Y @ X3 )
       => ( ( ord_le1029496871892194115at_nat @ Xs @ Ys3 )
         => ( ord_le1029496871892194115at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) ) ) ) ).

% lexordp.Cons_eq
thf(fact_460_lexordp_OCons__eq,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat,Ys3: list_nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ~ ( ord_less_nat @ Y @ X3 )
       => ( ( ord_lexordp_nat @ Xs @ Ys3 )
         => ( ord_lexordp_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys3 ) ) ) ) ) ).

% lexordp.Cons_eq
thf(fact_461_lexordp_OCons,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( ord_le1203424502768444845at_nat @ X3 @ Y )
     => ( ord_le1029496871892194115at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) ) ).

% lexordp.Cons
thf(fact_462_lexordp_OCons,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat,Ys3: list_nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ord_lexordp_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys3 ) ) ) ).

% lexordp.Cons
thf(fact_463_lexordp_ONil,axiom,
    ! [Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] : ( ord_le1029496871892194115at_nat @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) ).

% lexordp.Nil
thf(fact_464_lexordp__append__leftD,axiom,
    ! [Xs: list_nat,Us2: list_nat,Vs2: list_nat] :
      ( ( ord_lexordp_nat @ ( append_nat @ Xs @ Us2 ) @ ( append_nat @ Xs @ Vs2 ) )
     => ( ! [A5: nat] :
            ~ ( ord_less_nat @ A5 @ A5 )
       => ( ord_lexordp_nat @ Us2 @ Vs2 ) ) ) ).

% lexordp_append_leftD
thf(fact_465_lexordp_Ocases,axiom,
    ! [A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat] :
      ( ( ord_le1029496871892194115at_nat @ A1 @ A22 )
     => ( ( ( A1 = nil_Pr5478986624290739719at_nat )
         => ! [Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
              ( A22
             != ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) )
       => ( ! [X4: product_prod_nat_nat] :
              ( ? [Xs2: list_P6011104703257516679at_nat] :
                  ( A1
                  = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
             => ! [Y4: product_prod_nat_nat] :
                  ( ? [Ys: list_P6011104703257516679at_nat] :
                      ( A22
                      = ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
                 => ~ ( ord_le1203424502768444845at_nat @ X4 @ Y4 ) ) )
         => ~ ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                ( ( A1
                  = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
               => ! [Ys: list_P6011104703257516679at_nat] :
                    ( ( A22
                      = ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
                   => ( ~ ( ord_le1203424502768444845at_nat @ X4 @ Y4 )
                     => ( ~ ( ord_le1203424502768444845at_nat @ Y4 @ X4 )
                       => ~ ( ord_le1029496871892194115at_nat @ Xs2 @ Ys ) ) ) ) ) ) ) ) ).

% lexordp.cases
thf(fact_466_lexordp_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat] :
      ( ( ord_lexordp_nat @ A1 @ A22 )
     => ( ( ( A1 = nil_nat )
         => ! [Y4: nat,Ys: list_nat] :
              ( A22
             != ( cons_nat @ Y4 @ Ys ) ) )
       => ( ! [X4: nat] :
              ( ? [Xs2: list_nat] :
                  ( A1
                  = ( cons_nat @ X4 @ Xs2 ) )
             => ! [Y4: nat] :
                  ( ? [Ys: list_nat] :
                      ( A22
                      = ( cons_nat @ Y4 @ Ys ) )
                 => ~ ( ord_less_nat @ X4 @ Y4 ) ) )
         => ~ ! [X4: nat,Y4: nat,Xs2: list_nat] :
                ( ( A1
                  = ( cons_nat @ X4 @ Xs2 ) )
               => ! [Ys: list_nat] :
                    ( ( A22
                      = ( cons_nat @ Y4 @ Ys ) )
                   => ( ~ ( ord_less_nat @ X4 @ Y4 )
                     => ( ~ ( ord_less_nat @ Y4 @ X4 )
                       => ~ ( ord_lexordp_nat @ Xs2 @ Ys ) ) ) ) ) ) ) ) ).

% lexordp.cases
thf(fact_467_lexordp_Osimps,axiom,
    ( ord_le1029496871892194115at_nat
    = ( ^ [A12: list_P6011104703257516679at_nat,A23: list_P6011104703257516679at_nat] :
          ( ? [Y3: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
              ( ( A12 = nil_Pr5478986624290739719at_nat )
              & ( A23
                = ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) ) )
          | ? [X: product_prod_nat_nat,Y3: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
              ( ( A12
                = ( cons_P6512896166579812791at_nat @ X @ Xs3 ) )
              & ( A23
                = ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) )
              & ( ord_le1203424502768444845at_nat @ X @ Y3 ) )
          | ? [X: product_prod_nat_nat,Y3: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
              ( ( A12
                = ( cons_P6512896166579812791at_nat @ X @ Xs3 ) )
              & ( A23
                = ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) )
              & ~ ( ord_le1203424502768444845at_nat @ X @ Y3 )
              & ~ ( ord_le1203424502768444845at_nat @ Y3 @ X )
              & ( ord_le1029496871892194115at_nat @ Xs3 @ Ys2 ) ) ) ) ) ).

% lexordp.simps
thf(fact_468_lexordp_Osimps,axiom,
    ( ord_lexordp_nat
    = ( ^ [A12: list_nat,A23: list_nat] :
          ( ? [Y3: nat,Ys2: list_nat] :
              ( ( A12 = nil_nat )
              & ( A23
                = ( cons_nat @ Y3 @ Ys2 ) ) )
          | ? [X: nat,Y3: nat,Xs3: list_nat,Ys2: list_nat] :
              ( ( A12
                = ( cons_nat @ X @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y3 @ Ys2 ) )
              & ( ord_less_nat @ X @ Y3 ) )
          | ? [X: nat,Y3: nat,Xs3: list_nat,Ys2: list_nat] :
              ( ( A12
                = ( cons_nat @ X @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y3 @ Ys2 ) )
              & ~ ( ord_less_nat @ X @ Y3 )
              & ~ ( ord_less_nat @ Y3 @ X )
              & ( ord_lexordp_nat @ Xs3 @ Ys2 ) ) ) ) ) ).

% lexordp.simps
thf(fact_469_lexordp__cases,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( ord_le1029496871892194115at_nat @ Xs @ Ys3 )
     => ( ( ( Xs = nil_Pr5478986624290739719at_nat )
         => ! [Y4: product_prod_nat_nat,Ys5: list_P6011104703257516679at_nat] :
              ( Ys3
             != ( cons_P6512896166579812791at_nat @ Y4 @ Ys5 ) ) )
       => ( ! [X4: product_prod_nat_nat] :
              ( ? [Xs4: list_P6011104703257516679at_nat] :
                  ( Xs
                  = ( cons_P6512896166579812791at_nat @ X4 @ Xs4 ) )
             => ! [Y4: product_prod_nat_nat] :
                  ( ? [Ys5: list_P6011104703257516679at_nat] :
                      ( Ys3
                      = ( cons_P6512896166579812791at_nat @ Y4 @ Ys5 ) )
                 => ~ ( ord_le1203424502768444845at_nat @ X4 @ Y4 ) ) )
         => ~ ! [X4: product_prod_nat_nat,Xs4: list_P6011104703257516679at_nat] :
                ( ( Xs
                  = ( cons_P6512896166579812791at_nat @ X4 @ Xs4 ) )
               => ! [Ys5: list_P6011104703257516679at_nat] :
                    ( ( Ys3
                      = ( cons_P6512896166579812791at_nat @ X4 @ Ys5 ) )
                   => ~ ( ord_le1029496871892194115at_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% lexordp_cases
thf(fact_470_lexordp__cases,axiom,
    ! [Xs: list_nat,Ys3: list_nat] :
      ( ( ord_lexordp_nat @ Xs @ Ys3 )
     => ( ( ( Xs = nil_nat )
         => ! [Y4: nat,Ys5: list_nat] :
              ( Ys3
             != ( cons_nat @ Y4 @ Ys5 ) ) )
       => ( ! [X4: nat] :
              ( ? [Xs4: list_nat] :
                  ( Xs
                  = ( cons_nat @ X4 @ Xs4 ) )
             => ! [Y4: nat] :
                  ( ? [Ys5: list_nat] :
                      ( Ys3
                      = ( cons_nat @ Y4 @ Ys5 ) )
                 => ~ ( ord_less_nat @ X4 @ Y4 ) ) )
         => ~ ! [X4: nat,Xs4: list_nat] :
                ( ( Xs
                  = ( cons_nat @ X4 @ Xs4 ) )
               => ! [Ys5: list_nat] :
                    ( ( Ys3
                      = ( cons_nat @ X4 @ Ys5 ) )
                   => ~ ( ord_lexordp_nat @ Xs4 @ Ys5 ) ) ) ) ) ) ).

% lexordp_cases
thf(fact_471_lexordp__induct,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ord_le1029496871892194115at_nat @ Xs @ Ys3 )
     => ( ! [Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] : ( P @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
       => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
              ( ( ord_le1203424502768444845at_nat @ X4 @ Y4 )
             => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) )
         => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
                ( ( ord_le1029496871892194115at_nat @ Xs2 @ Ys )
               => ( ( P @ Xs2 @ Ys )
                 => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ X4 @ Ys ) ) ) )
           => ( P @ Xs @ Ys3 ) ) ) ) ) ).

% lexordp_induct
thf(fact_472_lexordp__induct,axiom,
    ! [Xs: list_nat,Ys3: list_nat,P: list_nat > list_nat > $o] :
      ( ( ord_lexordp_nat @ Xs @ Ys3 )
     => ( ! [Y4: nat,Ys: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y4 @ Ys ) )
       => ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys: list_nat] :
              ( ( ord_less_nat @ X4 @ Y4 )
             => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) ) )
         => ( ! [X4: nat,Xs2: list_nat,Ys: list_nat] :
                ( ( ord_lexordp_nat @ Xs2 @ Ys )
               => ( ( P @ Xs2 @ Ys )
                 => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ X4 @ Ys ) ) ) )
           => ( P @ Xs @ Ys3 ) ) ) ) ) ).

% lexordp_induct
thf(fact_473_lexordp__append__left__rightI,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,Us2: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( ord_le1203424502768444845at_nat @ X3 @ Y )
     => ( ord_le1029496871892194115at_nat @ ( append985823374593552924at_nat @ Us2 @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) @ ( append985823374593552924at_nat @ Us2 @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) ) ) ).

% lexordp_append_left_rightI
thf(fact_474_lexordp__append__left__rightI,axiom,
    ! [X3: nat,Y: nat,Us2: list_nat,Xs: list_nat,Ys3: list_nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ord_lexordp_nat @ ( append_nat @ Us2 @ ( cons_nat @ X3 @ Xs ) ) @ ( append_nat @ Us2 @ ( cons_nat @ Y @ Ys3 ) ) ) ) ).

% lexordp_append_left_rightI
thf(fact_475_lexordp__iff,axiom,
    ( ord_le1029496871892194115at_nat
    = ( ^ [Xs3: list_P6011104703257516679at_nat,Ys2: list_P6011104703257516679at_nat] :
          ( ? [X: product_prod_nat_nat,Vs3: list_P6011104703257516679at_nat] :
              ( Ys2
              = ( append985823374593552924at_nat @ Xs3 @ ( cons_P6512896166579812791at_nat @ X @ Vs3 ) ) )
          | ? [Us3: list_P6011104703257516679at_nat,A4: product_prod_nat_nat,B9: product_prod_nat_nat,Vs3: list_P6011104703257516679at_nat,Ws: list_P6011104703257516679at_nat] :
              ( ( ord_le1203424502768444845at_nat @ A4 @ B9 )
              & ( Xs3
                = ( append985823374593552924at_nat @ Us3 @ ( cons_P6512896166579812791at_nat @ A4 @ Vs3 ) ) )
              & ( Ys2
                = ( append985823374593552924at_nat @ Us3 @ ( cons_P6512896166579812791at_nat @ B9 @ Ws ) ) ) ) ) ) ) ).

% lexordp_iff
thf(fact_476_lexordp__iff,axiom,
    ( ord_lexordp_nat
    = ( ^ [Xs3: list_nat,Ys2: list_nat] :
          ( ? [X: nat,Vs3: list_nat] :
              ( Ys2
              = ( append_nat @ Xs3 @ ( cons_nat @ X @ Vs3 ) ) )
          | ? [Us3: list_nat,A4: nat,B9: nat,Vs3: list_nat,Ws: list_nat] :
              ( ( ord_less_nat @ A4 @ B9 )
              & ( Xs3
                = ( append_nat @ Us3 @ ( cons_nat @ A4 @ Vs3 ) ) )
              & ( Ys2
                = ( append_nat @ Us3 @ ( cons_nat @ B9 @ Ws ) ) ) ) ) ) ) ).

% lexordp_iff
thf(fact_477_pair__lessI1,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_478_measures__lesseq,axiom,
    ! [F2: nat > nat,X3: nat,Y: nat,Fs: list_nat_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ Fs ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_479_measures__lesseq,axiom,
    ! [F2: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_480_greater__bounded__Suc__iff,axiom,
    ! [X3: nat,N4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( suc @ X3 ) @ X3 ) @ ( refine6930732557055572328ounded @ N4 ) )
      = ( ord_less_eq_nat @ ( suc @ X3 ) @ N4 ) ) ).

% greater_bounded_Suc_iff
thf(fact_481_finite__psupset__def,axiom,
    ( refine3059638062392704385et_nat
    = ( ^ [S4: set_nat] :
          ( collec6662362479098859352et_nat
          @ ( produc6247414631856714078_nat_o
            @ ^ [Q5: set_nat,Q4: set_nat] :
                ( ( ord_less_set_nat @ Q4 @ Q5 )
                & ( ord_less_eq_set_nat @ Q5 @ S4 ) ) ) ) ) ) ).

% finite_psupset_def
thf(fact_482_splice_Opelims,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( splice938226673677627610at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P7052990409830227952at_nat @ splice425205989056337325at_nat @ ( produc5943733680697469783at_nat @ X3 @ Xa2 ) )
       => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
           => ( ( Y = Xa2 )
             => ~ ( accp_P7052990409830227952at_nat @ splice425205989056337325at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xa2 ) ) ) )
         => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
               => ( ( Y
                    = ( cons_P6512896166579812791at_nat @ X4 @ ( splice938226673677627610at_nat @ Xa2 @ Xs2 ) ) )
                 => ~ ( accp_P7052990409830227952at_nat @ splice425205989056337325at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ Xa2 ) ) ) ) ) ) ) ).

% splice.pelims
thf(fact_483_splice_Opsimps_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( accp_P7052990409830227952at_nat @ splice425205989056337325at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ Ys3 ) )
     => ( ( splice938226673677627610at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ Ys3 )
        = ( cons_P6512896166579812791at_nat @ X3 @ ( splice938226673677627610at_nat @ Ys3 @ Xs ) ) ) ) ).

% splice.psimps(2)
thf(fact_484_bounded__Max__nat,axiom,
    ! [P: nat > $o,X3: nat,M3: nat] :
      ( ( P @ X3 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M3 ) )
       => ~ ! [M: nat] :
              ( ( P @ M )
             => ~ ! [X6: nat] :
                    ( ( P @ X6 )
                   => ( ord_less_eq_nat @ X6 @ M ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_485_splice_Osimps_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( splice938226673677627610at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ Ys3 )
      = ( cons_P6512896166579812791at_nat @ X3 @ ( splice938226673677627610at_nat @ Ys3 @ Xs ) ) ) ).

% splice.simps(2)
thf(fact_486_splice_Oelims,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( splice938226673677627610at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
         => ( Y != Xa2 ) )
       => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( X3
                = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
             => ( Y
               != ( cons_P6512896166579812791at_nat @ X4 @ ( splice938226673677627610at_nat @ Xa2 @ Xs2 ) ) ) ) ) ) ).

% splice.elims
thf(fact_487_less__eq__set__def,axiom,
    ( ord_le3000389064537975527at_nat
    = ( ^ [A8: set_Pr8693737435421807431at_nat,B8: set_Pr8693737435421807431at_nat] :
          ( ord_le7858099551454983350_nat_o
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A8 )
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_488_less__eq__set__def,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A8: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( ord_le704812498762024988_nat_o
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A8 )
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_489_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A8: set_nat,B8: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A8 )
          @ ^ [X: nat] : ( member_nat @ X @ B8 ) ) ) ) ).

% less_eq_set_def
thf(fact_490_pair__leqI2,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S3 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_491_pair__leqI1,axiom,
    ! [A: nat,B: nat,S3: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S3 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_492_Set_Oempty__def,axiom,
    ( bot_bot_set_list_a
    = ( collect_list_a
      @ ^ [X: list_a] : $false ) ) ).

% Set.empty_def
thf(fact_493_Set_Oempty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% Set.empty_def
thf(fact_494_Collect__subset,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_495_Collect__subset,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_496_Collect__subset,axiom,
    ! [A3: set_list_a,P: list_a > $o] :
      ( ord_le8861187494160871172list_a
      @ ( collect_list_a
        @ ^ [X: list_a] :
            ( ( member_list_a @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_497_Collect__subset,axiom,
    ! [A3: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A3 )
            & ( P @ X ) ) )
      @ A3 ) ).

% Collect_subset
thf(fact_498_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A8: set_nat,B8: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A8 )
          @ ^ [X: nat] : ( member_nat @ X @ B8 ) ) ) ) ).

% less_set_def
thf(fact_499_less__set__def,axiom,
    ( ord_le6428140832669894131at_nat
    = ( ^ [A8: set_Pr8693737435421807431at_nat,B8: set_Pr8693737435421807431at_nat] :
          ( ord_le7432393201274343594_nat_o
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A8 )
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ B8 ) ) ) ) ).

% less_set_def
thf(fact_500_less__set__def,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A8: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
          ( ord_le549003669493604880_nat_o
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A8 )
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B8 ) ) ) ) ).

% less_set_def
thf(fact_501_quicksort__by__rel_Opelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > $o,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( quicks6042341746770278769at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P7097408270579464809at_nat @ quicks44083209960565206at_nat @ ( produc6665944465064540112at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ Xb ) ) )
       => ( ( ( Xb = nil_Pr5478986624290739719at_nat )
           => ( ( Y = Xa2 )
             => ~ ( accp_P7097408270579464809at_nat @ quicks44083209960565206at_nat @ ( produc6665944465064540112at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ nil_Pr5478986624290739719at_nat ) ) ) ) )
         => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
               => ( ( Y
                    = ( produc390734169182146075at_nat
                      @ ^ [Xs_s: list_P6011104703257516679at_nat,Xs_b: list_P6011104703257516679at_nat] : ( quicks6042341746770278769at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ X4 @ ( quicks6042341746770278769at_nat @ X3 @ Xa2 @ Xs_b ) ) @ Xs_s )
                      @ ( partit1285214537004407631at_nat
                        @ ^ [Y3: product_prod_nat_nat] : ( X3 @ Y3 @ X4 )
                        @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
                        @ Xs2 ) ) )
                 => ~ ( accp_P7097408270579464809at_nat @ quicks44083209960565206at_nat @ ( produc6665944465064540112at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ) ) ).

% quicksort_by_rel.pelims
thf(fact_502_quicksort__by__rel_Opsimps_I2_J,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o,Sl2: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( accp_P7097408270579464809at_nat @ quicks44083209960565206at_nat @ ( produc6665944465064540112at_nat @ R @ ( produc5943733680697469783at_nat @ Sl2 @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) ) )
     => ( ( quicks6042341746770278769at_nat @ R @ Sl2 @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
        = ( produc390734169182146075at_nat
          @ ^ [Xs_s: list_P6011104703257516679at_nat,Xs_b: list_P6011104703257516679at_nat] : ( quicks6042341746770278769at_nat @ R @ ( cons_P6512896166579812791at_nat @ X3 @ ( quicks6042341746770278769at_nat @ R @ Sl2 @ Xs_b ) ) @ Xs_s )
          @ ( partit1285214537004407631at_nat
            @ ^ [Y3: product_prod_nat_nat] : ( R @ Y3 @ X3 )
            @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
            @ Xs ) ) ) ) ).

% quicksort_by_rel.psimps(2)
thf(fact_503_quicksort__by__rel_Opinduct,axiom,
    ! [A0: product_prod_nat_nat > product_prod_nat_nat > $o,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P7097408270579464809at_nat @ quicks44083209960565206at_nat @ ( produc6665944465064540112at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,Sl: list_P6011104703257516679at_nat] :
            ( ( accp_P7097408270579464809at_nat @ quicks44083209960565206at_nat @ ( produc6665944465064540112at_nat @ R3 @ ( produc5943733680697469783at_nat @ Sl @ nil_Pr5478986624290739719at_nat ) ) )
           => ( P @ R3 @ Sl @ nil_Pr5478986624290739719at_nat ) )
       => ( ! [R3: product_prod_nat_nat > product_prod_nat_nat > $o,Sl: list_P6011104703257516679at_nat,X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( accp_P7097408270579464809at_nat @ quicks44083209960565206at_nat @ ( produc6665944465064540112at_nat @ R3 @ ( produc5943733680697469783at_nat @ Sl @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) )
             => ( ! [Xa: produc6392793444374437607at_nat,Xb2: list_P6011104703257516679at_nat,Y5: list_P6011104703257516679at_nat] :
                    ( ( Xa
                      = ( partit1285214537004407631at_nat
                        @ ^ [Z4: product_prod_nat_nat] : ( R3 @ Z4 @ X4 )
                        @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
                        @ Xs2 ) )
                   => ( ( ( produc5943733680697469783at_nat @ Xb2 @ Y5 )
                        = Xa )
                     => ( P @ R3 @ Sl @ Y5 ) ) )
               => ( ! [Xa: produc6392793444374437607at_nat,Xb2: list_P6011104703257516679at_nat,Y5: list_P6011104703257516679at_nat] :
                      ( ( Xa
                        = ( partit1285214537004407631at_nat
                          @ ^ [Z4: product_prod_nat_nat] : ( R3 @ Z4 @ X4 )
                          @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
                          @ Xs2 ) )
                     => ( ( ( produc5943733680697469783at_nat @ Xb2 @ Y5 )
                          = Xa )
                       => ( P @ R3 @ ( cons_P6512896166579812791at_nat @ X4 @ ( quicks6042341746770278769at_nat @ R3 @ Sl @ Y5 ) ) @ Xb2 ) ) )
                 => ( P @ R3 @ Sl @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ) )
         => ( P @ A0 @ A1 @ A22 ) ) ) ) ).

% quicksort_by_rel.pinduct
thf(fact_504_in__measure,axiom,
    ! [X3: nat,Y: nat,F2: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( measure_nat @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_505_in__measure,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,F2: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( measur8038558561449204169at_nat @ F2 ) )
      = ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) ) ).

% in_measure
thf(fact_506_in__finite__psubset,axiom,
    ! [A3: set_list_a,B3: set_list_a] :
      ( ( member2637305593812847696list_a @ ( produc161253244684412695list_a @ A3 @ B3 ) @ finite881665802039894343list_a )
      = ( ( ord_less_set_list_a @ A3 @ B3 )
        & ( finite_finite_list_a @ B3 ) ) ) ).

% in_finite_psubset
thf(fact_507_in__finite__psubset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A3 @ B3 ) @ finite_psubset_nat )
      = ( ( ord_less_set_nat @ A3 @ B3 )
        & ( finite_finite_nat @ B3 ) ) ) ).

% in_finite_psubset
thf(fact_508_finite__psubset__def,axiom,
    ( finite881665802039894343list_a
    = ( collec8610444156961935506list_a
      @ ( produc6571814993887497580st_a_o
        @ ^ [A8: set_list_a,B8: set_list_a] :
            ( ( ord_less_set_list_a @ A8 @ B8 )
            & ( finite_finite_list_a @ B8 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_509_finite__psubset__def,axiom,
    ( finite_psubset_nat
    = ( collec6662362479098859352et_nat
      @ ( produc6247414631856714078_nat_o
        @ ^ [A8: set_nat,B8: set_nat] :
            ( ( ord_less_set_nat @ A8 @ B8 )
            & ( finite_finite_nat @ B8 ) ) ) ) ) ).

% finite_psubset_def
thf(fact_510_sunion__with_Opinduct,axiom,
    ! [A0: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A1: list_P8469869581646625389at_nat,A22: list_P8469869581646625389at_nat,P: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o] :
      ( ( accp_P207528311475610904at_nat @ rBT_or1288022168724296696at_nat @ ( produc2486214636825388819at_nat @ A0 @ ( produc1338542795132623831at_nat @ A1 @ A22 ) ) )
     => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat,K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
            ( ( accp_P207528311475610904at_nat @ rBT_or1288022168724296696at_nat @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
           => ( ( ( ord_le1203424502768444845at_nat @ K2 @ K )
               => ( P @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( ord_le1203424502768444845at_nat @ K2 @ K )
                 => ( ( ord_le1203424502768444845at_nat @ K @ K2 )
                   => ( P @ F @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( ord_le1203424502768444845at_nat @ K2 @ K )
                   => ( ~ ( ord_le1203424502768444845at_nat @ K @ K2 )
                     => ( P @ F @ As @ Bs ) ) )
                 => ( P @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
              ( ( accp_P207528311475610904at_nat @ rBT_or1288022168724296696at_nat @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Bs ) ) )
             => ( P @ F @ nil_Pr2582115297535392877at_nat @ Bs ) )
         => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
                ( ( accp_P207528311475610904at_nat @ rBT_or1288022168724296696at_nat @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ As @ nil_Pr2582115297535392877at_nat ) ) )
               => ( P @ F @ As @ nil_Pr2582115297535392877at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% sunion_with.pinduct
thf(fact_511_sunion__with_Opinduct,axiom,
    ! [A0: nat > nat > nat > nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P: ( nat > nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [F: nat > nat > nat > nat,K: nat,V: nat,As: list_P6011104703257516679at_nat,K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
            ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
           => ( ( ( ord_less_nat @ K2 @ K )
               => ( P @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( ord_less_nat @ K2 @ K )
                 => ( ( ord_less_nat @ K @ K2 )
                   => ( P @ F @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( ord_less_nat @ K2 @ K )
                   => ( ~ ( ord_less_nat @ K @ K2 )
                     => ( P @ F @ As @ Bs ) ) )
                 => ( P @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
              ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
             => ( P @ F @ nil_Pr5478986624290739719at_nat @ Bs ) )
         => ( ! [F: nat > nat > nat > nat,As: list_P6011104703257516679at_nat] :
                ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ As @ nil_Pr5478986624290739719at_nat ) ) )
               => ( P @ F @ As @ nil_Pr5478986624290739719at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% sunion_with.pinduct
thf(fact_512_sinter__with_Opinduct,axiom,
    ! [A0: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A1: list_P8469869581646625389at_nat,A22: list_P8469869581646625389at_nat,P: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o] :
      ( ( accp_P207528311475610904at_nat @ rBT_or7061124992729039659at_nat @ ( produc2486214636825388819at_nat @ A0 @ ( produc1338542795132623831at_nat @ A1 @ A22 ) ) )
     => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat,K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
            ( ( accp_P207528311475610904at_nat @ rBT_or7061124992729039659at_nat @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
           => ( ( ( ord_le1203424502768444845at_nat @ K2 @ K )
               => ( P @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( ord_le1203424502768444845at_nat @ K2 @ K )
                 => ( ( ord_le1203424502768444845at_nat @ K @ K2 )
                   => ( P @ F @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( ord_le1203424502768444845at_nat @ K2 @ K )
                   => ( ~ ( ord_le1203424502768444845at_nat @ K @ K2 )
                     => ( P @ F @ As @ Bs ) ) )
                 => ( P @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uu: list_P8469869581646625389at_nat] :
              ( ( accp_P207528311475610904at_nat @ rBT_or7061124992729039659at_nat @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Uu ) ) )
             => ( P @ F @ nil_Pr2582115297535392877at_nat @ Uu ) )
         => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: list_P8469869581646625389at_nat] :
                ( ( accp_P207528311475610904at_nat @ rBT_or7061124992729039659at_nat @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ Uv @ nil_Pr2582115297535392877at_nat ) ) )
               => ( P @ F @ Uv @ nil_Pr2582115297535392877at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% sinter_with.pinduct
thf(fact_513_sinter__with_Opinduct,axiom,
    ! [A0: nat > nat > nat > nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P: ( nat > nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [F: nat > nat > nat > nat,K: nat,V: nat,As: list_P6011104703257516679at_nat,K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
            ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
           => ( ( ( ord_less_nat @ K2 @ K )
               => ( P @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( ord_less_nat @ K2 @ K )
                 => ( ( ord_less_nat @ K @ K2 )
                   => ( P @ F @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( ord_less_nat @ K2 @ K )
                   => ( ~ ( ord_less_nat @ K @ K2 )
                     => ( P @ F @ As @ Bs ) ) )
                 => ( P @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F: nat > nat > nat > nat,Uu: list_P6011104703257516679at_nat] :
              ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Uu ) ) )
             => ( P @ F @ nil_Pr5478986624290739719at_nat @ Uu ) )
         => ( ! [F: nat > nat > nat > nat,Uv: list_P6011104703257516679at_nat] :
                ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ Uv @ nil_Pr5478986624290739719at_nat ) ) )
               => ( P @ F @ Uv @ nil_Pr5478986624290739719at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% sinter_with.pinduct
thf(fact_514_ord_Osunion__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > nat > nat > nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P: ( nat > nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [F: nat > nat > nat > nat,K: nat,V: nat,As: list_P6011104703257516679at_nat,K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
            ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K2 @ K )
               => ( P @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K2 @ K )
                 => ( ( Less @ K @ K2 )
                   => ( P @ F @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K2 @ K )
                   => ( ~ ( Less @ K @ K2 )
                     => ( P @ F @ As @ Bs ) ) )
                 => ( P @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
              ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
             => ( P @ F @ nil_Pr5478986624290739719at_nat @ Bs ) )
         => ( ! [F: nat > nat > nat > nat,As: list_P6011104703257516679at_nat] :
                ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ As @ nil_Pr5478986624290739719at_nat ) ) )
               => ( P @ F @ As @ nil_Pr5478986624290739719at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_515_ord_Osunion__with_Opinduct,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,A0: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A1: list_P8469869581646625389at_nat,A22: list_P8469869581646625389at_nat,P: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o] :
      ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ A0 @ ( produc1338542795132623831at_nat @ A1 @ A22 ) ) )
     => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat,K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
            ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K2 @ K )
               => ( P @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K2 @ K )
                 => ( ( Less @ K @ K2 )
                   => ( P @ F @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K2 @ K )
                   => ( ~ ( Less @ K @ K2 )
                     => ( P @ F @ As @ Bs ) ) )
                 => ( P @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
              ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Bs ) ) )
             => ( P @ F @ nil_Pr2582115297535392877at_nat @ Bs ) )
         => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
                ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ As @ nil_Pr2582115297535392877at_nat ) ) )
               => ( P @ F @ As @ nil_Pr2582115297535392877at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_516_ord_Osinter__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > nat > nat > nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P: ( nat > nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [F: nat > nat > nat > nat,K: nat,V: nat,As: list_P6011104703257516679at_nat,K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
            ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K2 @ K )
               => ( P @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K2 @ K )
                 => ( ( Less @ K @ K2 )
                   => ( P @ F @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K2 @ K )
                   => ( ~ ( Less @ K @ K2 )
                     => ( P @ F @ As @ Bs ) ) )
                 => ( P @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F: nat > nat > nat > nat,Uu: list_P6011104703257516679at_nat] :
              ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Uu ) ) )
             => ( P @ F @ nil_Pr5478986624290739719at_nat @ Uu ) )
         => ( ! [F: nat > nat > nat > nat,Uv: list_P6011104703257516679at_nat] :
                ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ Uv @ nil_Pr5478986624290739719at_nat ) ) )
               => ( P @ F @ Uv @ nil_Pr5478986624290739719at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_517_ord_Osinter__with_Opinduct,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,A0: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A1: list_P8469869581646625389at_nat,A22: list_P8469869581646625389at_nat,P: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > list_P8469869581646625389at_nat > list_P8469869581646625389at_nat > $o] :
      ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ A0 @ ( produc1338542795132623831at_nat @ A1 @ A22 ) ) )
     => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat,K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
            ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
           => ( ( ( Less @ K2 @ K )
               => ( P @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) )
             => ( ( ~ ( Less @ K2 @ K )
                 => ( ( Less @ K @ K2 )
                   => ( P @ F @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K2 @ K )
                   => ( ~ ( Less @ K @ K2 )
                     => ( P @ F @ As @ Bs ) ) )
                 => ( P @ F @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) )
       => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uu: list_P8469869581646625389at_nat] :
              ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Uu ) ) )
             => ( P @ F @ nil_Pr2582115297535392877at_nat @ Uu ) )
         => ( ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: list_P8469869581646625389at_nat] :
                ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F @ ( produc1338542795132623831at_nat @ Uv @ nil_Pr2582115297535392877at_nat ) ) )
               => ( P @ F @ Uv @ nil_Pr2582115297535392877at_nat ) )
           => ( P @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_518_max__ext_Ocases,axiom,
    ! [A1: set_Pr8693737435421807431at_nat,A22: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ A1 @ A22 ) @ ( max_ex4511810952740877858at_nat @ R ) )
     => ~ ( ( finite4392333629123659920at_nat @ A1 )
         => ( ( finite4392333629123659920at_nat @ A22 )
           => ( ( A22 != bot_bo5327735625951526323at_nat )
             => ~ ! [X6: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ X6 @ A1 )
                   => ? [Xa3: produc859450856879609959at_nat] :
                        ( ( member8206827879206165904at_nat @ Xa3 @ A22 )
                        & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_519_max__ext_Ocases,axiom,
    ! [A1: set_list_a,A22: set_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member2637305593812847696list_a @ ( produc161253244684412695list_a @ A1 @ A22 ) @ ( max_ext_list_a @ R ) )
     => ~ ( ( finite_finite_list_a @ A1 )
         => ( ( finite_finite_list_a @ A22 )
           => ( ( A22 != bot_bot_set_list_a )
             => ~ ! [X6: list_a] :
                    ( ( member_list_a @ X6 @ A1 )
                   => ? [Xa3: list_a] :
                        ( ( member_list_a @ Xa3 @ A22 )
                        & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_520_max__ext_Ocases,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R ) )
     => ~ ( ( finite6177210948735845034at_nat @ A1 )
         => ( ( finite6177210948735845034at_nat @ A22 )
           => ( ( A22 != bot_bo2099793752762293965at_nat )
             => ~ ! [X6: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X6 @ A1 )
                   => ? [Xa3: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ Xa3 @ A22 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_521_max__ext_Ocases,axiom,
    ! [A1: set_nat,A22: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R ) )
     => ~ ( ( finite_finite_nat @ A1 )
         => ( ( finite_finite_nat @ A22 )
           => ( ( A22 != bot_bot_set_nat )
             => ~ ! [X6: nat] :
                    ( ( member_nat @ X6 @ A1 )
                   => ? [Xa3: nat] :
                        ( ( member_nat @ Xa3 @ A22 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Xa3 ) @ R ) ) ) ) ) ) ) ).

% max_ext.cases
thf(fact_522_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr8693737435421807431at_nat,Y6: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( finite4392333629123659920at_nat @ X5 )
     => ( ( finite4392333629123659920at_nat @ Y6 )
       => ( ( Y6 != bot_bo5327735625951526323at_nat )
         => ( ! [X4: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X4 @ X5 )
               => ? [Xa: produc859450856879609959at_nat] :
                    ( ( member8206827879206165904at_nat @ Xa @ Y6 )
                    & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ Xa ) @ R ) ) )
           => ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ X5 @ Y6 ) @ ( max_ex4511810952740877858at_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_523_max__ext_Omax__extI,axiom,
    ! [X5: set_list_a,Y6: set_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( finite_finite_list_a @ X5 )
     => ( ( finite_finite_list_a @ Y6 )
       => ( ( Y6 != bot_bot_set_list_a )
         => ( ! [X4: list_a] :
                ( ( member_list_a @ X4 @ X5 )
               => ? [Xa: list_a] :
                    ( ( member_list_a @ Xa @ Y6 )
                    & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Xa ) @ R ) ) )
           => ( member2637305593812847696list_a @ ( produc161253244684412695list_a @ X5 @ Y6 ) @ ( max_ext_list_a @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_524_max__ext_Omax__extI,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,Y6: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( finite6177210948735845034at_nat @ X5 )
     => ( ( finite6177210948735845034at_nat @ Y6 )
       => ( ( Y6 != bot_bo2099793752762293965at_nat )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ X5 )
               => ? [Xa: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa @ Y6 )
                    & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Xa ) @ R ) ) )
           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y6 ) @ ( max_ex8135407076693332796at_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_525_max__ext_Omax__extI,axiom,
    ! [X5: set_nat,Y6: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( finite_finite_nat @ X5 )
     => ( ( finite_finite_nat @ Y6 )
       => ( ( Y6 != bot_bot_set_nat )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ X5 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ Y6 )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Xa ) @ R ) ) )
           => ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X5 @ Y6 ) @ ( max_ext_nat @ R ) ) ) ) ) ) ).

% max_ext.max_extI
thf(fact_526_max__ext_Osimps,axiom,
    ! [A1: set_Pr8693737435421807431at_nat,A22: set_Pr8693737435421807431at_nat,R: set_Pr553994874890374343at_nat] :
      ( ( member5855424355840516880at_nat @ ( produc3236233026405413719at_nat @ A1 @ A22 ) @ ( max_ex4511810952740877858at_nat @ R ) )
      = ( ( finite4392333629123659920at_nat @ A1 )
        & ( finite4392333629123659920at_nat @ A22 )
        & ( A22 != bot_bo5327735625951526323at_nat )
        & ! [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ A1 )
           => ? [Y3: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ Y3 @ A22 )
                & ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X @ Y3 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_527_max__ext_Osimps,axiom,
    ! [A1: set_list_a,A22: set_list_a,R: set_Pr4048851178543822343list_a] :
      ( ( member2637305593812847696list_a @ ( produc161253244684412695list_a @ A1 @ A22 ) @ ( max_ext_list_a @ R ) )
      = ( ( finite_finite_list_a @ A1 )
        & ( finite_finite_list_a @ A22 )
        & ( A22 != bot_bot_set_list_a )
        & ! [X: list_a] :
            ( ( member_list_a @ X @ A1 )
           => ? [Y3: list_a] :
                ( ( member_list_a @ Y3 @ A22 )
                & ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y3 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_528_max__ext_Osimps,axiom,
    ! [A1: set_Pr1261947904930325089at_nat,A22: set_Pr1261947904930325089at_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A1 @ A22 ) @ ( max_ex8135407076693332796at_nat @ R ) )
      = ( ( finite6177210948735845034at_nat @ A1 )
        & ( finite6177210948735845034at_nat @ A22 )
        & ( A22 != bot_bo2099793752762293965at_nat )
        & ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A1 )
           => ? [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ A22 )
                & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_529_max__ext_Osimps,axiom,
    ! [A1: set_nat,A22: set_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A1 @ A22 ) @ ( max_ext_nat @ R ) )
      = ( ( finite_finite_nat @ A1 )
        & ( finite_finite_nat @ A22 )
        & ( A22 != bot_bot_set_nat )
        & ! [X: nat] :
            ( ( member_nat @ X @ A1 )
           => ? [Y3: nat] :
                ( ( member_nat @ Y3 @ A22 )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) ) ) ) ) ).

