%------------------------------------------------------------------------------
% 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    : Finite_Fields/0008_Card_Irreducible_Polynomials_Aux/prob_00072_002505__18358696_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1820 ( 314 unt; 548 typ;   0 def)
%            Number of atoms       : 4166 ( 930 equ;   0 cnn)
%            Maximal formula atoms :    8 (   3 avg)
%            Number of connectives : 19831 ( 346   ~;  29   |; 248   &;16959   @)
%                                         (   0 <=>;2249  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (   9 avg)
%            Number of types       :   68 (  67 usr)
%            Number of type conns  : 1411 (1411   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  484 ( 481 usr;  18 con; 0-4 aty)
%            Number of variables   : 3384 ( 609   ^;2634   !; 141   ?;3384   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 13:22:52.619
%------------------------------------------------------------------------------
% Could-be-implicit typings (67)
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (481)
thf(sy_c_AbelCoset_OA__RCOSETS_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oa__r__coset_001t__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_001t__Product____Type__Ounit,type,
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    subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).

thf(sy_c_Subrings_Osubcring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    subcri7783154434480317835t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).

thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
    subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).

thf(sy_c_Subrings_Osubdomain_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    subdom3220114454046903646t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).

thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
    subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_Subrings_Osubfield_001t__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_001t__Product____Type__Ounit,type,
    subfie2054853380319850694t_unit: set_li664707282716828624list_a > partia1831960736934606626t_unit > $o ).

thf(sy_c_Subrings_Osubfield_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    subfie4546268998243038636t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).

thf(sy_c_Subrings_Osubfield_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
    subfie868023609828841932t_unit: set_list_set_list_a > partia4556295656693239580t_unit > $o ).

thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).

thf(sy_c_Subrings_Osubfield_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    subfie4339374749748326226t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).

thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
    subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_Subrings_Osubring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    subrin3541368690557094692t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).

thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).

thf(sy_c_Subrings_Osubring_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
    subrin3011427358523593924t_unit: set_set_list_list_a > partia4960592913263135132t_unit > $o ).

thf(sy_c_Subrings_Osubring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    subrin5643252653130547402t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).

thf(sy_c_Subrings_Osubring_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
    subrin1511138061850335568t_unit: set_set_a > partia6043505979758434576t_unit > $o ).

thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
    subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
    bound_list_a: list_a > nat > ( nat > list_a ) > $o ).

thf(sy_c_UnivPoly_Obound_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    bound_set_list_a: set_list_a > nat > ( nat > set_list_a ) > $o ).

thf(sy_c_UnivPoly_Obound_001tf__a,type,
    bound_a: a > nat > ( nat > a ) > $o ).

thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_list_a ).

thf(sy_c_UnivPoly_Oup_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    up_set529185716248919906t_unit: partia7496981018696276118t_unit > set_nat_set_list_a ).

thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
    up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).

thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Set__Oset_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J,type,
    member6369349398066642652list_a: ( list_list_list_a > set_list_list_list_a ) > set_li8602073249893796731list_a > $o ).

thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J,type,
    member3350673534532848354list_a: ( list_list_a > set_list_list_list_a ) > set_li6975380915727400577list_a > $o ).

thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    member2776101604912167772list_a: ( list_list_a > set_list_list_a ) > set_li2966187450340367355list_a > $o ).

thf(sy_c_member_001_062_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_J,type,
    member2273002913129892700list_a: ( list_set_list_a > set_list_set_list_a ) > set_li979503603270634491list_a > $o ).

thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    member6714375691612171394list_a: ( list_a > list_list_a ) > set_li6773872926390105121list_a > $o ).

thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    member5222132204073479906list_a: ( list_a > set_list_list_a ) > set_li6586005042197886593list_a > $o ).

thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member4263473470251683292list_a: ( list_a > set_list_a ) > set_li1071299071675007611list_a > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
    member_nat_list_a: ( nat > list_a ) > set_nat_list_a > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member491565700723299188list_a: ( nat > set_list_a ) > set_nat_set_list_a > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
    member_nat_a: ( nat > a ) > set_nat_a > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_J,type,
    member3288041530688867042list_a: ( set_list_a > set_list_set_list_a ) > set_se7318717477362135681list_a > $o ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member1188894495758700418list_a: ( set_a > set_list_a ) > set_set_a_set_list_a > $o ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_Itf__a_J_J,type,
    member_set_a_set_a: ( set_a > set_a ) > set_set_a_set_a > $o ).

thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).

thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member_a_set_list_a: ( a > set_list_a ) > set_a_set_list_a > $o ).

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

thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    member352051402189872281list_a: list_list_set_list_a > set_li664707282716828624list_a > $o ).

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__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    member6124916891863447321list_a: list_set_list_list_a > set_li7845362039408639568list_a > $o ).

thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member5524387281408368019list_a: list_set_list_a > set_list_set_list_a > $o ).

thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
    member_list_set_a: list_set_a > set_list_set_a > $o ).

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__Set__Oset_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    member2758200387949059059list_a: set_list_set_list_a > set_se895765194286668842list_a > $o ).

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

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

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

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

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

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

thf(sy_v_K,type,
    k: set_a ).

thf(sy_v_R,type,
    r: partia2175431115845679010xt_a_b ).

thf(sy_v_f,type,
    f: list_a ).

% Relevant facts (1264)
thf(fact_0__092_060open_062d_Odimension_A_Idegree_Af_J_A_Irupture__surj_AK_Af_A_096_Apoly__of__const_A_096_AK_J_A_Icarrier_A_IRupt_AK_Af_J_J_092_060close_062,axiom,
    embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( minus_minus_nat @ ( size_size_list_a @ f ) @ one_one_nat ) @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ k ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ).

% \<open>d.dimension (degree f) (rupture_surj K f ` poly_of_const ` K) (carrier (Rupt K f))\<close>
thf(fact_1_I_Oa__rcos__const,axiom,
    ! [H: list_a] :
      ( ( member_list_a @ H @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) )
     => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ H )
        = ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) ) ).

% I.a_rcos_const
thf(fact_2_domain__axioms,axiom,
    domain_a_b @ r ).

% domain_axioms
thf(fact_3_assms_I1_J,axiom,
    subfield_a_b @ k @ r ).

% assms(1)
thf(fact_4_p_Ofactorial__domain__axioms,axiom,
    ring_f796907574329358751t_unit @ ( univ_poly_a_b @ r @ k ) ).

% p.factorial_domain_axioms
thf(fact_5_assms_I2_J,axiom,
    member_list_a @ f @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% assms(2)
thf(fact_6_p_Onoetherian__domain__axioms,axiom,
    ring_n4705423059119889713t_unit @ ( univ_poly_a_b @ r @ k ) ).

% p.noetherian_domain_axioms
thf(fact_7_d_Oonepideal,axiom,
    princi5566405997746891961t_unit @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ).

% d.onepideal
thf(fact_8_p_Onoetherian__ring__axioms,axiom,
    ring_n5188127996776581661t_unit @ ( univ_poly_a_b @ r @ k ) ).

% p.noetherian_ring_axioms
thf(fact_9_p_Oprincipal__domain__axioms,axiom,
    ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ k ) ).

% p.principal_domain_axioms
thf(fact_10_I_Ois__additive__subgroup,axiom,
    additi4714453376129182166t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ ( univ_poly_a_b @ r @ k ) ).

% I.is_additive_subgroup
thf(fact_11_d_Ocgenideal__self,axiom,
    ! [I: set_list_a] :
      ( ( member_set_list_a @ I @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( member_set_list_a @ I @ ( cgenid9032708300698165283t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ I ) ) ) ).

% d.cgenideal_self
thf(fact_12_b,axiom,
    subfie4339374749748326226t_unit @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ k ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ).

% b
thf(fact_13_d_Osemiring__axioms,axiom,
    semiri4000464634269493571t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ).

% d.semiring_axioms
thf(fact_14_d_Ofinite__dimensionE_H,axiom,
    ! [K: set_set_list_a,E: set_set_list_a] :
      ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ E )
     => ~ ! [N: nat] :
            ~ ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N @ K @ E ) ) ).

% d.finite_dimensionE'
thf(fact_15_d_Ofinite__dimensionI,axiom,
    ! [N2: nat,K: set_set_list_a,E: set_set_list_a] :
      ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N2 @ K @ E )
     => ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ E ) ) ).

% d.finite_dimensionI
thf(fact_16_d_Ofinite__dimension__def,axiom,
    ! [K: set_set_list_a,E: set_set_list_a] :
      ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ E )
      = ( ? [N3: nat] : ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N3 @ K @ E ) ) ) ).

% d.finite_dimension_def
thf(fact_17_I_Oa__subgroup__in__rcosets,axiom,
    member_set_list_a @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) ).

% I.a_subgroup_in_rcosets
thf(fact_18_univ__poly__is__principal,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_is_principal
thf(fact_19_p_Ocgenideal__self,axiom,
    ! [I: list_a] :
      ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ I ) ) ) ).

% p.cgenideal_self
thf(fact_20_I_Oa__Hcarr,axiom,
    ! [H: list_a] :
      ( ( member_list_a @ H @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) )
     => ( member_list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% I.a_Hcarr
thf(fact_21_I_OIcarr,axiom,
    ! [I: list_a] :
      ( ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) )
     => ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% I.Icarr
thf(fact_22_d_Odimension__is__inj,axiom,
    ! [K: set_set_list_a,N2: nat,E: set_set_list_a,M: nat] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N2 @ K @ E )
       => ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ M @ K @ E )
         => ( N2 = M ) ) ) ) ).

% d.dimension_is_inj
thf(fact_23_d_Otelescopic__base__dim_I1_J,axiom,
    ! [K: set_set_list_a,F: set_set_list_a,E: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( subfie4339374749748326226t_unit @ F @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
       => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ F )
         => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ F @ E )
           => ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ E ) ) ) ) ) ).

% d.telescopic_base_dim(1)
thf(fact_24_I_Orcos__const__imp__mem,axiom,
    ! [I: list_a] :
      ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ I )
          = ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) )
       => ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) ) ) ).

% I.rcos_const_imp_mem
thf(fact_25_I_Oa__repr__independenceD,axiom,
    ! [Y: list_a,X: list_a] :
      ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ X )
          = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ Y ) )
       => ( member_list_a @ Y @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ X ) ) ) ) ).