% max_ext.simps
thf(fact_530_sunion__with_Opelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_or6911406339441992897at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P207528311475610904at_nat @ rBT_or1288022168724296696at_nat @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ Xa2 @ Xb ) ) )
       => ( ! [K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
              ( ( Xa2
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) )
             => ! [K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                  ( ( Xb
                    = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) )
                 => ( ( ( ( ord_le1203424502768444845at_nat @ K2 @ K )
                       => ( Y
                          = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ ( rBT_or6911406339441992897at_nat @ X3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( ord_le1203424502768444845at_nat @ K2 @ K )
                       => ( ( ( ord_le1203424502768444845at_nat @ K @ K2 )
                           => ( Y
                              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ ( rBT_or6911406339441992897at_nat @ X3 @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( ord_le1203424502768444845at_nat @ K @ K2 )
                           => ( Y
                              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_or6911406339441992897at_nat @ X3 @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P207528311475610904at_nat @ rBT_or1288022168724296696at_nat @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa2 = nil_Pr2582115297535392877at_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P207528311475610904at_nat @ rBT_or1288022168724296696at_nat @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
               => ( ( Y = Xa2 )
                 => ~ ( accp_P207528311475610904at_nat @ rBT_or1288022168724296696at_nat @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ Xa2 @ nil_Pr2582115297535392877at_nat ) ) ) ) ) ) ) ) ) ).

% sunion_with.pelims
thf(fact_531_sunion__with_Opelims,axiom,
    ! [X3: nat > nat > nat > nat,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_or5195327883522051875at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ Xb ) ) )
       => ( ! [K: nat,V: nat,As: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) )
             => ! [K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) )
                 => ( ( ( ( ord_less_nat @ K2 @ K )
                       => ( Y
                          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ ( rBT_or5195327883522051875at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( ord_less_nat @ K2 @ K )
                       => ( ( ( ord_less_nat @ K @ K2 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( rBT_or5195327883522051875at_nat @ X3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( ord_less_nat @ K @ K2 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_or5195327883522051875at_nat @ X3 @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y = Xa2 )
                 => ~ ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ) ) ) ) ).

% sunion_with.pelims
thf(fact_532_sinter__with_Opelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_or7569865696902473166at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P207528311475610904at_nat @ rBT_or7061124992729039659at_nat @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ Xa2 @ Xb ) ) )
       => ( ! [K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
              ( ( Xa2
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) )
             => ! [K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                  ( ( Xb
                    = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) )
                 => ( ( ( ( ord_le1203424502768444845at_nat @ K2 @ K )
                       => ( Y
                          = ( rBT_or7569865696902473166at_nat @ X3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( ord_le1203424502768444845at_nat @ K2 @ K )
                       => ( ( ( ord_le1203424502768444845at_nat @ K @ K2 )
                           => ( Y
                              = ( rBT_or7569865696902473166at_nat @ X3 @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( ord_le1203424502768444845at_nat @ K @ K2 )
                           => ( Y
                              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_or7569865696902473166at_nat @ X3 @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P207528311475610904at_nat @ rBT_or7061124992729039659at_nat @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa2 = nil_Pr2582115297535392877at_nat )
             => ( ( Y = nil_Pr2582115297535392877at_nat )
               => ~ ( accp_P207528311475610904at_nat @ rBT_or7061124992729039659at_nat @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
               => ( ( Y = nil_Pr2582115297535392877at_nat )
                 => ~ ( accp_P207528311475610904at_nat @ rBT_or7061124992729039659at_nat @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ Xa2 @ nil_Pr2582115297535392877at_nat ) ) ) ) ) ) ) ) ) ).

% sinter_with.pelims
thf(fact_533_sinter__with_Opelims,axiom,
    ! [X3: nat > nat > nat > nat,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_or1698556862823036848at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ Xb ) ) )
       => ( ! [K: nat,V: nat,As: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) )
             => ! [K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) )
                 => ( ( ( ( ord_less_nat @ K2 @ K )
                       => ( Y
                          = ( rBT_or1698556862823036848at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( ord_less_nat @ K2 @ K )
                       => ( ( ( ord_less_nat @ K @ K2 )
                           => ( Y
                              = ( rBT_or1698556862823036848at_nat @ X3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( ord_less_nat @ K @ K2 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_or1698556862823036848at_nat @ X3 @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
             => ( ( Y = nil_Pr5478986624290739719at_nat )
               => ~ ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y = nil_Pr5478986624290739719at_nat )
                 => ~ ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ) ) ) ) ).

% sinter_with.pelims
thf(fact_534_ord_Osinter__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X3: nat > nat > nat > nat,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_si2101226886118949098at_nat @ Less @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ Xb ) ) )
       => ( ! [K: nat,V: nat,As: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) )
             => ! [K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K2 @ K )
                       => ( Y
                          = ( rBT_si2101226886118949098at_nat @ Less @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( Less @ K2 @ K )
                       => ( ( ( Less @ K @ K2 )
                           => ( Y
                              = ( rBT_si2101226886118949098at_nat @ Less @ X3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K @ K2 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ X3 @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
             => ( ( Y = nil_Pr5478986624290739719at_nat )
               => ~ ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y = nil_Pr5478986624290739719at_nat )
                 => ~ ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_535_ord_Osinter__with_Opelims,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_si2205054011348398088at_nat @ Less @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ Xa2 @ Xb ) ) )
       => ( ! [K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
              ( ( Xa2
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) )
             => ! [K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                  ( ( Xb
                    = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K2 @ K )
                       => ( Y
                          = ( rBT_si2205054011348398088at_nat @ Less @ X3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) ) )
                      & ( ~ ( Less @ K2 @ K )
                       => ( ( ( Less @ K @ K2 )
                           => ( Y
                              = ( rBT_si2205054011348398088at_nat @ Less @ X3 @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K @ K2 )
                           => ( Y
                              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_si2205054011348398088at_nat @ Less @ X3 @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa2 = nil_Pr2582115297535392877at_nat )
             => ( ( Y = nil_Pr2582115297535392877at_nat )
               => ~ ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
               => ( ( Y = nil_Pr2582115297535392877at_nat )
                 => ~ ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ Xa2 @ nil_Pr2582115297535392877at_nat ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_536_ord_Osunion__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X3: nat > nat > nat > nat,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_su5597997906817964125at_nat @ Less @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ Xb ) ) )
       => ( ! [K: nat,V: nat,As: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) )
             => ! [K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K2 @ K )
                       => ( Y
                          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ ( rBT_su5597997906817964125at_nat @ Less @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K2 @ K )
                       => ( ( ( Less @ K @ K2 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( rBT_su5597997906817964125at_nat @ Less @ X3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K @ K2 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ X3 @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y = Xa2 )
                 => ~ ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_537_ord_Osunion__with_Opelims,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_su1546594653887917819at_nat @ Less @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ Xa2 @ Xb ) ) )
       => ( ! [K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
              ( ( Xa2
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) )
             => ! [K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                  ( ( Xb
                    = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) )
                 => ( ( ( ( Less @ K2 @ K )
                       => ( Y
                          = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ ( rBT_su1546594653887917819at_nat @ Less @ X3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K2 @ K )
                       => ( ( ( Less @ K @ K2 )
                           => ( Y
                              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ ( rBT_su1546594653887917819at_nat @ Less @ X3 @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K @ K2 )
                           => ( Y
                              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_su1546594653887917819at_nat @ Less @ X3 @ As @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa2 = nil_Pr2582115297535392877at_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ nil_Pr2582115297535392877at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
               => ( ( Y = Xa2 )
                 => ~ ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ X3 @ ( produc1338542795132623831at_nat @ Xa2 @ nil_Pr2582115297535392877at_nat ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_538_sunion__with_Opsimps_I1_J,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K3: product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,K4: product_prod_nat_nat,V5: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( accp_P207528311475610904at_nat @ rBT_or1288022168724296696at_nat @ ( produc2486214636825388819at_nat @ F2 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
     => ( ( ( ord_le1203424502768444845at_nat @ K4 @ K3 )
         => ( ( rBT_or6911406339441992897at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ ( rBT_or6911406339441992897at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( ord_le1203424502768444845at_nat @ K4 @ K3 )
         => ( ( ( ord_le1203424502768444845at_nat @ K3 @ K4 )
             => ( ( rBT_or6911406339441992897at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ ( rBT_or6911406339441992897at_nat @ F2 @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) ) )
            & ( ~ ( ord_le1203424502768444845at_nat @ K3 @ K4 )
             => ( ( rBT_or6911406339441992897at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_or6911406339441992897at_nat @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% sunion_with.psimps(1)
thf(fact_539_sunion__with_Opsimps_I1_J,axiom,
    ! [F2: nat > nat > nat > nat,K3: nat,V3: nat,As2: list_P6011104703257516679at_nat,K4: nat,V5: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( accp_P8135403078564911670at_nat @ rBT_or197455409983884506at_nat @ ( produc8375354190041410609at_nat @ F2 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
     => ( ( ( ord_less_nat @ K4 @ K3 )
         => ( ( rBT_or5195327883522051875at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ ( rBT_or5195327883522051875at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( ord_less_nat @ K4 @ K3 )
         => ( ( ( ord_less_nat @ K3 @ K4 )
             => ( ( rBT_or5195327883522051875at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ ( rBT_or5195327883522051875at_nat @ F2 @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) ) )
            & ( ~ ( ord_less_nat @ K3 @ K4 )
             => ( ( rBT_or5195327883522051875at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_or5195327883522051875at_nat @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% sunion_with.psimps(1)
thf(fact_540_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K4: nat,K3: nat,F2: nat > nat > nat > nat,V3: nat,As2: list_P6011104703257516679at_nat,V5: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( ( Less @ K4 @ K3 )
       => ( ( rBT_su5597997906817964125at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ ( rBT_su5597997906817964125at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K3 )
       => ( ( ( Less @ K3 @ K4 )
           => ( ( rBT_su5597997906817964125at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ ( rBT_su5597997906817964125at_nat @ Less @ F2 @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K3 @ K4 )
           => ( ( rBT_su5597997906817964125at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_541_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,K4: product_prod_nat_nat,K3: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,V5: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( ( Less @ K4 @ K3 )
       => ( ( rBT_su1546594653887917819at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
          = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ ( rBT_su1546594653887917819at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K3 )
       => ( ( ( Less @ K3 @ K4 )
           => ( ( rBT_su1546594653887917819at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ ( rBT_su1546594653887917819at_nat @ Less @ F2 @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K3 @ K4 )
           => ( ( rBT_su1546594653887917819at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_su1546594653887917819at_nat @ Less @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_542_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K4: nat,K3: nat,F2: nat > nat > nat > nat,V3: nat,As2: list_P6011104703257516679at_nat,V5: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( ( Less @ K4 @ K3 )
       => ( ( rBT_si2101226886118949098at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
          = ( rBT_si2101226886118949098at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K3 )
       => ( ( ( Less @ K3 @ K4 )
           => ( ( rBT_si2101226886118949098at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( rBT_si2101226886118949098at_nat @ Less @ F2 @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K3 @ K4 )
           => ( ( rBT_si2101226886118949098at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_543_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,K4: product_prod_nat_nat,K3: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,V5: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( ( Less @ K4 @ K3 )
       => ( ( rBT_si2205054011348398088at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
          = ( rBT_si2205054011348398088at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K3 )
       => ( ( ( Less @ K3 @ K4 )
           => ( ( rBT_si2205054011348398088at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( rBT_si2205054011348398088at_nat @ Less @ F2 @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K3 @ K4 )
           => ( ( rBT_si2205054011348398088at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_si2205054011348398088at_nat @ Less @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_544_sinter__with_Osimps_I1_J,axiom,
    ! [K4: product_prod_nat_nat,K3: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,V5: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( ( ord_le1203424502768444845at_nat @ K4 @ K3 )
       => ( ( rBT_or7569865696902473166at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
          = ( rBT_or7569865696902473166at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( ord_le1203424502768444845at_nat @ K4 @ K3 )
       => ( ( ( ord_le1203424502768444845at_nat @ K3 @ K4 )
           => ( ( rBT_or7569865696902473166at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( rBT_or7569865696902473166at_nat @ F2 @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
          & ( ~ ( ord_le1203424502768444845at_nat @ K3 @ K4 )
           => ( ( rBT_or7569865696902473166at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_or7569865696902473166at_nat @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% sinter_with.simps(1)
thf(fact_545_sinter__with_Osimps_I1_J,axiom,
    ! [K4: nat,K3: nat,F2: nat > nat > nat > nat,V3: nat,As2: list_P6011104703257516679at_nat,V5: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( ( ord_less_nat @ K4 @ K3 )
       => ( ( rBT_or1698556862823036848at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
          = ( rBT_or1698556862823036848at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) )
      & ( ~ ( ord_less_nat @ K4 @ K3 )
       => ( ( ( ord_less_nat @ K3 @ K4 )
           => ( ( rBT_or1698556862823036848at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( rBT_or1698556862823036848at_nat @ F2 @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
          & ( ~ ( ord_less_nat @ K3 @ K4 )
           => ( ( rBT_or1698556862823036848at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_or1698556862823036848at_nat @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% sinter_with.simps(1)
thf(fact_546_sunion__with_Osimps_I1_J,axiom,
    ! [K4: product_prod_nat_nat,K3: product_prod_nat_nat,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,V5: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( ( ord_le1203424502768444845at_nat @ K4 @ K3 )
       => ( ( rBT_or6911406339441992897at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
          = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ ( rBT_or6911406339441992897at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( ord_le1203424502768444845at_nat @ K4 @ K3 )
       => ( ( ( ord_le1203424502768444845at_nat @ K3 @ K4 )
           => ( ( rBT_or6911406339441992897at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ ( rBT_or6911406339441992897at_nat @ F2 @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) ) )
          & ( ~ ( ord_le1203424502768444845at_nat @ K3 @ K4 )
           => ( ( rBT_or6911406339441992897at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_or6911406339441992897at_nat @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% sunion_with.simps(1)
thf(fact_547_sunion__with_Osimps_I1_J,axiom,
    ! [K4: nat,K3: nat,F2: nat > nat > nat > nat,V3: nat,As2: list_P6011104703257516679at_nat,V5: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( ( ord_less_nat @ K4 @ K3 )
       => ( ( rBT_or5195327883522051875at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ ( rBT_or5195327883522051875at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) ) )
      & ( ~ ( ord_less_nat @ K4 @ K3 )
       => ( ( ( ord_less_nat @ K3 @ K4 )
           => ( ( rBT_or5195327883522051875at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ ( rBT_or5195327883522051875at_nat @ F2 @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) ) )
          & ( ~ ( ord_less_nat @ K3 @ K4 )
           => ( ( rBT_or5195327883522051875at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_or5195327883522051875at_nat @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ).

% sunion_with.simps(1)
thf(fact_548_ord_Osunion__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X3: nat > nat > nat > nat,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_su5597997906817964125at_nat @ Less @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ! [K: nat,V: nat,As: list_P6011104703257516679at_nat] :
            ( ( Xa2
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) )
           => ! [K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K2 @ K )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ ( rBT_su5597997906817964125at_nat @ Less @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K2 @ K )
                     => ( ( ( Less @ K @ K2 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( rBT_su5597997906817964125at_nat @ Less @ X3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K @ K2 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ X3 @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y != Xa2 ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_549_ord_Osunion__with_Oelims,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_su1546594653887917819at_nat @ Less @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ! [K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
            ( ( Xa2
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) )
           => ! [K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                ( ( Xb
                  = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K2 @ K )
                     => ( Y
                        = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ ( rBT_su1546594653887917819at_nat @ Less @ X3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K2 @ K )
                     => ( ( ( Less @ K @ K2 )
                         => ( Y
                            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ ( rBT_su1546594653887917819at_nat @ Less @ X3 @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K @ K2 )
                         => ( Y
                            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_su1546594653887917819at_nat @ Less @ X3 @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr2582115297535392877at_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
             => ( Y != Xa2 ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_550_ord_Osinter__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X3: nat > nat > nat > nat,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_si2101226886118949098at_nat @ Less @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ! [K: nat,V: nat,As: list_P6011104703257516679at_nat] :
            ( ( Xa2
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) )
           => ! [K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K2 @ K )
                     => ( Y
                        = ( rBT_si2101226886118949098at_nat @ Less @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K2 @ K )
                     => ( ( ( Less @ K @ K2 )
                         => ( Y
                            = ( rBT_si2101226886118949098at_nat @ Less @ X3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K @ K2 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ X3 @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( Y != nil_Pr5478986624290739719at_nat ) )
         => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y != nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_551_ord_Osinter__with_Oelims,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_si2205054011348398088at_nat @ Less @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ! [K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
            ( ( Xa2
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) )
           => ! [K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                ( ( Xb
                  = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) )
               => ~ ( ( ( Less @ K2 @ K )
                     => ( Y
                        = ( rBT_si2205054011348398088at_nat @ Less @ X3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( Less @ K2 @ K )
                     => ( ( ( Less @ K @ K2 )
                         => ( Y
                            = ( rBT_si2205054011348398088at_nat @ Less @ X3 @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K @ K2 )
                         => ( Y
                            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_si2205054011348398088at_nat @ Less @ X3 @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr2582115297535392877at_nat )
           => ( Y != nil_Pr2582115297535392877at_nat ) )
         => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
             => ( Y != nil_Pr2582115297535392877at_nat ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_552_sinter__with_Oelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_or7569865696902473166at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ! [K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
            ( ( Xa2
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) )
           => ! [K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                ( ( Xb
                  = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) )
               => ~ ( ( ( ord_le1203424502768444845at_nat @ K2 @ K )
                     => ( Y
                        = ( rBT_or7569865696902473166at_nat @ X3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( ord_le1203424502768444845at_nat @ K2 @ K )
                     => ( ( ( ord_le1203424502768444845at_nat @ K @ K2 )
                         => ( Y
                            = ( rBT_or7569865696902473166at_nat @ X3 @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( ord_le1203424502768444845at_nat @ K @ K2 )
                         => ( Y
                            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_or7569865696902473166at_nat @ X3 @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr2582115297535392877at_nat )
           => ( Y != nil_Pr2582115297535392877at_nat ) )
         => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
             => ( Y != nil_Pr2582115297535392877at_nat ) ) ) ) ) ).

% sinter_with.elims
thf(fact_553_sinter__with_Oelims,axiom,
    ! [X3: nat > nat > nat > nat,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_or1698556862823036848at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ! [K: nat,V: nat,As: list_P6011104703257516679at_nat] :
            ( ( Xa2
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) )
           => ! [K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) )
               => ~ ( ( ( ord_less_nat @ K2 @ K )
                     => ( Y
                        = ( rBT_or1698556862823036848at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) ) )
                    & ( ~ ( ord_less_nat @ K2 @ K )
                     => ( ( ( ord_less_nat @ K @ K2 )
                         => ( Y
                            = ( rBT_or1698556862823036848at_nat @ X3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) )
                        & ( ~ ( ord_less_nat @ K @ K2 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_or1698556862823036848at_nat @ X3 @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( Y != nil_Pr5478986624290739719at_nat ) )
         => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y != nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% sinter_with.elims
thf(fact_554_sunion__with_Oelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: list_P8469869581646625389at_nat,Xb: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( ( rBT_or6911406339441992897at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ! [K: product_prod_nat_nat,V: product_prod_nat_nat,As: list_P8469869581646625389at_nat] :
            ( ( Xa2
              = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) )
           => ! [K2: product_prod_nat_nat,V2: product_prod_nat_nat,Bs: list_P8469869581646625389at_nat] :
                ( ( Xb
                  = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) )
               => ~ ( ( ( ord_le1203424502768444845at_nat @ K2 @ K )
                     => ( Y
                        = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ ( rBT_or6911406339441992897at_nat @ X3 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( ord_le1203424502768444845at_nat @ K2 @ K )
                     => ( ( ( ord_le1203424502768444845at_nat @ K @ K2 )
                         => ( Y
                            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ V ) @ ( rBT_or6911406339441992897at_nat @ X3 @ As @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K2 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( ord_le1203424502768444845at_nat @ K @ K2 )
                         => ( Y
                            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_or6911406339441992897at_nat @ X3 @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr2582115297535392877at_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr2582115297535392877at_nat )
             => ( Y != Xa2 ) ) ) ) ) ).

% sunion_with.elims
thf(fact_555_sunion__with_Oelims,axiom,
    ! [X3: nat > nat > nat > nat,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_or5195327883522051875at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ! [K: nat,V: nat,As: list_P6011104703257516679at_nat] :
            ( ( Xa2
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) )
           => ! [K2: nat,V2: nat,Bs: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) )
               => ~ ( ( ( ord_less_nat @ K2 @ K )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ ( rBT_or5195327883522051875at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As ) @ Bs ) ) ) )
                    & ( ~ ( ord_less_nat @ K2 @ K )
                     => ( ( ( ord_less_nat @ K @ K2 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( rBT_or5195327883522051875at_nat @ X3 @ As @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ Bs ) ) ) ) )
                        & ( ~ ( ord_less_nat @ K @ K2 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( X3 @ K @ V @ V2 ) ) @ ( rBT_or5195327883522051875at_nat @ X3 @ As @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y != Xa2 ) ) ) ) ) ).

% sunion_with.elims
thf(fact_556_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F2: nat > nat > nat > nat,K3: nat,V3: nat,As2: list_P6011104703257516679at_nat,K4: nat,V5: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F2 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K3 )
         => ( ( rBT_su5597997906817964125at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ ( rBT_su5597997906817964125at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K4 @ K3 )
         => ( ( ( Less @ K3 @ K4 )
             => ( ( rBT_su5597997906817964125at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ ( rBT_su5597997906817964125at_nat @ Less @ F2 @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K3 @ K4 )
             => ( ( rBT_su5597997906817964125at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_557_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K3: product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,K4: product_prod_nat_nat,V5: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( accp_P207528311475610904at_nat @ ( rBT_su1423084493071127102at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F2 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K3 )
         => ( ( rBT_su1546594653887917819at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
            = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ ( rBT_su1546594653887917819at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K4 @ K3 )
         => ( ( ( Less @ K3 @ K4 )
             => ( ( rBT_su1546594653887917819at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ ( rBT_su1546594653887917819at_nat @ Less @ F2 @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K3 @ K4 )
             => ( ( rBT_su1546594653887917819at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_su1546594653887917819at_nat @ Less @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_558_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F2: nat > nat > nat > nat,K3: nat,V3: nat,As2: list_P6011104703257516679at_nat,K4: nat,V5: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F2 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K3 )
         => ( ( rBT_si2101226886118949098at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
            = ( rBT_si2101226886118949098at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K4 @ K3 )
         => ( ( ( Less @ K3 @ K4 )
             => ( ( rBT_si2101226886118949098at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( rBT_si2101226886118949098at_nat @ Less @ F2 @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K3 @ K4 )
             => ( ( rBT_si2101226886118949098at_nat @ Less @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_559_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: product_prod_nat_nat > product_prod_nat_nat > $o,F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K3: product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,K4: product_prod_nat_nat,V5: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( accp_P207528311475610904at_nat @ ( rBT_si7196187317075870065at_nat @ Less ) @ ( produc2486214636825388819at_nat @ F2 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K3 )
         => ( ( rBT_si2205054011348398088at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
            = ( rBT_si2205054011348398088at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K4 @ K3 )
         => ( ( ( Less @ K3 @ K4 )
             => ( ( rBT_si2205054011348398088at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( rBT_si2205054011348398088at_nat @ Less @ F2 @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K3 @ K4 )
             => ( ( rBT_si2205054011348398088at_nat @ Less @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_si2205054011348398088at_nat @ Less @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_560_sinter__with_Opsimps_I1_J,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,K3: product_prod_nat_nat,V3: product_prod_nat_nat,As2: list_P8469869581646625389at_nat,K4: product_prod_nat_nat,V5: product_prod_nat_nat,Bs2: list_P8469869581646625389at_nat] :
      ( ( accp_P207528311475610904at_nat @ rBT_or7061124992729039659at_nat @ ( produc2486214636825388819at_nat @ F2 @ ( produc1338542795132623831at_nat @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
     => ( ( ( ord_le1203424502768444845at_nat @ K4 @ K3 )
         => ( ( rBT_or7569865696902473166at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
            = ( rBT_or7569865696902473166at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( ord_le1203424502768444845at_nat @ K4 @ K3 )
         => ( ( ( ord_le1203424502768444845at_nat @ K3 @ K4 )
             => ( ( rBT_or7569865696902473166at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( rBT_or7569865696902473166at_nat @ F2 @ As2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
            & ( ~ ( ord_le1203424502768444845at_nat @ K3 @ K4 )
             => ( ( rBT_or7569865696902473166at_nat @ F2 @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_or7569865696902473166at_nat @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% sinter_with.psimps(1)
thf(fact_561_sinter__with_Opsimps_I1_J,axiom,
    ! [F2: nat > nat > nat > nat,K3: nat,V3: nat,As2: list_P6011104703257516679at_nat,K4: nat,V5: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( accp_P8135403078564911670at_nat @ rBT_or3742350075172107405at_nat @ ( produc8375354190041410609at_nat @ F2 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
     => ( ( ( ord_less_nat @ K4 @ K3 )
         => ( ( rBT_or1698556862823036848at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
            = ( rBT_or1698556862823036848at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ Bs2 ) ) )
        & ( ~ ( ord_less_nat @ K4 @ K3 )
         => ( ( ( ord_less_nat @ K3 @ K4 )
             => ( ( rBT_or1698556862823036848at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( rBT_or1698556862823036848at_nat @ F2 @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) ) ) )
            & ( ~ ( ord_less_nat @ K3 @ K4 )
             => ( ( rBT_or1698556862823036848at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V5 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ ( F2 @ K3 @ V3 @ V5 ) ) @ ( rBT_or1698556862823036848at_nat @ F2 @ As2 @ Bs2 ) ) ) ) ) ) ) ) ).

% sinter_with.psimps(1)
thf(fact_562_max__ext__def,axiom,
    ( max_ext_nat
    = ( ^ [R6: set_Pr1261947904930325089at_nat] :
          ( collec6662362479098859352et_nat
          @ ( produc6247414631856714078_nat_o
            @ ( max_extp_nat
              @ ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R6 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_563_max__ext__def,axiom,
    ( max_ex8135407076693332796at_nat
    = ( ^ [R6: set_Pr8693737435421807431at_nat] :
          ( collec6321179662152712658at_nat
          @ ( produc410239310623530412_nat_o
            @ ( max_ex4864111882549613972at_nat
              @ ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R6 ) ) ) ) ) ) ).

% max_ext_def
thf(fact_564_max__extp__max__ext__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat] :
      ( ( max_extp_nat
        @ ^ [X: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R ) )
      = ( ^ [X: set_nat,Y3: set_nat] : ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ X @ Y3 ) @ ( max_ext_nat @ R ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_565_max__extp__max__ext__eq,axiom,
    ! [R: set_Pr8693737435421807431at_nat] :
      ( ( max_ex4864111882549613972at_nat
        @ ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R ) )
      = ( ^ [X: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y3 ) @ ( max_ex8135407076693332796at_nat @ R ) ) ) ) ).

% max_extp_max_ext_eq
thf(fact_566_map__tailrec__rev_Opelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( map_ta8000144545886931765at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P7357970148062374190at_nat @ map_ta4366553869267992196at_nat @ ( produc8396950466146349609at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ Xb ) ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( ( Y = Xb )
             => ~ ( accp_P7357970148062374190at_nat @ map_ta4366553869267992196at_nat @ ( produc8396950466146349609at_nat @ X3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
         => ~ ! [A5: product_prod_nat_nat,As: list_P6011104703257516679at_nat] :
                ( ( Xa2
                  = ( cons_P6512896166579812791at_nat @ A5 @ As ) )
               => ( ( Y
                    = ( map_ta8000144545886931765at_nat @ X3 @ As @ ( cons_P6512896166579812791at_nat @ ( X3 @ A5 ) @ Xb ) ) )
                 => ~ ( accp_P7357970148062374190at_nat @ map_ta4366553869267992196at_nat @ ( produc8396950466146349609at_nat @ X3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ A5 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_567_mlex__eq,axiom,
    ( mlex_prod_nat
    = ( ^ [F4: nat > nat,R6: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y3: nat] :
                ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y3 ) )
                | ( ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y3 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R6 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_568_mlex__eq,axiom,
    ( mlex_p6366001652026297872at_nat
    = ( ^ [F4: product_prod_nat_nat > nat,R6: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
                ( ( ord_less_nat @ ( F4 @ X ) @ ( F4 @ Y3 ) )
                | ( ( ord_less_eq_nat @ ( F4 @ X ) @ ( F4 @ Y3 ) )
                  & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R6 ) ) ) ) ) ) ) ).

% mlex_eq
thf(fact_569_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,As2: list_P6011104703257516679at_nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( map_ta8000144545886931765at_nat @ F2 @ ( cons_P6512896166579812791at_nat @ A @ As2 ) @ Bs2 )
      = ( map_ta8000144545886931765at_nat @ F2 @ As2 @ ( cons_P6512896166579812791at_nat @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_570_map__tailrec__rev_Oelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( map_ta8000144545886931765at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
         => ( Y != Xb ) )
       => ~ ! [A5: product_prod_nat_nat,As: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ A5 @ As ) )
             => ( Y
               != ( map_ta8000144545886931765at_nat @ X3 @ As @ ( cons_P6512896166579812791at_nat @ ( X3 @ A5 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_571_mlex__less,axiom,
    ! [F2: nat > nat,X3: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( mlex_prod_nat @ F2 @ R ) ) ) ).

% mlex_less
thf(fact_572_mlex__less,axiom,
    ! [F2: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( mlex_p6366001652026297872at_nat @ F2 @ R ) ) ) ).

% mlex_less
thf(fact_573_mlex__iff,axiom,
    ! [X3: nat,Y: nat,F2: nat > nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( mlex_prod_nat @ F2 @ R ) )
      = ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X3 )
            = ( F2 @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_574_mlex__iff,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,F2: product_prod_nat_nat > nat,R: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( mlex_p6366001652026297872at_nat @ F2 @ R ) )
      = ( ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X3 )
            = ( F2 @ Y ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R ) ) ) ) ).

% mlex_iff
thf(fact_575_mlex__leq,axiom,
    ! [F2: nat > nat,X3: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( mlex_prod_nat @ F2 @ R ) ) ) ) ).

% mlex_leq
thf(fact_576_mlex__leq,axiom,
    ! [F2: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( mlex_p6366001652026297872at_nat @ F2 @ R ) ) ) ) ).

% mlex_leq
thf(fact_577_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( linord2614967742042102400et_nat @ A3 )
          = nil_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_578_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ( ( linord2614967742042102400et_nat @ bot_bot_set_nat )
    = nil_nat ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_579_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( linord2614967742042102400et_nat @ A3 )
        = nil_nat ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
thf(fact_580_filter_Osimps_I2_J,axiom,
    ! [P: product_prod_nat_nat > $o,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( P @ X3 )
       => ( ( filter6372491115368938494at_nat @ P @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
          = ( cons_P6512896166579812791at_nat @ X3 @ ( filter6372491115368938494at_nat @ P @ Xs ) ) ) )
      & ( ~ ( P @ X3 )
       => ( ( filter6372491115368938494at_nat @ P @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
          = ( filter6372491115368938494at_nat @ P @ Xs ) ) ) ) ).

% filter.simps(2)
thf(fact_581_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( linord2614967742042102400et_nat @ A3 )
        = ( linord2614967742042102400et_nat @ B3 ) )
     => ( ( finite_finite_nat @ A3 )
       => ( ( finite_finite_nat @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_582_quicksort_Oelims,axiom,
    ! [X3: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( linord6276275005637476481at_nat @ X3 )
        = Y )
     => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( X3
                = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
             => ( Y
               != ( append985823374593552924at_nat
                  @ ( linord6276275005637476481at_nat
                    @ ( filter6372491115368938494at_nat
                      @ ^ [Y3: product_prod_nat_nat] :
                          ~ ( ord_le8460144461188290721at_nat @ X4 @ Y3 )
                      @ Xs2 ) )
                  @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) @ ( linord6276275005637476481at_nat @ ( filter6372491115368938494at_nat @ ( ord_le8460144461188290721at_nat @ X4 ) @ Xs2 ) ) ) ) ) ) ) ) ).

% quicksort.elims
thf(fact_583_quicksort_Oelims,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( ( linord3817693808505947216rt_nat @ X3 )
        = Y )
     => ( ( ( X3 = nil_nat )
         => ( Y != nil_nat ) )
       => ~ ! [X4: nat,Xs2: list_nat] :
              ( ( X3
                = ( cons_nat @ X4 @ Xs2 ) )
             => ( Y
               != ( append_nat
                  @ ( linord3817693808505947216rt_nat
                    @ ( filter_nat
                      @ ^ [Y3: nat] :
                          ~ ( ord_less_eq_nat @ X4 @ Y3 )
                      @ Xs2 ) )
                  @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ ( linord3817693808505947216rt_nat @ ( filter_nat @ ( ord_less_eq_nat @ X4 ) @ Xs2 ) ) ) ) ) ) ) ) ).

% quicksort.elims
thf(fact_584_quicksort_Osimps_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( linord6276275005637476481at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( append985823374593552924at_nat
        @ ( linord6276275005637476481at_nat
          @ ( filter6372491115368938494at_nat
            @ ^ [Y3: product_prod_nat_nat] :
                ~ ( ord_le8460144461188290721at_nat @ X3 @ Y3 )
            @ Xs ) )
        @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) @ ( linord6276275005637476481at_nat @ ( filter6372491115368938494at_nat @ ( ord_le8460144461188290721at_nat @ X3 ) @ Xs ) ) ) ) ) ).

% quicksort.simps(2)
thf(fact_585_quicksort_Osimps_I2_J,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( linord3817693808505947216rt_nat @ ( cons_nat @ X3 @ Xs ) )
      = ( append_nat
        @ ( linord3817693808505947216rt_nat
          @ ( filter_nat
            @ ^ [Y3: nat] :
                ~ ( ord_less_eq_nat @ X3 @ Y3 )
            @ Xs ) )
        @ ( append_nat @ ( cons_nat @ X3 @ nil_nat ) @ ( linord3817693808505947216rt_nat @ ( filter_nat @ ( ord_less_eq_nat @ X3 ) @ Xs ) ) ) ) ) ).

% quicksort.simps(2)
thf(fact_586_quicksort_Opelims,axiom,
    ! [X3: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( linord6276275005637476481at_nat @ X3 )
        = Y )
     => ( ( accp_l244970489926305168at_nat @ linord1280469666291201734at_nat @ X3 )
       => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
           => ( ( Y = nil_Pr5478986624290739719at_nat )
             => ~ ( accp_l244970489926305168at_nat @ linord1280469666291201734at_nat @ nil_Pr5478986624290739719at_nat ) ) )
         => ~ ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
               => ( ( Y
                    = ( append985823374593552924at_nat
                      @ ( linord6276275005637476481at_nat
                        @ ( filter6372491115368938494at_nat
                          @ ^ [Y3: product_prod_nat_nat] :
                              ~ ( ord_le8460144461188290721at_nat @ X4 @ Y3 )
                          @ Xs2 ) )
                      @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) @ ( linord6276275005637476481at_nat @ ( filter6372491115368938494at_nat @ ( ord_le8460144461188290721at_nat @ X4 ) @ Xs2 ) ) ) ) )
                 => ~ ( accp_l244970489926305168at_nat @ linord1280469666291201734at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ).

% quicksort.pelims
thf(fact_587_quicksort_Opelims,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( ( linord3817693808505947216rt_nat @ X3 )
        = Y )
     => ( ( accp_list_nat @ linord2871863056034072651el_nat @ X3 )
       => ( ( ( X3 = nil_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_list_nat @ linord2871863056034072651el_nat @ nil_nat ) ) )
         => ~ ! [X4: nat,Xs2: list_nat] :
                ( ( X3
                  = ( cons_nat @ X4 @ Xs2 ) )
               => ( ( Y
                    = ( append_nat
                      @ ( linord3817693808505947216rt_nat
                        @ ( filter_nat
                          @ ^ [Y3: nat] :
                              ~ ( ord_less_eq_nat @ X4 @ Y3 )
                          @ Xs2 ) )
                      @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ ( linord3817693808505947216rt_nat @ ( filter_nat @ ( ord_less_eq_nat @ X4 ) @ Xs2 ) ) ) ) )
                 => ~ ( accp_list_nat @ linord2871863056034072651el_nat @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ).

% quicksort.pelims
thf(fact_588_less__than__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ less_than )
      = ( ord_less_nat @ X3 @ Y ) ) ).

% less_than_iff
thf(fact_589_in__lex__prod,axiom,
    ! [A: nat,B: product_prod_nat_nat,A2: nat,B2: product_prod_nat_nat,R4: set_Pr1261947904930325089at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ( member3048279782668807382at_nat @ ( produc2653010282673554021at_nat @ ( produc487386426758144856at_nat @ A @ B ) @ ( produc487386426758144856at_nat @ A2 @ B2 ) ) @ ( lex_pr8029265285556086080at_nat @ R4 @ S3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R4 )
        | ( ( A = A2 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_590_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: nat,A2: product_prod_nat_nat,B2: nat,R4: set_Pr8693737435421807431at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( member1995966531042493578at_nat @ ( produc7904928797850150681at_nat @ ( produc6350711070570205562at_nat @ A @ B ) @ ( produc6350711070570205562at_nat @ A2 @ B2 ) ) @ ( lex_pr4669217892513370978at_nat @ R4 @ S3 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R4 )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_591_in__lex__prod,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat,S3: set_Pr8693737435421807431at_nat] :
      ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( produc6161850002892822231at_nat @ A2 @ B2 ) ) @ ( lex_pr8801849515957261039at_nat @ R4 @ S3 ) )
      = ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A2 ) @ R4 )
        | ( ( A = A2 )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_592_in__lex__prod,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat,R4: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) @ ( lex_prod_nat_nat @ R4 @ S3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A2 ) @ R4 )
        | ( ( A = A2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ B2 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_593_lex__prod__def,axiom,
    ( lex_prod_nat_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ( produc8739625826339149834_nat_o
              @ ^ [A4: nat,B9: nat] :
                  ( produc6081775807080527818_nat_o
                  @ ^ [A9: nat,B10: nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A9 ) @ Ra )
                      | ( ( A4 = A9 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B9 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_594_lex__prod__def,axiom,
    ( lex_pr8029265285556086080at_nat
    = ( ^ [Ra: set_Pr1261947904930325089at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec3253044014973990424at_nat
          @ ( produc2869316064735525022_nat_o
            @ ( produc7480869177107369060_nat_o
              @ ^ [A4: nat,B9: product_prod_nat_nat] :
                  ( produc5864757623865647827_nat_o
                  @ ^ [A9: nat,B10: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A9 ) @ Ra )
                      | ( ( A4 = A9 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B9 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_595_lex__prod__def,axiom,
    ( lex_pr4669217892513370978at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr1261947904930325089at_nat] :
          ( collec2200730763347676620at_nat
          @ ( produc3600351907215576810_nat_o
            @ ( produc6198819664464039784_nat_o
              @ ^ [A4: product_prod_nat_nat,B9: nat] :
                  ( produc8758269395934548017_nat_o
                  @ ^ [A9: product_prod_nat_nat,B10: nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ A9 ) @ Ra )
                      | ( ( A4 = A9 )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B9 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_596_lex__prod__def,axiom,
    ( lex_pr8801849515957261039at_nat
    = ( ^ [Ra: set_Pr8693737435421807431at_nat,Rb: set_Pr8693737435421807431at_nat] :
          ( collec7334067512558549330at_nat
          @ ( produc3603361296207914284_nat_o
            @ ( produc5077327844045112642_nat_o
              @ ^ [A4: product_prod_nat_nat,B9: product_prod_nat_nat] :
                  ( produc6590410687421337004_nat_o
                  @ ^ [A9: product_prod_nat_nat,B10: product_prod_nat_nat] :
                      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A4 @ A9 ) @ Ra )
                      | ( ( A4 = A9 )
                        & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B9 @ B10 ) @ Rb ) ) ) ) ) ) ) ) ) ).

% lex_prod_def
thf(fact_597_mlex__prod__def,axiom,
    ( mlex_prod_nat
    = ( ^ [F4: nat > nat,R6: set_Pr1261947904930325089at_nat] :
          ( inv_im8721129524585065343at_nat @ ( lex_prod_nat_nat @ less_than @ R6 )
          @ ^ [X: nat] : ( product_Pair_nat_nat @ ( F4 @ X ) @ X ) ) ) ) ).

% mlex_prod_def
thf(fact_598_diff__sorted_Opelims,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( sorted5599871463696753131at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P7052990409830227952at_nat @ sorted8542721443386759644at_nat @ ( produc5943733680697469783at_nat @ X3 @ Xa2 ) )
       => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
           => ( ( Y = nil_Pr5478986624290739719at_nat )
             => ~ ( accp_P7052990409830227952at_nat @ sorted8542721443386759644at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xa2 ) ) ) )
         => ( ! [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ V @ Va ) )
               => ( ( Xa2 = nil_Pr5478986624290739719at_nat )
                 => ( ( Y
                      = ( cons_P6512896166579812791at_nat @ V @ Va ) )
                   => ~ ( accp_P7052990409830227952at_nat @ sorted8542721443386759644at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) @ nil_Pr5478986624290739719at_nat ) ) ) ) )
           => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
                  ( ( X3
                    = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
                 => ! [X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
                      ( ( Xa2
                        = ( cons_P6512896166579812791at_nat @ X23 @ L22 ) )
                     => ( ( ( ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ X12 @ ( sorted5599871463696753131at_nat @ L1 @ ( cons_P6512896166579812791at_nat @ X23 @ L22 ) ) ) ) )
                          & ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                           => ( ( ( X12 = X23 )
                               => ( Y
                                  = ( sorted5599871463696753131at_nat @ L1 @ L22 ) ) )
                              & ( ( X12 != X23 )
                               => ( Y
                                  = ( sorted5599871463696753131at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) )
                       => ~ ( accp_P7052990409830227952at_nat @ sorted8542721443386759644at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ ( cons_P6512896166579812791at_nat @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ).

% diff_sorted.pelims
thf(fact_599_diff__sorted_Opelims,axiom,
    ! [X3: list_nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( sorted636521546038734694ed_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P8037286306265792042st_nat @ sorted1291390330255423669el_nat @ ( produc2694037385005941721st_nat @ X3 @ Xa2 ) )
       => ( ( ( X3 = nil_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_P8037286306265792042st_nat @ sorted1291390330255423669el_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa2 ) ) ) )
         => ( ! [V: nat,Va: list_nat] :
                ( ( X3
                  = ( cons_nat @ V @ Va ) )
               => ( ( Xa2 = nil_nat )
                 => ( ( Y
                      = ( cons_nat @ V @ Va ) )
                   => ~ ( accp_P8037286306265792042st_nat @ sorted1291390330255423669el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) ) ) )
           => ~ ! [X12: nat,L1: list_nat] :
                  ( ( X3
                    = ( cons_nat @ X12 @ L1 ) )
                 => ! [X23: nat,L22: list_nat] :
                      ( ( Xa2
                        = ( cons_nat @ X23 @ L22 ) )
                     => ( ( ( ( ord_less_nat @ X12 @ X23 )
                           => ( Y
                              = ( cons_nat @ X12 @ ( sorted636521546038734694ed_nat @ L1 @ ( cons_nat @ X23 @ L22 ) ) ) ) )
                          & ( ~ ( ord_less_nat @ X12 @ X23 )
                           => ( ( ( X12 = X23 )
                               => ( Y
                                  = ( sorted636521546038734694ed_nat @ L1 @ L22 ) ) )
                              & ( ( X12 != X23 )
                               => ( Y
                                  = ( sorted636521546038734694ed_nat @ ( cons_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) )
                       => ~ ( accp_P8037286306265792042st_nat @ sorted1291390330255423669el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X12 @ L1 ) @ ( cons_nat @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ).

% diff_sorted.pelims
thf(fact_600_conj__subset__def,axiom,
    ! [A3: set_list_a,P: list_a > $o,Q: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ A3
        @ ( collect_list_a
          @ ^ [X: list_a] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_le8861187494160871172list_a @ A3 @ ( collect_list_a @ P ) )
        & ( ord_le8861187494160871172list_a @ A3 @ ( collect_list_a @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_601_conj__subset__def,axiom,
    ! [A3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) )
      = ( ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ P ) )
        & ( ord_less_eq_set_nat @ A3 @ ( collect_nat @ Q ) ) ) ) ).

% conj_subset_def
thf(fact_602_prop__restrict,axiom,
    ! [X3: produc859450856879609959at_nat,Z5: set_Pr8693737435421807431at_nat,X5: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X3 @ Z5 )
     => ( ( ord_le3000389064537975527at_nat @ Z5
          @ ( collec7088162979684241874at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_603_prop__restrict,axiom,
    ! [X3: product_prod_nat_nat,Z5: set_Pr1261947904930325089at_nat,X5: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X3 @ Z5 )
     => ( ( ord_le3146513528884898305at_nat @ Z5
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_604_prop__restrict,axiom,
    ! [X3: list_a,Z5: set_list_a,X5: set_list_a,P: list_a > $o] :
      ( ( member_list_a @ X3 @ Z5 )
     => ( ( ord_le8861187494160871172list_a @ Z5
          @ ( collect_list_a
            @ ^ [X: list_a] :
                ( ( member_list_a @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_605_prop__restrict,axiom,
    ! [X3: nat,Z5: set_nat,X5: set_nat,P: nat > $o] :
      ( ( member_nat @ X3 @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X5 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_606_in__inv__image,axiom,
    ! [X3: nat,Y: nat,R4: set_Pr1261947904930325089at_nat,F2: nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( inv_image_nat_nat @ R4 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R4 ) ) ).

% in_inv_image
thf(fact_607_in__inv__image,axiom,
    ! [X3: nat,Y: nat,R4: set_Pr8693737435421807431at_nat,F2: nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( inv_im8721129524585065343at_nat @ R4 @ F2 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R4 ) ) ).

% in_inv_image
thf(fact_608_in__inv__image,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R4: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( inv_im2857804880773004637at_nat @ R4 @ F2 ) )
      = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R4 ) ) ).

% in_inv_image
thf(fact_609_in__inv__image,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat,F2: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( inv_im5896302226278515346at_nat @ R4 @ F2 ) )
      = ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) ) @ R4 ) ) ).

% in_inv_image
thf(fact_610_diff__sorted_Osimps_I3_J,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,L12: list_P6011104703257516679at_nat,L23: list_P6011104703257516679at_nat] :
      ( ( ( ord_le1203424502768444845at_nat @ X1 @ X2 )
       => ( ( sorted5599871463696753131at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ ( cons_P6512896166579812791at_nat @ X2 @ L23 ) )
          = ( cons_P6512896166579812791at_nat @ X1 @ ( sorted5599871463696753131at_nat @ L12 @ ( cons_P6512896166579812791at_nat @ X2 @ L23 ) ) ) ) )
      & ( ~ ( ord_le1203424502768444845at_nat @ X1 @ X2 )
       => ( ( ( X1 = X2 )
           => ( ( sorted5599871463696753131at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ ( cons_P6512896166579812791at_nat @ X2 @ L23 ) )
              = ( sorted5599871463696753131at_nat @ L12 @ L23 ) ) )
          & ( ( X1 != X2 )
           => ( ( sorted5599871463696753131at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ ( cons_P6512896166579812791at_nat @ X2 @ L23 ) )
              = ( sorted5599871463696753131at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ).

% diff_sorted.simps(3)
thf(fact_611_diff__sorted_Osimps_I3_J,axiom,
    ! [X1: nat,X2: nat,L12: list_nat,L23: list_nat] :
      ( ( ( ord_less_nat @ X1 @ X2 )
       => ( ( sorted636521546038734694ed_nat @ ( cons_nat @ X1 @ L12 ) @ ( cons_nat @ X2 @ L23 ) )
          = ( cons_nat @ X1 @ ( sorted636521546038734694ed_nat @ L12 @ ( cons_nat @ X2 @ L23 ) ) ) ) )
      & ( ~ ( ord_less_nat @ X1 @ X2 )
       => ( ( ( X1 = X2 )
           => ( ( sorted636521546038734694ed_nat @ ( cons_nat @ X1 @ L12 ) @ ( cons_nat @ X2 @ L23 ) )
              = ( sorted636521546038734694ed_nat @ L12 @ L23 ) ) )
          & ( ( X1 != X2 )
           => ( ( sorted636521546038734694ed_nat @ ( cons_nat @ X1 @ L12 ) @ ( cons_nat @ X2 @ L23 ) )
              = ( sorted636521546038734694ed_nat @ ( cons_nat @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ).

% diff_sorted.simps(3)
thf(fact_612_diff__sorted_Osimps_I2_J,axiom,
    ! [V3: product_prod_nat_nat,Va2: list_P6011104703257516679at_nat] :
      ( ( sorted5599871463696753131at_nat @ ( cons_P6512896166579812791at_nat @ V3 @ Va2 ) @ nil_Pr5478986624290739719at_nat )
      = ( cons_P6512896166579812791at_nat @ V3 @ Va2 ) ) ).

% diff_sorted.simps(2)
thf(fact_613_ssubst__Pair__rhs,axiom,
    ! [R4: nat,S3: nat,R: set_Pr1261947904930325089at_nat,S5: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R4 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R4 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_614_ssubst__Pair__rhs,axiom,
    ! [R4: product_prod_nat_nat,S3: product_prod_nat_nat,R: set_Pr8693737435421807431at_nat,S5: product_prod_nat_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R4 @ S3 ) @ R )
     => ( ( S5 = S3 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R4 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_615_diff__sorted_Oelims,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( sorted5599871463696753131at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ( ! [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
              ( ( X3
                = ( cons_P6512896166579812791at_nat @ V @ Va ) )
             => ( ( Xa2 = nil_Pr5478986624290739719at_nat )
               => ( Y
                 != ( cons_P6512896166579812791at_nat @ V @ Va ) ) ) )
         => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
               => ! [X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
                    ( ( Xa2
                      = ( cons_P6512896166579812791at_nat @ X23 @ L22 ) )
                   => ~ ( ( ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ X12 @ ( sorted5599871463696753131at_nat @ L1 @ ( cons_P6512896166579812791at_nat @ X23 @ L22 ) ) ) ) )
                        & ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                         => ( ( ( X12 = X23 )
                             => ( Y
                                = ( sorted5599871463696753131at_nat @ L1 @ L22 ) ) )
                            & ( ( X12 != X23 )
                             => ( Y
                                = ( sorted5599871463696753131at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ).

% diff_sorted.elims
thf(fact_616_diff__sorted_Oelims,axiom,
    ! [X3: list_nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( sorted636521546038734694ed_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = nil_nat )
         => ( Y != nil_nat ) )
       => ( ! [V: nat,Va: list_nat] :
              ( ( X3
                = ( cons_nat @ V @ Va ) )
             => ( ( Xa2 = nil_nat )
               => ( Y
                 != ( cons_nat @ V @ Va ) ) ) )
         => ~ ! [X12: nat,L1: list_nat] :
                ( ( X3
                  = ( cons_nat @ X12 @ L1 ) )
               => ! [X23: nat,L22: list_nat] :
                    ( ( Xa2
                      = ( cons_nat @ X23 @ L22 ) )
                   => ~ ( ( ( ord_less_nat @ X12 @ X23 )
                         => ( Y
                            = ( cons_nat @ X12 @ ( sorted636521546038734694ed_nat @ L1 @ ( cons_nat @ X23 @ L22 ) ) ) ) )
                        & ( ~ ( ord_less_nat @ X12 @ X23 )
                         => ( ( ( X12 = X23 )
                             => ( Y
                                = ( sorted636521546038734694ed_nat @ L1 @ L22 ) ) )
                            & ( ( X12 != X23 )
                             => ( Y
                                = ( sorted636521546038734694ed_nat @ ( cons_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ).

% diff_sorted.elims
thf(fact_617_Collect__restrict,axiom,
    ! [X5: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ord_le3000389064537975527at_nat
      @ ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_618_Collect__restrict,axiom,
    ! [X5: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_619_Collect__restrict,axiom,
    ! [X5: set_list_a,P: list_a > $o] :
      ( ord_le8861187494160871172list_a
      @ ( collect_list_a
        @ ^ [X: list_a] :
            ( ( member_list_a @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_620_Collect__restrict,axiom,
    ! [X5: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X5 )
            & ( P @ X ) ) )
      @ X5 ) ).

% Collect_restrict
thf(fact_621_subset__Collect__iff,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B3 @ A3 )
     => ( ( ord_le3000389064537975527at_nat @ B3
          @ ( collec7088162979684241874at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_622_subset__Collect__iff,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ B3
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_623_subset__Collect__iff,axiom,
    ! [B3: set_list_a,A3: set_list_a,P: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ B3 @ A3 )
     => ( ( ord_le8861187494160871172list_a @ B3
          @ ( collect_list_a
            @ ^ [X: list_a] :
                ( ( member_list_a @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: list_a] :
              ( ( member_list_a @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_624_subset__Collect__iff,axiom,
    ! [B3: set_nat,A3: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( ord_less_eq_set_nat @ B3
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ B3 )
             => ( P @ X ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_625_subset__CollectI,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat,Q: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat > $o] :
      ( ( ord_le3000389064537975527at_nat @ B3 @ A3 )
     => ( ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ B3 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le3000389064537975527at_nat
          @ ( collec7088162979684241874at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collec7088162979684241874at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_626_subset__CollectI,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,Q: product_prod_nat_nat > $o,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ B3 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le3146513528884898305at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_627_subset__CollectI,axiom,
    ! [B3: set_list_a,A3: set_list_a,Q: list_a > $o,P: list_a > $o] :
      ( ( ord_le8861187494160871172list_a @ B3 @ A3 )
     => ( ! [X4: list_a] :
            ( ( member_list_a @ X4 @ B3 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_le8861187494160871172list_a
          @ ( collect_list_a
            @ ^ [X: list_a] :
                ( ( member_list_a @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collect_list_a
            @ ^ [X: list_a] :
                ( ( member_list_a @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_628_subset__CollectI,axiom,
    ! [B3: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ B3 )
           => ( ( Q @ X4 )
             => ( P @ X4 ) ) )
       => ( ord_less_eq_set_nat
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ B3 )
                & ( Q @ X ) ) )
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_629_same__fstI,axiom,
    ! [P: nat > $o,X3: nat,Y7: nat,Y: nat,R: nat > set_Pr1261947904930325089at_nat] :
      ( ( P @ X3 )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y7 @ Y ) @ ( R @ X3 ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y7 ) @ ( product_Pair_nat_nat @ X3 @ Y ) ) @ ( same_fst_nat_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_630_same__fstI,axiom,
    ! [P: product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y7: product_prod_nat_nat,Y: product_prod_nat_nat,R: product_prod_nat_nat > set_Pr8693737435421807431at_nat] :
      ( ( P @ X3 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y7 @ Y ) @ ( R @ X3 ) )
       => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y7 ) @ ( produc6161850002892822231at_nat @ X3 @ Y ) ) @ ( same_f4956014544515070124at_nat @ P @ R ) ) ) ) ).

% same_fstI
thf(fact_631_Succ__def,axiom,
    ( bNF_Gr6352880689984616693cc_nat
    = ( ^ [Kl: set_list_nat,Kl2: list_nat] :
          ( collect_nat
          @ ^ [K5: nat] : ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K5 @ nil_nat ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_632_Succ__def,axiom,
    ( bNF_Gr4634511371912843295list_a
    = ( ^ [Kl: set_list_list_a,Kl2: list_list_a] :
          ( collect_list_a
          @ ^ [K5: list_a] : ( member_list_list_a @ ( append_list_a @ Kl2 @ ( cons_list_a @ K5 @ nil_list_a ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_633_Succ__def,axiom,
    ( bNF_Gr5363859321595349404at_nat
    = ( ^ [Kl: set_li5450038453877631591at_nat,Kl2: list_P6011104703257516679at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [K5: product_prod_nat_nat] : ( member3067507820990806192at_nat @ ( append985823374593552924at_nat @ Kl2 @ ( cons_P6512896166579812791at_nat @ K5 @ nil_Pr5478986624290739719at_nat ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_634_SuccI,axiom,
    ! [Kl3: list_nat,K3: nat,Kl4: set_list_nat] :
      ( ( member_list_nat @ ( append_nat @ Kl3 @ ( cons_nat @ K3 @ nil_nat ) ) @ Kl4 )
     => ( member_nat @ K3 @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_635_SuccI,axiom,
    ! [Kl3: list_P8469869581646625389at_nat,K3: produc859450856879609959at_nat,Kl4: set_li3197816953174176717at_nat] :
      ( ( member3799944675974059798at_nat @ ( append8751754712269456642at_nat @ Kl3 @ ( cons_P8732206157123786781at_nat @ K3 @ nil_Pr2582115297535392877at_nat ) ) @ Kl4 )
     => ( member8206827879206165904at_nat @ K3 @ ( bNF_Gr8603829661560577154at_nat @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_636_SuccI,axiom,
    ! [Kl3: list_P6011104703257516679at_nat,K3: product_prod_nat_nat,Kl4: set_li5450038453877631591at_nat] :
      ( ( member3067507820990806192at_nat @ ( append985823374593552924at_nat @ Kl3 @ ( cons_P6512896166579812791at_nat @ K3 @ nil_Pr5478986624290739719at_nat ) ) @ Kl4 )
     => ( member8440522571783428010at_nat @ K3 @ ( bNF_Gr5363859321595349404at_nat @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_637_SuccD,axiom,
    ! [K3: nat,Kl4: set_list_nat,Kl3: list_nat] :
      ( ( member_nat @ K3 @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ Kl3 ) )
     => ( member_list_nat @ ( append_nat @ Kl3 @ ( cons_nat @ K3 @ nil_nat ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_638_SuccD,axiom,
    ! [K3: produc859450856879609959at_nat,Kl4: set_li3197816953174176717at_nat,Kl3: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ K3 @ ( bNF_Gr8603829661560577154at_nat @ Kl4 @ Kl3 ) )
     => ( member3799944675974059798at_nat @ ( append8751754712269456642at_nat @ Kl3 @ ( cons_P8732206157123786781at_nat @ K3 @ nil_Pr2582115297535392877at_nat ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_639_SuccD,axiom,
    ! [K3: product_prod_nat_nat,Kl4: set_li5450038453877631591at_nat,Kl3: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ K3 @ ( bNF_Gr5363859321595349404at_nat @ Kl4 @ Kl3 ) )
     => ( member3067507820990806192at_nat @ ( append985823374593552924at_nat @ Kl3 @ ( cons_P6512896166579812791at_nat @ K3 @ nil_Pr5478986624290739719at_nat ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_640_empty__Shift,axiom,
    ! [Kl4: set_list_nat,K3: nat] :
      ( ( member_list_nat @ nil_nat @ Kl4 )
     => ( ( member_nat @ K3 @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ nil_nat ) )
       => ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl4 @ K3 ) ) ) ) ).

% empty_Shift
thf(fact_641_empty__Shift,axiom,
    ! [Kl4: set_li3197816953174176717at_nat,K3: produc859450856879609959at_nat] :
      ( ( member3799944675974059798at_nat @ nil_Pr2582115297535392877at_nat @ Kl4 )
     => ( ( member8206827879206165904at_nat @ K3 @ ( bNF_Gr8603829661560577154at_nat @ Kl4 @ nil_Pr2582115297535392877at_nat ) )
       => ( member3799944675974059798at_nat @ nil_Pr2582115297535392877at_nat @ ( bNF_Gr9155129491085760262at_nat @ Kl4 @ K3 ) ) ) ) ).

% empty_Shift
thf(fact_642_empty__Shift,axiom,
    ! [Kl4: set_li5450038453877631591at_nat,K3: product_prod_nat_nat] :
      ( ( member3067507820990806192at_nat @ nil_Pr5478986624290739719at_nat @ Kl4 )
     => ( ( member8440522571783428010at_nat @ K3 @ ( bNF_Gr5363859321595349404at_nat @ Kl4 @ nil_Pr5478986624290739719at_nat ) )
       => ( member3067507820990806192at_nat @ nil_Pr5478986624290739719at_nat @ ( bNF_Gr3130287167067265568at_nat @ Kl4 @ K3 ) ) ) ) ).

% empty_Shift
thf(fact_643_Succ__Shift,axiom,
    ! [Kl4: set_li5450038453877631591at_nat,K3: product_prod_nat_nat,Kl3: list_P6011104703257516679at_nat] :
      ( ( bNF_Gr5363859321595349404at_nat @ ( bNF_Gr3130287167067265568at_nat @ Kl4 @ K3 ) @ Kl3 )
      = ( bNF_Gr5363859321595349404at_nat @ Kl4 @ ( cons_P6512896166579812791at_nat @ K3 @ Kl3 ) ) ) ).

% Succ_Shift
thf(fact_644_inter__sorted_Opelims,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( sorted4403996544162776090at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P7052990409830227952at_nat @ sorted8602531354981012077at_nat @ ( produc5943733680697469783at_nat @ X3 @ Xa2 ) )
       => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
           => ( ( Y = nil_Pr5478986624290739719at_nat )
             => ~ ( accp_P7052990409830227952at_nat @ sorted8602531354981012077at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xa2 ) ) ) )
         => ( ! [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ V @ Va ) )
               => ( ( Xa2 = nil_Pr5478986624290739719at_nat )
                 => ( ( Y = nil_Pr5478986624290739719at_nat )
                   => ~ ( accp_P7052990409830227952at_nat @ sorted8602531354981012077at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) @ nil_Pr5478986624290739719at_nat ) ) ) ) )
           => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
                  ( ( X3
                    = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
                 => ! [X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
                      ( ( Xa2
                        = ( cons_P6512896166579812791at_nat @ X23 @ L22 ) )
                     => ( ( ( ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                           => ( Y
                              = ( sorted4403996544162776090at_nat @ L1 @ ( cons_P6512896166579812791at_nat @ X23 @ L22 ) ) ) )
                          & ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                           => ( ( ( X12 = X23 )
                               => ( Y
                                  = ( cons_P6512896166579812791at_nat @ X12 @ ( sorted4403996544162776090at_nat @ L1 @ L22 ) ) ) )
                              & ( ( X12 != X23 )
                               => ( Y
                                  = ( sorted4403996544162776090at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) )
                       => ~ ( accp_P7052990409830227952at_nat @ sorted8602531354981012077at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ ( cons_P6512896166579812791at_nat @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ).

% inter_sorted.pelims
thf(fact_645_inter__sorted_Opelims,axiom,
    ! [X3: list_nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( sorted6641401135340794871ed_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P8037286306265792042st_nat @ sorted6532095975127779684el_nat @ ( produc2694037385005941721st_nat @ X3 @ Xa2 ) )
       => ( ( ( X3 = nil_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_P8037286306265792042st_nat @ sorted6532095975127779684el_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa2 ) ) ) )
         => ( ! [V: nat,Va: list_nat] :
                ( ( X3
                  = ( cons_nat @ V @ Va ) )
               => ( ( Xa2 = nil_nat )
                 => ( ( Y = nil_nat )
                   => ~ ( accp_P8037286306265792042st_nat @ sorted6532095975127779684el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) ) ) ) )
           => ~ ! [X12: nat,L1: list_nat] :
                  ( ( X3
                    = ( cons_nat @ X12 @ L1 ) )
                 => ! [X23: nat,L22: list_nat] :
                      ( ( Xa2
                        = ( cons_nat @ X23 @ L22 ) )
                     => ( ( ( ( ord_less_nat @ X12 @ X23 )
                           => ( Y
                              = ( sorted6641401135340794871ed_nat @ L1 @ ( cons_nat @ X23 @ L22 ) ) ) )
                          & ( ~ ( ord_less_nat @ X12 @ X23 )
                           => ( ( ( X12 = X23 )
                               => ( Y
                                  = ( cons_nat @ X12 @ ( sorted6641401135340794871ed_nat @ L1 @ L22 ) ) ) )
                              & ( ( X12 != X23 )
                               => ( Y
                                  = ( sorted6641401135340794871ed_nat @ ( cons_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) )
                       => ~ ( accp_P8037286306265792042st_nat @ sorted6532095975127779684el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X12 @ L1 ) @ ( cons_nat @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ).

% inter_sorted.pelims
thf(fact_646_inter__sorted_Osimps_I3_J,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,L12: list_P6011104703257516679at_nat,L23: list_P6011104703257516679at_nat] :
      ( ( ( ord_le1203424502768444845at_nat @ X1 @ X2 )
       => ( ( sorted4403996544162776090at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ ( cons_P6512896166579812791at_nat @ X2 @ L23 ) )
          = ( sorted4403996544162776090at_nat @ L12 @ ( cons_P6512896166579812791at_nat @ X2 @ L23 ) ) ) )
      & ( ~ ( ord_le1203424502768444845at_nat @ X1 @ X2 )
       => ( ( ( X1 = X2 )
           => ( ( sorted4403996544162776090at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ ( cons_P6512896166579812791at_nat @ X2 @ L23 ) )
              = ( cons_P6512896166579812791at_nat @ X1 @ ( sorted4403996544162776090at_nat @ L12 @ L23 ) ) ) )
          & ( ( X1 != X2 )
           => ( ( sorted4403996544162776090at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ ( cons_P6512896166579812791at_nat @ X2 @ L23 ) )
              = ( sorted4403996544162776090at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ).

% inter_sorted.simps(3)
thf(fact_647_inter__sorted_Osimps_I3_J,axiom,
    ! [X1: nat,X2: nat,L12: list_nat,L23: list_nat] :
      ( ( ( ord_less_nat @ X1 @ X2 )
       => ( ( sorted6641401135340794871ed_nat @ ( cons_nat @ X1 @ L12 ) @ ( cons_nat @ X2 @ L23 ) )
          = ( sorted6641401135340794871ed_nat @ L12 @ ( cons_nat @ X2 @ L23 ) ) ) )
      & ( ~ ( ord_less_nat @ X1 @ X2 )
       => ( ( ( X1 = X2 )
           => ( ( sorted6641401135340794871ed_nat @ ( cons_nat @ X1 @ L12 ) @ ( cons_nat @ X2 @ L23 ) )
              = ( cons_nat @ X1 @ ( sorted6641401135340794871ed_nat @ L12 @ L23 ) ) ) )
          & ( ( X1 != X2 )
           => ( ( sorted6641401135340794871ed_nat @ ( cons_nat @ X1 @ L12 ) @ ( cons_nat @ X2 @ L23 ) )
              = ( sorted6641401135340794871ed_nat @ ( cons_nat @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ).

% inter_sorted.simps(3)
thf(fact_648_inter__sorted_Osimps_I2_J,axiom,
    ! [V3: product_prod_nat_nat,Va2: list_P6011104703257516679at_nat] :
      ( ( sorted4403996544162776090at_nat @ ( cons_P6512896166579812791at_nat @ V3 @ Va2 ) @ nil_Pr5478986624290739719at_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% inter_sorted.simps(2)
thf(fact_649_ShiftD,axiom,
    ! [Kl3: list_P6011104703257516679at_nat,Kl4: set_li5450038453877631591at_nat,K3: product_prod_nat_nat] :
      ( ( member3067507820990806192at_nat @ Kl3 @ ( bNF_Gr3130287167067265568at_nat @ Kl4 @ K3 ) )
     => ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ K3 @ Kl3 ) @ Kl4 ) ) ).

% ShiftD
thf(fact_650_Shift__def,axiom,
    ( bNF_Greatest_Shift_a
    = ( ^ [Kl: set_list_a,K5: a] :
          ( collect_list_a
          @ ^ [Kl2: list_a] : ( member_list_a @ ( cons_a @ K5 @ Kl2 ) @ Kl ) ) ) ) ).

% Shift_def
thf(fact_651_Shift__def,axiom,
    ( bNF_Gr3130287167067265568at_nat
    = ( ^ [Kl: set_li5450038453877631591at_nat,K5: product_prod_nat_nat] :
          ( collec3343600615725829874at_nat
          @ ^ [Kl2: list_P6011104703257516679at_nat] : ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ K5 @ Kl2 ) @ Kl ) ) ) ) ).

% Shift_def
thf(fact_652_inter__sorted_Oelims,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( sorted4403996544162776090at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ( ( ? [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
                ( X3
                = ( cons_P6512896166579812791at_nat @ V @ Va ) )
           => ( ( Xa2 = nil_Pr5478986624290739719at_nat )
             => ( Y != nil_Pr5478986624290739719at_nat ) ) )
         => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
               => ! [X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
                    ( ( Xa2
                      = ( cons_P6512896166579812791at_nat @ X23 @ L22 ) )
                   => ~ ( ( ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                         => ( Y
                            = ( sorted4403996544162776090at_nat @ L1 @ ( cons_P6512896166579812791at_nat @ X23 @ L22 ) ) ) )
                        & ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                         => ( ( ( X12 = X23 )
                             => ( Y
                                = ( cons_P6512896166579812791at_nat @ X12 @ ( sorted4403996544162776090at_nat @ L1 @ L22 ) ) ) )
                            & ( ( X12 != X23 )
                             => ( Y
                                = ( sorted4403996544162776090at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ).

% inter_sorted.elims
thf(fact_653_inter__sorted_Oelims,axiom,
    ! [X3: list_nat,Xa2: list_nat,Y: list_nat] :
      ( ( ( sorted6641401135340794871ed_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = nil_nat )
         => ( Y != nil_nat ) )
       => ( ( ? [V: nat,Va: list_nat] :
                ( X3
                = ( cons_nat @ V @ Va ) )
           => ( ( Xa2 = nil_nat )
             => ( Y != nil_nat ) ) )
         => ~ ! [X12: nat,L1: list_nat] :
                ( ( X3
                  = ( cons_nat @ X12 @ L1 ) )
               => ! [X23: nat,L22: list_nat] :
                    ( ( Xa2
                      = ( cons_nat @ X23 @ L22 ) )
                   => ~ ( ( ( ord_less_nat @ X12 @ X23 )
                         => ( Y
                            = ( sorted6641401135340794871ed_nat @ L1 @ ( cons_nat @ X23 @ L22 ) ) ) )
                        & ( ~ ( ord_less_nat @ X12 @ X23 )
                         => ( ( ( X12 = X23 )
                             => ( Y
                                = ( cons_nat @ X12 @ ( sorted6641401135340794871ed_nat @ L1 @ L22 ) ) ) )
                            & ( ( X12 != X23 )
                             => ( Y
                                = ( sorted6641401135340794871ed_nat @ ( cons_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ).

% inter_sorted.elims
thf(fact_654_revg_Opelims,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( revg_P4849834641995823428at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P7052990409830227952at_nat @ revg_r4913911500435558851at_nat @ ( produc5943733680697469783at_nat @ X3 @ Xa2 ) )
       => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
           => ( ( Y = Xa2 )
             => ~ ( accp_P7052990409830227952at_nat @ revg_r4913911500435558851at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xa2 ) ) ) )
         => ~ ! [A5: product_prod_nat_nat,As: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ A5 @ As ) )
               => ( ( Y
                    = ( revg_P4849834641995823428at_nat @ As @ ( cons_P6512896166579812791at_nat @ A5 @ Xa2 ) ) )
                 => ~ ( accp_P7052990409830227952at_nat @ revg_r4913911500435558851at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ A5 @ As ) @ Xa2 ) ) ) ) ) ) ) ).

% revg.pelims
thf(fact_655_mergesort__by__rel__merge_Opelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > $o,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( merges7462883909168810252at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( accp_P7097408270579464809at_nat @ merges3489886619874950139at_nat @ ( produc6665944465064540112at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ Xb ) ) )
       => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( ( Xa2
                = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
             => ! [Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
                 => ( ( ( ( X3 @ X4 @ Y4 )
                       => ( Y
                          = ( cons_P6512896166579812791at_nat @ X4 @ ( merges7462883909168810252at_nat @ X3 @ Xs2 @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) ) )
                      & ( ~ ( X3 @ X4 @ Y4 )
                       => ( Y
                          = ( cons_P6512896166579812791at_nat @ Y4 @ ( merges7462883909168810252at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ Ys ) ) ) ) )
                   => ~ ( accp_P7097408270579464809at_nat @ merges3489886619874950139at_nat @ ( produc6665944465064540112at_nat @ X3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) ) ) ) )
         => ( ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( ( Y = Xa2 )
               => ~ ( accp_P7097408270579464809at_nat @ merges3489886619874950139at_nat @ ( produc6665944465064540112at_nat @ X3 @ ( produc5943733680697469783at_nat @ Xa2 @ nil_Pr5478986624290739719at_nat ) ) ) ) )
           => ~ ( ( Xa2 = nil_Pr5478986624290739719at_nat )
               => ! [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
                    ( ( Xb
                      = ( cons_P6512896166579812791at_nat @ V @ Va ) )
                   => ( ( Y
                        = ( cons_P6512896166579812791at_nat @ V @ Va ) )
                     => ~ ( accp_P7097408270579464809at_nat @ merges3489886619874950139at_nat @ ( produc6665944465064540112at_nat @ X3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ V @ Va ) ) ) ) ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.pelims
thf(fact_656_some__elem,axiom,
    ! [S: set_Pr8693737435421807431at_nat] :
      ( ( S != bot_bo5327735625951526323at_nat )
     => ( member8206827879206165904at_nat
        @ ( fChoic6673916560228149179at_nat
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ S ) )
        @ S ) ) ).

% some_elem
thf(fact_657_some__elem,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( S != bot_bo2099793752762293965at_nat )
     => ( member8440522571783428010at_nat
        @ ( fChoic6978938873391328853at_nat
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S ) )
        @ S ) ) ).

% some_elem
thf(fact_658_some__elem,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ( member_nat
        @ ( fChoice_nat
          @ ^ [X: nat] : ( member_nat @ X @ S ) )
        @ S ) ) ).

% some_elem
thf(fact_659_mergesort__by__rel__merge__simps_I1_J,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( ( R @ X3 @ Y )
       => ( ( merges7462883909168810252at_nat @ R @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) )
          = ( cons_P6512896166579812791at_nat @ X3 @ ( merges7462883909168810252at_nat @ R @ Xs @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) ) ) )
      & ( ~ ( R @ X3 @ Y )
       => ( ( merges7462883909168810252at_nat @ R @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) )
          = ( cons_P6512896166579812791at_nat @ Y @ ( merges7462883909168810252at_nat @ R @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ Ys3 ) ) ) ) ) ).

% mergesort_by_rel_merge_simps(1)
thf(fact_660_revg_Osimps_I2_J,axiom,
    ! [A: product_prod_nat_nat,As2: list_P6011104703257516679at_nat,B: list_P6011104703257516679at_nat] :
      ( ( revg_P4849834641995823428at_nat @ ( cons_P6512896166579812791at_nat @ A @ As2 ) @ B )
      = ( revg_P4849834641995823428at_nat @ As2 @ ( cons_P6512896166579812791at_nat @ A @ B ) ) ) ).

% revg.simps(2)
thf(fact_661_mergesort__by__rel__merge_Osimps_I3_J,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o,V3: product_prod_nat_nat,Va2: list_P6011104703257516679at_nat] :
      ( ( merges7462883909168810252at_nat @ R @ nil_Pr5478986624290739719at_nat @ ( cons_P6512896166579812791at_nat @ V3 @ Va2 ) )
      = ( cons_P6512896166579812791at_nat @ V3 @ Va2 ) ) ).