% I.a_repr_independenceD
thf(fact_26_I_Oa__repr__independence_H,axiom,
    ! [Y: list_a,X: list_a] :
      ( ( member_list_a @ Y @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ X ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ X )
          = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ Y ) ) ) ) ).

% I.a_repr_independence'
thf(fact_27_I_Oa__rcos__self,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( member_list_a @ X @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ X ) ) ) ).

% I.a_rcos_self
thf(fact_28_I_Oa__elemrcos__carrier,axiom,
    ! [A: list_a,A2: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A2 @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ A ) )
       => ( member_list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% I.a_elemrcos_carrier
thf(fact_29_d_Otelescopic__base__aux,axiom,
    ! [K: set_set_list_a,F: set_set_list_a,N2: nat,E: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( subfie4339374749748326226t_unit @ F @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
       => ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N2 @ K @ F )
         => ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ one_one_nat @ F @ E )
           => ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N2 @ K @ E ) ) ) ) ) ).

% d.telescopic_base_aux
thf(fact_30_d_Ounique__dimension,axiom,
    ! [K: set_set_list_a,E: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ E )
       => ? [X2: nat] :
            ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X2 @ K @ E )
            & ! [Y2: nat] :
                ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ Y2 @ K @ E )
               => ( Y2 = X2 ) ) ) ) ) ).

% d.unique_dimension
thf(fact_31_d_Odimension__one,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ one_one_nat @ K @ K ) ) ).

% d.dimension_one
thf(fact_32_I_Oa__setinv__closed,axiom,
    ! [K: set_list_a] :
      ( ( member_set_list_a @ K @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) )
     => ( member_set_list_a @ ( a_SET_7640846956710366103t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) ) ) ).

% I.a_setinv_closed
thf(fact_33_univ__poly__is__euclidean,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ring_e7478897652244013592t_unit @ ( univ_poly_a_b @ r @ K )
        @ ^ [P: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).

% univ_poly_is_euclidean
thf(fact_34_p_Ocgenideal__is__principalideal,axiom,
    ! [I: list_a] :
      ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ I ) @ ( univ_poly_a_b @ r @ k ) ) ) ).

% p.cgenideal_is_principalideal
thf(fact_35_p_Oonepideal,axiom,
    princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).

% p.onepideal
thf(fact_36_poly__of__const__over__subfield,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K )
        = ( collect_list_a
          @ ^ [P: list_a] :
              ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
              & ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
                = zero_zero_nat ) ) ) ) ) ).

% poly_of_const_over_subfield
thf(fact_37_I_Oa__coset__eq,axiom,
    ! [X3: list_a] :
      ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ X3 )
        = ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ X3 @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) ) ) ).

% I.a_coset_eq
thf(fact_38_d_Ofinite__dimension__imp__subalgebra,axiom,
    ! [K: set_set_list_a,E: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ E )
       => ( embedd3586486045337765042t_unit @ K @ E @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.finite_dimension_imp_subalgebra
thf(fact_39_I_Oa__rcosets__carrier,axiom,
    ! [X4: set_list_a] :
      ( ( member_set_list_a @ X4 @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) )
     => ( ord_le8861187494160871172list_a @ X4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% I.a_rcosets_carrier
thf(fact_40_d_Ospace__subgroup__props_I1_J,axiom,
    ! [K: set_set_list_a,N2: nat,E: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N2 @ K @ E )
       => ( ord_le8877086941679407844list_a @ E @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ).

% d.space_subgroup_props(1)
thf(fact_41_long__division__closed_I1_J,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% long_division_closed(1)
thf(fact_42_univ__poly__infinite__dimension,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).