% mergesort_by_rel_merge.simps(3)
thf(fact_662_mergesort__by__rel__merge_Oelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > $o,Xa2: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( merges7462883909168810252at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( ( Xa2
              = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
           => ! [Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) )
               => ~ ( ( ( X3 @ X4 @ Y4 )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ X4 @ ( merges7462883909168810252at_nat @ X3 @ Xs2 @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) ) )
                    & ( ~ ( X3 @ X4 @ Y4 )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ Y4 @ ( merges7462883909168810252at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ Ys ) ) ) ) ) ) )
       => ( ( ( Xb = nil_Pr5478986624290739719at_nat )
           => ( Y != Xa2 ) )
         => ~ ( ( Xa2 = nil_Pr5478986624290739719at_nat )
             => ! [V: product_prod_nat_nat,Va: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ V @ Va ) )
                 => ( Y
                   != ( cons_P6512896166579812791at_nat @ V @ Va ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.elims
thf(fact_663_revg_Oelims,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( revg_P4849834641995823428at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
         => ( Y != Xa2 ) )
       => ~ ! [A5: product_prod_nat_nat,As: list_P6011104703257516679at_nat] :
              ( ( X3
                = ( cons_P6512896166579812791at_nat @ A5 @ As ) )
             => ( Y
               != ( revg_P4849834641995823428at_nat @ As @ ( cons_P6512896166579812791at_nat @ A5 @ Xa2 ) ) ) ) ) ) ).

% revg.elims
thf(fact_664_Eps__case__prod__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( fChoic6978938873391328853at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X7: nat,Y8: nat] :
              ( ( X3 = X7 )
              & ( Y = Y8 ) ) ) )
      = ( product_Pair_nat_nat @ X3 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_665_Eps__case__prod__eq,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( fChoic6673916560228149179at_nat
        @ ( produc6590410687421337004_nat_o
          @ ^ [X7: product_prod_nat_nat,Y8: product_prod_nat_nat] :
              ( ( X3 = X7 )
              & ( Y = Y8 ) ) ) )
      = ( produc6161850002892822231at_nat @ X3 @ Y ) ) ).

% Eps_case_prod_eq
thf(fact_666_someI2__bex,axiom,
    ! [A3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ A3 )
          & ( P @ X6 ) )
     => ( ! [X4: nat] :
            ( ( ( member_nat @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoice_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_667_someI2__bex,axiom,
    ! [A3: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
      ( ? [X6: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X6 @ A3 )
          & ( P @ X6 ) )
     => ( ! [X4: produc859450856879609959at_nat] :
            ( ( ( member8206827879206165904at_nat @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoic6673916560228149179at_nat
            @ ^ [X: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_668_someI2__bex,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X6 @ A3 )
          & ( P @ X6 ) )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( ( member8440522571783428010at_nat @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoic6978938873391328853at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A3 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_669_split__paired__Eps,axiom,
    ( fChoic6978938873391328853at_nat
    = ( ^ [P6: product_prod_nat_nat > $o] :
          ( fChoic6978938873391328853at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [A4: nat,B9: nat] : ( P6 @ ( product_Pair_nat_nat @ A4 @ B9 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_670_split__paired__Eps,axiom,
    ( fChoic6673916560228149179at_nat
    = ( ^ [P6: produc859450856879609959at_nat > $o] :
          ( fChoic6673916560228149179at_nat
          @ ( produc6590410687421337004_nat_o
            @ ^ [A4: product_prod_nat_nat,B9: product_prod_nat_nat] : ( P6 @ ( produc6161850002892822231at_nat @ A4 @ B9 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_671_some__in__eq,axiom,
    ! [A3: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat
        @ ( fChoic6673916560228149179at_nat
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A3 ) )
        @ A3 )
      = ( A3 != bot_bo5327735625951526323at_nat ) ) ).

% some_in_eq
thf(fact_672_some__in__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat
        @ ( fChoic6978938873391328853at_nat
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A3 ) )
        @ A3 )
      = ( A3 != bot_bo2099793752762293965at_nat ) ) ).

% some_in_eq
thf(fact_673_some__in__eq,axiom,
    ! [A3: set_nat] :
      ( ( member_nat
        @ ( fChoice_nat
          @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
        @ A3 )
      = ( A3 != bot_bot_set_nat ) ) ).

% some_in_eq
thf(fact_674_fun__of__rel__def,axiom,
    ( fun_of_rel_nat_nat
    = ( ^ [R6: set_Pr1261947904930325089at_nat,X: nat] :
          ( fChoice_nat
          @ ^ [Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R6 ) ) ) ) ).

% fun_of_rel_def
thf(fact_675_fun__of__rel__def,axiom,
    ( fun_of7254032675221162843at_nat
    = ( ^ [R6: set_Pr8693737435421807431at_nat,X: product_prod_nat_nat] :
          ( fChoic6978938873391328853at_nat
          @ ^ [Y3: product_prod_nat_nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R6 ) ) ) ) ).

% fun_of_rel_def
thf(fact_676_enumerate__mono__iff,axiom,
    ! [S: set_nat,M4: nat,N: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M4 ) @ ( infini8530281810654367211te_nat @ S @ N ) )
        = ( ord_less_nat @ M4 @ N ) ) ) ).

% enumerate_mono_iff
thf(fact_677_enumerate__mono__le__iff,axiom,
    ! [S: set_nat,M4: nat,N: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ( ord_less_eq_nat @ ( infini8530281810654367211te_nat @ S @ M4 ) @ ( infini8530281810654367211te_nat @ S @ N ) )
        = ( ord_less_eq_nat @ M4 @ N ) ) ) ).

% enumerate_mono_le_iff
thf(fact_678_symcl__def,axiom,
    ( symcl_nat
    = ( ^ [R5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y3: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R5 )
                | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ X ) @ R5 ) ) ) ) ) ) ).

% symcl_def
thf(fact_679_symcl__def,axiom,
    ( symcl_1453671975044266200at_nat
    = ( ^ [R5: set_Pr8693737435421807431at_nat] :
          ( collec7088162979684241874at_nat
          @ ( produc6590410687421337004_nat_o
            @ ^ [X: product_prod_nat_nat,Y3: product_prod_nat_nat] :
                ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y3 ) @ R5 )
                | ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ X ) @ R5 ) ) ) ) ) ) ).

% symcl_def
thf(fact_680_enumerate__in__set,axiom,
    ! [S: set_Pr8693737435421807431at_nat,N: nat] :
      ( ~ ( finite4392333629123659920at_nat @ S )
     => ( member8206827879206165904at_nat @ ( infini1315406684128855308at_nat @ S @ N ) @ S ) ) ).

% enumerate_in_set
thf(fact_681_enumerate__in__set,axiom,
    ! [S: set_Pr1261947904930325089at_nat,N: nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ( member8440522571783428010at_nat @ ( infini875302759086991654at_nat @ S @ N ) @ S ) ) ).

% enumerate_in_set
thf(fact_682_enumerate__in__set,axiom,
    ! [S: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( member_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ).

% enumerate_in_set
thf(fact_683_enumerate__Ex,axiom,
    ! [S: set_nat,S3: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ( member_nat @ S3 @ S )
       => ? [N9: nat] :
            ( ( infini8530281810654367211te_nat @ S @ N9 )
            = S3 ) ) ) ).

% enumerate_Ex
thf(fact_684_in__symclI_I2_J,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( symcl_nat @ R4 ) ) ) ).

% in_symclI(2)
thf(fact_685_in__symclI_I2_J,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R4 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ B @ A ) @ ( symcl_1453671975044266200at_nat @ R4 ) ) ) ).

% in_symclI(2)
thf(fact_686_in__symclI_I1_J,axiom,
    ! [A: nat,B: nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R4 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( symcl_nat @ R4 ) ) ) ).

% in_symclI(1)
thf(fact_687_in__symclI_I1_J,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ R4 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( symcl_1453671975044266200at_nat @ R4 ) ) ) ).

% in_symclI(1)
thf(fact_688_le__enumerate,axiom,
    ! [S: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).

% le_enumerate
thf(fact_689_enumerate__step,axiom,
    ! [S: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ).

% enumerate_step
thf(fact_690_enumerate__mono,axiom,
    ! [M4: nat,N: nat,S: set_nat] :
      ( ( ord_less_nat @ M4 @ N )
     => ( ~ ( finite_finite_nat @ S )
       => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M4 ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).

% enumerate_mono
thf(fact_691_enumerate__Suc_H_H,axiom,
    ! [S: set_Pr8693737435421807431at_nat,N: nat] :
      ( ~ ( finite4392333629123659920at_nat @ S )
     => ( ( infini1315406684128855308at_nat @ S @ ( suc @ N ) )
        = ( ord_Le1799159837870784375at_nat
          @ ^ [S2: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ S2 @ S )
              & ( ord_le9033551061567896339at_nat @ ( infini1315406684128855308at_nat @ S @ N ) @ S2 ) ) ) ) ) ).

% enumerate_Suc''
thf(fact_692_enumerate__Suc_H_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,N: nat] :
      ( ~ ( finite6177210948735845034at_nat @ S )
     => ( ( infini875302759086991654at_nat @ S @ ( suc @ N ) )
        = ( ord_Le3216233473801179153at_nat
          @ ^ [S2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ S2 @ S )
              & ( ord_le1203424502768444845at_nat @ ( infini875302759086991654at_nat @ S @ N ) @ S2 ) ) ) ) ) ).

% enumerate_Suc''
thf(fact_693_enumerate__Suc_H_H,axiom,
    ! [S: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( ( infini8530281810654367211te_nat @ S @ ( suc @ N ) )
        = ( ord_Least_nat
          @ ^ [S2: nat] :
              ( ( member_nat @ S2 @ S )
              & ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S2 ) ) ) ) ) ).

% enumerate_Suc''
thf(fact_694_finite__enum__subset,axiom,
    ! [X5: set_nat,Y6: set_nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( finite_card_nat @ X5 ) )
         => ( ( infini8530281810654367211te_nat @ X5 @ I4 )
            = ( infini8530281810654367211te_nat @ Y6 @ I4 ) ) )
     => ( ( finite_finite_nat @ X5 )
       => ( ( finite_finite_nat @ Y6 )
         => ( ( ord_less_eq_nat @ ( finite_card_nat @ X5 ) @ ( finite_card_nat @ Y6 ) )
           => ( ord_less_eq_set_nat @ X5 @ Y6 ) ) ) ) ) ).

% finite_enum_subset
thf(fact_695_finite__enumerate__step,axiom,
    ! [S: set_nat,N: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S ) )
       => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ) ).

% finite_enumerate_step
thf(fact_696_card__Collect__less__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I5: nat] : ( ord_less_nat @ I5 @ N ) ) )
      = N ) ).

% card_Collect_less_nat
thf(fact_697_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I5: nat] : ( ord_less_eq_nat @ I5 @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_698_finite__enumerate__mono__iff,axiom,
    ! [S: set_nat,M4: nat,N: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ M4 @ ( finite_card_nat @ S ) )
       => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
         => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M4 ) @ ( infini8530281810654367211te_nat @ S @ N ) )
            = ( ord_less_nat @ M4 @ N ) ) ) ) ) ).

% finite_enumerate_mono_iff
thf(fact_699_infinite__arbitrarily__large,axiom,
    ! [A3: set_list_a,N: nat] :
      ( ~ ( finite_finite_list_a @ A3 )
     => ? [B11: set_list_a] :
          ( ( finite_finite_list_a @ B11 )
          & ( ( finite_card_list_a @ B11 )
            = N )
          & ( ord_le8861187494160871172list_a @ B11 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_700_infinite__arbitrarily__large,axiom,
    ! [A3: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ? [B11: set_nat] :
          ( ( finite_finite_nat @ B11 )
          & ( ( finite_card_nat @ B11 )
            = N )
          & ( ord_less_eq_set_nat @ B11 @ A3 ) ) ) ).

% infinite_arbitrarily_large
thf(fact_701_card__subset__eq,axiom,
    ! [B3: set_list_a,A3: set_list_a] :
      ( ( finite_finite_list_a @ B3 )
     => ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
       => ( ( ( finite_card_list_a @ A3 )
            = ( finite_card_list_a @ B3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_702_card__subset__eq,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ( ( finite_card_nat @ A3 )
            = ( finite_card_nat @ B3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_703_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A3: set_nat,R4: nat > nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A3 )
           => ? [B5: nat] :
                ( ( member_nat @ B5 @ B3 )
                & ( R4 @ A5 @ B5 ) ) )
       => ( ! [A13: nat,A24: nat,B4: nat] :
              ( ( member_nat @ A13 @ A3 )
             => ( ( member_nat @ A24 @ A3 )
               => ( ( member_nat @ B4 @ B3 )
                 => ( ( R4 @ A13 @ B4 )
                   => ( ( R4 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_704_card__le__if__inj__on__rel,axiom,
    ! [B3: set_list_a,A3: set_nat,R4: nat > list_a > $o] :
      ( ( finite_finite_list_a @ B3 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A3 )
           => ? [B5: list_a] :
                ( ( member_list_a @ B5 @ B3 )
                & ( R4 @ A5 @ B5 ) ) )
       => ( ! [A13: nat,A24: nat,B4: list_a] :
              ( ( member_nat @ A13 @ A3 )
             => ( ( member_nat @ A24 @ A3 )
               => ( ( member_list_a @ B4 @ B3 )
                 => ( ( R4 @ A13 @ B4 )
                   => ( ( R4 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_list_a @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_705_card__le__if__inj__on__rel,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_nat,R4: nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A3 )
           => ? [B5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ B5 @ B3 )
                & ( R4 @ A5 @ B5 ) ) )
       => ( ! [A13: nat,A24: nat,B4: product_prod_nat_nat] :
              ( ( member_nat @ A13 @ A3 )
             => ( ( member_nat @ A24 @ A3 )
               => ( ( member8440522571783428010at_nat @ B4 @ B3 )
                 => ( ( R4 @ A13 @ B4 )
                   => ( ( R4 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite711546835091564841at_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_706_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A3: set_Pr1261947904930325089at_nat,R4: product_prod_nat_nat > nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ! [A5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A3 )
           => ? [B5: nat] :
                ( ( member_nat @ B5 @ B3 )
                & ( R4 @ A5 @ B5 ) ) )
       => ( ! [A13: product_prod_nat_nat,A24: product_prod_nat_nat,B4: nat] :
              ( ( member8440522571783428010at_nat @ A13 @ A3 )
             => ( ( member8440522571783428010at_nat @ A24 @ A3 )
               => ( ( member_nat @ B4 @ B3 )
                 => ( ( R4 @ A13 @ B4 )
                   => ( ( R4 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_707_card__le__if__inj__on__rel,axiom,
    ! [B3: set_list_a,A3: set_Pr1261947904930325089at_nat,R4: product_prod_nat_nat > list_a > $o] :
      ( ( finite_finite_list_a @ B3 )
     => ( ! [A5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A3 )
           => ? [B5: list_a] :
                ( ( member_list_a @ B5 @ B3 )
                & ( R4 @ A5 @ B5 ) ) )
       => ( ! [A13: product_prod_nat_nat,A24: product_prod_nat_nat,B4: list_a] :
              ( ( member8440522571783428010at_nat @ A13 @ A3 )
             => ( ( member8440522571783428010at_nat @ A24 @ A3 )
               => ( ( member_list_a @ B4 @ B3 )
                 => ( ( R4 @ A13 @ B4 )
                   => ( ( R4 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite_card_list_a @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_708_card__le__if__inj__on__rel,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,R4: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ! [A5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ A5 @ A3 )
           => ? [B5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ B5 @ B3 )
                & ( R4 @ A5 @ B5 ) ) )
       => ( ! [A13: product_prod_nat_nat,A24: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ A13 @ A3 )
             => ( ( member8440522571783428010at_nat @ A24 @ A3 )
               => ( ( member8440522571783428010at_nat @ B4 @ B3 )
                 => ( ( R4 @ A13 @ B4 )
                   => ( ( R4 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_709_card__le__if__inj__on__rel,axiom,
    ! [B3: set_Pr8693737435421807431at_nat,A3: set_nat,R4: nat > produc859450856879609959at_nat > $o] :
      ( ( finite4392333629123659920at_nat @ B3 )
     => ( ! [A5: nat] :
            ( ( member_nat @ A5 @ A3 )
           => ? [B5: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ B5 @ B3 )
                & ( R4 @ A5 @ B5 ) ) )
       => ( ! [A13: nat,A24: nat,B4: produc859450856879609959at_nat] :
              ( ( member_nat @ A13 @ A3 )
             => ( ( member_nat @ A24 @ A3 )
               => ( ( member8206827879206165904at_nat @ B4 @ B3 )
                 => ( ( R4 @ A13 @ B4 )
                   => ( ( R4 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_710_card__le__if__inj__on__rel,axiom,
    ! [B3: set_nat,A3: set_Pr8693737435421807431at_nat,R4: produc859450856879609959at_nat > nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ! [A5: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ A5 @ A3 )
           => ? [B5: nat] :
                ( ( member_nat @ B5 @ B3 )
                & ( R4 @ A5 @ B5 ) ) )
       => ( ! [A13: produc859450856879609959at_nat,A24: produc859450856879609959at_nat,B4: nat] :
              ( ( member8206827879206165904at_nat @ A13 @ A3 )
             => ( ( member8206827879206165904at_nat @ A24 @ A3 )
               => ( ( member_nat @ B4 @ B3 )
                 => ( ( R4 @ A13 @ B4 )
                   => ( ( R4 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_711_card__le__if__inj__on__rel,axiom,
    ! [B3: set_list_a,A3: set_Pr8693737435421807431at_nat,R4: produc859450856879609959at_nat > list_a > $o] :
      ( ( finite_finite_list_a @ B3 )
     => ( ! [A5: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ A5 @ A3 )
           => ? [B5: list_a] :
                ( ( member_list_a @ B5 @ B3 )
                & ( R4 @ A5 @ B5 ) ) )
       => ( ! [A13: produc859450856879609959at_nat,A24: produc859450856879609959at_nat,B4: list_a] :
              ( ( member8206827879206165904at_nat @ A13 @ A3 )
             => ( ( member8206827879206165904at_nat @ A24 @ A3 )
               => ( ( member_list_a @ B4 @ B3 )
                 => ( ( R4 @ A13 @ B4 )
                   => ( ( R4 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite_card_list_a @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_712_card__le__if__inj__on__rel,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr8693737435421807431at_nat,R4: produc859450856879609959at_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B3 )
     => ( ! [A5: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ A5 @ A3 )
           => ? [B5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ B5 @ B3 )
                & ( R4 @ A5 @ B5 ) ) )
       => ( ! [A13: produc859450856879609959at_nat,A24: produc859450856879609959at_nat,B4: product_prod_nat_nat] :
              ( ( member8206827879206165904at_nat @ A13 @ A3 )
             => ( ( member8206827879206165904at_nat @ A24 @ A3 )
               => ( ( member8440522571783428010at_nat @ B4 @ B3 )
                 => ( ( R4 @ A13 @ B4 )
                   => ( ( R4 @ A24 @ B4 )
                     => ( A13 = A24 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B3 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_713_finite__enumerate__Suc_H_H,axiom,
    ! [S: set_Pr8693737435421807431at_nat,N: nat] :
      ( ( finite4392333629123659920at_nat @ S )
     => ( ( ord_less_nat @ ( suc @ N ) @ ( finite1207074278014112911at_nat @ S ) )
       => ( ( infini1315406684128855308at_nat @ S @ ( suc @ N ) )
          = ( ord_Le1799159837870784375at_nat
            @ ^ [S2: produc859450856879609959at_nat] :
                ( ( member8206827879206165904at_nat @ S2 @ S )
                & ( ord_le9033551061567896339at_nat @ ( infini1315406684128855308at_nat @ S @ N ) @ S2 ) ) ) ) ) ) ).

% finite_enumerate_Suc''
thf(fact_714_finite__enumerate__Suc_H_H,axiom,
    ! [S: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( ord_less_nat @ ( suc @ N ) @ ( finite711546835091564841at_nat @ S ) )
       => ( ( infini875302759086991654at_nat @ S @ ( suc @ N ) )
          = ( ord_Le3216233473801179153at_nat
            @ ^ [S2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ S2 @ S )
                & ( ord_le1203424502768444845at_nat @ ( infini875302759086991654at_nat @ S @ N ) @ S2 ) ) ) ) ) ) ).

% finite_enumerate_Suc''
thf(fact_715_finite__enumerate__Suc_H_H,axiom,
    ! [S: set_nat,N: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S ) )
       => ( ( infini8530281810654367211te_nat @ S @ ( suc @ N ) )
          = ( ord_Least_nat
            @ ^ [S2: nat] :
                ( ( member_nat @ S2 @ S )
                & ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S2 ) ) ) ) ) ) ).

% finite_enumerate_Suc''
thf(fact_716_card__mono,axiom,
    ! [B3: set_list_a,A3: set_list_a] :
      ( ( finite_finite_list_a @ B3 )
     => ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
       => ( ord_less_eq_nat @ ( finite_card_list_a @ A3 ) @ ( finite_card_list_a @ B3 ) ) ) ) ).

% card_mono
thf(fact_717_card__mono,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ).

% card_mono
thf(fact_718_card__seteq,axiom,
    ! [B3: set_list_a,A3: set_list_a] :
      ( ( finite_finite_list_a @ B3 )
     => ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_a @ B3 ) @ ( finite_card_list_a @ A3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
thf(fact_719_card__seteq,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B3 ) @ ( finite_card_nat @ A3 ) )
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
thf(fact_720_exists__subset__between,axiom,
    ! [A3: set_list_a,N: nat,C4: set_list_a] :
      ( ( ord_less_eq_nat @ ( finite_card_list_a @ A3 ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ C4 ) )
       => ( ( ord_le8861187494160871172list_a @ A3 @ C4 )
         => ( ( finite_finite_list_a @ C4 )
           => ? [B11: set_list_a] :
                ( ( ord_le8861187494160871172list_a @ A3 @ B11 )
                & ( ord_le8861187494160871172list_a @ B11 @ C4 )
                & ( ( finite_card_list_a @ B11 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_721_exists__subset__between,axiom,
    ! [A3: set_nat,N: nat,C4: set_nat] :
      ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C4 ) )
       => ( ( ord_less_eq_set_nat @ A3 @ C4 )
         => ( ( finite_finite_nat @ C4 )
           => ? [B11: set_nat] :
                ( ( ord_less_eq_set_nat @ A3 @ B11 )
                & ( ord_less_eq_set_nat @ B11 @ C4 )
                & ( ( finite_card_nat @ B11 )
                  = N ) ) ) ) ) ) ).

% exists_subset_between
thf(fact_722_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_list_a] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_list_a @ S ) )
     => ~ ! [T3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ T3 @ S )
           => ( ( ( finite_card_list_a @ T3 )
                = N )
             => ~ ( finite_finite_list_a @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_723_obtain__subset__with__card__n,axiom,
    ! [N: nat,S: set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
     => ~ ! [T3: set_nat] :
            ( ( ord_less_eq_set_nat @ T3 @ S )
           => ( ( ( finite_card_nat @ T3 )
                = N )
             => ~ ( finite_finite_nat @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_724_finite__if__finite__subsets__card__bdd,axiom,
    ! [F5: set_list_a,C4: nat] :
      ( ! [G: set_list_a] :
          ( ( ord_le8861187494160871172list_a @ G @ F5 )
         => ( ( finite_finite_list_a @ G )
           => ( ord_less_eq_nat @ ( finite_card_list_a @ G ) @ C4 ) ) )
     => ( ( finite_finite_list_a @ F5 )
        & ( ord_less_eq_nat @ ( finite_card_list_a @ F5 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_725_finite__if__finite__subsets__card__bdd,axiom,
    ! [F5: set_nat,C4: nat] :
      ( ! [G: set_nat] :
          ( ( ord_less_eq_set_nat @ G @ F5 )
         => ( ( finite_finite_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C4 ) ) )
     => ( ( finite_finite_nat @ F5 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F5 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_726_psubset__card__mono,axiom,
    ! [B3: set_list_a,A3: set_list_a] :
      ( ( finite_finite_list_a @ B3 )
     => ( ( ord_less_set_list_a @ A3 @ B3 )
       => ( ord_less_nat @ ( finite_card_list_a @ A3 ) @ ( finite_card_list_a @ B3 ) ) ) ) ).

% psubset_card_mono
thf(fact_727_psubset__card__mono,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_set_nat @ A3 @ B3 )
       => ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ).

% psubset_card_mono
thf(fact_728_finite__le__enumerate,axiom,
    ! [S: set_nat,N: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
       => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).

% finite_le_enumerate
thf(fact_729_finite__enum__ext,axiom,
    ! [X5: set_nat,Y6: set_nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( finite_card_nat @ X5 ) )
         => ( ( infini8530281810654367211te_nat @ X5 @ I4 )
            = ( infini8530281810654367211te_nat @ Y6 @ I4 ) ) )
     => ( ( finite_finite_nat @ X5 )
       => ( ( finite_finite_nat @ Y6 )
         => ( ( ( finite_card_nat @ X5 )
              = ( finite_card_nat @ Y6 ) )
           => ( X5 = Y6 ) ) ) ) ) ).

% finite_enum_ext
thf(fact_730_finite__enumerate__Ex,axiom,
    ! [S: set_Pr8693737435421807431at_nat,S3: produc859450856879609959at_nat] :
      ( ( finite4392333629123659920at_nat @ S )
     => ( ( member8206827879206165904at_nat @ S3 @ S )
       => ? [N9: nat] :
            ( ( ord_less_nat @ N9 @ ( finite1207074278014112911at_nat @ S ) )
            & ( ( infini1315406684128855308at_nat @ S @ N9 )
              = S3 ) ) ) ) ).

% finite_enumerate_Ex
thf(fact_731_finite__enumerate__Ex,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S3: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( member8440522571783428010at_nat @ S3 @ S )
       => ? [N9: nat] :
            ( ( ord_less_nat @ N9 @ ( finite711546835091564841at_nat @ S ) )
            & ( ( infini875302759086991654at_nat @ S @ N9 )
              = S3 ) ) ) ) ).

% finite_enumerate_Ex
thf(fact_732_finite__enumerate__Ex,axiom,
    ! [S: set_nat,S3: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( member_nat @ S3 @ S )
       => ? [N9: nat] :
            ( ( ord_less_nat @ N9 @ ( finite_card_nat @ S ) )
            & ( ( infini8530281810654367211te_nat @ S @ N9 )
              = S3 ) ) ) ) ).

% finite_enumerate_Ex
thf(fact_733_finite__enumerate__in__set,axiom,
    ! [S: set_Pr8693737435421807431at_nat,N: nat] :
      ( ( finite4392333629123659920at_nat @ S )
     => ( ( ord_less_nat @ N @ ( finite1207074278014112911at_nat @ S ) )
       => ( member8206827879206165904at_nat @ ( infini1315406684128855308at_nat @ S @ N ) @ S ) ) ) ).

% finite_enumerate_in_set
thf(fact_734_finite__enumerate__in__set,axiom,
    ! [S: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ S )
     => ( ( ord_less_nat @ N @ ( finite711546835091564841at_nat @ S ) )
       => ( member8440522571783428010at_nat @ ( infini875302759086991654at_nat @ S @ N ) @ S ) ) ) ).

% finite_enumerate_in_set
thf(fact_735_finite__enumerate__in__set,axiom,
    ! [S: set_nat,N: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
       => ( member_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ) ).

% finite_enumerate_in_set
thf(fact_736_card__psubset,axiom,
    ! [B3: set_list_a,A3: set_list_a] :
      ( ( finite_finite_list_a @ B3 )
     => ( ( ord_le8861187494160871172list_a @ A3 @ B3 )
       => ( ( ord_less_nat @ ( finite_card_list_a @ A3 ) @ ( finite_card_list_a @ B3 ) )
         => ( ord_less_set_list_a @ A3 @ B3 ) ) ) ) ).

% card_psubset
thf(fact_737_card__psubset,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) )
         => ( ord_less_set_nat @ A3 @ B3 ) ) ) ) ).

% card_psubset
thf(fact_738_finite__enumerate__mono,axiom,
    ! [M4: nat,N: nat,S: set_nat] :
      ( ( ord_less_nat @ M4 @ N )
     => ( ( finite_finite_nat @ S )
       => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
         => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M4 ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ) ).

% finite_enumerate_mono
thf(fact_739_not__less__Least,axiom,
    ! [K3: nat,P: nat > $o] :
      ( ( ord_less_nat @ K3 @ ( ord_Least_nat @ P ) )
     => ~ ( P @ K3 ) ) ).

% not_less_Least
thf(fact_740_Least__le,axiom,
    ! [P: nat > $o,K3: nat] :
      ( ( P @ K3 )
     => ( ord_less_eq_nat @ ( ord_Least_nat @ P ) @ K3 ) ) ).

% Least_le
thf(fact_741_Bleast__def,axiom,
    ( bleast3078576362778749909at_nat
    = ( ^ [S4: set_Pr8693737435421807431at_nat,P6: produc859450856879609959at_nat > $o] :
          ( ord_Le1799159837870784375at_nat
          @ ^ [X: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X @ S4 )
              & ( P6 @ X ) ) ) ) ) ).

% Bleast_def
thf(fact_742_Bleast__def,axiom,
    ( bleast198245214120542575at_nat
    = ( ^ [S4: set_Pr1261947904930325089at_nat,P6: product_prod_nat_nat > $o] :
          ( ord_Le3216233473801179153at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ S4 )
              & ( P6 @ X ) ) ) ) ) ).

% Bleast_def
thf(fact_743_Bleast__def,axiom,
    ( bleast_nat
    = ( ^ [S4: set_nat,P6: nat > $o] :
          ( ord_Least_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ S4 )
              & ( P6 @ X ) ) ) ) ) ).

% Bleast_def
thf(fact_744_LeastI,axiom,
    ! [P: nat > $o,K3: nat] :
      ( ( P @ K3 )
     => ( P @ ( ord_Least_nat @ P ) ) ) ).

% LeastI
thf(fact_745_abort__Bleast__def,axiom,
    ( abort_8368135353768099641at_nat
    = ( ^ [S4: set_Pr8693737435421807431at_nat,P6: produc859450856879609959at_nat > $o] :
          ( ord_Le1799159837870784375at_nat
          @ ^ [X: produc859450856879609959at_nat] :
              ( ( member8206827879206165904at_nat @ X @ S4 )
              & ( P6 @ X ) ) ) ) ) ).

% abort_Bleast_def
thf(fact_746_abort__Bleast__def,axiom,
    ( abort_5709735807243680467at_nat
    = ( ^ [S4: set_Pr1261947904930325089at_nat,P6: product_prod_nat_nat > $o] :
          ( ord_Le3216233473801179153at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ S4 )
              & ( P6 @ X ) ) ) ) ) ).

% abort_Bleast_def
thf(fact_747_abort__Bleast__def,axiom,
    ( abort_Bleast_nat
    = ( ^ [S4: set_nat,P6: nat > $o] :
          ( ord_Least_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ S4 )
              & ( P6 @ X ) ) ) ) ) ).

% abort_Bleast_def
thf(fact_748_bot_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X )
    @ ^ [X: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X )
    @ bot_bot_nat ) ).

% bot.ordering_top_axioms
thf(fact_749_bot_Oordering__top__axioms,axiom,
    ( ordering_top_set_nat
    @ ^ [X: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ X )
    @ ^ [X: set_nat,Y3: set_nat] : ( ord_less_set_nat @ Y3 @ X )
    @ bot_bot_set_nat ) ).

% bot.ordering_top_axioms
thf(fact_750_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( ord_less_eq_nat @ ( finite_card_list_a @ A3 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X: list_a] :
              ( ( member_list_a @ X @ A3 )
             => ! [Y3: list_a] :
                  ( ( member_list_a @ Y3 @ A3 )
                 => ( X = Y3 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_751_card__le__Suc0__iff__eq,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ A3 )
             => ! [Y3: nat] :
                  ( ( member_nat @ Y3 @ A3 )
                 => ( X = Y3 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_752_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_753_card_Oinfinite,axiom,
    ! [A3: set_list_a] :
      ( ~ ( finite_finite_list_a @ A3 )
     => ( ( finite_card_list_a @ A3 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_754_card_Oinfinite,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_card_nat @ A3 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_755_card__0__eq,axiom,
    ! [A3: set_list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( ( finite_card_list_a @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_list_a ) ) ) ).

% card_0_eq
thf(fact_756_card__0__eq,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( finite_card_nat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_757_list__decode_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ~ ! [N9: nat] :
            ( X3
           != ( suc @ N9 ) ) ) ).

% list_decode.cases
thf(fact_758_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N9: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N9 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N9 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct'
thf(fact_759_nat__compl__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N9: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N9 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N9 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct
thf(fact_760_filter__preserves__multiset,axiom,
    ! [M3: list_a > nat,P: list_a > $o] :
      ( ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X: list_a] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X ) ) ) )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X: list_a] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M3 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_761_filter__preserves__multiset,axiom,
    ! [M3: nat > nat,P: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X ) @ ( M3 @ X ) @ zero_zero_nat ) ) ) ) ) ).

% filter_preserves_multiset
thf(fact_762_enumerate__0,axiom,
    ! [S: set_Pr8693737435421807431at_nat] :
      ( ( infini1315406684128855308at_nat @ S @ zero_zero_nat )
      = ( ord_Le1799159837870784375at_nat
        @ ^ [N3: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ N3 @ S ) ) ) ).

% enumerate_0
thf(fact_763_enumerate__0,axiom,
    ! [S: set_Pr1261947904930325089at_nat] :
      ( ( infini875302759086991654at_nat @ S @ zero_zero_nat )
      = ( ord_Le3216233473801179153at_nat
        @ ^ [N3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ N3 @ S ) ) ) ).

% enumerate_0
thf(fact_764_enumerate__0,axiom,
    ! [S: set_nat] :
      ( ( infini8530281810654367211te_nat @ S @ zero_zero_nat )
      = ( ord_Least_nat
        @ ^ [N3: nat] : ( member_nat @ N3 @ S ) ) ) ).

% enumerate_0
thf(fact_765_card__eq__0__iff,axiom,
    ! [A3: set_list_a] :
      ( ( ( finite_card_list_a @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_list_a )
        | ~ ( finite_finite_list_a @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_766_card__eq__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( ( finite_card_nat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_767_card__ge__0__finite,axiom,
    ! [A3: set_list_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_a @ A3 ) )
     => ( finite_finite_list_a @ A3 ) ) ).

% card_ge_0_finite
thf(fact_768_card__ge__0__finite,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
     => ( finite_finite_nat @ A3 ) ) ).

% card_ge_0_finite
thf(fact_769_add__mset__in__multiset,axiom,
    ! [M3: list_a > nat,A: list_a] :
      ( ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X: list_a] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X ) ) ) )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X: list_a] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A ) @ ( suc @ ( M3 @ X ) ) @ ( M3 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_770_add__mset__in__multiset,axiom,
    ! [M3: nat > nat,A: nat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( M3 @ X ) ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( X = A ) @ ( suc @ ( M3 @ X ) ) @ ( M3 @ X ) ) ) ) ) ) ).

% add_mset_in_multiset
thf(fact_771_card__less,axiom,
    ! [M3: set_nat,I3: nat] :
      ( ( member_nat @ zero_zero_nat @ M3 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K5: nat] :
                ( ( member_nat @ K5 @ M3 )
                & ( ord_less_nat @ K5 @ ( suc @ I3 ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_772_card__less__Suc,axiom,
    ! [M3: set_nat,I3: nat] :
      ( ( member_nat @ zero_zero_nat @ M3 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K5: nat] :
                  ( ( member_nat @ ( suc @ K5 ) @ M3 )
                  & ( ord_less_nat @ K5 @ I3 ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K5: nat] :
                ( ( member_nat @ K5 @ M3 )
                & ( ord_less_nat @ K5 @ ( suc @ I3 ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_773_card__less__Suc2,axiom,
    ! [M3: set_nat,I3: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M3 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K5: nat] :
                ( ( member_nat @ ( suc @ K5 ) @ M3 )
                & ( ord_less_nat @ K5 @ I3 ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K5: nat] :
                ( ( member_nat @ K5 @ M3 )
                & ( ord_less_nat @ K5 @ ( suc @ I3 ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_774_card__gt__0__iff,axiom,
    ! [A3: set_list_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_a @ A3 ) )
      = ( ( A3 != bot_bot_set_list_a )
        & ( finite_finite_list_a @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_775_card__gt__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
      = ( ( A3 != bot_bot_set_nat )
        & ( finite_finite_nat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_776_remove__nth_Ocases,axiom,
    ! [X3: produc8472197452120411308at_nat] :
      ( ! [I4: nat] :
          ( X3
         != ( produc6109913384486294878at_nat @ I4 @ nil_Pr5478986624290739719at_nat ) )
     => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
            ( X3
           != ( produc6109913384486294878at_nat @ zero_zero_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) )
       => ~ ! [I4: nat,X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
              ( X3
             != ( produc6109913384486294878at_nat @ ( suc @ I4 ) @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ) ) ).

% remove_nth.cases
thf(fact_777_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X )
    @ ^ [X: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_778_Least__Suc,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_779_term_Osize__gen_I2_J,axiom,
    ! [X3: a > nat,X2: nat] :
      ( ( relati1355589192283164855term_a @ X3 @ ( relational_Var_a @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% term.size_gen(2)
thf(fact_780_nths__Cons,axiom,
    ! [X3: product_prod_nat_nat,L: list_P6011104703257516679at_nat,A3: set_nat] :
      ( ( nths_P6079298444859966469at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ L ) @ A3 )
      = ( append985823374593552924at_nat @ ( if_lis9186351972506106189at_nat @ ( member_nat @ zero_zero_nat @ A3 ) @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) @ nil_Pr5478986624290739719at_nat )
        @ ( nths_P6079298444859966469at_nat @ L
          @ ( collect_nat
            @ ^ [J: nat] : ( member_nat @ ( suc @ J ) @ A3 ) ) ) ) ) ).

% nths_Cons
thf(fact_781_remove__nth_Oelims,axiom,
    ! [X3: nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( list_r415652507956852618at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ( ( ( X3 = zero_zero_nat )
           => ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                ( ( Xa2
                  = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
               => ( Y != Xs2 ) ) )
         => ~ ! [I4: nat] :
                ( ( X3
                  = ( suc @ I4 ) )
               => ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                    ( ( Xa2
                      = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
                   => ( Y
                     != ( cons_P6512896166579812791at_nat @ X4 @ ( list_r415652507956852618at_nat @ I4 @ Xs2 ) ) ) ) ) ) ) ) ).

% remove_nth.elims
thf(fact_782_card__def,axiom,
    ( finite_card_nat
    = ( finite3337649020256630879at_nat
      @ ^ [Uu2: nat] : suc
      @ zero_zero_nat ) ) ).