% univ_poly_infinite_dimension
thf(fact_43_I_Oa__rcosets__part__G,axiom,
    ( ( comple6928918032620976721list_a @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) )
    = ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% I.a_rcosets_part_G
thf(fact_44_mem__Collect__eq,axiom,
    ! [A: nat > set_list_a,P3: ( nat > set_list_a ) > $o] :
      ( ( member491565700723299188list_a @ A @ ( collec7638719012365873078list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_45_mem__Collect__eq,axiom,
    ! [A: nat > a,P3: ( nat > a ) > $o] :
      ( ( member_nat_a @ A @ ( collect_nat_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
    ! [A: a > list_a,P3: ( a > list_a ) > $o] :
      ( ( member_a_list_a @ A @ ( collect_a_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
    ! [A: a > set_list_a,P3: ( a > set_list_a ) > $o] :
      ( ( member_a_set_list_a @ A @ ( collect_a_set_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A: list_a,P3: list_a > $o] :
      ( ( member_list_a @ A @ ( collect_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
    ! [A: list_list_a,P3: list_list_a > $o] :
      ( ( member_list_list_a @ A @ ( collect_list_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
    ! [A: list_set_list_a,P3: list_set_list_a > $o] :
      ( ( member5524387281408368019list_a @ A @ ( collec5381118732811369429list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
    ! [A: nat,P3: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
    ! [A: a,P3: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_53_Collect__mem__eq,axiom,
    ! [A3: set_nat_set_list_a] :
      ( ( collec7638719012365873078list_a
        @ ^ [X5: nat > set_list_a] : ( member491565700723299188list_a @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
    ! [A3: set_nat_a] :
      ( ( collect_nat_a
        @ ^ [X5: nat > a] : ( member_nat_a @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
    ! [A3: set_a_list_a] :
      ( ( collect_a_list_a
        @ ^ [X5: a > list_a] : ( member_a_list_a @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
    ! [A3: set_a_set_list_a] :
      ( ( collect_a_set_list_a
        @ ^ [X5: a > set_list_a] : ( member_a_set_list_a @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
    ! [A3: set_list_a] :
      ( ( collect_list_a
        @ ^ [X5: list_a] : ( member_list_a @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
    ! [A3: set_list_list_a] :
      ( ( collect_list_list_a
        @ ^ [X5: list_list_a] : ( member_list_list_a @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
    ! [A3: set_list_set_list_a] :
      ( ( collec5381118732811369429list_a
        @ ^ [X5: list_set_list_a] : ( member5524387281408368019list_a @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X5: nat] : ( member_nat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
    ! [A3: set_a] :
      ( ( collect_a
        @ ^ [X5: a] : ( member_a @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_62_Collect__cong,axiom,
    ! [P3: list_a > $o,Q2: list_a > $o] :
      ( ! [X2: list_a] :
          ( ( P3 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collect_list_a @ P3 )
        = ( collect_list_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_63_Collect__cong,axiom,
    ! [P3: list_list_a > $o,Q2: list_list_a > $o] :
      ( ! [X2: list_list_a] :
          ( ( P3 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collect_list_list_a @ P3 )
        = ( collect_list_list_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_64_Collect__cong,axiom,
    ! [P3: list_set_list_a > $o,Q2: list_set_list_a > $o] :
      ( ! [X2: list_set_list_a] :
          ( ( P3 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collec5381118732811369429list_a @ P3 )
        = ( collec5381118732811369429list_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_65_Collect__cong,axiom,
    ! [P3: nat > $o,Q2: nat > $o] :
      ( ! [X2: nat] :
          ( ( P3 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collect_nat @ P3 )
        = ( collect_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_66_Collect__cong,axiom,
    ! [P3: a > $o,Q2: a > $o] :
      ( ! [X2: a] :
          ( ( P3 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collect_a @ P3 )
        = ( collect_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_67__092_060open_062inj__on_A_Irupture__surj_AK_Af_J_A_Ipoly__of__const_A_096_AK_J_092_060close_062,axiom,
    inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ k ) ).

% \<open>inj_on (rupture_surj K f) (poly_of_const ` K)\<close>
thf(fact_68_d_Osubring__props_I1_J,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ord_le8877086941679407844list_a @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.subring_props(1)
thf(fact_69_p_Oa__r__coset__subset__G,axiom,
    ! [H2: set_list_a,X: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ord_le8861187494160871172list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.a_r_coset_subset_G
thf(fact_70_p_Oa__l__coset__subset__G,axiom,
    ! [H2: set_list_a,X: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ H2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.a_l_coset_subset_G
thf(fact_71_p_Oa__rcosetsI,axiom,
    ! [H2: set_list_a,X: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( member_set_list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ X ) @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 ) ) ) ) ).

% p.a_rcosetsI
thf(fact_72_I_Oa__subset,axiom,
    ord_le8861187494160871172list_a @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% I.a_subset
thf(fact_73_d_Ocarrier__is__subalgebra,axiom,
    ! [K: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( embedd3586486045337765042t_unit @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% d.carrier_is_subalgebra
thf(fact_74_d_Osubalgebra__in__carrier,axiom,
    ! [K: set_set_list_a,V: set_set_list_a] :
      ( ( embedd3586486045337765042t_unit @ K @ V @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ord_le8877086941679407844list_a @ V @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.subalgebra_in_carrier
thf(fact_75_d_Osubalbegra__incl__imp__finite__dimension,axiom,
    ! [K: set_set_list_a,E: set_set_list_a,V: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ E )
       => ( ( embedd3586486045337765042t_unit @ K @ V @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
         => ( ( ord_le8877086941679407844list_a @ V @ E )
           => ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ V ) ) ) ) ) ).

% d.subalbegra_incl_imp_finite_dimension
thf(fact_76_assms_I3_J,axiom,
    ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ f ) @ one_one_nat ) ).

% assms(3)
thf(fact_77_rupture__dimension,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) )
         => ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) ) ) ) ) ) ).

% rupture_dimension
thf(fact_78_h_Opoly__of__const__over__subfield,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( image_a_list_a
          @ ( poly_of_const_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) )
          @ K )
        = ( collect_list_a
          @ ^ [P: list_a] :
              ( ( member_list_a @ P
                @ ( partia5361259788508890537t_unit
                  @ ( univ_poly_a_b
                    @ ( partia8674076737563717228xt_a_b
                      @ ^ [Uu: set_a] : k
                      @ r )
                    @ K ) ) )
              & ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
                = zero_zero_nat ) ) ) ) ) ).

% h.poly_of_const_over_subfield
thf(fact_79_domain_Ouniv__poly__is__euclidean,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ring_e6072138977393874316t_unit @ ( univ_p2250591967980070728t_unit @ R @ K )
          @ ^ [P: list_list_list_a] : ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat ) ) ) ) ).