% card_def
thf(fact_783_nths__singleton,axiom,
    ! [A3: set_nat,X3: product_prod_nat_nat] :
      ( ( ( member_nat @ zero_zero_nat @ A3 )
       => ( ( nths_P6079298444859966469at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) @ A3 )
          = ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) )
      & ( ~ ( member_nat @ zero_zero_nat @ A3 )
       => ( ( nths_P6079298444859966469at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) @ A3 )
          = nil_Pr5478986624290739719at_nat ) ) ) ).

% nths_singleton
thf(fact_784_remove__nth_Osimps_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( list_r415652507956852618at_nat @ zero_zero_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = Xs ) ).

% remove_nth.simps(2)
thf(fact_785_remove__nth_Osimps_I3_J,axiom,
    ! [I3: nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( list_r415652507956852618at_nat @ ( suc @ I3 ) @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ X3 @ ( list_r415652507956852618at_nat @ I3 @ Xs ) ) ) ).

% remove_nth.simps(3)
thf(fact_786_remove__nth_Opelims,axiom,
    ! [X3: nat,Xa2: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( list_r415652507956852618at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P1391465874720445027at_nat @ list_r7910557391251434237at_nat @ ( produc6109913384486294878at_nat @ X3 @ Xa2 ) )
       => ( ( ( Xa2 = nil_Pr5478986624290739719at_nat )
           => ( ( Y = nil_Pr5478986624290739719at_nat )
             => ~ ( accp_P1391465874720445027at_nat @ list_r7910557391251434237at_nat @ ( produc6109913384486294878at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) )
         => ( ( ( X3 = zero_zero_nat )
             => ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                  ( ( Xa2
                    = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
                 => ( ( Y = Xs2 )
                   => ~ ( accp_P1391465874720445027at_nat @ list_r7910557391251434237at_nat @ ( produc6109913384486294878at_nat @ zero_zero_nat @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ) ) )
           => ~ ! [I4: nat] :
                  ( ( X3
                    = ( suc @ I4 ) )
                 => ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
                      ( ( Xa2
                        = ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) )
                     => ( ( Y
                          = ( cons_P6512896166579812791at_nat @ X4 @ ( list_r415652507956852618at_nat @ I4 @ Xs2 ) ) )
                       => ~ ( accp_P1391465874720445027at_nat @ list_r7910557391251434237at_nat @ ( produc6109913384486294878at_nat @ ( suc @ I4 ) @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).

% remove_nth.pelims
thf(fact_787_term_Osize_I4_J,axiom,
    ! [X2: nat] :
      ( ( size_s49661629988129973term_a @ ( relational_Var_a @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% term.size(4)
thf(fact_788_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_789_length__ge__1__conv,axiom,
    ! [L: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ L ) )
      = ( L != nil_nat ) ) ).

% length_ge_1_conv
thf(fact_790_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys3: list_nat,Us2: list_nat,Vs2: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys3 ) )
        | ( ( size_size_list_nat @ Us2 )
          = ( size_size_list_nat @ Vs2 ) ) )
     => ( ( ( append_nat @ Xs @ Us2 )
          = ( append_nat @ Ys3 @ Vs2 ) )
        = ( ( Xs = Ys3 )
          & ( Us2 = Vs2 ) ) ) ) ).

% append_eq_append_conv
thf(fact_791_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_792_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_793_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A3: set_nat] :
      ( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A3 ) )
      = ( finite_card_nat @ A3 ) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_794_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_795_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys6: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys6 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys6 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_796_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_797_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys3: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys3 ) )
     => ( Xs != Ys3 ) ) ).

% neq_if_length_neq
thf(fact_798_list__induct2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
       => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
              ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                = ( size_s5460976970255530739at_nat @ Ys ) )
             => ( ( P @ Xs2 @ Ys )
               => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) )
         => ( P @ Xs @ Ys3 ) ) ) ) ).

% list_induct2
thf(fact_799_list__induct2,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_nat,P: list_P6011104703257516679at_nat > list_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat )
       => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: nat,Ys: list_nat] :
              ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs2 @ Ys )
               => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) ) ) )
         => ( P @ Xs @ Ys3 ) ) ) ) ).

% list_induct2
thf(fact_800_list__induct2,axiom,
    ! [Xs: list_nat,Ys3: list_P6011104703257516679at_nat,P: list_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat )
       => ( ! [X4: nat,Xs2: list_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_s5460976970255530739at_nat @ Ys ) )
             => ( ( P @ Xs2 @ Ys )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) ) ) )
         => ( P @ Xs @ Ys3 ) ) ) ) ).

% list_induct2
thf(fact_801_list__induct2,axiom,
    ! [Xs: list_nat,Ys3: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys: list_nat] :
              ( ( ( size_size_list_nat @ Xs2 )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs2 @ Ys )
               => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) ) ) )
         => ( P @ Xs @ Ys3 ) ) ) ) ).

% list_induct2
thf(fact_802_list__induct3,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( ( ( size_s5460976970255530739at_nat @ Ys3 )
          = ( size_s5460976970255530739at_nat @ Zs ) )
       => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
         => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
                ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                  = ( size_s5460976970255530739at_nat @ Ys ) )
               => ( ( ( size_s5460976970255530739at_nat @ Ys )
                    = ( size_s5460976970255530739at_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys @ Zs2 )
                   => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_803_list__induct3,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( ( ( size_s5460976970255530739at_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
         => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Z: nat,Zs2: list_nat] :
                ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                  = ( size_s5460976970255530739at_nat @ Ys ) )
               => ( ( ( size_s5460976970255530739at_nat @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys @ Zs2 )
                   => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_804_list__induct3,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_nat,Zs: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( ( size_size_list_nat @ Ys3 )
          = ( size_s5460976970255530739at_nat @ Zs ) )
       => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
         => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: nat,Ys: list_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
                ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_s5460976970255530739at_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys @ Zs2 )
                   => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_805_list__induct3,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_nat,Zs: list_nat,P: list_P6011104703257516679at_nat > list_nat > list_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( ( size_size_list_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat )
         => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: nat,Ys: list_nat,Z: nat,Zs2: list_nat] :
                ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys @ Zs2 )
                   => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_806_list__induct3,axiom,
    ! [Xs: list_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,P: list_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( ( ( size_s5460976970255530739at_nat @ Ys3 )
          = ( size_s5460976970255530739at_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat )
         => ( ! [X4: nat,Xs2: list_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_s5460976970255530739at_nat @ Ys ) )
               => ( ( ( size_s5460976970255530739at_nat @ Ys )
                    = ( size_s5460976970255530739at_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_807_list__induct3,axiom,
    ! [Xs: list_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_nat,P: list_nat > list_P6011104703257516679at_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( ( ( size_s5460976970255530739at_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
         => ( ! [X4: nat,Xs2: list_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Z: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_s5460976970255530739at_nat @ Ys ) )
               => ( ( ( size_s5460976970255530739at_nat @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_808_list__induct3,axiom,
    ! [Xs: list_nat,Ys3: list_nat,Zs: list_P6011104703257516679at_nat,P: list_nat > list_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( ( size_size_list_nat @ Ys3 )
          = ( size_s5460976970255530739at_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
         => ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys: list_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_s5460976970255530739at_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_809_list__induct3,axiom,
    ! [Xs: list_nat,Ys3: list_nat,Zs: list_nat,P: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( ( size_size_list_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys: list_nat,Z: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs2 )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs2 @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys3 @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_810_list__induct4,axiom,
    ! [Xs: list_nat,Ys3: list_nat,Zs: list_nat,Ws2: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( ( size_size_list_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws3: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys3 @ Zs @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_811_list__induct4,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_nat,Zs: list_nat,Ws2: list_nat,P: list_P6011104703257516679at_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( ( size_size_list_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws2 ) )
         => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: nat,Ys: list_nat,Z: nat,Zs2: list_nat,W: nat,Ws3: list_nat] :
                  ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys @ Zs2 @ Ws3 )
                       => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys3 @ Zs @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_812_list__induct4,axiom,
    ! [Xs: list_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_nat,Ws2: list_nat,P: list_nat > list_P6011104703257516679at_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( ( ( size_s5460976970255530739at_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Z: nat,Zs2: list_nat,W: nat,Ws3: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_s5460976970255530739at_nat @ Ys ) )
                 => ( ( ( size_s5460976970255530739at_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys3 @ Zs @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_813_list__induct4,axiom,
    ! [Xs: list_nat,Ys3: list_nat,Zs: list_P6011104703257516679at_nat,Ws2: list_nat,P: list_nat > list_nat > list_P6011104703257516679at_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( ( size_size_list_nat @ Ys3 )
          = ( size_s5460976970255530739at_nat @ Zs ) )
       => ( ( ( size_s5460976970255530739at_nat @ Zs )
            = ( size_size_list_nat @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys: list_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat,W: nat,Ws3: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_s5460976970255530739at_nat @ Zs2 ) )
                   => ( ( ( size_s5460976970255530739at_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys3 @ Zs @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_814_list__induct4,axiom,
    ! [Xs: list_nat,Ys3: list_nat,Zs: list_nat,Ws2: list_P6011104703257516679at_nat,P: list_nat > list_nat > list_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( ( size_size_list_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s5460976970255530739at_nat @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y4: nat,Ys: list_nat,Z: nat,Zs2: list_nat,W: product_prod_nat_nat,Ws3: list_P6011104703257516679at_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s5460976970255530739at_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_P6512896166579812791at_nat @ W @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys3 @ Zs @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_815_list__induct4,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_nat,Ws2: list_nat,P: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_nat > list_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( ( ( size_s5460976970255530739at_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws2 ) )
         => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat )
           => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Z: nat,Zs2: list_nat,W: nat,Ws3: list_nat] :
                  ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                    = ( size_s5460976970255530739at_nat @ Ys ) )
                 => ( ( ( size_s5460976970255530739at_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys @ Zs2 @ Ws3 )
                       => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys3 @ Zs @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_816_list__induct4,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_nat,Zs: list_P6011104703257516679at_nat,Ws2: list_nat,P: list_P6011104703257516679at_nat > list_nat > list_P6011104703257516679at_nat > list_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( ( size_size_list_nat @ Ys3 )
          = ( size_s5460976970255530739at_nat @ Zs ) )
       => ( ( ( size_s5460976970255530739at_nat @ Zs )
            = ( size_size_list_nat @ Ws2 ) )
         => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
           => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: nat,Ys: list_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat,W: nat,Ws3: list_nat] :
                  ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_s5460976970255530739at_nat @ Zs2 ) )
                   => ( ( ( size_s5460976970255530739at_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys @ Zs2 @ Ws3 )
                       => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys3 @ Zs @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_817_list__induct4,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_nat,Zs: list_nat,Ws2: list_P6011104703257516679at_nat,P: list_P6011104703257516679at_nat > list_nat > list_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ( ( size_size_list_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s5460976970255530739at_nat @ Ws2 ) )
         => ( ( P @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
           => ( ! [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y4: nat,Ys: list_nat,Z: nat,Zs2: list_nat,W: product_prod_nat_nat,Ws3: list_P6011104703257516679at_nat] :
                  ( ( ( size_s5460976970255530739at_nat @ Xs2 )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s5460976970255530739at_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys @ Zs2 @ Ws3 )
                       => ( P @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_P6512896166579812791at_nat @ W @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys3 @ Zs @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_818_list__induct4,axiom,
    ! [Xs: list_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,Ws2: list_nat,P: list_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( ( ( size_s5460976970255530739at_nat @ Ys3 )
          = ( size_s5460976970255530739at_nat @ Zs ) )
       => ( ( ( size_s5460976970255530739at_nat @ Zs )
            = ( size_size_list_nat @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_Pr5478986624290739719at_nat @ nil_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Z: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat,W: nat,Ws3: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_s5460976970255530739at_nat @ Ys ) )
                 => ( ( ( size_s5460976970255530739at_nat @ Ys )
                      = ( size_s5460976970255530739at_nat @ Zs2 ) )
                   => ( ( ( size_s5460976970255530739at_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) @ ( cons_P6512896166579812791at_nat @ Z @ Zs2 ) @ ( cons_nat @ W @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys3 @ Zs @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_819_list__induct4,axiom,
    ! [Xs: list_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_nat,Ws2: list_P6011104703257516679at_nat,P: list_nat > list_P6011104703257516679at_nat > list_nat > list_P6011104703257516679at_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( ( ( size_s5460976970255530739at_nat @ Ys3 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s5460976970255530739at_nat @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_Pr5478986624290739719at_nat @ nil_nat @ nil_Pr5478986624290739719at_nat )
           => ( ! [X4: nat,Xs2: list_nat,Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Z: nat,Zs2: list_nat,W: product_prod_nat_nat,Ws3: list_P6011104703257516679at_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_s5460976970255530739at_nat @ Ys ) )
                 => ( ( ( size_s5460976970255530739at_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s5460976970255530739at_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y4 @ Ys ) @ ( cons_nat @ Z @ Zs2 ) @ ( cons_P6512896166579812791at_nat @ W @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys3 @ Zs @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_820_length__Cons,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( size_s5460976970255530739at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( suc @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).

% length_Cons
thf(fact_821_length__Cons,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X3 @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_Cons
thf(fact_822_length__Suc__conv,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N: nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) )
            & ( ( size_s5460976970255530739at_nat @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_823_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys2 ) )
            & ( ( size_size_list_nat @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_824_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( suc @ N )
        = ( size_s5460976970255530739at_nat @ Xs ) )
      = ( ? [Y3: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) )
            & ( ( size_s5460976970255530739at_nat @ Ys2 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_825_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs ) )
      = ( ? [Y3: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys2 ) )
            & ( ( size_size_list_nat @ Ys2 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_826_len__greater__imp__nonempty,axiom,
    ! [X3: nat,L: list_nat] :
      ( ( ord_less_nat @ X3 @ ( size_size_list_nat @ L ) )
     => ( L != nil_nat ) ) ).

% len_greater_imp_nonempty
thf(fact_827_impossible__Cons,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
      ( ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ ( size_s5460976970255530739at_nat @ Ys3 ) )
     => ( Xs
       != ( cons_P6512896166579812791at_nat @ X3 @ Ys3 ) ) ) ).

% impossible_Cons
thf(fact_828_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys3: list_nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys3 ) )
     => ( Xs
       != ( cons_nat @ X3 @ Ys3 ) ) ) ).

% impossible_Cons
thf(fact_829_list__rest__coinc,axiom,
    ! [S22: list_nat,S1: list_nat,R1: list_nat,R2: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ S22 ) @ ( size_size_list_nat @ S1 ) )
     => ( ( ( append_nat @ S1 @ R1 )
          = ( append_nat @ S22 @ R2 ) )
       => ? [R1p: list_nat] :
            ( R2
            = ( append_nat @ R1p @ R1 ) ) ) ) ).

% list_rest_coinc
thf(fact_830_set__encode__eq,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ( nat_set_encode @ A3 )
            = ( nat_set_encode @ B3 ) )
          = ( A3 = B3 ) ) ) ) ).

% set_encode_eq
thf(fact_831_length__filter__le,axiom,
    ! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% length_filter_le
thf(fact_832_finite__maxlen,axiom,
    ! [M3: set_list_a] :
      ( ( finite_finite_list_a @ M3 )
     => ? [N9: nat] :
        ! [X6: list_a] :
          ( ( member_list_a @ X6 @ M3 )
         => ( ord_less_nat @ ( size_size_list_a @ X6 ) @ N9 ) ) ) ).

% finite_maxlen
thf(fact_833_finite__maxlen,axiom,
    ! [M3: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M3 )
     => ? [N9: nat] :
        ! [X6: list_nat] :
          ( ( member_list_nat @ X6 @ M3 )
         => ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N9 ) ) ) ).

% finite_maxlen
thf(fact_834_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( listrel1_nat @ R4 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) ) ) ).

% listrel1_eq_len
thf(fact_835_nths__all,axiom,
    ! [Xs: list_nat,I: set_nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
         => ( member_nat @ I4 @ I ) )
     => ( ( nths_nat @ Xs @ I )
        = Xs ) ) ).

% nths_all
thf(fact_836_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( listrel_nat_nat @ R4 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) ) ) ).

% listrel_eq_len
thf(fact_837_length__code,axiom,
    ( size_size_list_nat
    = ( gen_length_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_838_length__compl__induct,axiom,
    ! [P: list_P6011104703257516679at_nat > $o,L: list_P6011104703257516679at_nat] :
      ( ( P @ nil_Pr5478986624290739719at_nat )
     => ( ! [E: product_prod_nat_nat,L2: list_P6011104703257516679at_nat] :
            ( ! [Ll2: list_P6011104703257516679at_nat] :
                ( ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Ll2 ) @ ( size_s5460976970255530739at_nat @ L2 ) )
               => ( P @ Ll2 ) )
           => ( P @ ( cons_P6512896166579812791at_nat @ E @ L2 ) ) )
       => ( P @ L ) ) ) ).

% length_compl_induct
thf(fact_839_length__compl__induct,axiom,
    ! [P: list_nat > $o,L: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [E: nat,L2: list_nat] :
            ( ! [Ll2: list_nat] :
                ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ll2 ) @ ( size_size_list_nat @ L2 ) )
               => ( P @ Ll2 ) )
           => ( P @ ( cons_nat @ E @ L2 ) ) )
       => ( P @ L ) ) ) ).

% length_compl_induct
thf(fact_840_same__length__different,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( Xs != Ys3 )
     => ( ( ( size_s5460976970255530739at_nat @ Xs )
          = ( size_s5460976970255530739at_nat @ Ys3 ) )
       => ? [Pre: list_P6011104703257516679at_nat,X4: product_prod_nat_nat,Xs4: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat,Ys5: list_P6011104703257516679at_nat] :
            ( ( X4 != Y4 )
            & ( Xs
              = ( append985823374593552924at_nat @ Pre @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ X4 @ nil_Pr5478986624290739719at_nat ) @ Xs4 ) ) )
            & ( Ys3
              = ( append985823374593552924at_nat @ Pre @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_841_same__length__different,axiom,
    ! [Xs: list_nat,Ys3: list_nat] :
      ( ( Xs != Ys3 )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys3 ) )
       => ? [Pre: list_nat,X4: nat,Xs4: list_nat,Y4: nat,Ys5: list_nat] :
            ( ( X4 != Y4 )
            & ( Xs
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ Xs4 ) ) )
            & ( Ys3
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ Ys5 ) ) ) ) ) ) ).

% same_length_different
thf(fact_842_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s5460976970255530739at_nat @ Xs ) )
      = ( ? [X: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( cons_P6512896166579812791at_nat @ X @ Ys2 ) )
            & ( ord_less_eq_nat @ N @ ( size_s5460976970255530739at_nat @ Ys2 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_843_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X: nat,Ys2: list_nat] :
            ( ( Xs
              = ( cons_nat @ X @ Ys2 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_844_lexord__sufE,axiom,
    ! [Xs: list_nat,Zs: list_nat,Ys3: list_nat,Qs: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Zs ) @ ( append_nat @ Ys3 @ Qs ) ) @ ( lexord_nat @ R4 ) )
     => ( ( Xs != Ys3 )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
         => ( ( ( size_size_list_nat @ Zs )
              = ( size_size_list_nat @ Qs ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( lexord_nat @ R4 ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_845_set__encode__inf,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( nat_set_encode @ A3 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_846_length__nths,axiom,
    ! [Xs: list_nat,I: set_nat] :
      ( ( size_size_list_nat @ ( nths_nat @ Xs @ I ) )
      = ( finite_card_nat
        @ ( collect_nat
          @ ^ [I5: nat] :
              ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
              & ( member_nat @ I5 @ I ) ) ) ) ) ).

% length_nths
thf(fact_847_length__append__singleton,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
      ( ( size_s5460976970255530739at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) )
      = ( suc @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).

% length_append_singleton
thf(fact_848_length__append__singleton,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X3 @ nil_nat ) ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_append_singleton
thf(fact_849_length__Suc__rev__conv,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N: nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,Y3: product_prod_nat_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ Y3 @ nil_Pr5478986624290739719at_nat ) ) )
            & ( ( size_s5460976970255530739at_nat @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_850_length__Suc__rev__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Ys2: list_nat,Y3: nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_851_length__Suc__conv__rev,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N: nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ Y3 @ nil_Pr5478986624290739719at_nat ) ) )
            & ( ( size_s5460976970255530739at_nat @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_852_length__Suc__conv__rev,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: nat,Ys2: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys2 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_853_length__compl__rev__induct,axiom,
    ! [P: list_P6011104703257516679at_nat > $o,L: list_P6011104703257516679at_nat] :
      ( ( P @ nil_Pr5478986624290739719at_nat )
     => ( ! [L2: list_P6011104703257516679at_nat,E: product_prod_nat_nat] :
            ( ! [Ll2: list_P6011104703257516679at_nat] :
                ( ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Ll2 ) @ ( size_s5460976970255530739at_nat @ L2 ) )
               => ( P @ Ll2 ) )
           => ( P @ ( append985823374593552924at_nat @ L2 @ ( cons_P6512896166579812791at_nat @ E @ nil_Pr5478986624290739719at_nat ) ) ) )
       => ( P @ L ) ) ) ).

% length_compl_rev_induct
thf(fact_854_length__compl__rev__induct,axiom,
    ! [P: list_nat > $o,L: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [L2: list_nat,E: nat] :
            ( ! [Ll2: list_nat] :
                ( ( ord_less_eq_nat @ ( size_size_list_nat @ Ll2 ) @ ( size_size_list_nat @ L2 ) )
               => ( P @ Ll2 ) )
           => ( P @ ( append_nat @ L2 @ ( cons_nat @ E @ nil_nat ) ) ) )
       => ( P @ L ) ) ) ).

% length_compl_rev_induct
thf(fact_855_lexord__sufI,axiom,
    ! [U: list_nat,W3: list_nat,R4: set_Pr1261947904930325089at_nat,V3: list_nat,Z3: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ W3 ) @ ( lexord_nat @ R4 ) )
     => ( ( ord_less_eq_nat @ ( size_size_list_nat @ W3 ) @ ( size_size_list_nat @ U ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ V3 ) @ ( append_nat @ W3 @ Z3 ) ) @ ( lexord_nat @ R4 ) ) ) ) ).

% lexord_sufI
thf(fact_856_length__nth__simps_I1_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% length_nth_simps(1)
thf(fact_857_length__nth__simps_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( size_s5460976970255530739at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( suc @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).

% length_nth_simps(2)
thf(fact_858_length__nth__simps_I2_J,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X3 @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_nth_simps(2)
thf(fact_859_Cons__lenlex__iff,axiom,
    ! [M4: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,N: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ M4 @ Ms ) @ ( cons_P6512896166579812791at_nat @ N @ Ns ) ) @ ( lenlex325483962726685836at_nat @ R4 ) )
      = ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ms ) @ ( size_s5460976970255530739at_nat @ Ns ) )
        | ( ( ( size_s5460976970255530739at_nat @ Ms )
            = ( size_s5460976970255530739at_nat @ Ns ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ M4 @ N ) @ R4 ) )
        | ( ( M4 = N )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ms @ Ns ) @ ( lenlex325483962726685836at_nat @ R4 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_860_Cons__lenlex__iff,axiom,
    ! [M4: nat,Ms: list_nat,N: nat,Ns: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M4 @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R4 ) )
      = ( ( 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 @ M4 @ N ) @ R4 ) )
        | ( ( M4 = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R4 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_861_lenlex__irreflexive,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R4 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R4 ) ) ) ).

% lenlex_irreflexive
thf(fact_862_lenlex__irreflexive,axiom,
    ! [R4: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R4 )
     => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Xs ) @ ( lenlex325483962726685836at_nat @ R4 ) ) ) ).

% lenlex_irreflexive
thf(fact_863_lenlex__length,axiom,
    ! [Ms: list_nat,Ns: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R4 ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_864_lenlex__append1,axiom,
    ! [Us2: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat,Vs2: list_nat,Ys3: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us2 @ Xs ) @ ( lenlex_nat @ R ) )
     => ( ( ( size_size_list_nat @ Vs2 )
          = ( size_size_list_nat @ Ys3 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Us2 @ Vs2 ) @ ( append_nat @ Xs @ Ys3 ) ) @ ( lenlex_nat @ R ) ) ) ) ).

% lenlex_append1
thf(fact_865_lenlex__conv,axiom,
    ( lenlex_nat
    = ( ^ [R5: set_Pr1261947904930325089at_nat] :
          ( collec1570431334306492044st_nat
          @ ( produc6382697403170970154_nat_o
            @ ^ [Xs3: list_nat,Ys2: list_nat] :
                ( ( ord_less_nat @ ( size_size_list_nat @ Xs3 ) @ ( size_size_list_nat @ Ys2 ) )
                | ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys2 ) )
                  & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys2 ) @ ( lex_nat @ R5 ) ) ) ) ) ) ) ) ).

% lenlex_conv
thf(fact_866_lenlex__def,axiom,
    ( lenlex_nat
    = ( ^ [R5: set_Pr1261947904930325089at_nat] :
          ( inv_im1258040488774060191st_nat @ ( lex_pr42628451598655713st_nat @ less_than @ ( lex_nat @ R5 ) )
          @ ^ [Xs3: list_nat] : ( produc8282810413953273033st_nat @ ( size_size_list_nat @ Xs3 ) @ Xs3 ) ) ) ) ).

% lenlex_def
thf(fact_867_Cons__in__lex,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) @ ( lex_Pr8571645452597969515at_nat @ R4 ) )
      = ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R4 )
          & ( ( size_s5460976970255530739at_nat @ Xs )
            = ( size_s5460976970255530739at_nat @ Ys3 ) ) )
        | ( ( X3 = Y )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys3 ) @ ( lex_Pr8571645452597969515at_nat @ R4 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_868_Cons__in__lex,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys3 ) ) @ ( lex_nat @ R4 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R4 )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) ) )
        | ( ( X3 = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( lex_nat @ R4 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_869_lex__append__leftD,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R4 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys3 ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R4 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys3 @ Zs ) @ ( lex_nat @ R4 ) ) ) ) ).

% lex_append_leftD
thf(fact_870_lex__append__leftD,axiom,
    ! [R4: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R4 )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs @ Ys3 ) @ ( append985823374593552924at_nat @ Xs @ Zs ) ) @ ( lex_Pr8571645452597969515at_nat @ R4 ) )
       => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys3 @ Zs ) @ ( lex_Pr8571645452597969515at_nat @ R4 ) ) ) ) ).

% lex_append_leftD
thf(fact_871_lex__append__left__iff,axiom,
    ! [R4: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
      ( ! [X4: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ X4 ) @ R4 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys3 ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R4 ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys3 @ Zs ) @ ( lex_nat @ R4 ) ) ) ) ).

% lex_append_left_iff
thf(fact_872_lex__append__left__iff,axiom,
    ! [R4: set_Pr8693737435421807431at_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ X4 ) @ R4 )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs @ Ys3 ) @ ( append985823374593552924at_nat @ Xs @ Zs ) ) @ ( lex_Pr8571645452597969515at_nat @ R4 ) )
        = ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys3 @ Zs ) @ ( lex_Pr8571645452597969515at_nat @ R4 ) ) ) ) ).

% lex_append_left_iff
thf(fact_873_lex__append__rightI,axiom,
    ! [Xs: list_nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat,Vs2: list_nat,Us2: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( lex_nat @ R4 ) )
     => ( ( ( size_size_list_nat @ Vs2 )
          = ( size_size_list_nat @ Us2 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us2 ) @ ( append_nat @ Ys3 @ Vs2 ) ) @ ( lex_nat @ R4 ) ) ) ) ).

% lex_append_rightI
thf(fact_874_lexord__lex,axiom,
    ! [X3: list_nat,Y: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ ( lex_nat @ R4 ) )
      = ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ ( lexord_nat @ R4 ) )
        & ( ( size_size_list_nat @ X3 )
          = ( size_size_list_nat @ Y ) ) ) ) ).

% lexord_lex
thf(fact_875_finite__lists__length__le,axiom,
    ! [A3: set_a,N: nat] :
      ( ( finite_finite_a @ A3 )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Xs3: list_a] :
              ( ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_a @ Xs3 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_876_finite__lists__length__le,axiom,
    ! [A3: set_list_a,N: nat] :
      ( ( finite_finite_list_a @ A3 )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [Xs3: list_list_a] :
              ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs3 ) @ A3 )
              & ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs3 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_877_finite__lists__length__le,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A3 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs3 ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_878_set__encode__inverse,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
        = A3 ) ) ).

% set_encode_inverse
thf(fact_879_eval__empty__close,axiom,
    ! [Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
      ( ( ( relational_eval_a_b @ ( relati7004044288120072026TS_a_b @ ( linord2614967742042102400et_nat @ ( relational_fv_a_b @ Q ) ) @ Q ) @ I )
        = bot_bot_set_list_a )
      = ( ! [Sigma: nat > a] :
            ~ ( relational_sat_a_b @ Q @ I @ Sigma ) ) ) ).

% eval_empty_close
thf(fact_880_List_Ofinite__set,axiom,
    ! [Xs: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs ) ) ).

% List.finite_set
thf(fact_881_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_882_in__set__insert,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X3 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_883_in__set__insert,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ( insert4532235091570160003at_nat @ X3 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_884_in__set__insert,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( insert8944034826898310173at_nat @ X3 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_885_set__filter,axiom,
    ! [P: produc859450856879609959at_nat > $o,Xs: list_P8469869581646625389at_nat] :
      ( ( set_Pr5518436109238095868at_nat @ ( filter1540621253233046500at_nat @ P @ Xs ) )
      = ( collec7088162979684241874at_nat
        @ ^ [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) )
            & ( P @ X ) ) ) ) ).

% set_filter
thf(fact_886_set__filter,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( filter6372491115368938494at_nat @ P @ Xs ) )
      = ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
            & ( P @ X ) ) ) ) ).

% set_filter
thf(fact_887_set__filter,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( set_nat2 @ ( filter_nat @ P @ Xs ) )
      = ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
            & ( P @ X ) ) ) ) ).

% set_filter
thf(fact_888_set__filter,axiom,
    ! [P: list_a > $o,Xs: list_list_a] :
      ( ( set_list_a2 @ ( filter_list_a @ P @ Xs ) )
      = ( collect_list_a
        @ ^ [X: list_a] :
            ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
            & ( P @ X ) ) ) ) ).

% set_filter
thf(fact_889_set__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_set_encode @ ( nat_set_decode @ N ) )
      = N ) ).

% set_decode_inverse
thf(fact_890_List_Oset__empty,axiom,
    ! [Xs: list_nat] :
      ( ( ( set_nat2 @ Xs )
        = bot_bot_set_nat )
      = ( Xs = nil_nat ) ) ).

% List.set_empty
thf(fact_891_set__empty2,axiom,
    ! [Xs: list_nat] :
      ( ( bot_bot_set_nat
        = ( set_nat2 @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% set_empty2
thf(fact_892_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( set_nat2 @ ( linord2614967742042102400et_nat @ A3 ) )
        = A3 ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_893_not__in__set__insert,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X3 @ Xs )
        = ( cons_nat @ X3 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_894_not__in__set__insert,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ( insert4532235091570160003at_nat @ X3 @ Xs )
        = ( cons_P8732206157123786781at_nat @ X3 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_895_not__in__set__insert,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( insert8944034826898310173at_nat @ X3 @ Xs )
        = ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_896_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_897_notin__set__nthsI,axiom,
    ! [X3: nat,Xs: list_nat,I: set_nat] :
      ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ~ ( member_nat @ X3 @ ( set_nat2 @ ( nths_nat @ Xs @ I ) ) ) ) ).

% notin_set_nthsI
thf(fact_898_notin__set__nthsI,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat,I: set_nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( nths_P7654847071924024683at_nat @ Xs @ I ) ) ) ) ).

% notin_set_nthsI
thf(fact_899_notin__set__nthsI,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,I: set_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( nths_P6079298444859966469at_nat @ Xs @ I ) ) ) ) ).

% notin_set_nthsI
thf(fact_900_in__set__nthsD,axiom,
    ! [X3: nat,Xs: list_nat,I: set_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( nths_nat @ Xs @ I ) ) )
     => ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% in_set_nthsD
thf(fact_901_in__set__nthsD,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat,I: set_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( nths_P7654847071924024683at_nat @ Xs @ I ) ) )
     => ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% in_set_nthsD
thf(fact_902_in__set__nthsD,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,I: set_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( nths_P6079298444859966469at_nat @ Xs @ I ) ) )
     => ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% in_set_nthsD
thf(fact_903_filter__cong,axiom,
    ! [Xs: list_nat,Ys3: list_nat,P: nat > $o,Q: nat > $o] :
      ( ( Xs = Ys3 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys3 ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( filter_nat @ P @ Xs )
          = ( filter_nat @ Q @ Ys3 ) ) ) ) ).

% filter_cong
thf(fact_904_filter__cong,axiom,
    ! [Xs: list_P8469869581646625389at_nat,Ys3: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
      ( ( Xs = Ys3 )
     => ( ! [X4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X4 @ ( set_Pr5518436109238095868at_nat @ Ys3 ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( filter1540621253233046500at_nat @ P @ Xs )
          = ( filter1540621253233046500at_nat @ Q @ Ys3 ) ) ) ) ).

% filter_cong
thf(fact_905_filter__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( Xs = Ys3 )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys3 ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( filter6372491115368938494at_nat @ P @ Xs )
          = ( filter6372491115368938494at_nat @ Q @ Ys3 ) ) ) ) ).

% filter_cong
thf(fact_906_set__ConsD,axiom,
    ! [Y: nat,X3: nat,Xs: list_nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
     => ( ( Y = X3 )
        | ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_907_set__ConsD,axiom,
    ! [Y: produc859450856879609959at_nat,X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X3 @ Xs ) ) )
     => ( ( Y = X3 )
        | ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_908_set__ConsD,axiom,
    ! [Y: product_prod_nat_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) )
     => ( ( Y = X3 )
        | ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_909_list_Oset__cases,axiom,
    ! [E3: nat,A: list_nat] :
      ( ( member_nat @ E3 @ ( set_nat2 @ A ) )
     => ( ! [Z22: list_nat] :
            ( A
           != ( cons_nat @ E3 @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E3 @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_910_list_Oset__cases,axiom,
    ! [E3: produc859450856879609959at_nat,A: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ E3 @ ( set_Pr5518436109238095868at_nat @ A ) )
     => ( ! [Z22: list_P8469869581646625389at_nat] :
            ( A
           != ( cons_P8732206157123786781at_nat @ E3 @ Z22 ) )
       => ~ ! [Z1: produc859450856879609959at_nat,Z22: list_P8469869581646625389at_nat] :
              ( ( A
                = ( cons_P8732206157123786781at_nat @ Z1 @ Z22 ) )
             => ~ ( member8206827879206165904at_nat @ E3 @ ( set_Pr5518436109238095868at_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_911_list_Oset__cases,axiom,
    ! [E3: product_prod_nat_nat,A: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ E3 @ ( set_Pr5648618587558075414at_nat @ A ) )
     => ( ! [Z22: list_P6011104703257516679at_nat] :
            ( A
           != ( cons_P6512896166579812791at_nat @ E3 @ Z22 ) )
       => ~ ! [Z1: product_prod_nat_nat,Z22: list_P6011104703257516679at_nat] :
              ( ( A
                = ( cons_P6512896166579812791at_nat @ Z1 @ Z22 ) )
             => ~ ( member8440522571783428010at_nat @ E3 @ ( set_Pr5648618587558075414at_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_912_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_913_list_Oset__intros_I1_J,axiom,
    ! [X21: produc859450856879609959at_nat,X22: list_P8469869581646625389at_nat] : ( member8206827879206165904at_nat @ X21 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_914_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] : ( member8440522571783428010at_nat @ X21 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_915_list_Oset__intros_I2_J,axiom,
    ! [Y: nat,X22: list_nat,X21: nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
     => ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_916_list_Oset__intros_I2_J,axiom,
    ! [Y: produc859450856879609959at_nat,X22: list_P8469869581646625389at_nat,X21: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ X22 ) )
     => ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_917_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,X21: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ X22 ) )
     => ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_918_subset__code_I1_J,axiom,
    ! [Xs: list_P8469869581646625389at_nat,B3: set_Pr8693737435421807431at_nat] :
      ( ( ord_le3000389064537975527at_nat @ ( set_Pr5518436109238095868at_nat @ Xs ) @ B3 )
      = ( ! [X: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( member8206827879206165904at_nat @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_919_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B3 )
      = ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_920_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_921_finite__list,axiom,
    ! [A3: set_list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ? [Xs2: list_list_a] :
          ( ( set_list_a2 @ Xs2 )
          = A3 ) ) ).

% finite_list
thf(fact_922_finite__list,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [Xs2: list_nat] :
          ( ( set_nat2 @ Xs2 )
          = A3 ) ) ).

% finite_list
thf(fact_923_sat_Osimps_I5_J,axiom,
    ! [Phi: relational_fmla_a_b,Psi: relational_fmla_a_b,I: product_prod_b_nat > set_list_a,Sigma2: nat > a] :
      ( ( relational_sat_a_b @ ( relational_Conj_a_b @ Phi @ Psi ) @ I @ Sigma2 )
      = ( ( relational_sat_a_b @ Phi @ I @ Sigma2 )
        & ( relational_sat_a_b @ Psi @ I @ Sigma2 ) ) ) ).

% sat.simps(5)
thf(fact_924_list__ex1__iff,axiom,
    ( list_ex1_nat
    = ( ^ [P6: nat > $o,Xs3: list_nat] :
        ? [X: nat] :
          ( ( member_nat @ X @ ( set_nat2 @ Xs3 ) )
          & ( P6 @ X )
          & ! [Y3: nat] :
              ( ( ( member_nat @ Y3 @ ( set_nat2 @ Xs3 ) )
                & ( P6 @ Y3 ) )
             => ( Y3 = X ) ) ) ) ) ).

% list_ex1_iff
thf(fact_925_list__ex1__iff,axiom,
    ( list_e2327910815390407600at_nat
    = ( ^ [P6: produc859450856879609959at_nat > $o,Xs3: list_P8469869581646625389at_nat] :
        ? [X: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs3 ) )
          & ( P6 @ X )
          & ! [Y3: produc859450856879609959at_nat] :
              ( ( ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ Xs3 ) )
                & ( P6 @ Y3 ) )
             => ( Y3 = X ) ) ) ) ) ).

% list_ex1_iff
thf(fact_926_list__ex1__iff,axiom,
    ( list_e8644085759156585930at_nat
    = ( ^ [P6: product_prod_nat_nat > $o,Xs3: list_P6011104703257516679at_nat] :
        ? [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs3 ) )
          & ( P6 @ X )
          & ! [Y3: product_prod_nat_nat] :
              ( ( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs3 ) )
                & ( P6 @ Y3 ) )
             => ( Y3 = X ) ) ) ) ) ).