% domain.univ_poly_is_euclidean
thf(fact_80_domain_Ouniv__poly__is__euclidean,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ring_e1884229898437606828t_unit @ ( univ_p2555602637952293736t_unit @ R @ K )
          @ ^ [P: list_list_set_list_a] : ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P ) @ one_one_nat ) ) ) ) ).

% domain.univ_poly_is_euclidean
thf(fact_81_domain_Ouniv__poly__is__euclidean,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ring_e499914632588284338t_unit @ ( univ_p863672496597069550t_unit @ R @ K )
          @ ^ [P: list_set_list_a] : ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P ) @ one_one_nat ) ) ) ) ).

% domain.univ_poly_is_euclidean
thf(fact_82_domain_Ouniv__poly__is__euclidean,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ring_e7478897652244013592t_unit @ ( univ_poly_a_b @ R @ K )
          @ ^ [P: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).

% domain.univ_poly_is_euclidean
thf(fact_83_domain_Ouniv__poly__is__euclidean,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ring_e6434146001954145682t_unit @ ( univ_p7953238456130426574t_unit @ R @ K )
          @ ^ [P: list_list_a] : ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ).

% domain.univ_poly_is_euclidean
thf(fact_84_d_Oa__l__coset__subset__G,axiom,
    ! [H2: set_set_list_a,X: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( ord_le8877086941679407844list_a @ ( a_l_co4135707798524667186t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X @ H2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ).

% d.a_l_coset_subset_G
thf(fact_85_d_Ogenideal__self,axiom,
    ! [S: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ S @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ord_le8877086941679407844list_a @ S @ ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ S ) ) ) ).

% d.genideal_self
thf(fact_86_d_Osubset__Idl__subset,axiom,
    ! [I2: set_set_list_a,H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ I2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( ord_le8877086941679407844list_a @ H2 @ I2 )
       => ( ord_le8877086941679407844list_a @ ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ H2 ) @ ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ I2 ) ) ) ) ).

% d.subset_Idl_subset
thf(fact_87_domain_Ouniv__poly__infinite__dimension,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ~ ( embedd5776004836630637299t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ ( image_1156962946714028939list_a @ ( poly_o1617770581224298896t_unit @ R ) @ K ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) ) ) ) ).

% domain.univ_poly_infinite_dimension
thf(fact_88_domain_Ouniv__poly__infinite__dimension,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ~ ( embedd7099302156544194323t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ ( image_3300606519373034379list_a @ ( poly_o1450351294650543536t_unit @ R ) @ K ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) ) ) ) ).

% domain.univ_poly_infinite_dimension
thf(fact_89_domain_Ouniv__poly__infinite__dimension,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ~ ( embedd2331549258725911193t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( image_3509949574358669579list_a @ ( poly_o3535013565302768950t_unit @ R ) @ K ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) ) ) ) ).

% domain.univ_poly_infinite_dimension
thf(fact_90_domain_Ouniv__poly__infinite__dimension,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ R @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ R ) @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ).

% domain.univ_poly_infinite_dimension
thf(fact_91_domain_Ouniv__poly__infinite__dimension,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ~ ( embedd2411333406617385593t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ R ) @ K ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ).

% domain.univ_poly_infinite_dimension
thf(fact_92_I_Oa__cosets__finite,axiom,
    ! [C: set_list_a] :
      ( ( member_set_list_a @ C @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( finite_finite_list_a @ C ) ) ) ) ).

% I.a_cosets_finite
thf(fact_93_d_Oline__extension__in__carrier,axiom,
    ! [K: set_set_list_a,A: set_list_a,E: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( ( ord_le8877086941679407844list_a @ E @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
         => ( ord_le8877086941679407844list_a @ ( embedd5951720228509767443t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ A @ E ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ) ).

% d.line_extension_in_carrier
thf(fact_94_p_Ouniv__poly__infinite__dimension,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ~ ( embedd2411333406617385593t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ).

% p.univ_poly_infinite_dimension
thf(fact_95_p_Ouniv__poly__subfield__of__consts,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% p.univ_poly_subfield_of_consts
thf(fact_96_p_Odomain__axioms,axiom,
    domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) ).

% p.domain_axioms
thf(fact_97_p_Otelescopic__base__dim_I1_J,axiom,
    ! [K: set_list_a,F: set_list_a,E: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( subfie1779122896746047282t_unit @ F @ ( univ_poly_a_b @ r @ k ) )
       => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ F )
         => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ F @ E )
           => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E ) ) ) ) ) ).

% p.telescopic_base_dim(1)
thf(fact_98__092_060open_062subfield_AK_A_IR_092_060lparr_062carrier_A_058_061_AK_092_060rparr_062_J_092_060close_062,axiom,
    ( subfield_a_b @ k
    @ ( partia8674076737563717228xt_a_b
      @ ^ [Uu: set_a] : k
      @ r ) ) ).

% \<open>subfield K (R\<lparr>carrier := K\<rparr>)\<close>
thf(fact_99_p_Osubring__props_I1_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.subring_props(1)
thf(fact_100_univ__poly__subfield__of__consts,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_subfield_of_consts
thf(fact_101_p_Ouniv__poly__is__euclidean,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ring_e6434146001954145682t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K )
        @ ^ [P: list_list_a] : ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ).