% list_ex1_iff
thf(fact_927_split__list__first__prop__iff,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o] :
      ( ( ? [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
            ( ? [Zs3: list_P6011104703257516679at_nat] :
                ( Xs
                = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X @ Zs3 ) ) )
            & ( P @ X )
            & ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Ys2 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_928_split__list__last__prop__iff,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o] :
      ( ( ? [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,X: product_prod_nat_nat,Zs3: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X @ Zs3 ) ) )
            & ( P @ X )
            & ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Zs3 ) )
               => ~ ( P @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_929_in__set__conv__decomp__first,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
            & ~ ( member_nat @ X3 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_930_in__set__conv__decomp__first,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
      = ( ? [Ys2: list_P8469869581646625389at_nat,Zs3: list_P8469869581646625389at_nat] :
            ( ( Xs
              = ( append8751754712269456642at_nat @ Ys2 @ ( cons_P8732206157123786781at_nat @ X3 @ Zs3 ) ) )
            & ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_931_in__set__conv__decomp__first,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X3 @ Zs3 ) ) )
            & ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_932_in__set__conv__decomp__last,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
            & ~ ( member_nat @ X3 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_933_in__set__conv__decomp__last,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
      = ( ? [Ys2: list_P8469869581646625389at_nat,Zs3: list_P8469869581646625389at_nat] :
            ( ( Xs
              = ( append8751754712269456642at_nat @ Ys2 @ ( cons_P8732206157123786781at_nat @ X3 @ Zs3 ) ) )
            & ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_934_in__set__conv__decomp__last,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X3 @ Zs3 ) ) )
            & ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_935_split__list__first__propE,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys: list_P6011104703257516679at_nat,X4: product_prod_nat_nat] :
            ( ? [Zs2: list_P6011104703257516679at_nat] :
                ( Xs
                = ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X4 @ Zs2 ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa @ ( set_Pr5648618587558075414at_nat @ Ys ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_936_split__list__last__propE,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys: list_P6011104703257516679at_nat,X4: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X4 @ Zs2 ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa @ ( set_Pr5648618587558075414at_nat @ Zs2 ) )
                   => ~ ( P @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_937_split__list__first__prop,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys: list_P6011104703257516679at_nat,X4: product_prod_nat_nat] :
          ( ? [Zs2: list_P6011104703257516679at_nat] :
              ( Xs
              = ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X4 @ Zs2 ) ) )
          & ( P @ X4 )
          & ! [Xa: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Xa @ ( set_Pr5648618587558075414at_nat @ Ys ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_938_split__list__last__prop,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys: list_P6011104703257516679at_nat,X4: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
          ( ( Xs
            = ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X4 @ Zs2 ) ) )
          & ( P @ X4 )
          & ! [Xa: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Xa @ ( set_Pr5648618587558075414at_nat @ Zs2 ) )
             => ~ ( P @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_939_in__set__conv__decomp,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( Xs
            = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_940_in__set__conv__decomp,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
      = ( ? [Ys2: list_P8469869581646625389at_nat,Zs3: list_P8469869581646625389at_nat] :
            ( Xs
            = ( append8751754712269456642at_nat @ Ys2 @ ( cons_P8732206157123786781at_nat @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_941_in__set__conv__decomp,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
            ( Xs
            = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_942_append__Cons__eq__iff,axiom,
    ! [X3: nat,Xs: list_nat,Ys3: list_nat,Xs5: list_nat,Ys7: list_nat] :
      ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ( ~ ( member_nat @ X3 @ ( set_nat2 @ Ys3 ) )
       => ( ( ( append_nat @ Xs @ ( cons_nat @ X3 @ Ys3 ) )
            = ( append_nat @ Xs5 @ ( cons_nat @ X3 @ Ys7 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys3 = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_943_append__Cons__eq__iff,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat,Ys3: list_P8469869581646625389at_nat,Xs5: list_P8469869581646625389at_nat,Ys7: list_P8469869581646625389at_nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Ys3 ) )
       => ( ( ( append8751754712269456642at_nat @ Xs @ ( cons_P8732206157123786781at_nat @ X3 @ Ys3 ) )
            = ( append8751754712269456642at_nat @ Xs5 @ ( cons_P8732206157123786781at_nat @ X3 @ Ys7 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys3 = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_944_append__Cons__eq__iff,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,Xs5: list_P6011104703257516679at_nat,Ys7: list_P6011104703257516679at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Ys3 ) )
       => ( ( ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X3 @ Ys3 ) )
            = ( append985823374593552924at_nat @ Xs5 @ ( cons_P6512896166579812791at_nat @ X3 @ Ys7 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys3 = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_945_split__list__propE,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys: list_P6011104703257516679at_nat,X4: product_prod_nat_nat] :
            ( ? [Zs2: list_P6011104703257516679at_nat] :
                ( Xs
                = ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X4 @ Zs2 ) ) )
           => ~ ( P @ X4 ) ) ) ).

% split_list_propE
thf(fact_946_split__list__first,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ? [Ys: list_nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs2 ) ) )
          & ~ ( member_nat @ X3 @ ( set_nat2 @ Ys ) ) ) ) ).

% split_list_first
thf(fact_947_split__list__first,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ? [Ys: list_P8469869581646625389at_nat,Zs2: list_P8469869581646625389at_nat] :
          ( ( Xs
            = ( append8751754712269456642at_nat @ Ys @ ( cons_P8732206157123786781at_nat @ X3 @ Zs2 ) ) )
          & ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Ys ) ) ) ) ).

% split_list_first
thf(fact_948_split__list__first,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ? [Ys: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
          ( ( Xs
            = ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X3 @ Zs2 ) ) )
          & ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Ys ) ) ) ) ).

% split_list_first
thf(fact_949_split__list__prop,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o] :
      ( ? [X6: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys: list_P6011104703257516679at_nat,X4: product_prod_nat_nat] :
          ( ? [Zs2: list_P6011104703257516679at_nat] :
              ( Xs
              = ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X4 @ Zs2 ) ) )
          & ( P @ X4 ) ) ) ).

% split_list_prop
thf(fact_950_split__list__last,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ? [Ys: list_nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs2 ) ) )
          & ~ ( member_nat @ X3 @ ( set_nat2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_951_split__list__last,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ? [Ys: list_P8469869581646625389at_nat,Zs2: list_P8469869581646625389at_nat] :
          ( ( Xs
            = ( append8751754712269456642at_nat @ Ys @ ( cons_P8732206157123786781at_nat @ X3 @ Zs2 ) ) )
          & ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_952_split__list__last,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ? [Ys: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
          ( ( Xs
            = ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X3 @ Zs2 ) ) )
          & ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_953_split__list,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ? [Ys: list_nat,Zs2: list_nat] :
          ( Xs
          = ( append_nat @ Ys @ ( cons_nat @ X3 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_954_split__list,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ? [Ys: list_P8469869581646625389at_nat,Zs2: list_P8469869581646625389at_nat] :
          ( Xs
          = ( append8751754712269456642at_nat @ Ys @ ( cons_P8732206157123786781at_nat @ X3 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_955_split__list,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ? [Ys: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
          ( Xs
          = ( append985823374593552924at_nat @ Ys @ ( cons_P6512896166579812791at_nat @ X3 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_956_xy__in__set__cases,axiom,
    ! [X3: nat,L: list_nat,Y: nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ L ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ L ) )
       => ( ( ( X3 = Y )
           => ! [L1: list_nat,L22: list_nat] :
                ( L
               != ( append_nat @ L1 @ ( cons_nat @ Y @ L22 ) ) ) )
         => ( ( ( X3 != Y )
             => ! [L1: list_nat,L22: list_nat,L32: list_nat] :
                  ( L
                 != ( append_nat @ L1 @ ( cons_nat @ X3 @ ( append_nat @ L22 @ ( cons_nat @ Y @ L32 ) ) ) ) ) )
           => ~ ( ( X3 != Y )
               => ! [L1: list_nat,L22: list_nat,L32: list_nat] :
                    ( L
                   != ( append_nat @ L1 @ ( cons_nat @ Y @ ( append_nat @ L22 @ ( cons_nat @ X3 @ L32 ) ) ) ) ) ) ) ) ) ) ).

% xy_in_set_cases
thf(fact_957_xy__in__set__cases,axiom,
    ! [X3: produc859450856879609959at_nat,L: list_P8469869581646625389at_nat,Y: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ L ) )
     => ( ( member8206827879206165904at_nat @ Y @ ( set_Pr5518436109238095868at_nat @ L ) )
       => ( ( ( X3 = Y )
           => ! [L1: list_P8469869581646625389at_nat,L22: list_P8469869581646625389at_nat] :
                ( L
               != ( append8751754712269456642at_nat @ L1 @ ( cons_P8732206157123786781at_nat @ Y @ L22 ) ) ) )
         => ( ( ( X3 != Y )
             => ! [L1: list_P8469869581646625389at_nat,L22: list_P8469869581646625389at_nat,L32: list_P8469869581646625389at_nat] :
                  ( L
                 != ( append8751754712269456642at_nat @ L1 @ ( cons_P8732206157123786781at_nat @ X3 @ ( append8751754712269456642at_nat @ L22 @ ( cons_P8732206157123786781at_nat @ Y @ L32 ) ) ) ) ) )
           => ~ ( ( X3 != Y )
               => ! [L1: list_P8469869581646625389at_nat,L22: list_P8469869581646625389at_nat,L32: list_P8469869581646625389at_nat] :
                    ( L
                   != ( append8751754712269456642at_nat @ L1 @ ( cons_P8732206157123786781at_nat @ Y @ ( append8751754712269456642at_nat @ L22 @ ( cons_P8732206157123786781at_nat @ X3 @ L32 ) ) ) ) ) ) ) ) ) ) ).

% xy_in_set_cases
thf(fact_958_xy__in__set__cases,axiom,
    ! [X3: product_prod_nat_nat,L: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ L ) )
     => ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ L ) )
       => ( ( ( X3 = Y )
           => ! [L1: list_P6011104703257516679at_nat,L22: list_P6011104703257516679at_nat] :
                ( L
               != ( append985823374593552924at_nat @ L1 @ ( cons_P6512896166579812791at_nat @ Y @ L22 ) ) ) )
         => ( ( ( X3 != Y )
             => ! [L1: list_P6011104703257516679at_nat,L22: list_P6011104703257516679at_nat,L32: list_P6011104703257516679at_nat] :
                  ( L
                 != ( append985823374593552924at_nat @ L1 @ ( cons_P6512896166579812791at_nat @ X3 @ ( append985823374593552924at_nat @ L22 @ ( cons_P6512896166579812791at_nat @ Y @ L32 ) ) ) ) ) )
           => ~ ( ( X3 != Y )
               => ! [L1: list_P6011104703257516679at_nat,L22: list_P6011104703257516679at_nat,L32: list_P6011104703257516679at_nat] :
                    ( L
                   != ( append985823374593552924at_nat @ L1 @ ( cons_P6512896166579812791at_nat @ Y @ ( append985823374593552924at_nat @ L22 @ ( cons_P6512896166579812791at_nat @ X3 @ L32 ) ) ) ) ) ) ) ) ) ) ).

% xy_in_set_cases
thf(fact_959_in__set__list__format,axiom,
    ! [E3: nat,L: list_nat] :
      ( ( member_nat @ E3 @ ( set_nat2 @ L ) )
     => ~ ! [L1: list_nat,L22: list_nat] :
            ( L
           != ( append_nat @ L1 @ ( cons_nat @ E3 @ L22 ) ) ) ) ).

% in_set_list_format
thf(fact_960_in__set__list__format,axiom,
    ! [E3: produc859450856879609959at_nat,L: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ E3 @ ( set_Pr5518436109238095868at_nat @ L ) )
     => ~ ! [L1: list_P8469869581646625389at_nat,L22: list_P8469869581646625389at_nat] :
            ( L
           != ( append8751754712269456642at_nat @ L1 @ ( cons_P8732206157123786781at_nat @ E3 @ L22 ) ) ) ) ).

% in_set_list_format
thf(fact_961_in__set__list__format,axiom,
    ! [E3: product_prod_nat_nat,L: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ E3 @ ( set_Pr5648618587558075414at_nat @ L ) )
     => ~ ! [L1: list_P6011104703257516679at_nat,L22: list_P6011104703257516679at_nat] :
            ( L
           != ( append985823374593552924at_nat @ L1 @ ( cons_P6512896166579812791at_nat @ E3 @ L22 ) ) ) ) ).

% in_set_list_format
thf(fact_962_eval__empty__alt,axiom,
    ! [Q: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
      ( ( ( relational_eval_a_b @ Q @ I )
        = bot_bot_set_list_a )
      = ( ! [Sigma: nat > a] :
            ~ ( relational_sat_a_b @ Q @ I @ Sigma ) ) ) ).

% eval_empty_alt
thf(fact_963_set__subset__Cons,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_964_set__subset__Cons,axiom,
    ! [Xs: list_nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_965_empty__set,axiom,
    ( bot_bot_set_nat
    = ( set_nat2 @ nil_nat ) ) ).

% empty_set
thf(fact_966_eval__cong,axiom,
    ! [Q: relational_fmla_a_b,Q3: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
      ( ( ( relational_fv_a_b @ Q )
        = ( relational_fv_a_b @ Q3 ) )
     => ( ! [Sigma3: nat > a] :
            ( ( relational_sat_a_b @ Q @ I @ Sigma3 )
            = ( relational_sat_a_b @ Q3 @ I @ Sigma3 ) )
       => ( ( relational_eval_a_b @ Q @ I )
          = ( relational_eval_a_b @ Q3 @ I ) ) ) ) ).

% eval_cong
thf(fact_967_finite__set__decode,axiom,
    ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).

% finite_set_decode
thf(fact_968_set__n__lists,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( set_list_a2 @ ( n_lists_a @ N @ Xs ) )
      = ( collect_list_a
        @ ^ [Ys2: list_a] :
            ( ( ( size_size_list_a @ Ys2 )
              = N )
            & ( ord_less_eq_set_a @ ( set_a2 @ Ys2 ) @ ( set_a2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_969_set__n__lists,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
      = ( collect_list_nat
        @ ^ [Ys2: list_nat] :
            ( ( ( size_size_list_nat @ Ys2 )
              = N )
            & ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_970_filter__is__subset,axiom,
    ! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( filter_nat @ P @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).

% filter_is_subset
thf(fact_971_set__nths__subset,axiom,
    ! [Xs: list_nat,I: set_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( nths_nat @ Xs @ I ) ) @ ( set_nat2 @ Xs ) ) ).

% set_nths_subset
thf(fact_972_List_Oinsert__def,axiom,
    ( insert_nat
    = ( ^ [X: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_973_List_Oinsert__def,axiom,
    ( insert4532235091570160003at_nat
    = ( ^ [X: produc859450856879609959at_nat,Xs3: list_P8469869581646625389at_nat] : ( if_lis7763640049307703347at_nat @ ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ Xs3 ) ) @ Xs3 @ ( cons_P8732206157123786781at_nat @ X @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_974_List_Oinsert__def,axiom,
    ( insert8944034826898310173at_nat
    = ( ^ [X: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat] : ( if_lis9186351972506106189at_nat @ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs3 ) ) @ Xs3 @ ( cons_P6512896166579812791at_nat @ X @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_975_Cons__in__subseqsD,axiom,
    ! [Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member3067507820990806192at_nat @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) @ ( set_li2962362208618583836at_nat @ ( subseq4535541509918465494at_nat @ Xs ) ) )
     => ( member3067507820990806192at_nat @ Ys3 @ ( set_li2962362208618583836at_nat @ ( subseq4535541509918465494at_nat @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_976_in__set__product__lists__length,axiom,
    ! [Xs: list_nat,Xss2: list_list_nat] :
      ( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_977_length__n__lists__elem,axiom,
    ! [Ys3: list_nat,N: nat,Xs: list_nat] :
      ( ( member_list_nat @ Ys3 @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
     => ( ( size_size_list_nat @ Ys3 )
        = N ) ) ).

% length_n_lists_elem
thf(fact_978_length__pos__if__in__set,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3679842834875189465at_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_979_length__pos__if__in__set,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_980_length__pos__if__in__set,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_981_card__length,axiom,
    ! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).

% card_length
thf(fact_982_filter__eq__Cons__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Ys3: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( filter6372491115368938494at_nat @ P @ Ys3 )
        = ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( ? [Us3: list_P6011104703257516679at_nat,Vs3: list_P6011104703257516679at_nat] :
            ( ( Ys3
              = ( append985823374593552924at_nat @ Us3 @ ( cons_P6512896166579812791at_nat @ X3 @ Vs3 ) ) )
            & ! [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Us3 ) )
               => ~ ( P @ X ) )
            & ( P @ X3 )
            & ( Xs
              = ( filter6372491115368938494at_nat @ P @ Vs3 ) ) ) ) ) ).

% filter_eq_Cons_iff
thf(fact_983_Cons__eq__filter__iff,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,Ys3: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ X3 @ Xs )
        = ( filter6372491115368938494at_nat @ P @ Ys3 ) )
      = ( ? [Us3: list_P6011104703257516679at_nat,Vs3: list_P6011104703257516679at_nat] :
            ( ( Ys3
              = ( append985823374593552924at_nat @ Us3 @ ( cons_P6512896166579812791at_nat @ X3 @ Vs3 ) ) )
            & ! [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Us3 ) )
               => ~ ( P @ X ) )
            & ( P @ X3 )
            & ( Xs
              = ( filter6372491115368938494at_nat @ P @ Vs3 ) ) ) ) ) ).

% Cons_eq_filter_iff
thf(fact_984_filter__eq__ConsD,axiom,
    ! [P: product_prod_nat_nat > $o,Ys3: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( filter6372491115368938494at_nat @ P @ Ys3 )
        = ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
     => ? [Us: list_P6011104703257516679at_nat,Vs: list_P6011104703257516679at_nat] :
          ( ( Ys3
            = ( append985823374593552924at_nat @ Us @ ( cons_P6512896166579812791at_nat @ X3 @ Vs ) ) )
          & ! [X6: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Us ) )
             => ~ ( P @ X6 ) )
          & ( P @ X3 )
          & ( Xs
            = ( filter6372491115368938494at_nat @ P @ Vs ) ) ) ) ).

% filter_eq_ConsD
thf(fact_985_Cons__eq__filterD,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,Ys3: list_P6011104703257516679at_nat] :
      ( ( ( cons_P6512896166579812791at_nat @ X3 @ Xs )
        = ( filter6372491115368938494at_nat @ P @ Ys3 ) )
     => ? [Us: list_P6011104703257516679at_nat,Vs: list_P6011104703257516679at_nat] :
          ( ( Ys3
            = ( append985823374593552924at_nat @ Us @ ( cons_P6512896166579812791at_nat @ X3 @ Vs ) ) )
          & ! [X6: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Us ) )
             => ~ ( P @ X6 ) )
          & ( P @ X3 )
          & ( Xs
            = ( filter6372491115368938494at_nat @ P @ Vs ) ) ) ) ).

% Cons_eq_filterD
thf(fact_986_length__filter__less,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ~ ( P @ X3 )
       => ( ord_less_nat @ ( size_s3679842834875189465at_nat @ ( filter1540621253233046500at_nat @ P @ Xs ) ) @ ( size_s3679842834875189465at_nat @ Xs ) ) ) ) ).

% length_filter_less
thf(fact_987_length__filter__less,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ~ ( P @ X3 )
       => ( ord_less_nat @ ( size_s5460976970255530739at_nat @ ( filter6372491115368938494at_nat @ P @ Xs ) ) @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ).

% length_filter_less
thf(fact_988_length__filter__less,axiom,
    ! [X3: nat,Xs: list_nat,P: nat > $o] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ( ~ ( P @ X3 )
       => ( ord_less_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ) ).

% length_filter_less
thf(fact_989_lexord__partial__trans,axiom,
    ! [Xs: list_P8469869581646625389at_nat,R4: set_Pr553994874890374343at_nat,Ys3: list_P8469869581646625389at_nat,Zs: list_P8469869581646625389at_nat] :
      ( ! [X4: produc859450856879609959at_nat,Y4: produc859450856879609959at_nat,Z: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
         => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ Y4 ) @ R4 )
           => ( ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ Y4 @ Z ) @ R4 )
             => ( member8062223511168850704at_nat @ ( produc4662710985925991255at_nat @ X4 @ Z ) @ R4 ) ) ) )
     => ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs @ Ys3 ) @ ( lexord5831005462426227802at_nat @ R4 ) )
       => ( ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Ys3 @ Zs ) @ ( lexord5831005462426227802at_nat @ R4 ) )
         => ( member4574794575480667280at_nat @ ( produc1338542795132623831at_nat @ Xs @ Zs ) @ ( lexord5831005462426227802at_nat @ R4 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_990_lexord__partial__trans,axiom,
    ! [Xs: list_nat,R4: set_Pr1261947904930325089at_nat,Ys3: list_nat,Zs: list_nat] :
      ( ! [X4: nat,Y4: nat,Z: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R4 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z ) @ R4 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Z ) @ R4 ) ) ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( lexord_nat @ R4 ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys3 @ Zs ) @ ( lexord_nat @ R4 ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( lexord_nat @ R4 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_991_lexord__partial__trans,axiom,
    ! [Xs: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat,Z: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Y4 ) @ R4 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y4 @ Z ) @ R4 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X4 @ Z ) @ R4 ) ) ) )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys3 ) @ ( lexord2841853652668343668at_nat @ R4 ) )
       => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys3 @ Zs ) @ ( lexord2841853652668343668at_nat @ R4 ) )
         => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Zs ) @ ( lexord2841853652668343668at_nat @ R4 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_992_subset__decode__imp__le,axiom,
    ! [M4: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M4 ) @ ( nat_set_decode @ N ) )
     => ( ord_less_eq_nat @ M4 @ N ) ) ).

% subset_decode_imp_le
thf(fact_993_filter__eq__snocD,axiom,
    ! [P: nat > $o,L: list_nat,L4: list_nat,X3: nat] :
      ( ( ( filter_nat @ P @ L )
        = ( append_nat @ L4 @ ( cons_nat @ X3 @ nil_nat ) ) )
     => ( ( member_nat @ X3 @ ( set_nat2 @ L ) )
        & ( P @ X3 ) ) ) ).

% filter_eq_snocD
thf(fact_994_filter__eq__snocD,axiom,
    ! [P: produc859450856879609959at_nat > $o,L: list_P8469869581646625389at_nat,L4: list_P8469869581646625389at_nat,X3: produc859450856879609959at_nat] :
      ( ( ( filter1540621253233046500at_nat @ P @ L )
        = ( append8751754712269456642at_nat @ L4 @ ( cons_P8732206157123786781at_nat @ X3 @ nil_Pr2582115297535392877at_nat ) ) )
     => ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ L ) )
        & ( P @ X3 ) ) ) ).

% filter_eq_snocD
thf(fact_995_filter__eq__snocD,axiom,
    ! [P: product_prod_nat_nat > $o,L: list_P6011104703257516679at_nat,L4: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
      ( ( ( filter6372491115368938494at_nat @ P @ L )
        = ( append985823374593552924at_nat @ L4 @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) )
     => ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ L ) )
        & ( P @ X3 ) ) ) ).

% filter_eq_snocD
thf(fact_996_lexord__same__pref__iff,axiom,
    ! [Xs: list_nat,Ys3: list_nat,Zs: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys3 ) @ ( append_nat @ Xs @ Zs ) ) @ ( lexord_nat @ R4 ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R4 ) )
        | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys3 @ Zs ) @ ( lexord_nat @ R4 ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_997_lexord__same__pref__iff,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( append985823374593552924at_nat @ Xs @ Ys3 ) @ ( append985823374593552924at_nat @ Xs @ Zs ) ) @ ( lexord2841853652668343668at_nat @ R4 ) )
      = ( ? [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ X ) @ R4 ) )
        | ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys3 @ Zs ) @ ( lexord2841853652668343668at_nat @ R4 ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_998_finite__lists__length__eq,axiom,
    ! [A3: set_a,N: nat] :
      ( ( finite_finite_a @ A3 )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Xs3: list_a] :
              ( ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A3 )
              & ( ( size_size_list_a @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_999_finite__lists__length__eq,axiom,
    ! [A3: set_list_a,N: nat] :
      ( ( finite_finite_list_a @ A3 )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [Xs3: list_list_a] :
              ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs3 ) @ A3 )
              & ( ( size_s349497388124573686list_a @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1000_finite__lists__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A3 )
              & ( ( size_size_list_nat @ Xs3 )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_1001_in__set__simps_I2_J,axiom,
    ! [X3: nat,Y: nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( X3 = Y ) ) ).

% in_set_simps(2)
thf(fact_1002_in__set__simps_I2_J,axiom,
    ! [X3: produc859450856879609959at_nat,Y: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ Y @ nil_Pr2582115297535392877at_nat ) ) )
      = ( X3 = Y ) ) ).

% in_set_simps(2)
thf(fact_1003_in__set__simps_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) )
      = ( X3 = Y ) ) ).

% in_set_simps(2)
thf(fact_1004_the__elem__set,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( the_el2281957884133575798at_nat @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) )
      = X3 ) ).

% the_elem_set
thf(fact_1005_can__select__def,axiom,
    ( can_select_nat
    = ( ^ [P6: nat > $o,A8: set_nat] :
        ? [X: nat] :
          ( ( member_nat @ X @ A8 )
          & ( P6 @ X )
          & ! [Y3: nat] :
              ( ( ( member_nat @ Y3 @ A8 )
                & ( P6 @ Y3 ) )
             => ( Y3 = X ) ) ) ) ) ).

% can_select_def
thf(fact_1006_can__select__def,axiom,
    ( can_se2810612172434690276at_nat
    = ( ^ [P6: produc859450856879609959at_nat > $o,A8: set_Pr8693737435421807431at_nat] :
        ? [X: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X @ A8 )
          & ( P6 @ X )
          & ! [Y3: produc859450856879609959at_nat] :
              ( ( ( member8206827879206165904at_nat @ Y3 @ A8 )
                & ( P6 @ Y3 ) )
             => ( Y3 = X ) ) ) ) ) ).

% can_select_def
thf(fact_1007_can__select__def,axiom,
    ( can_se4754832747099445502at_nat
    = ( ^ [P6: product_prod_nat_nat > $o,A8: set_Pr1261947904930325089at_nat] :
        ? [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ A8 )
          & ( P6 @ X )
          & ! [Y3: product_prod_nat_nat] :
              ( ( ( member8440522571783428010at_nat @ Y3 @ A8 )
                & ( P6 @ Y3 ) )
             => ( Y3 = X ) ) ) ) ) ).

% can_select_def
thf(fact_1008_in__set__simps_I1_J,axiom,
    ! [X3: nat,Y: nat,Z3: nat,Ys3: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( cons_nat @ Y @ ( cons_nat @ Z3 @ Ys3 ) ) ) )
      = ( ( X3 = Y )
        | ( member_nat @ X3 @ ( set_nat2 @ ( cons_nat @ Z3 @ Ys3 ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_1009_in__set__simps_I1_J,axiom,
    ! [X3: produc859450856879609959at_nat,Y: produc859450856879609959at_nat,Z3: produc859450856879609959at_nat,Ys3: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ Y @ ( cons_P8732206157123786781at_nat @ Z3 @ Ys3 ) ) ) )
      = ( ( X3 = Y )
        | ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( cons_P8732206157123786781at_nat @ Z3 @ Ys3 ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_1010_in__set__simps_I1_J,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,Z3: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ Y @ ( cons_P6512896166579812791at_nat @ Z3 @ Ys3 ) ) ) )
      = ( ( X3 = Y )
        | ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ Z3 @ Ys3 ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_1011_in__set__simps_I3_J,axiom,
    ! [X3: nat] :
      ~ ( member_nat @ X3 @ ( set_nat2 @ nil_nat ) ) ).

% in_set_simps(3)
thf(fact_1012_in__set__simps_I3_J,axiom,
    ! [X3: produc859450856879609959at_nat] :
      ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ nil_Pr2582115297535392877at_nat ) ) ).

% in_set_simps(3)
thf(fact_1013_in__set__simps_I3_J,axiom,
    ! [X3: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ nil_Pr5478986624290739719at_nat ) ) ).

% in_set_simps(3)
thf(fact_1014_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_a,N: nat] :
      ( ( finite_finite_a @ A3 )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Xs3: list_a] :
              ( ( ( size_size_list_a @ Xs3 )
                = N )
              & ( distinct_a @ Xs3 )
              & ( ord_less_eq_set_a @ ( set_a2 @ Xs3 ) @ A3 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_1015_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_list_a,N: nat] :
      ( ( finite_finite_list_a @ A3 )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [Xs3: list_list_a] :
              ( ( ( size_s349497388124573686list_a @ Xs3 )
                = N )
              & ( distinct_list_a @ Xs3 )
              & ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs3 ) @ A3 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_1016_finite__lists__distinct__length__eq,axiom,
    ! [A3: set_nat,N: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs3: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = N )
              & ( distinct_nat @ Xs3 )
              & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs3 ) @ A3 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_1017_distinct_Osimps_I2_J,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( cons_nat @ X3 @ Xs ) )
      = ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
        & ( distinct_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_1018_distinct_Osimps_I2_J,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( distin6906083803243959008at_nat @ ( cons_P8732206157123786781at_nat @ X3 @ Xs ) )
      = ( ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
        & ( distin6906083803243959008at_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_1019_distinct_Osimps_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( distin6923225563576452346at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
        & ( distin6923225563576452346at_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_1020_finite__distinct__list,axiom,
    ! [A3: set_list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ? [Xs2: list_list_a] :
          ( ( ( set_list_a2 @ Xs2 )
            = A3 )
          & ( distinct_list_a @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_1021_finite__distinct__list,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [Xs2: list_nat] :
          ( ( ( set_nat2 @ Xs2 )
            = A3 )
          & ( distinct_nat @ Xs2 ) ) ) ).

% finite_distinct_list
thf(fact_1022_finite__eval__Disj2D,axiom,
    ! [Q1: relational_fmla_a_b,Q2: relational_fmla_a_b,I: product_prod_b_nat > set_list_a] :
      ( ( finite_finite_list_a @ ( relational_eval_a_b @ ( relational_Disj_a_b @ Q1 @ Q2 ) @ I ) )
     => ( finite_finite_list_a @ ( relational_eval_a_b @ Q2 @ I ) ) ) ).

% finite_eval_Disj2D
thf(fact_1023_Conjs__reorder,axiom,
    ! [Xys: list_P6011104703257516679at_nat,Xys3: list_P6011104703257516679at_nat,Q: relational_fmla_a_b] :
      ( ( distin6923225563576452346at_nat @ Xys )
     => ( ( distin6923225563576452346at_nat @ Xys3 )
       => ( ( ( set_Pr5648618587558075414at_nat @ Xys )
            = ( set_Pr5648618587558075414at_nat @ Xys3 ) )
         => ( relational_equiv_a_b @ ( restrict_Conjs_a_b @ Q @ Xys ) @ ( restrict_Conjs_a_b @ Q @ Xys3 ) ) ) ) ) ).

% Conjs_reorder
thf(fact_1024_distinct__length__2__or__more,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( distin6923225563576452346at_nat @ ( cons_P6512896166579812791at_nat @ A @ ( cons_P6512896166579812791at_nat @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distin6923225563576452346at_nat @ ( cons_P6512896166579812791at_nat @ A @ Xs ) )
        & ( distin6923225563576452346at_nat @ ( cons_P6512896166579812791at_nat @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_1025_fmla_Odistinct_I27_J,axiom,
    ! [X31: nat,X32: relational_term_a,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(27)
thf(fact_1026_fmla_Odistinct_I37_J,axiom,
    ! [X51: relational_fmla_a_b,X52: relational_fmla_a_b,X61: relational_fmla_a_b,X62: relational_fmla_a_b] :
      ( ( relational_Conj_a_b @ X51 @ X52 )
     != ( relational_Disj_a_b @ X61 @ X62 ) ) ).

% fmla.distinct(37)
thf(fact_1027_distinct__match,axiom,
    ! [Al: list_P6011104703257516679at_nat,E3: product_prod_nat_nat,Bl: list_P6011104703257516679at_nat,Al2: list_P6011104703257516679at_nat,Bl2: list_P6011104703257516679at_nat] :
      ( ( distin6923225563576452346at_nat @ ( append985823374593552924at_nat @ Al @ ( cons_P6512896166579812791at_nat @ E3 @ Bl ) ) )
     => ( ( ( append985823374593552924at_nat @ Al @ ( cons_P6512896166579812791at_nat @ E3 @ Bl ) )
          = ( append985823374593552924at_nat @ Al2 @ ( cons_P6512896166579812791at_nat @ E3 @ Bl2 ) ) )
        = ( ( Al = Al2 )
          & ( Bl = Bl2 ) ) ) ) ).

% distinct_match
thf(fact_1028_distinct__singleton,axiom,
    ! [X3: product_prod_nat_nat] : ( distin6923225563576452346at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ).

% distinct_singleton
thf(fact_1029_distinct__finite__set,axiom,
    ! [X3: set_a] :
      ( finite_finite_list_a
      @ ( collect_list_a
        @ ^ [Ys2: list_a] :
            ( ( ( set_a2 @ Ys2 )
              = X3 )
            & ( distinct_a @ Ys2 ) ) ) ) ).

% distinct_finite_set
thf(fact_1030_not__distinct__decomp,axiom,
    ! [Ws2: list_P6011104703257516679at_nat] :
      ( ~ ( distin6923225563576452346at_nat @ Ws2 )
     => ? [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat,Y4: product_prod_nat_nat] :
          ( Ws2
          = ( append985823374593552924at_nat @ Xs2 @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) @ ( append985823374593552924at_nat @ Ys @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) @ Zs2 ) ) ) ) ) ) ).

% not_distinct_decomp
thf(fact_1031_not__distinct__conv__prefix,axiom,
    ! [As2: list_nat] :
      ( ( ~ ( distinct_nat @ As2 ) )
      = ( ? [Xs3: list_nat,Y3: nat,Ys2: list_nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs3 ) )
            & ( distinct_nat @ Xs3 )
            & ( As2
              = ( append_nat @ Xs3 @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1032_not__distinct__conv__prefix,axiom,
    ! [As2: list_P8469869581646625389at_nat] :
      ( ( ~ ( distin6906083803243959008at_nat @ As2 ) )
      = ( ? [Xs3: list_P8469869581646625389at_nat,Y3: produc859450856879609959at_nat,Ys2: list_P8469869581646625389at_nat] :
            ( ( member8206827879206165904at_nat @ Y3 @ ( set_Pr5518436109238095868at_nat @ Xs3 ) )
            & ( distin6906083803243959008at_nat @ Xs3 )
            & ( As2
              = ( append8751754712269456642at_nat @ Xs3 @ ( cons_P8732206157123786781at_nat @ Y3 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1033_not__distinct__conv__prefix,axiom,
    ! [As2: list_P6011104703257516679at_nat] :
      ( ( ~ ( distin6923225563576452346at_nat @ As2 ) )
      = ( ? [Xs3: list_P6011104703257516679at_nat,Y3: product_prod_nat_nat,Ys2: list_P6011104703257516679at_nat] :
            ( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs3 ) )
            & ( distin6923225563576452346at_nat @ Xs3 )
            & ( As2
              = ( append985823374593552924at_nat @ Xs3 @ ( cons_P6512896166579812791at_nat @ Y3 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1034_distinct__length__le,axiom,
    ! [Ys3: list_nat,Xs: list_nat] :
      ( ( distinct_nat @ Ys3 )
     => ( ( ( set_nat2 @ Ys3 )
          = ( set_nat2 @ Xs ) )
       => ( ord_less_eq_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs ) ) ) ) ).

% distinct_length_le
thf(fact_1035_distinct__card,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
        = ( size_size_list_nat @ Xs ) ) ) ).

% distinct_card
thf(fact_1036_card__distinct,axiom,
    ! [Xs: list_nat] :
      ( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
        = ( size_size_list_nat @ Xs ) )
     => ( distinct_nat @ Xs ) ) ).

% card_distinct
thf(fact_1037_filter__in__nths,axiom,
    ! [Xs: list_nat,S3: set_nat] :
      ( ( distinct_nat @ Xs )
     => ( ( filter_nat
          @ ^ [X: nat] : ( member_nat @ X @ ( set_nat2 @ ( nths_nat @ Xs @ S3 ) ) )
          @ Xs )
        = ( nths_nat @ Xs @ S3 ) ) ) ).

% filter_in_nths
thf(fact_1038_filter__in__nths,axiom,
    ! [Xs: list_P8469869581646625389at_nat,S3: set_nat] :
      ( ( distin6906083803243959008at_nat @ Xs )
     => ( ( filter1540621253233046500at_nat
          @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ ( set_Pr5518436109238095868at_nat @ ( nths_P7654847071924024683at_nat @ Xs @ S3 ) ) )
          @ Xs )
        = ( nths_P7654847071924024683at_nat @ Xs @ S3 ) ) ) ).

% filter_in_nths
thf(fact_1039_filter__in__nths,axiom,
    ! [Xs: list_P6011104703257516679at_nat,S3: set_nat] :
      ( ( distin6923225563576452346at_nat @ Xs )
     => ( ( filter6372491115368938494at_nat
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( nths_P6079298444859966469at_nat @ Xs @ S3 ) ) )
          @ Xs )
        = ( nths_P6079298444859966469at_nat @ Xs @ S3 ) ) ) ).

% filter_in_nths
thf(fact_1040_not__distinct__split__distinct,axiom,
    ! [Xs: list_nat] :
      ( ~ ( distinct_nat @ Xs )
     => ~ ! [Y4: nat,Ys: list_nat] :
            ( ( distinct_nat @ Ys )
           => ( ( member_nat @ Y4 @ ( set_nat2 @ Ys ) )
             => ! [Zs2: list_nat] :
                  ( Xs
                 != ( append_nat @ Ys @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ Zs2 ) ) ) ) ) ) ).

% not_distinct_split_distinct
thf(fact_1041_not__distinct__split__distinct,axiom,
    ! [Xs: list_P8469869581646625389at_nat] :
      ( ~ ( distin6906083803243959008at_nat @ Xs )
     => ~ ! [Y4: produc859450856879609959at_nat,Ys: list_P8469869581646625389at_nat] :
            ( ( distin6906083803243959008at_nat @ Ys )
           => ( ( member8206827879206165904at_nat @ Y4 @ ( set_Pr5518436109238095868at_nat @ Ys ) )
             => ! [Zs2: list_P8469869581646625389at_nat] :
                  ( Xs
                 != ( append8751754712269456642at_nat @ Ys @ ( append8751754712269456642at_nat @ ( cons_P8732206157123786781at_nat @ Y4 @ nil_Pr2582115297535392877at_nat ) @ Zs2 ) ) ) ) ) ) ).

% not_distinct_split_distinct
thf(fact_1042_not__distinct__split__distinct,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ~ ( distin6923225563576452346at_nat @ Xs )
     => ~ ! [Y4: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
            ( ( distin6923225563576452346at_nat @ Ys )
           => ( ( member8440522571783428010at_nat @ Y4 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
             => ! [Zs2: list_P6011104703257516679at_nat] :
                  ( Xs
                 != ( append985823374593552924at_nat @ Ys @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ Y4 @ nil_Pr5478986624290739719at_nat ) @ Zs2 ) ) ) ) ) ) ).

% not_distinct_split_distinct
thf(fact_1043_distinct__finite__subset,axiom,
    ! [X3: set_a] :
      ( ( finite_finite_a @ X3 )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [Ys2: list_a] :
              ( ( ord_less_eq_set_a @ ( set_a2 @ Ys2 ) @ X3 )
              & ( distinct_a @ Ys2 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_1044_distinct__finite__subset,axiom,
    ! [X3: set_list_a] :
      ( ( finite_finite_list_a @ X3 )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [Ys2: list_list_a] :
              ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ys2 ) @ X3 )
              & ( distinct_list_a @ Ys2 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_1045_distinct__finite__subset,axiom,
    ! [X3: set_nat] :
      ( ( finite_finite_nat @ X3 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Ys2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ys2 ) @ X3 )
              & ( distinct_nat @ Ys2 ) ) ) ) ) ).

% distinct_finite_subset
thf(fact_1046_card__set__1__iff__replicate,axiom,
    ! [Xs: list_nat] :
      ( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
        = ( suc @ zero_zero_nat ) )
      = ( ( Xs != nil_nat )
        & ? [X: nat] :
            ( Xs
            = ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X ) ) ) ) ).

% card_set_1_iff_replicate
thf(fact_1047_length__replicate,axiom,
    ! [N: nat,X3: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N @ X3 ) )
      = N ) ).

% length_replicate
thf(fact_1048_in__set__replicate,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
      = ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1049_in__set__replicate,axiom,
    ! [X3: produc859450856879609959at_nat,N: nat,Y: produc859450856879609959at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( replic6713244433751818279at_nat @ N @ Y ) ) )
      = ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1050_in__set__replicate,axiom,
    ! [X3: product_prod_nat_nat,N: nat,Y: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ Y ) ) )
      = ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_1051_replicate__app__Cons__same,axiom,
    ! [N: nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( append985823374593552924at_nat @ ( replic4235873036481779905at_nat @ N @ X3 ) @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ X3 @ ( append985823374593552924at_nat @ ( replic4235873036481779905at_nat @ N @ X3 ) @ Xs ) ) ) ).

% replicate_app_Cons_same
thf(fact_1052_replicate__Suc,axiom,
    ! [N: nat,X3: product_prod_nat_nat] :
      ( ( replic4235873036481779905at_nat @ ( suc @ N ) @ X3 )
      = ( cons_P6512896166579812791at_nat @ X3 @ ( replic4235873036481779905at_nat @ N @ X3 ) ) ) ).

% replicate_Suc
thf(fact_1053_replicate__length__filter,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( replicate_nat
        @ ( size_size_list_nat
          @ ( filter_nat
            @ ( ^ [Y9: nat,Z6: nat] : ( Y9 = Z6 )
              @ X3 )
            @ Xs ) )
        @ X3 )
      = ( filter_nat
        @ ( ^ [Y9: nat,Z6: nat] : ( Y9 = Z6 )
          @ X3 )
        @ Xs ) ) ).

% replicate_length_filter
thf(fact_1054_replicate__eqI,axiom,
    ! [Xs: list_P8469869581646625389at_nat,N: nat,X3: produc859450856879609959at_nat] :
      ( ( ( size_s3679842834875189465at_nat @ Xs )
        = N )
     => ( ! [Y4: produc859450856879609959at_nat] :
            ( ( member8206827879206165904at_nat @ Y4 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( Y4 = X3 ) )
       => ( Xs
          = ( replic6713244433751818279at_nat @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_1055_replicate__eqI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N: nat,X3: product_prod_nat_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = N )
     => ( ! [Y4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ Y4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( Y4 = X3 ) )
       => ( Xs
          = ( replic4235873036481779905at_nat @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_1056_replicate__eqI,axiom,
    ! [Xs: list_nat,N: nat,X3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = N )
     => ( ! [Y4: nat] :
            ( ( member_nat @ Y4 @ ( set_nat2 @ Xs ) )
           => ( Y4 = X3 ) )
       => ( Xs
          = ( replicate_nat @ N @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_1057_replicate__length__same,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( X4 = X3 ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X3 )
        = Xs ) ) ).

% replicate_length_same
thf(fact_1058_replicate__append__same,axiom,
    ! [I3: nat,X3: product_prod_nat_nat] :
      ( ( append985823374593552924at_nat @ ( replic4235873036481779905at_nat @ I3 @ X3 ) @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) )
      = ( cons_P6512896166579812791at_nat @ X3 @ ( replic4235873036481779905at_nat @ I3 @ X3 ) ) ) ).

% replicate_append_same
thf(fact_1059_replicate__Suc__conv__snoc,axiom,
    ! [N: nat,X3: product_prod_nat_nat] :
      ( ( replic4235873036481779905at_nat @ ( suc @ N ) @ X3 )
      = ( append985823374593552924at_nat @ ( replic4235873036481779905at_nat @ N @ X3 ) @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ) ).

% replicate_Suc_conv_snoc
thf(fact_1060_ex__Conjs__disjoint__eq__Conjs,axiom,
    ! [Xys: list_P6011104703257516679at_nat,Q: relational_fmla_a_b] :
      ( ( distin6923225563576452346at_nat @ Xys )
     => ? [Xys4: list_P6011104703257516679at_nat] :
          ( ( distin6923225563576452346at_nat @ Xys4 )
          & ( ( set_Pr5648618587558075414at_nat @ Xys )
            = ( set_Pr5648618587558075414at_nat @ Xys4 ) )
          & ( ( restri3786999866973890729nt_a_b @ Q @ Xys )
            = ( restrict_Conjs_a_b @ Q @ Xys4 ) ) ) ) ).

% ex_Conjs_disjoint_eq_Conjs
thf(fact_1061_Conjs__disjoint__equiv__Conjs,axiom,
    ! [Xys: list_P6011104703257516679at_nat,Q: relational_fmla_a_b] :
      ( ( distin6923225563576452346at_nat @ Xys )
     => ( relational_equiv_a_b @ ( restri3786999866973890729nt_a_b @ Q @ Xys ) @ ( restrict_Conjs_a_b @ Q @ Xys ) ) ) ).

% Conjs_disjoint_equiv_Conjs
thf(fact_1062_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A3: set_nat,L: list_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( sorted_wrt_nat @ ord_less_nat @ L )
          & ( ( set_nat2 @ L )
            = A3 )
          & ( ( size_size_list_nat @ L )
            = ( finite_card_nat @ A3 ) ) )
        = ( ( linord2614967742042102400et_nat @ A3 )
          = L ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_1063_sorted__wrt__mergesort__by__rel__merge,axiom,
    ! [R: nat > nat > $o,Xs: list_nat,Ys3: list_nat] :
      ( ! [X4: nat,Y4: nat] :
          ( ( R @ X4 @ Y4 )
          | ( R @ Y4 @ X4 ) )
     => ( ! [X4: nat,Y4: nat,Z: nat] :
            ( ( R @ X4 @ Y4 )
           => ( ( R @ Y4 @ Z )
             => ( R @ X4 @ Z ) ) )
       => ( ( sorted_wrt_nat @ R @ ( merges2309793660682651013ge_nat @ R @ Xs @ Ys3 ) )
          = ( ( sorted_wrt_nat @ R @ Xs )
            & ( sorted_wrt_nat @ R @ Ys3 ) ) ) ) ) ).

% sorted_wrt_mergesort_by_rel_merge
thf(fact_1064_sorted1,axiom,
    ! [X3: product_prod_nat_nat] : ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ).

% sorted1
thf(fact_1065_sorted1,axiom,
    ! [X3: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X3 @ nil_nat ) ) ).

% sorted1
thf(fact_1066_sorted__simps_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Ys3 ) )
      = ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Ys3 ) )
           => ( ord_le8460144461188290721at_nat @ X3 @ X ) )
        & ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Ys3 ) ) ) ).

% sorted_simps(2)
thf(fact_1067_sorted__simps_I2_J,axiom,
    ! [X3: nat,Ys3: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X3 @ Ys3 ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys3 ) )
           => ( ord_less_eq_nat @ X3 @ X ) )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ Ys3 ) ) ) ).

% sorted_simps(2)
thf(fact_1068_strict__sorted__simps_I2_J,axiom,
    ! [X3: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le1203424502768444845at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Ys3 ) )
      = ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Ys3 ) )
           => ( ord_le1203424502768444845at_nat @ X3 @ X ) )
        & ( sorted5214655850825725294at_nat @ ord_le1203424502768444845at_nat @ Ys3 ) ) ) ).

% strict_sorted_simps(2)
thf(fact_1069_strict__sorted__simps_I2_J,axiom,
    ! [X3: nat,Ys3: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ ( cons_nat @ X3 @ Ys3 ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys3 ) )
           => ( ord_less_nat @ X3 @ X ) )
        & ( sorted_wrt_nat @ ord_less_nat @ Ys3 ) ) ) ).

% strict_sorted_simps(2)
thf(fact_1070_strict__sorted__iff,axiom,
    ! [L: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ L )
      = ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
        & ( distinct_nat @ L ) ) ) ).

% strict_sorted_iff
thf(fact_1071_sorted__append,axiom,
    ! [Xs: list_nat,Ys3: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ Ys3 ) )
      = ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ Ys3 )
        & ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( set_nat2 @ Ys3 ) )
               => ( ord_less_eq_nat @ X @ Y3 ) ) ) ) ) ).

% sorted_append
thf(fact_1072_set__eq__sorted__correct,axiom,
    ! [L12: list_nat,L23: list_nat] :
      ( ( ( distinct_nat @ L12 )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ L12 ) )
     => ( ( ( distinct_nat @ L23 )
          & ( sorted_wrt_nat @ ord_less_eq_nat @ L23 ) )
       => ( ( L12 = L23 )
          = ( ( set_nat2 @ L12 )
            = ( set_nat2 @ L23 ) ) ) ) ) ).

% set_eq_sorted_correct
thf(fact_1073_sorted__distinct__set__unique,axiom,
    ! [Xs: list_nat,Ys3: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( distinct_nat @ Xs )
       => ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys3 )
         => ( ( distinct_nat @ Ys3 )
           => ( ( ( set_nat2 @ Xs )
                = ( set_nat2 @ Ys3 ) )
             => ( Xs = Ys3 ) ) ) ) ) ) ).

% sorted_distinct_set_unique
thf(fact_1074_sorted__quicksort,axiom,
    ! [Xs: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord3817693808505947216rt_nat @ Xs ) ) ).

% sorted_quicksort
thf(fact_1075_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A3: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A3 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_1076_sorted__nths,axiom,
    ! [Xs: list_nat,I: set_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( nths_nat @ Xs @ I ) ) ) ).

% sorted_nths
thf(fact_1077_sorted__wrt__quicksort__by__rel,axiom,
    ! [R: nat > nat > $o,Xs: list_nat] :
      ( ! [X4: nat,Y4: nat] :
          ( ( R @ X4 @ Y4 )
          | ( R @ Y4 @ X4 ) )
     => ( ! [X4: nat,Y4: nat,Z: nat] :
            ( ( R @ X4 @ Y4 )
           => ( ( R @ Y4 @ Z )
             => ( R @ X4 @ Z ) ) )
       => ( sorted_wrt_nat @ R @ ( quicksort_by_rel_nat @ R @ nil_nat @ Xs ) ) ) ) ).

% sorted_wrt_quicksort_by_rel
thf(fact_1078_sorted2,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,Zs: list_P6011104703257516679at_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Zs ) ) )
      = ( ( ord_le8460144461188290721at_nat @ X3 @ Y )
        & ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ ( cons_P6512896166579812791at_nat @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_1079_sorted2,axiom,
    ! [X3: nat,Y: nat,Zs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X3 @ ( cons_nat @ Y @ Zs ) ) )
      = ( ( ord_less_eq_nat @ X3 @ Y )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs ) ) ) ) ).

% sorted2
thf(fact_1080_sorted0,axiom,
    sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).

% sorted0
thf(fact_1081_sorted__wrt1,axiom,
    ! [P: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat] : ( sorted5214655850825725294at_nat @ P @ ( cons_P6512896166579812791at_nat @ X3 @ nil_Pr5478986624290739719at_nat ) ) ).

% sorted_wrt1
thf(fact_1082_sorted__wrt1,axiom,
    ! [P: nat > nat > $o,X3: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X3 @ nil_nat ) ) ).

% sorted_wrt1
thf(fact_1083_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).

% sorted_wrt.simps(1)
thf(fact_1084_fmla_Odistinct_I31_J,axiom,
    ! [X42: relational_fmla_a_b,X51: relational_fmla_a_b,X52: relational_fmla_a_b] :
      ( ( relational_Neg_a_b @ X42 )
     != ( relational_Conj_a_b @ X51 @ X52 ) ) ).

% fmla.distinct(31)
thf(fact_1085_fmla_Odistinct_I23_J,axiom,
    ! [X31: nat,X32: relational_term_a,X42: relational_fmla_a_b] :
      ( ( relational_Eq_a_b @ X31 @ X32 )
     != ( relational_Neg_a_b @ X42 ) ) ).

% fmla.distinct(23)
thf(fact_1086_sorted__wrt__filter,axiom,
    ! [F2: nat > nat > $o,Xs: list_nat,P: nat > $o] :
      ( ( sorted_wrt_nat @ F2 @ Xs )
     => ( sorted_wrt_nat @ F2 @ ( filter_nat @ P @ Xs ) ) ) ).

% sorted_wrt_filter
thf(fact_1087_sorted__wrt__true,axiom,
    ! [Xs: list_nat] :
      ( sorted_wrt_nat
      @ ^ [Uu2: nat,Uv2: nat] : $true
      @ Xs ) ).

% sorted_wrt_true
thf(fact_1088_sorted__filter_H,axiom,
    ! [L: list_nat,P: nat > $o] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( filter_nat @ P @ L ) ) ) ).

% sorted_filter'
thf(fact_1089_sorted__same,axiom,
    ! [G2: list_nat > nat,Xs: list_nat] :
      ( sorted_wrt_nat @ ord_less_eq_nat
      @ ( filter_nat
        @ ^ [X: nat] :
            ( X
            = ( G2 @ Xs ) )
        @ Xs ) ) ).

% sorted_same
thf(fact_1090_strict__sorted__simps_I1_J,axiom,
    sorted_wrt_nat @ ord_less_nat @ nil_nat ).

% strict_sorted_simps(1)
thf(fact_1091_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_1092_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A3: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A3 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_1093_strict__sorted__equal,axiom,
    ! [Xs: list_nat,Ys3: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Xs )
     => ( ( sorted_wrt_nat @ ord_less_nat @ Ys3 )
       => ( ( ( set_nat2 @ Ys3 )
            = ( set_nat2 @ Xs ) )
         => ( Ys3 = Xs ) ) ) ) ).

% strict_sorted_equal
thf(fact_1094_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > produc859450856879609959at_nat > $o] :
      ( ! [X4: produc859450856879609959at_nat,Y4: produc859450856879609959at_nat] :
          ( ( member8206827879206165904at_nat @ X4 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
         => ( ( member8206827879206165904at_nat @ Y4 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
           => ( ( P @ X4 @ Y4 )
             => ( Q @ X4 @ Y4 ) ) ) )
     => ( ( sorted8717129596388558164at_nat @ P @ Xs )
       => ( sorted8717129596388558164at_nat @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_1095_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > product_prod_nat_nat > $o,Q: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X4: product_prod_nat_nat,Y4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( ( member8440522571783428010at_nat @ Y4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( ( P @ X4 @ Y4 )
             => ( Q @ X4 @ Y4 ) ) ) )
     => ( ( sorted5214655850825725294at_nat @ P @ Xs )
       => ( sorted5214655850825725294at_nat @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_1096_sorted__wrt__mono__rel,axiom,
    ! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
      ( ! [X4: nat,Y4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( member_nat @ Y4 @ ( set_nat2 @ Xs ) )
           => ( ( P @ X4 @ Y4 )
             => ( Q @ X4 @ Y4 ) ) ) )
     => ( ( sorted_wrt_nat @ P @ Xs )
       => ( sorted_wrt_nat @ Q @ Xs ) ) ) ).

% sorted_wrt_mono_rel
thf(fact_1097_sorted__wrt__append,axiom,
    ! [P: nat > nat > $o,Xs: list_nat,Ys3: list_nat] :
      ( ( sorted_wrt_nat @ P @ ( append_nat @ Xs @ Ys3 ) )
      = ( ( sorted_wrt_nat @ P @ Xs )
        & ( sorted_wrt_nat @ P @ Ys3 )
        & ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ ( set_nat2 @ Ys3 ) )
               => ( P @ X @ Y3 ) ) ) ) ) ).

% sorted_wrt_append
thf(fact_1098_sorted__replicate,axiom,
    ! [N: nat,X3: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( replicate_nat @ N @ X3 ) ) ).

% sorted_replicate
thf(fact_1099_sorted__quicksort__by__rel,axiom,
    ! [Xs: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( quicksort_by_rel_nat @ ord_less_eq_nat @ nil_nat @ Xs ) ) ).

% sorted_quicksort_by_rel
thf(fact_1100_finite__sorted__distinct__unique,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [X4: list_nat] :
          ( ( ( set_nat2 @ X4 )
            = A3 )
          & ( sorted_wrt_nat @ ord_less_eq_nat @ X4 )
          & ( distinct_nat @ X4 )
          & ! [Y5: list_nat] :
              ( ( ( ( set_nat2 @ Y5 )
                  = A3 )
                & ( sorted_wrt_nat @ ord_less_eq_nat @ Y5 )
                & ( distinct_nat @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ).

% finite_sorted_distinct_unique
thf(fact_1101_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( distinct_nat @ Xs )
       => ( ( linord2614967742042102400et_nat @ ( set_nat2 @ Xs ) )
          = Xs ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_1102_sorted__append__bigger,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Xs )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( ord_le8460144461188290721at_nat @ X4 @ Y ) )
       => ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% sorted_append_bigger
thf(fact_1103_sorted__append__bigger,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( ord_less_eq_nat @ X4 @ Y ) )
       => ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ ( cons_nat @ Y @ nil_nat ) ) ) ) ) ).

% sorted_append_bigger
thf(fact_1104_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ~ ! [L2: list_nat] :
            ( ( sorted_wrt_nat @ ord_less_nat @ L2 )
           => ( ( ( set_nat2 @ L2 )
                = A3 )
             => ( ( size_size_list_nat @ L2 )
               != ( finite_card_nat @ A3 ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
thf(fact_1105_insertion__sort__correct,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Xs )
     => ( ( distin6923225563576452346at_nat @ Xs )
       => ( ( distin6923225563576452346at_nat @ ( sorted252080234835893926at_nat @ X3 @ Xs ) )
          & ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ ( sorted252080234835893926at_nat @ X3 @ Xs ) )
          & ( ( set_Pr5648618587558075414at_nat @ ( sorted252080234835893926at_nat @ X3 @ Xs ) )
            = ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) ) ) ) ) ).

% insertion_sort_correct
thf(fact_1106_insertion__sort__correct,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( distinct_nat @ Xs )
       => ( ( distinct_nat @ ( sorted4128666710326294635rt_nat @ X3 @ Xs ) )
          & ( sorted_wrt_nat @ ord_less_eq_nat @ ( sorted4128666710326294635rt_nat @ X3 @ Xs ) )
          & ( ( set_nat2 @ ( sorted4128666710326294635rt_nat @ X3 @ Xs ) )
            = ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) ) ) ) ) ).

% insertion_sort_correct
thf(fact_1107_distinct__sortet__list__app,axiom,
    ! [Xs: list_P6011104703257516679at_nat,As2: list_P6011104703257516679at_nat,B: product_prod_nat_nat,Cs: list_P6011104703257516679at_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Xs )
     => ( ( distin6923225563576452346at_nat @ Xs )
       => ( ( Xs
            = ( append985823374593552924at_nat @ As2 @ ( cons_P6512896166579812791at_nat @ B @ Cs ) ) )
         => ! [X6: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X6 @ ( set_Pr5648618587558075414at_nat @ Cs ) )
             => ( ord_le1203424502768444845at_nat @ B @ X6 ) ) ) ) ) ).

% distinct_sortet_list_app
thf(fact_1108_distinct__sortet__list__app,axiom,
    ! [Xs: list_nat,As2: list_nat,B: nat,Cs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( distinct_nat @ Xs )
       => ( ( Xs
            = ( append_nat @ As2 @ ( cons_nat @ B @ Cs ) ) )
         => ! [X6: nat] :
              ( ( member_nat @ X6 @ ( set_nat2 @ Cs ) )
             => ( ord_less_nat @ B @ X6 ) ) ) ) ) ).

% distinct_sortet_list_app
thf(fact_1109_distinct__sorted__list__lem2_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,E3: product_prod_nat_nat,E2: product_prod_nat_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Xs )
     => ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Ys3 )
       => ( ( distin6923225563576452346at_nat @ Xs )
         => ( ( distin6923225563576452346at_nat @ Ys3 )
           => ( ( ord_le1203424502768444845at_nat @ E3 @ E2 )
             => ( ! [X4: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
                   => ( ord_le1203424502768444845at_nat @ X4 @ E3 ) )
               => ( ! [X4: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys3 ) )
                     => ( ord_le1203424502768444845at_nat @ E2 @ X4 ) )
                 => ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ E3 @ ( cons_P6512896166579812791at_nat @ E2 @ Ys3 ) ) ) ) ) ) ) ) ) ) ) ).

% distinct_sorted_list_lem2(1)
thf(fact_1110_distinct__sorted__list__lem2_I1_J,axiom,
    ! [Xs: list_nat,Ys3: list_nat,E3: nat,E2: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys3 )
       => ( ( distinct_nat @ Xs )
         => ( ( distinct_nat @ Ys3 )
           => ( ( ord_less_nat @ E3 @ E2 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
                   => ( ord_less_nat @ X4 @ E3 ) )
               => ( ! [X4: nat] :
                      ( ( member_nat @ X4 @ ( set_nat2 @ Ys3 ) )
                     => ( ord_less_nat @ E2 @ X4 ) )
                 => ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ ( cons_nat @ E3 @ ( cons_nat @ E2 @ Ys3 ) ) ) ) ) ) ) ) ) ) ) ).

% distinct_sorted_list_lem2(1)
thf(fact_1111_distinct__sorted__list__lem1_I2_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,E3: product_prod_nat_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Xs )
     => ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Ys3 )
       => ( ( distin6923225563576452346at_nat @ Xs )
         => ( ( distin6923225563576452346at_nat @ Ys3 )
           => ( ! [X4: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
                 => ( ord_le1203424502768444845at_nat @ X4 @ E3 ) )
             => ( ! [X4: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys3 ) )
                   => ( ord_le1203424502768444845at_nat @ E3 @ X4 ) )
               => ( distin6923225563576452346at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ E3 @ Ys3 ) ) ) ) ) ) ) ) ) ).

% distinct_sorted_list_lem1(2)
thf(fact_1112_distinct__sorted__list__lem1_I2_J,axiom,
    ! [Xs: list_nat,Ys3: list_nat,E3: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys3 )
       => ( ( distinct_nat @ Xs )
         => ( ( distinct_nat @ Ys3 )
           => ( ! [X4: nat] :
                  ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
                 => ( ord_less_nat @ X4 @ E3 ) )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ ( set_nat2 @ Ys3 ) )
                   => ( ord_less_nat @ E3 @ X4 ) )
               => ( distinct_nat @ ( append_nat @ Xs @ ( cons_nat @ E3 @ Ys3 ) ) ) ) ) ) ) ) ) ).

% distinct_sorted_list_lem1(2)
thf(fact_1113_distinct__sorted__list__lem1_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,E3: product_prod_nat_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Xs )
     => ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Ys3 )
       => ( ( distin6923225563576452346at_nat @ Xs )
         => ( ( distin6923225563576452346at_nat @ Ys3 )
           => ( ! [X4: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
                 => ( ord_le1203424502768444845at_nat @ X4 @ E3 ) )
             => ( ! [X4: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys3 ) )
                   => ( ord_le1203424502768444845at_nat @ E3 @ X4 ) )
               => ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ E3 @ Ys3 ) ) ) ) ) ) ) ) ) ).

% distinct_sorted_list_lem1(1)
thf(fact_1114_distinct__sorted__list__lem1_I1_J,axiom,
    ! [Xs: list_nat,Ys3: list_nat,E3: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys3 )
       => ( ( distinct_nat @ Xs )
         => ( ( distinct_nat @ Ys3 )
           => ( ! [X4: nat] :
                  ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
                 => ( ord_less_nat @ X4 @ E3 ) )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ ( set_nat2 @ Ys3 ) )
                   => ( ord_less_nat @ E3 @ X4 ) )
               => ( sorted_wrt_nat @ ord_less_eq_nat @ ( append_nat @ Xs @ ( cons_nat @ E3 @ Ys3 ) ) ) ) ) ) ) ) ) ).

% distinct_sorted_list_lem1(1)
thf(fact_1115_distinct__sorted__list__lem2_I2_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,E3: product_prod_nat_nat,E2: product_prod_nat_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Xs )
     => ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Ys3 )
       => ( ( distin6923225563576452346at_nat @ Xs )
         => ( ( distin6923225563576452346at_nat @ Ys3 )
           => ( ( ord_le1203424502768444845at_nat @ E3 @ E2 )
             => ( ! [X4: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
                   => ( ord_le1203424502768444845at_nat @ X4 @ E3 ) )
               => ( ! [X4: product_prod_nat_nat] :
                      ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys3 ) )
                     => ( ord_le1203424502768444845at_nat @ E2 @ X4 ) )
                 => ( distin6923225563576452346at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ E3 @ ( cons_P6512896166579812791at_nat @ E2 @ Ys3 ) ) ) ) ) ) ) ) ) ) ) ).

% distinct_sorted_list_lem2(2)
thf(fact_1116_distinct__sorted__list__lem2_I2_J,axiom,
    ! [Xs: list_nat,Ys3: list_nat,E3: nat,E2: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys3 )
       => ( ( distinct_nat @ Xs )
         => ( ( distinct_nat @ Ys3 )
           => ( ( ord_less_nat @ E3 @ E2 )
             => ( ! [X4: nat] :
                    ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
                   => ( ord_less_nat @ X4 @ E3 ) )
               => ( ! [X4: nat] :
                      ( ( member_nat @ X4 @ ( set_nat2 @ Ys3 ) )
                     => ( ord_less_nat @ E2 @ X4 ) )
                 => ( distinct_nat @ ( append_nat @ Xs @ ( cons_nat @ E3 @ ( cons_nat @ E2 @ Ys3 ) ) ) ) ) ) ) ) ) ) ) ).

% distinct_sorted_list_lem2(2)
thf(fact_1117_mergesort__remdups__correct,axiom,
    ! [L: list_nat] :
      ( ( distinct_nat @ ( merges1656613366846331073ps_nat @ L ) )
      & ( sorted_wrt_nat @ ord_less_eq_nat @ ( merges1656613366846331073ps_nat @ L ) )
      & ( ( set_nat2 @ ( merges1656613366846331073ps_nat @ L ) )
        = ( set_nat2 @ L ) ) ) ).

% mergesort_remdups_correct
thf(fact_1118_subset__sorted__correct,axiom,
    ! [L12: list_nat,L23: list_nat] :
      ( ( ( distinct_nat @ L12 )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ L12 ) )
     => ( ( ( distinct_nat @ L23 )
          & ( sorted_wrt_nat @ ord_less_eq_nat @ L23 ) )
       => ( ( sorted5575391298698439683ed_nat @ L12 @ L23 )
          = ( ord_less_eq_set_nat @ ( set_nat2 @ L12 ) @ ( set_nat2 @ L23 ) ) ) ) ) ).

% subset_sorted_correct
thf(fact_1119_subset__sorted_Osimps_I3_J,axiom,
    ! [X1: product_prod_nat_nat,L12: list_P6011104703257516679at_nat,X2: product_prod_nat_nat,L23: list_P6011104703257516679at_nat] :
      ( ( sorted4866863899924483342at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ ( cons_P6512896166579812791at_nat @ X2 @ L23 ) )
      = ( ~ ( ord_le1203424502768444845at_nat @ X1 @ X2 )
        & ( ~ ( ord_le1203424502768444845at_nat @ X1 @ X2 )
         => ( ( ( X1 = X2 )
             => ( sorted4866863899924483342at_nat @ L12 @ L23 ) )
            & ( ( X1 != X2 )
             => ( sorted4866863899924483342at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ).

% subset_sorted.simps(3)
thf(fact_1120_subset__sorted_Osimps_I3_J,axiom,
    ! [X1: nat,L12: list_nat,X2: nat,L23: list_nat] :
      ( ( sorted5575391298698439683ed_nat @ ( cons_nat @ X1 @ L12 ) @ ( cons_nat @ X2 @ L23 ) )
      = ( ~ ( ord_less_nat @ X1 @ X2 )
        & ( ~ ( ord_less_nat @ X1 @ X2 )
         => ( ( ( X1 = X2 )
             => ( sorted5575391298698439683ed_nat @ L12 @ L23 ) )
            & ( ( X1 != X2 )
             => ( sorted5575391298698439683ed_nat @ ( cons_nat @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ).

% subset_sorted.simps(3)
thf(fact_1121_subset__sorted_Osimps_I2_J,axiom,
    ! [X1: product_prod_nat_nat,L12: list_P6011104703257516679at_nat] :
      ~ ( sorted4866863899924483342at_nat @ ( cons_P6512896166579812791at_nat @ X1 @ L12 ) @ nil_Pr5478986624290739719at_nat ) ).

% subset_sorted.simps(2)
thf(fact_1122_subset__sorted_Oelims_I1_J,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat,Y: $o] :
      ( ( ( sorted4866863899924483342at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
         => ~ Y )
       => ( ( ? [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
                ( X3
                = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
           => ( ( Xa2 = nil_Pr5478986624290739719at_nat )
             => Y ) )
         => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
               => ! [X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
                    ( ( Xa2
                      = ( cons_P6512896166579812791at_nat @ X23 @ L22 ) )
                   => ( Y
                      = ( ~ ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                            & ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                             => ( ( ( X12 = X23 )
                                 => ( sorted4866863899924483342at_nat @ L1 @ L22 ) )
                                & ( ( X12 != X23 )
                                 => ( sorted4866863899924483342at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ).

% subset_sorted.elims(1)
thf(fact_1123_subset__sorted_Oelims_I1_J,axiom,
    ! [X3: list_nat,Xa2: list_nat,Y: $o] :
      ( ( ( sorted5575391298698439683ed_nat @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = nil_nat )
         => ~ Y )
       => ( ( ? [X12: nat,L1: list_nat] :
                ( X3
                = ( cons_nat @ X12 @ L1 ) )
           => ( ( Xa2 = nil_nat )
             => Y ) )
         => ~ ! [X12: nat,L1: list_nat] :
                ( ( X3
                  = ( cons_nat @ X12 @ L1 ) )
               => ! [X23: nat,L22: list_nat] :
                    ( ( Xa2
                      = ( cons_nat @ X23 @ L22 ) )
                   => ( Y
                      = ( ~ ( ~ ( ord_less_nat @ X12 @ X23 )
                            & ( ~ ( ord_less_nat @ X12 @ X23 )
                             => ( ( ( X12 = X23 )
                                 => ( sorted5575391298698439683ed_nat @ L1 @ L22 ) )
                                & ( ( X12 != X23 )
                                 => ( sorted5575391298698439683ed_nat @ ( cons_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ).

% subset_sorted.elims(1)
thf(fact_1124_subset__sorted_Oelims_I2_J,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat] :
      ( ( sorted4866863899924483342at_nat @ X3 @ Xa2 )
     => ( ( X3 != nil_Pr5478986624290739719at_nat )
       => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
              ( ( X3
                = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
             => ! [X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
                  ( ( Xa2
                    = ( cons_P6512896166579812791at_nat @ X23 @ L22 ) )
                 => ~ ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                      & ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                       => ( ( ( X12 = X23 )
                           => ( sorted4866863899924483342at_nat @ L1 @ L22 ) )
                          & ( ( X12 != X23 )
                           => ( sorted4866863899924483342at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ).

% subset_sorted.elims(2)
thf(fact_1125_subset__sorted_Oelims_I2_J,axiom,
    ! [X3: list_nat,Xa2: list_nat] :
      ( ( sorted5575391298698439683ed_nat @ X3 @ Xa2 )
     => ( ( X3 != nil_nat )
       => ~ ! [X12: nat,L1: list_nat] :
              ( ( X3
                = ( cons_nat @ X12 @ L1 ) )
             => ! [X23: nat,L22: list_nat] :
                  ( ( Xa2
                    = ( cons_nat @ X23 @ L22 ) )
                 => ~ ( ~ ( ord_less_nat @ X12 @ X23 )
                      & ( ~ ( ord_less_nat @ X12 @ X23 )
                       => ( ( ( X12 = X23 )
                           => ( sorted5575391298698439683ed_nat @ L1 @ L22 ) )
                          & ( ( X12 != X23 )
                           => ( sorted5575391298698439683ed_nat @ ( cons_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ).

% subset_sorted.elims(2)
thf(fact_1126_subset__sorted_Oelims_I3_J,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat] :
      ( ~ ( sorted4866863899924483342at_nat @ X3 @ Xa2 )
     => ( ( ? [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
              ( X3
              = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
         => ( Xa2 != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
              ( ( X3
                = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
             => ! [X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
                  ( ( Xa2
                    = ( cons_P6512896166579812791at_nat @ X23 @ L22 ) )
                 => ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                    & ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                     => ( ( ( X12 = X23 )
                         => ( sorted4866863899924483342at_nat @ L1 @ L22 ) )
                        & ( ( X12 != X23 )
                         => ( sorted4866863899924483342at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ).

% subset_sorted.elims(3)
thf(fact_1127_subset__sorted_Oelims_I3_J,axiom,
    ! [X3: list_nat,Xa2: list_nat] :
      ( ~ ( sorted5575391298698439683ed_nat @ X3 @ Xa2 )
     => ( ( ? [X12: nat,L1: list_nat] :
              ( X3
              = ( cons_nat @ X12 @ L1 ) )
         => ( Xa2 != nil_nat ) )
       => ~ ! [X12: nat,L1: list_nat] :
              ( ( X3
                = ( cons_nat @ X12 @ L1 ) )
             => ! [X23: nat,L22: list_nat] :
                  ( ( Xa2
                    = ( cons_nat @ X23 @ L22 ) )
                 => ( ~ ( ord_less_nat @ X12 @ X23 )
                    & ( ~ ( ord_less_nat @ X12 @ X23 )
                     => ( ( ( X12 = X23 )
                         => ( sorted5575391298698439683ed_nat @ L1 @ L22 ) )
                        & ( ( X12 != X23 )
                         => ( sorted5575391298698439683ed_nat @ ( cons_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ).

% subset_sorted.elims(3)
thf(fact_1128_subset__sorted_Opelims_I1_J,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat,Y: $o] :
      ( ( ( sorted4866863899924483342at_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P7052990409830227952at_nat @ sorted2568869543759264377at_nat @ ( produc5943733680697469783at_nat @ X3 @ Xa2 ) )
       => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
           => ( Y
             => ~ ( accp_P7052990409830227952at_nat @ sorted2568869543759264377at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xa2 ) ) ) )
         => ( ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
               => ( ( Xa2 = nil_Pr5478986624290739719at_nat )
                 => ( ~ Y
                   => ~ ( accp_P7052990409830227952at_nat @ sorted2568869543759264377at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ nil_Pr5478986624290739719at_nat ) ) ) ) )
           => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
                  ( ( X3
                    = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
                 => ! [X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
                      ( ( Xa2
                        = ( cons_P6512896166579812791at_nat @ X23 @ L22 ) )
                     => ( ( Y
                          = ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                            & ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                             => ( ( ( X12 = X23 )
                                 => ( sorted4866863899924483342at_nat @ L1 @ L22 ) )
                                & ( ( X12 != X23 )
                                 => ( sorted4866863899924483342at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) )
                       => ~ ( accp_P7052990409830227952at_nat @ sorted2568869543759264377at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ ( cons_P6512896166579812791at_nat @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ).

% subset_sorted.pelims(1)
thf(fact_1129_subset__sorted_Opelims_I1_J,axiom,
    ! [X3: list_nat,Xa2: list_nat,Y: $o] :
      ( ( ( sorted5575391298698439683ed_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P8037286306265792042st_nat @ sorted8773635453300400216el_nat @ ( produc2694037385005941721st_nat @ X3 @ Xa2 ) )
       => ( ( ( X3 = nil_nat )
           => ( Y
             => ~ ( accp_P8037286306265792042st_nat @ sorted8773635453300400216el_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa2 ) ) ) )
         => ( ! [X12: nat,L1: list_nat] :
                ( ( X3
                  = ( cons_nat @ X12 @ L1 ) )
               => ( ( Xa2 = nil_nat )
                 => ( ~ Y
                   => ~ ( accp_P8037286306265792042st_nat @ sorted8773635453300400216el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X12 @ L1 ) @ nil_nat ) ) ) ) )
           => ~ ! [X12: nat,L1: list_nat] :
                  ( ( X3
                    = ( cons_nat @ X12 @ L1 ) )
                 => ! [X23: nat,L22: list_nat] :
                      ( ( Xa2
                        = ( cons_nat @ X23 @ L22 ) )
                     => ( ( Y
                          = ( ~ ( ord_less_nat @ X12 @ X23 )
                            & ( ~ ( ord_less_nat @ X12 @ X23 )
                             => ( ( ( X12 = X23 )
                                 => ( sorted5575391298698439683ed_nat @ L1 @ L22 ) )
                                & ( ( X12 != X23 )
                                 => ( sorted5575391298698439683ed_nat @ ( cons_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) )
                       => ~ ( accp_P8037286306265792042st_nat @ sorted8773635453300400216el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X12 @ L1 ) @ ( cons_nat @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ).