% p.univ_poly_is_euclidean
thf(fact_102_d_Ouniv__poly__subfield__of__consts,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( subfie868023609828841932t_unit @ ( image_3509949574358669579list_a @ ( poly_o3535013565302768950t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ K ) @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K ) ) ) ).

% d.univ_poly_subfield_of_consts
thf(fact_103_p_Opoly__of__const__over__subfield,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K )
        = ( collect_list_list_a
          @ ^ [P: list_list_a] :
              ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
              & ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
                = zero_zero_nat ) ) ) ) ) ).

% p.poly_of_const_over_subfield
thf(fact_104_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_105_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_106_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: multiset_list_a] :
      ( ( minus_7431248565939055793list_a @ A @ A )
      = zero_z4454100511807792257list_a ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_107_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ A @ A )
      = zero_zero_multiset_a ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_108_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_109_diff__zero,axiom,
    ! [A: multiset_list_a] :
      ( ( minus_7431248565939055793list_a @ A @ zero_z4454100511807792257list_a )
      = A ) ).

% diff_zero
thf(fact_110_diff__zero,axiom,
    ! [A: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ A @ zero_zero_multiset_a )
      = A ) ).

% diff_zero
thf(fact_111_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_112_zero__diff,axiom,
    ! [A: multiset_list_a] :
      ( ( minus_7431248565939055793list_a @ zero_z4454100511807792257list_a @ A )
      = zero_z4454100511807792257list_a ) ).

% zero_diff
thf(fact_113_zero__diff,axiom,
    ! [A: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ A )
      = zero_zero_multiset_a ) ).

% zero_diff
thf(fact_114_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_115_h_Ouniv__poly__subfield__of__consts,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( subfie1779122896746047282t_unit
        @ ( image_a_list_a
          @ ( poly_of_const_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) )
          @ K )
        @ ( univ_poly_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ K ) ) ) ).

% h.univ_poly_subfield_of_consts
thf(fact_116_d_Opoly__of__const__over__subfield,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( image_3509949574358669579list_a @ ( poly_o3535013565302768950t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ K )
        = ( collec5381118732811369429list_a
          @ ^ [P: list_set_list_a] :
              ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K ) ) )
              & ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P ) @ one_one_nat )
                = zero_zero_nat ) ) ) ) ) ).

% d.poly_of_const_over_subfield
thf(fact_117_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_118_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_119_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_120_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_121_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_122_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_123_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_124_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_125_zero__reorient,axiom,
    ! [X: multiset_list_a] :
      ( ( zero_z4454100511807792257list_a = X )
      = ( X = zero_z4454100511807792257list_a ) ) ).

% zero_reorient
thf(fact_126_zero__reorient,axiom,
    ! [X: multiset_a] :
      ( ( zero_zero_multiset_a = X )
      = ( X = zero_zero_multiset_a ) ) ).

% zero_reorient
thf(fact_127_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_128_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_129_ring_Opoly__of__const_Ocong,axiom,
    poly_of_const_a_b = poly_of_const_a_b ).

% ring.poly_of_const.cong
thf(fact_130_ring_Opoly__of__const_Ocong,axiom,
    poly_o8716471131768098070t_unit = poly_o8716471131768098070t_unit ).

% ring.poly_of_const.cong
thf(fact_131_ring_Opoly__of__const_Ocong,axiom,
    poly_o3535013565302768950t_unit = poly_o3535013565302768950t_unit ).

% ring.poly_of_const.cong
thf(fact_132_ring_Opdiv_Ocong,axiom,
    polynomial_pdiv_a_b = polynomial_pdiv_a_b ).

% ring.pdiv.cong
thf(fact_133_ring_Opdiv_Ocong,axiom,
    polyno5893782122288709345t_unit = polyno5893782122288709345t_unit ).

% ring.pdiv.cong
thf(fact_134_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_135_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_136_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_137_domain_Orupture__dimension,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat ) )
           => ( embedd2654441813954712322t_unit @ ( polyno8931960069169623149t_unit @ R @ K @ P2 ) @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat ) @ ( image_5096927308667620453list_a @ ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 ) ) @ ( image_1156962946714028939list_a @ ( poly_o1617770581224298896t_unit @ R ) @ K ) ) @ ( partia7210328371471923989t_unit @ ( polyno8931960069169623149t_unit @ R @ K @ P2 ) ) ) ) ) ) ) ).

% domain.rupture_dimension
thf(fact_138_domain_Orupture__dimension,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P2 ) @ one_one_nat ) )
           => ( embedd6718854712678180642t_unit @ ( polyno1128427740506820749t_unit @ R @ K @ P2 ) @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P2 ) @ one_one_nat ) @ ( image_3381647565828645989list_a @ ( a_r_co6659872811071116064t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ ( cgenid3857712956745501987t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ P2 ) ) @ ( image_3300606519373034379list_a @ ( poly_o1450351294650543536t_unit @ R ) @ K ) ) @ ( partia5834437165032953589t_unit @ ( polyno1128427740506820749t_unit @ R @ K @ P2 ) ) ) ) ) ) ) ).

% domain.rupture_dimension
thf(fact_139_domain_Orupture__dimension,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat ) )
           => ( embedd8319361068390300840t_unit @ ( polyno7054224608333822611t_unit @ R @ K @ P2 ) @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat ) @ ( image_1309896914751589349list_a @ ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 ) ) @ ( image_3509949574358669579list_a @ ( poly_o3535013565302768950t_unit @ R ) @ K ) ) @ ( partia5585283898957912943t_unit @ ( polyno7054224608333822611t_unit @ R @ K @ P2 ) ) ) ) ) ) ) ).

% domain.rupture_dimension
thf(fact_140_domain_Orupture__dimension,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) )
           => ( embedd7670639971858547848t_unit @ ( polyno859807163042199155t_unit @ R @ K @ P2 ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( image_551801017575455717list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) ) @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ R ) @ K ) ) @ ( partia3317168157747563407t_unit @ ( polyno859807163042199155t_unit @ R @ K @ P2 ) ) ) ) ) ) ) ).

% domain.rupture_dimension
thf(fact_141_domain_Orupture__dimension,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) )
           => ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ R @ K @ P2 ) @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) @ ( image_a_list_a @ ( poly_of_const_a_b @ R ) @ K ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ R @ K @ P2 ) ) ) ) ) ) ) ).

% domain.rupture_dimension
thf(fact_142_domain_Ouniv__poly__is__principal,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ring_p8404492108403472412t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_principal
thf(fact_143_domain_Ouniv__poly__is__principal,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ring_p1388813177073033276t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_principal
thf(fact_144_domain_Ouniv__poly__is__principal,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ring_p4603458870916348994t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_principal
thf(fact_145_domain_Ouniv__poly__is__principal,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.univ_poly_is_principal
thf(fact_146_domain_Ouniv__poly__is__principal,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_principal
thf(fact_147_domain_Olong__division__closed_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
           => ( member5342144027231129785list_a @ ( polyno4115915122720352731t_unit @ R @ P2 @ Q ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(1)
thf(fact_148_domain_Olong__division__closed_I1_J,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
           => ( member352051402189872281list_a @ ( polyno5933693564459536891t_unit @ R @ P2 @ Q ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(1)
thf(fact_149_domain_Olong__division__closed_I1_J,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
           => ( member5524387281408368019list_a @ ( polyno6856256803263433473t_unit @ R @ P2 @ Q ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(1)
thf(fact_150_domain_Olong__division__closed_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( member_list_a @ ( polynomial_pdiv_a_b @ R @ P2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(1)
thf(fact_151_domain_Olong__division__closed_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( member_list_list_a @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(1)
thf(fact_152_p_Odegree__zero__imp__splitted,axiom,
    ! [P2: list_list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
          = zero_zero_nat )
       => ( polyno6259083269128200473t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) ) ).

% p.degree_zero_imp_splitted
thf(fact_153_p_Ouniv__poly__is__principal,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% p.univ_poly_is_principal
thf(fact_154_p_Oorder__gt__0__iff__finite,axiom,
    ( ( ord_less_nat @ zero_zero_nat @ ( order_3240872107759947550t_unit @ ( univ_poly_a_b @ r @ k ) ) )
    = ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.order_gt_0_iff_finite
thf(fact_155_p_Odegree__zero__imp__not__is__root,axiom,
    ! [P2: list_list_a,X: list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
          = zero_zero_nat )
       => ~ ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X ) ) ) ).

% p.degree_zero_imp_not_is_root
thf(fact_156_d_Opirreducible__degree,axiom,
    ! [K: set_set_list_a,P2: list_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K ) ) )
       => ( ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K ) @ P2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).

% d.pirreducible_degree
thf(fact_157_p_Ofinite__number__of__roots,axiom,
    ! [P2: list_list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( finite_finite_list_a @ ( collect_list_a @ ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) ) ) ).

% p.finite_number_of_roots
thf(fact_158_finite__UN__I,axiom,
    ! [A3: set_nat,B2: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( finite_finite_nat @ ( B2 @ A4 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_159_finite__UN__I,axiom,
    ! [A3: set_a,B2: a > set_nat] :
      ( ( finite_finite_a @ A3 )
     => ( ! [A4: a] :
            ( ( member_a @ A4 @ A3 )
           => ( finite_finite_nat @ ( B2 @ A4 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B2 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_160_finite__UN__I,axiom,
    ! [A3: set_nat,B2: nat > set_a] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( finite_finite_a @ ( B2 @ A4 ) ) )
       => ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B2 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_161_finite__UN__I,axiom,
    ! [A3: set_a,B2: a > set_a] :
      ( ( finite_finite_a @ A3 )
     => ( ! [A4: a] :
            ( ( member_a @ A4 @ A3 )
           => ( finite_finite_a @ ( B2 @ A4 ) ) )
       => ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_162_finite__UN__I,axiom,
    ! [A3: set_list_a,B2: list_a > set_nat] :
      ( ( finite_finite_list_a @ A3 )
     => ( ! [A4: list_a] :
            ( ( member_list_a @ A4 @ A3 )
           => ( finite_finite_nat @ ( B2 @ A4 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B2 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_163_finite__UN__I,axiom,
    ! [A3: set_nat,B2: nat > set_list_a] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A4: nat] :
            ( ( member_nat @ A4 @ A3 )
           => ( finite_finite_list_a @ ( B2 @ A4 ) ) )
       => ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B2 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_164_finite__UN__I,axiom,
    ! [A3: set_a,B2: a > set_list_a] :
      ( ( finite_finite_a @ A3 )
     => ( ! [A4: a] :
            ( ( member_a @ A4 @ A3 )
           => ( finite_finite_list_a @ ( B2 @ A4 ) ) )
       => ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_a_set_list_a @ B2 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_165_finite__UN__I,axiom,
    ! [A3: set_list_a,B2: list_a > set_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( ! [A4: list_a] :
            ( ( member_list_a @ A4 @ A3 )
           => ( finite_finite_a @ ( B2 @ A4 ) ) )
       => ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ B2 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_166_finite__UN__I,axiom,
    ! [A3: set_nat_a,B2: ( nat > a ) > set_nat] :
      ( ( finite_finite_nat_a @ A3 )
     => ( ! [A4: nat > a] :
            ( ( member_nat_a @ A4 @ A3 )
           => ( finite_finite_nat @ ( B2 @ A4 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_a_set_nat @ B2 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_167_finite__UN__I,axiom,
    ! [A3: set_set_list_a,B2: set_list_a > set_nat] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ( ! [A4: set_list_a] :
            ( ( member_set_list_a @ A4 @ A3 )
           => ( finite_finite_nat @ ( B2 @ A4 ) ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_7346808305986241189et_nat @ B2 @ A3 ) ) ) ) ) ).