% subset_sorted.pelims(1)
thf(fact_1130_subset__sorted_Opelims_I2_J,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat] :
      ( ( sorted4866863899924483342at_nat @ X3 @ Xa2 )
     => ( ( accp_P7052990409830227952at_nat @ sorted2568869543759264377at_nat @ ( produc5943733680697469783at_nat @ X3 @ Xa2 ) )
       => ( ( ( X3 = nil_Pr5478986624290739719at_nat )
           => ~ ( accp_P7052990409830227952at_nat @ sorted2568869543759264377at_nat @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xa2 ) ) )
         => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
               => ! [X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
                    ( ( Xa2
                      = ( cons_P6512896166579812791at_nat @ X23 @ L22 ) )
                   => ( ( accp_P7052990409830227952at_nat @ sorted2568869543759264377at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ ( cons_P6512896166579812791at_nat @ X23 @ L22 ) ) )
                     => ~ ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                          & ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                           => ( ( ( X12 = X23 )
                               => ( sorted4866863899924483342at_nat @ L1 @ L22 ) )
                              & ( ( X12 != X23 )
                               => ( sorted4866863899924483342at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ).

% subset_sorted.pelims(2)
thf(fact_1131_subset__sorted_Opelims_I2_J,axiom,
    ! [X3: list_nat,Xa2: list_nat] :
      ( ( sorted5575391298698439683ed_nat @ X3 @ Xa2 )
     => ( ( accp_P8037286306265792042st_nat @ sorted8773635453300400216el_nat @ ( produc2694037385005941721st_nat @ X3 @ Xa2 ) )
       => ( ( ( X3 = nil_nat )
           => ~ ( accp_P8037286306265792042st_nat @ sorted8773635453300400216el_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xa2 ) ) )
         => ~ ! [X12: nat,L1: list_nat] :
                ( ( X3
                  = ( cons_nat @ X12 @ L1 ) )
               => ! [X23: nat,L22: list_nat] :
                    ( ( Xa2
                      = ( cons_nat @ X23 @ L22 ) )
                   => ( ( accp_P8037286306265792042st_nat @ sorted8773635453300400216el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X12 @ L1 ) @ ( cons_nat @ X23 @ L22 ) ) )
                     => ~ ( ~ ( ord_less_nat @ X12 @ X23 )
                          & ( ~ ( ord_less_nat @ X12 @ X23 )
                           => ( ( ( X12 = X23 )
                               => ( sorted5575391298698439683ed_nat @ L1 @ L22 ) )
                              & ( ( X12 != X23 )
                               => ( sorted5575391298698439683ed_nat @ ( cons_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ).

% subset_sorted.pelims(2)
thf(fact_1132_subset__sorted_Opelims_I3_J,axiom,
    ! [X3: list_P6011104703257516679at_nat,Xa2: list_P6011104703257516679at_nat] :
      ( ~ ( sorted4866863899924483342at_nat @ X3 @ Xa2 )
     => ( ( accp_P7052990409830227952at_nat @ sorted2568869543759264377at_nat @ ( produc5943733680697469783at_nat @ X3 @ Xa2 ) )
       => ( ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
              ( ( X3
                = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
             => ( ( Xa2 = nil_Pr5478986624290739719at_nat )
               => ~ ( accp_P7052990409830227952at_nat @ sorted2568869543759264377at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ nil_Pr5478986624290739719at_nat ) ) ) )
         => ~ ! [X12: product_prod_nat_nat,L1: list_P6011104703257516679at_nat] :
                ( ( X3
                  = ( cons_P6512896166579812791at_nat @ X12 @ L1 ) )
               => ! [X23: product_prod_nat_nat,L22: list_P6011104703257516679at_nat] :
                    ( ( Xa2
                      = ( cons_P6512896166579812791at_nat @ X23 @ L22 ) )
                   => ( ( accp_P7052990409830227952at_nat @ sorted2568869543759264377at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ ( cons_P6512896166579812791at_nat @ X23 @ L22 ) ) )
                     => ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                        & ( ~ ( ord_le1203424502768444845at_nat @ X12 @ X23 )
                         => ( ( ( X12 = X23 )
                             => ( sorted4866863899924483342at_nat @ L1 @ L22 ) )
                            & ( ( X12 != X23 )
                             => ( sorted4866863899924483342at_nat @ ( cons_P6512896166579812791at_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ).

% subset_sorted.pelims(3)
thf(fact_1133_subset__sorted_Opelims_I3_J,axiom,
    ! [X3: list_nat,Xa2: list_nat] :
      ( ~ ( sorted5575391298698439683ed_nat @ X3 @ Xa2 )
     => ( ( accp_P8037286306265792042st_nat @ sorted8773635453300400216el_nat @ ( produc2694037385005941721st_nat @ X3 @ Xa2 ) )
       => ( ! [X12: nat,L1: list_nat] :
              ( ( X3
                = ( cons_nat @ X12 @ L1 ) )
             => ( ( Xa2 = nil_nat )
               => ~ ( accp_P8037286306265792042st_nat @ sorted8773635453300400216el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X12 @ L1 ) @ nil_nat ) ) ) )
         => ~ ! [X12: nat,L1: list_nat] :
                ( ( X3
                  = ( cons_nat @ X12 @ L1 ) )
               => ! [X23: nat,L22: list_nat] :
                    ( ( Xa2
                      = ( cons_nat @ X23 @ L22 ) )
                   => ( ( accp_P8037286306265792042st_nat @ sorted8773635453300400216el_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X12 @ L1 ) @ ( cons_nat @ X23 @ L22 ) ) )
                     => ( ~ ( ord_less_nat @ X12 @ X23 )
                        & ( ~ ( ord_less_nat @ X12 @ X23 )
                         => ( ( ( X12 = X23 )
                             => ( sorted5575391298698439683ed_nat @ L1 @ L22 ) )
                            & ( ( X12 != X23 )
                             => ( sorted5575391298698439683ed_nat @ ( cons_nat @ X12 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ).

% subset_sorted.pelims(3)
thf(fact_1134_sorted__insort__is__snoc,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A: product_prod_nat_nat] :
      ( ( sorted5214655850825725294at_nat @ ord_le8460144461188290721at_nat @ Xs )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( ord_le8460144461188290721at_nat @ X4 @ A ) )
       => ( ( linord8970147294467115835at_nat
            @ ^ [X: product_prod_nat_nat] : X
            @ A
            @ Xs )
          = ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ A @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% sorted_insort_is_snoc
thf(fact_1135_sorted__insort__is__snoc,axiom,
    ! [Xs: list_nat,A: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( ord_less_eq_nat @ X4 @ A ) )
       => ( ( linord8961336180081300637at_nat
            @ ^ [X: nat] : X
            @ A
            @ Xs )
          = ( append_nat @ Xs @ ( cons_nat @ A @ nil_nat ) ) ) ) ) ).

% sorted_insort_is_snoc
thf(fact_1136_distinct__sorted__mono__iff,axiom,
    ! [L: list_nat,I3: nat,J2: nat] :
      ( ( distinct_nat @ L )
     => ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
       => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ L ) )
         => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ L ) )
           => ( ( ord_less_eq_nat @ ( nth_nat @ L @ I3 ) @ ( nth_nat @ L @ J2 ) )
              = ( ord_less_eq_nat @ I3 @ J2 ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
thf(fact_1137_length__nth__simps_I3_J,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ zero_zero_nat )
      = X3 ) ).

% length_nth_simps(3)
thf(fact_1138_length__nth__simps_I3_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ zero_zero_nat )
      = X3 ) ).

% length_nth_simps(3)
thf(fact_1139_nth__Cons__0,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ zero_zero_nat )
      = X3 ) ).

% nth_Cons_0
thf(fact_1140_nth__Cons__0,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ zero_zero_nat )
      = X3 ) ).

% nth_Cons_0
thf(fact_1141_length__nth__simps_I4_J,axiom,
    ! [X3: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% length_nth_simps(4)
thf(fact_1142_length__nth__simps_I4_J,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,N: nat] :
      ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( suc @ N ) )
      = ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) ).

% length_nth_simps(4)
thf(fact_1143_nth__Cons__Suc,axiom,
    ! [X3: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_1144_nth__Cons__Suc,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,N: nat] :
      ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ ( suc @ N ) )
      = ( nth_Pr7617993195940197384at_nat @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_1145_nth__replicate,axiom,
    ! [I3: nat,N: nat,X3: nat] :
      ( ( ord_less_nat @ I3 @ N )
     => ( ( nth_nat @ ( replicate_nat @ N @ X3 ) @ I3 )
        = X3 ) ) ).

% nth_replicate
thf(fact_1146_nth__append__length,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( nth_Pr7617993195940197384at_nat @ ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X3 @ Ys3 ) ) @ ( size_s5460976970255530739at_nat @ Xs ) )
      = X3 ) ).

% nth_append_length
thf(fact_1147_nth__append__length,axiom,
    ! [Xs: list_nat,X3: nat,Ys3: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X3 @ Ys3 ) ) @ ( size_size_list_nat @ Xs ) )
      = X3 ) ).

% nth_append_length
thf(fact_1148_nth__append__first,axiom,
    ! [I3: nat,L: list_nat,L4: list_nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ L ) )
     => ( ( nth_nat @ ( append_nat @ L @ L4 ) @ I3 )
        = ( nth_nat @ L @ I3 ) ) ) ).

% nth_append_first
thf(fact_1149_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I5 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1150_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8469869581646625389at_nat,P: produc859450856879609959at_nat > $o,X3: produc859450856879609959at_nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s3679842834875189465at_nat @ Xs ) )
         => ( P @ ( nth_Pr6744343527793145070at_nat @ Xs @ I4 ) ) )
     => ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_1151_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) ) )
     => ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_1152_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X3: nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I4 ) ) )
     => ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_1153_in__set__conv__nth,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s3679842834875189465at_nat @ Xs ) )
            & ( ( nth_Pr6744343527793145070at_nat @ Xs @ I5 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_1154_in__set__conv__nth,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( ( nth_Pr7617993195940197384at_nat @ Xs @ I5 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_1155_in__set__conv__nth,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I5 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_1156_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_1157_nth__mem,axiom,
    ! [N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( ord_less_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
     => ( member8206827879206165904at_nat @ ( nth_Pr6744343527793145070at_nat @ Xs @ N ) @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_1158_nth__mem,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_1159_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_1160_all__set__conv__nth,axiom,
    ! [L: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ L ) )
           => ( P @ X ) ) )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ L ) )
           => ( P @ ( nth_nat @ L @ I5 ) ) ) ) ) ).

% all_set_conv_nth
thf(fact_1161_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs3: list_nat] :
        ! [I5: nat] :
          ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs3 ) )
         => ! [J: nat] :
              ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs3 ) )
             => ( ( I5 != J )
               => ( ( nth_nat @ Xs3 @ I5 )
                 != ( nth_nat @ Xs3 @ J ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_1162_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I3: nat,J2: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
         => ( ( ( nth_nat @ Xs @ I3 )
              = ( nth_nat @ Xs @ J2 ) )
            = ( I3 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_1163_insort__key_Osimps_I2_J,axiom,
    ! [F2: product_prod_nat_nat > nat,X3: product_prod_nat_nat,Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
       => ( ( linord2653521316007143958at_nat @ F2 @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) )
          = ( cons_P6512896166579812791at_nat @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y ) )
       => ( ( linord2653521316007143958at_nat @ F2 @ X3 @ ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) )
          = ( cons_P6512896166579812791at_nat @ Y @ ( linord2653521316007143958at_nat @ F2 @ X3 @ Ys3 ) ) ) ) ) ).

% insort_key.simps(2)
thf(fact_1164_obtain__list__from__elements,axiom,
    ! [N: nat,P: nat > nat > $o] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N )
         => ? [Li: nat] : ( P @ Li @ I4 ) )
     => ~ ! [L2: list_nat] :
            ( ( ( size_size_list_nat @ L2 )
              = N )
           => ~ ! [I6: nat] :
                  ( ( ord_less_nat @ I6 @ N )
                 => ( P @ ( nth_nat @ L2 @ I6 ) @ I6 ) ) ) ) ).

% obtain_list_from_elements
thf(fact_1165_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys3: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys3 ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I4 )
              = ( nth_nat @ Ys3 @ I4 ) ) )
       => ( Xs = Ys3 ) ) ) ).

% nth_equalityI
thf(fact_1166_Skolem__list__nth,axiom,
    ! [K3: nat,P: nat > nat > $o] :
      ( ( ! [I5: nat] :
            ( ( ord_less_nat @ I5 @ K3 )
           => ? [X8: nat] : ( P @ I5 @ X8 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K3 )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K3 )
               => ( P @ I5 @ ( nth_nat @ Xs3 @ I5 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_1167_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y9: list_nat,Z6: list_nat] : ( Y9 = Z6 ) )
    = ( ^ [Xs3: list_nat,Ys2: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys2 ) )
          & ! [I5: nat] :
              ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I5 )
                = ( nth_nat @ Ys2 @ I5 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_1168_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I3: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I3 @ ( nth_nat @ Ns @ I3 ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_1169_sorted__insort,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat
        @ ( linord8961336180081300637at_nat
          @ ^ [X: nat] : X
          @ X3
          @ Xs ) )
      = ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).

% sorted_insort
thf(fact_1170_sorted__wrt__nth__less,axiom,
    ! [P: nat > nat > $o,Xs: list_nat,I3: nat,J2: nat] :
      ( ( sorted_wrt_nat @ P @ Xs )
     => ( ( ord_less_nat @ I3 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_1171_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_nat
    = ( ^ [P6: nat > nat > $o,Xs3: list_nat] :
        ! [I5: nat,J: nat] :
          ( ( ord_less_nat @ I5 @ J )
         => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs3 ) )
           => ( P6 @ ( nth_nat @ Xs3 @ I5 ) @ ( nth_nat @ Xs3 @ J ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_1172_insort__is__Cons,axiom,
    ! [Xs: list_P6011104703257516679at_nat,F2: product_prod_nat_nat > nat,A: product_prod_nat_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ ( F2 @ X4 ) ) )
     => ( ( linord2653521316007143958at_nat @ F2 @ A @ Xs )
        = ( cons_P6512896166579812791at_nat @ A @ Xs ) ) ) ).

% insort_is_Cons
thf(fact_1173_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I5: nat,J: nat] :
            ( ( ord_less_nat @ I5 @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I5 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_1174_distinct__Ex1,axiom,
    ! [Xs: list_P8469869581646625389at_nat,X3: produc859450856879609959at_nat] :
      ( ( distin6906083803243959008at_nat @ Xs )
     => ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s3679842834875189465at_nat @ Xs ) )
            & ( ( nth_Pr6744343527793145070at_nat @ Xs @ X4 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s3679842834875189465at_nat @ Xs ) )
                  & ( ( nth_Pr6744343527793145070at_nat @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_1175_distinct__Ex1,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
      ( ( distin6923225563576452346at_nat @ Xs )
     => ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( ( nth_Pr7617993195940197384at_nat @ Xs @ X4 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s5460976970255530739at_nat @ Xs ) )
                  & ( ( nth_Pr7617993195940197384at_nat @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_1176_distinct__Ex1,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X4 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_1177_length__filter__conv__card,axiom,
    ! [P2: nat > $o,Xs: list_nat] :
      ( ( size_size_list_nat @ ( filter_nat @ P2 @ Xs ) )
      = ( finite_card_nat
        @ ( collect_nat
          @ ^ [I5: nat] :
              ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs ) )
              & ( P2 @ ( nth_nat @ Xs @ I5 ) ) ) ) ) ) ).

% length_filter_conv_card
thf(fact_1178_filter__eq__nths,axiom,
    ( filter_nat
    = ( ^ [P6: nat > $o,Xs3: list_nat] :
          ( nths_nat @ Xs3
          @ ( collect_nat
            @ ^ [I5: nat] :
                ( ( ord_less_nat @ I5 @ ( size_size_list_nat @ Xs3 ) )
                & ( P6 @ ( nth_nat @ Xs3 @ I5 ) ) ) ) ) ) ) ).

% filter_eq_nths
thf(fact_1179_nth__equal__first__eq,axiom,
    ! [X3: produc859450856879609959at_nat,Xs: list_P8469869581646625389at_nat,N: nat] :
      ( ~ ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3679842834875189465at_nat @ Xs ) )
       => ( ( ( nth_Pr6744343527793145070at_nat @ ( cons_P8732206157123786781at_nat @ X3 @ Xs ) @ N )
            = X3 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1180_nth__equal__first__eq,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,N: nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
       => ( ( ( nth_Pr7617993195940197384at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) @ N )
            = X3 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1181_nth__equal__first__eq,axiom,
    ! [X3: nat,Xs: list_nat,N: nat] :
      ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( ( nth_nat @ ( cons_nat @ X3 @ Xs ) @ N )
            = X3 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1182_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I5: nat] :
            ( ( ord_less_nat @ ( suc @ I5 ) @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I5 ) @ ( nth_nat @ Xs @ ( suc @ I5 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_1183_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I5: nat,J: nat] :
            ( ( ord_less_eq_nat @ I5 @ J )
           => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I5 ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_1184_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I3: nat,J2: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( ord_less_eq_nat @ I3 @ J2 )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_1185_listrel__iff__nth,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys3 ) @ ( listre818007680106770737at_nat @ R4 ) )
      = ( ( ( size_s5460976970255530739at_nat @ Xs )
          = ( size_s5460976970255530739at_nat @ Ys3 ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s5460976970255530739at_nat @ Xs ) )
           => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N3 ) @ ( nth_Pr7617993195940197384at_nat @ Ys3 @ N3 ) ) @ R4 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1186_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( listrel_nat_nat @ R4 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys3 ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ ( nth_nat @ Ys3 @ N3 ) ) @ R4 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1187_distinct__sorted__strict__mono__iff,axiom,
    ! [L: list_nat,I3: nat,J2: nat] :
      ( ( distinct_nat @ L )
     => ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
       => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ L ) )
         => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ L ) )
           => ( ( ord_less_nat @ ( nth_nat @ L @ I3 ) @ ( nth_nat @ L @ J2 ) )
              = ( ord_less_nat @ I3 @ J2 ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
thf(fact_1188_distinct__sorted__mono,axiom,
    ! [L: list_nat,I3: nat,J2: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ L )
     => ( ( distinct_nat @ L )
       => ( ( ord_less_nat @ I3 @ J2 )
         => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ L ) )
           => ( ord_less_nat @ ( nth_nat @ L @ I3 ) @ ( nth_nat @ L @ J2 ) ) ) ) ) ) ).

% distinct_sorted_mono
thf(fact_1189_take__Suc__conv__app__nth,axiom,
    ! [I3: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( ( take_P2173866234530122223at_nat @ ( suc @ I3 ) @ Xs )
        = ( append985823374593552924at_nat @ ( take_P2173866234530122223at_nat @ I3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) @ nil_Pr5478986624290739719at_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_1190_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_1191_lex__take__index,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys3: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys3 ) @ ( lex_Pr8571645452597969515at_nat @ R4 ) )
     => ~ ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
           => ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Ys3 ) )
             => ( ( ( take_P2173866234530122223at_nat @ I4 @ Xs )
                  = ( take_P2173866234530122223at_nat @ I4 @ Ys3 ) )
               => ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) @ ( nth_Pr7617993195940197384at_nat @ Ys3 @ I4 ) ) @ R4 ) ) ) ) ) ).

% lex_take_index
thf(fact_1192_lex__take__index,axiom,
    ! [Xs: list_nat,Ys3: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys3 ) @ ( lex_nat @ R4 ) )
     => ~ ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
           => ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys3 ) )
             => ( ( ( take_nat @ I4 @ Xs )
                  = ( take_nat @ I4 @ Ys3 ) )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Ys3 @ I4 ) ) @ R4 ) ) ) ) ) ).

% lex_take_index
thf(fact_1193_take__Suc__Cons,axiom,
    ! [N: nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( take_P2173866234530122223at_nat @ ( suc @ N ) @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( cons_P6512896166579812791at_nat @ X3 @ ( take_P2173866234530122223at_nat @ N @ Xs ) ) ) ).

% take_Suc_Cons
thf(fact_1194_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_1195_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_1196_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_1197_sorted__wrt__take,axiom,
    ! [F2: nat > nat > $o,Xs: list_nat,N: nat] :
      ( ( sorted_wrt_nat @ F2 @ Xs )
     => ( sorted_wrt_nat @ F2 @ ( take_nat @ N @ Xs ) ) ) ).

% sorted_wrt_take
thf(fact_1198_sorted__take,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( take_nat @ N @ Xs ) ) ) ).

% sorted_take
thf(fact_1199_in__set__takeD,axiom,
    ! [X3: nat,N: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
     => ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1200_in__set__takeD,axiom,
    ! [X3: produc859450856879609959at_nat,N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( take_P5254422574997664853at_nat @ N @ Xs ) ) )
     => ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1201_in__set__takeD,axiom,
    ! [X3: product_prod_nat_nat,N: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( take_P2173866234530122223at_nat @ N @ Xs ) ) )
     => ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_1202_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_1203_set__take__subset__set__take,axiom,
    ! [M4: nat,N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ M4 @ N )
     => ( ord_less_eq_set_nat @ ( set_nat2 @ ( take_nat @ M4 @ Xs ) ) @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1204_nth__take__lemma,axiom,
    ! [K3: nat,Xs: list_nat,Ys3: list_nat] :
      ( ( ord_less_eq_nat @ K3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ K3 @ ( size_size_list_nat @ Ys3 ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ K3 )
             => ( ( nth_nat @ Xs @ I4 )
                = ( nth_nat @ Ys3 @ I4 ) ) )
         => ( ( take_nat @ K3 @ Xs )
            = ( take_nat @ K3 @ Ys3 ) ) ) ) ) ).

% nth_take_lemma
thf(fact_1205_filter__nth__ex__nth,axiom,
    ! [N: nat,P: nat > $o,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) ) )
     => ? [M: nat] :
          ( ( ord_less_eq_nat @ N @ M )
          & ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
          & ( ( nth_nat @ ( filter_nat @ P @ Xs ) @ N )
            = ( nth_nat @ Xs @ M ) )
          & ( ( filter_nat @ P @ ( take_nat @ M @ Xs ) )
            = ( take_nat @ N @ ( filter_nat @ P @ Xs ) ) ) ) ) ).

% filter_nth_ex_nth
thf(fact_1206_lexord__take__index__conv,axiom,
    ! [X3: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat,R4: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ X3 @ Y ) @ ( lexord2841853652668343668at_nat @ R4 ) )
      = ( ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ X3 ) @ ( size_s5460976970255530739at_nat @ Y ) )
          & ( ( take_P2173866234530122223at_nat @ ( size_s5460976970255530739at_nat @ X3 ) @ Y )
            = X3 ) )
        | ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( ord_min_nat @ ( size_s5460976970255530739at_nat @ X3 ) @ ( size_s5460976970255530739at_nat @ Y ) ) )
            & ( ( take_P2173866234530122223at_nat @ I5 @ X3 )
              = ( take_P2173866234530122223at_nat @ I5 @ Y ) )
            & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( nth_Pr7617993195940197384at_nat @ X3 @ I5 ) @ ( nth_Pr7617993195940197384at_nat @ Y @ I5 ) ) @ R4 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1207_lexord__take__index__conv,axiom,
    ! [X3: list_nat,Y: list_nat,R4: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X3 @ Y ) @ ( lexord_nat @ R4 ) )
      = ( ( ( ord_less_nat @ ( size_size_list_nat @ X3 ) @ ( size_size_list_nat @ Y ) )
          & ( ( take_nat @ ( size_size_list_nat @ X3 ) @ Y )
            = X3 ) )
        | ? [I5: nat] :
            ( ( ord_less_nat @ I5 @ ( ord_min_nat @ ( size_size_list_nat @ X3 ) @ ( size_size_list_nat @ Y ) ) )
            & ( ( take_nat @ I5 @ X3 )
              = ( take_nat @ I5 @ Y ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ X3 @ I5 ) @ ( nth_nat @ Y @ I5 ) ) @ R4 ) ) ) ) ).

% lexord_take_index_conv
thf(fact_1208_id__take__nth__drop,axiom,
    ! [I3: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( Xs
        = ( append985823374593552924at_nat @ ( take_P2173866234530122223at_nat @ I3 @ Xs ) @ ( cons_P6512896166579812791at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) @ ( drop_P8868858903918902087at_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_1209_id__take__nth__drop,axiom,
    ! [I3: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( Xs
        = ( append_nat @ ( take_nat @ I3 @ Xs ) @ ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ ( drop_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_1210_min__arg__le_I2_J,axiom,
    ! [M4: nat,N: nat] :
      ( ( ord_less_eq_nat @ M4 @ ( ord_min_nat @ M4 @ N ) )
      = ( ( ord_min_nat @ M4 @ N )
        = M4 ) ) ).

% min_arg_le(2)
thf(fact_1211_min__arg__le_I2_J,axiom,
    ! [M4: set_nat,N: set_nat] :
      ( ( ord_less_eq_set_nat @ M4 @ ( ord_min_set_nat @ M4 @ N ) )
      = ( ( ord_min_set_nat @ M4 @ N )
        = M4 ) ) ).

% min_arg_le(2)
thf(fact_1212_min__arg__le_I1_J,axiom,
    ! [N: nat,M4: nat] :
      ( ( ord_less_eq_nat @ N @ ( ord_min_nat @ M4 @ N ) )
      = ( ( ord_min_nat @ M4 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_1213_min__arg__le_I1_J,axiom,
    ! [N: set_nat,M4: set_nat] :
      ( ( ord_less_eq_set_nat @ N @ ( ord_min_set_nat @ M4 @ N ) )
      = ( ( ord_min_set_nat @ M4 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_1214_min__eq__arg_I2_J,axiom,
    ! [M4: nat,N: nat] :
      ( ( ( ord_min_nat @ M4 @ N )
        = N )
      = ( ord_less_eq_nat @ N @ M4 ) ) ).

% min_eq_arg(2)
thf(fact_1215_min__eq__arg_I1_J,axiom,
    ! [M4: nat,N: nat] :
      ( ( ( ord_min_nat @ M4 @ N )
        = M4 )
      = ( ord_less_eq_nat @ M4 @ N ) ) ).

% min_eq_arg(1)
thf(fact_1216_min__arg__not__ge_I1_J,axiom,
    ! [M4: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ ( ord_min_nat @ M4 @ N ) @ M4 ) )
      = ( ( ord_min_nat @ M4 @ N )
        = M4 ) ) ).

% min_arg_not_ge(1)
thf(fact_1217_min__arg__not__ge_I2_J,axiom,
    ! [M4: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ ( ord_min_nat @ M4 @ N ) @ N ) )
      = ( ( ord_min_nat @ M4 @ N )
        = N ) ) ).

% min_arg_not_ge(2)
thf(fact_1218_min__less__self__conv_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ A )
      = ( ord_less_nat @ B @ A ) ) ).

% min_less_self_conv(1)
thf(fact_1219_min__less__self__conv_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ B ) ) ).

% min_less_self_conv(2)
thf(fact_1220_min__simps_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ A @ B )
        = A ) ) ).

% min_simps(1)
thf(fact_1221_min__simps_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_min_nat @ A @ B )
        = B ) ) ).

% min_simps(2)
thf(fact_1222_drop__Suc__Cons,axiom,
    ! [N: nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( drop_P8868858903918902087at_nat @ ( suc @ N ) @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) )
      = ( drop_P8868858903918902087at_nat @ N @ Xs ) ) ).

% drop_Suc_Cons
thf(fact_1223_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ ( suc @ A ) @ B )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(1)
thf(fact_1224_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ B @ ( suc @ A ) )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(2)
thf(fact_1225_length__take,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( take_nat @ N @ Xs ) )
      = ( ord_min_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% length_take
thf(fact_1226_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( nil_nat
        = ( drop_nat @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_1227_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( drop_nat @ N @ Xs )
        = nil_nat )
      = ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_1228_drop__all,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
     => ( ( drop_nat @ N @ Xs )
        = nil_nat ) ) ).

% drop_all
thf(fact_1229_nth__via__drop,axiom,
    ! [N: nat,Xs: list_nat,Y: nat,Ys3: list_nat] :
      ( ( ( drop_nat @ N @ Xs )
        = ( cons_nat @ Y @ Ys3 ) )
     => ( ( nth_nat @ Xs @ N )
        = Y ) ) ).

% nth_via_drop
thf(fact_1230_nth__via__drop,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys3: list_P6011104703257516679at_nat] :
      ( ( ( drop_P8868858903918902087at_nat @ N @ Xs )
        = ( cons_P6512896166579812791at_nat @ Y @ Ys3 ) )
     => ( ( nth_Pr7617993195940197384at_nat @ Xs @ N )
        = Y ) ) ).

% nth_via_drop
thf(fact_1231_sorted__wrt__drop,axiom,
    ! [F2: nat > nat > $o,Xs: list_nat,N: nat] :
      ( ( sorted_wrt_nat @ F2 @ Xs )
     => ( sorted_wrt_nat @ F2 @ ( drop_nat @ N @ Xs ) ) ) ).

% sorted_wrt_drop
thf(fact_1232_sorted__drop,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( drop_nat @ N @ Xs ) ) ) ).

% sorted_drop
thf(fact_1233_append__eq__conv__conj,axiom,
    ! [Xs: list_nat,Ys3: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys3 )
        = Zs )
      = ( ( Xs
          = ( take_nat @ ( size_size_list_nat @ Xs ) @ Zs ) )
        & ( Ys3
          = ( drop_nat @ ( size_size_list_nat @ Xs ) @ Zs ) ) ) ) ).

% append_eq_conv_conj
thf(fact_1234_set__drop__subset__set__drop,axiom,
    ! [N: nat,M4: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ N @ M4 )
     => ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ M4 @ Xs ) ) @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_1235_drop__eq__ConsD,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat,Xs5: list_P6011104703257516679at_nat] :
      ( ( ( drop_P8868858903918902087at_nat @ N @ Xs )
        = ( cons_P6512896166579812791at_nat @ X3 @ Xs5 ) )
     => ( ( drop_P8868858903918902087at_nat @ ( suc @ N ) @ Xs )
        = Xs5 ) ) ).

% drop_eq_ConsD
thf(fact_1236_min__def__raw,axiom,
    ( ord_min_nat
    = ( ^ [A4: nat,B9: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B9 ) @ A4 @ B9 ) ) ) ).

% min_def_raw
thf(fact_1237_min__def__raw,axiom,
    ( ord_min_set_nat
    = ( ^ [A4: set_nat,B9: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A4 @ B9 ) @ A4 @ B9 ) ) ) ).

% min_def_raw
thf(fact_1238_set__drop__subset,axiom,
    ! [N: nat,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).

% set_drop_subset
thf(fact_1239_in__set__dropD,axiom,
    ! [X3: nat,N: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
     => ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_1240_in__set__dropD,axiom,
    ! [X3: produc859450856879609959at_nat,N: nat,Xs: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( drop_P4867432431663989165at_nat @ N @ Xs ) ) )
     => ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ Xs ) ) ) ).

% in_set_dropD
thf(fact_1241_in__set__dropD,axiom,
    ! [X3: product_prod_nat_nat,N: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( drop_P8868858903918902087at_nat @ N @ Xs ) ) )
     => ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% in_set_dropD
thf(fact_1242_append__eq__append__conv__if,axiom,
    ! [Xs_1: list_nat,Xs_2: list_nat,Ys_1: list_nat,Ys_2: list_nat] :
      ( ( ( append_nat @ Xs_1 @ Xs_2 )
        = ( append_nat @ Ys_1 @ Ys_2 ) )
      = ( ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
         => ( ( Xs_1
              = ( take_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) )
            & ( Xs_2
              = ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ ( size_size_list_nat @ Xs_1 ) @ ( size_size_list_nat @ Ys_1 ) )
         => ( ( ( take_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 )
              = Ys_1 )
            & ( ( append_nat @ ( drop_nat @ ( size_size_list_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
              = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
thf(fact_1243_Cons__nth__drop__Suc,axiom,
    ! [I3: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( ( cons_P6512896166579812791at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) @ ( drop_P8868858903918902087at_nat @ ( suc @ I3 ) @ Xs ) )
        = ( drop_P8868858903918902087at_nat @ I3 @ Xs ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_1244_Cons__nth__drop__Suc,axiom,
    ! [I3: nat,Xs: list_nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( cons_nat @ ( nth_nat @ Xs @ I3 ) @ ( drop_nat @ ( suc @ I3 ) @ Xs ) )
        = ( drop_nat @ I3 @ Xs ) ) ) ).

% Cons_nth_drop_Suc
thf(fact_1245_in__set__drop__conv__nth,axiom,
    ! [X3: produc859450856879609959at_nat,N: nat,L: list_P8469869581646625389at_nat] :
      ( ( member8206827879206165904at_nat @ X3 @ ( set_Pr5518436109238095868at_nat @ ( drop_P4867432431663989165at_nat @ N @ L ) ) )
      = ( ? [I5: nat] :
            ( ( ord_less_eq_nat @ N @ I5 )
            & ( ord_less_nat @ I5 @ ( size_s3679842834875189465at_nat @ L ) )
            & ( X3
              = ( nth_Pr6744343527793145070at_nat @ L @ I5 ) ) ) ) ) ).

% in_set_drop_conv_nth
thf(fact_1246_in__set__drop__conv__nth,axiom,
    ! [X3: product_prod_nat_nat,N: nat,L: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( drop_P8868858903918902087at_nat @ N @ L ) ) )
      = ( ? [I5: nat] :
            ( ( ord_less_eq_nat @ N @ I5 )
            & ( ord_less_nat @ I5 @ ( size_s5460976970255530739at_nat @ L ) )
            & ( X3
              = ( nth_Pr7617993195940197384at_nat @ L @ I5 ) ) ) ) ) ).

% in_set_drop_conv_nth
thf(fact_1247_in__set__drop__conv__nth,axiom,
    ! [X3: nat,N: nat,L: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( drop_nat @ N @ L ) ) )
      = ( ? [I5: nat] :
            ( ( ord_less_eq_nat @ N @ I5 )
            & ( ord_less_nat @ I5 @ ( size_size_list_nat @ L ) )
            & ( X3
              = ( nth_nat @ L @ I5 ) ) ) ) ) ).

% in_set_drop_conv_nth
thf(fact_1248_insert__nth__take__drop,axiom,
    ( list_i4302290065034157023at_nat
    = ( ^ [N3: nat,X: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat] : ( append985823374593552924at_nat @ ( take_P2173866234530122223at_nat @ N3 @ Xs3 ) @ ( append985823374593552924at_nat @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) @ ( drop_P8868858903918902087at_nat @ N3 @ Xs3 ) ) ) ) ) ).

% insert_nth_take_drop
thf(fact_1249_take__hd__drop,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( ( append985823374593552924at_nat @ ( take_P2173866234530122223at_nat @ N @ Xs ) @ ( cons_P6512896166579812791at_nat @ ( hd_Pro3460610213475200108at_nat @ ( drop_P8868858903918902087at_nat @ N @ Xs ) ) @ nil_Pr5478986624290739719at_nat ) )
        = ( take_P2173866234530122223at_nat @ ( suc @ N ) @ Xs ) ) ) ).

% take_hd_drop
thf(fact_1250_take__hd__drop,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( append_nat @ ( take_nat @ N @ Xs ) @ ( cons_nat @ ( hd_nat @ ( drop_nat @ N @ Xs ) ) @ nil_nat ) )
        = ( take_nat @ ( suc @ N ) @ Xs ) ) ) ).

% take_hd_drop
thf(fact_1251_list_Oset__sel_I1_J,axiom,
    ! [A: list_nat] :
      ( ( A != nil_nat )
     => ( member_nat @ ( hd_nat @ A ) @ ( set_nat2 @ A ) ) ) ).

% list.set_sel(1)
thf(fact_1252_list_Oset__sel_I1_J,axiom,
    ! [A: list_P8469869581646625389at_nat] :
      ( ( A != nil_Pr2582115297535392877at_nat )
     => ( member8206827879206165904at_nat @ ( hd_Pro8462011474880202578at_nat @ A ) @ ( set_Pr5518436109238095868at_nat @ A ) ) ) ).

% list.set_sel(1)
thf(fact_1253_list_Oset__sel_I1_J,axiom,
    ! [A: list_P6011104703257516679at_nat] :
      ( ( A != nil_Pr5478986624290739719at_nat )
     => ( member8440522571783428010at_nat @ ( hd_Pro3460610213475200108at_nat @ A ) @ ( set_Pr5648618587558075414at_nat @ A ) ) ) ).

% list.set_sel(1)

% Helper facts (18)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ( if_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ( if_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( fChoice_nat @ P ) )
      = ( ? [X8: nat] : ( P @ X8 ) ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( P @ ( fChoic6978938873391328853at_nat @ P ) )
      = ( ? [X8: product_prod_nat_nat] : ( P @ X8 ) ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X3: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( if_lis9186351972506106189at_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X3: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( if_lis9186351972506106189at_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [P: produc859450856879609959at_nat > $o] :
      ( ( P @ ( fChoic6673916560228149179at_nat @ P ) )
      = ( ? [X8: produc859450856879609959at_nat] : ( P @ X8 ) ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X3: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( if_lis7763640049307703347at_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X3: list_P8469869581646625389at_nat,Y: list_P8469869581646625389at_nat] :
      ( ( if_lis7763640049307703347at_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X3: produc6392793444374437607at_nat,Y: produc6392793444374437607at_nat] :
      ( ( if_Pro969932733043319213at_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_T,axiom,
    ! [X3: produc6392793444374437607at_nat,Y: produc6392793444374437607at_nat] :
      ( ( if_Pro969932733043319213at_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X3: produc5956105832733135175at_nat,Y: produc5956105832733135175at_nat] :
      ( ( if_Pro5163289783556962317at_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_T,axiom,
    ! [X3: produc5956105832733135175at_nat,Y: produc5956105832733135175at_nat] :
      ( ( if_Pro5163289783556962317at_nat @ $true @ X3 @ Y )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ~ ( finite_finite_list_a @ ( relational_eval_a_b @ ( restrict_Conjs_a_b @ qa @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ x @ y ) @ xysa ) ) @ i ) ) ).

%------------------------------------------------------------------------------