% finite_UN_I
thf(fact_168_pmod__const_I1_J,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
           => ( ( polynomial_pdiv_a_b @ r @ P2 @ Q )
              = nil_a ) ) ) ) ) ).

% pmod_const(1)
thf(fact_169_rupture__carrier__as__pmod__image,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) )
            @ ( image_list_a_list_a
              @ ^ [Q3: list_a] : ( polynomial_pmod_a_b @ r @ Q3 @ P2 )
              @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) )
          = ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) ) ) ) ) ).

% rupture_carrier_as_pmod_image
thf(fact_170__092_060open_062poly__of__const_A_096_AK_A_092_060subseteq_062_A_I_092_060lambda_062q_O_Aq_Apmod_Af_J_A_096_Acarrier_A_IK_A_091X_093_J_092_060close_062,axiom,
    ( ord_le8861187494160871172list_a @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ k )
    @ ( image_list_a_list_a
      @ ^ [Q3: list_a] : ( polynomial_pmod_a_b @ r @ Q3 @ f )
      @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% \<open>poly_of_const ` K \<subseteq> (\<lambda>q. q pmod f) ` carrier (K [X])\<close>
thf(fact_171_finite__Diff2,axiom,
    ! [B2: set_list_set_list_a,A3: set_list_set_list_a] :
      ( ( finite3202133031812794515list_a @ B2 )
     => ( ( finite3202133031812794515list_a @ ( minus_4926175736359902257list_a @ A3 @ B2 ) )
        = ( finite3202133031812794515list_a @ A3 ) ) ) ).

% finite_Diff2
thf(fact_172_finite__Diff2,axiom,
    ! [B2: set_list_a,A3: set_list_a] :
      ( ( finite_finite_list_a @ B2 )
     => ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A3 @ B2 ) )
        = ( finite_finite_list_a @ A3 ) ) ) ).

% finite_Diff2
thf(fact_173_finite__Diff2,axiom,
    ! [B2: set_set_list_a,A3: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ B2 )
     => ( ( finite5282473924520328461list_a @ ( minus_4782336368215558443list_a @ A3 @ B2 ) )
        = ( finite5282473924520328461list_a @ A3 ) ) ) ).

% finite_Diff2
thf(fact_174_finite__Diff2,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
        = ( finite_finite_nat @ A3 ) ) ) ).

% finite_Diff2
thf(fact_175_finite__Diff2,axiom,
    ! [B2: set_a,A3: set_a] :
      ( ( finite_finite_a @ B2 )
     => ( ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B2 ) )
        = ( finite_finite_a @ A3 ) ) ) ).

% finite_Diff2
thf(fact_176_finite__Diff,axiom,
    ! [A3: set_list_set_list_a,B2: set_list_set_list_a] :
      ( ( finite3202133031812794515list_a @ A3 )
     => ( finite3202133031812794515list_a @ ( minus_4926175736359902257list_a @ A3 @ B2 ) ) ) ).

% finite_Diff
thf(fact_177_finite__Diff,axiom,
    ! [A3: set_list_a,B2: set_list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A3 @ B2 ) ) ) ).

% finite_Diff
thf(fact_178_finite__Diff,axiom,
    ! [A3: set_set_list_a,B2: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ( finite5282473924520328461list_a @ ( minus_4782336368215558443list_a @ A3 @ B2 ) ) ) ).

% finite_Diff
thf(fact_179_finite__Diff,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).

% finite_Diff
thf(fact_180_finite__Diff,axiom,
    ! [A3: set_a,B2: set_a] :
      ( ( finite_finite_a @ A3 )
     => ( finite_finite_a @ ( minus_minus_set_a @ A3 @ B2 ) ) ) ).

% finite_Diff
thf(fact_181_long__division__closed_I2_J,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( member_list_a @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% long_division_closed(2)
thf(fact_182_finite__Collect__disjI,axiom,
    ! [P3: list_list_a > $o,Q2: list_list_a > $o] :
      ( ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [X5: list_list_a] :
              ( ( P3 @ X5 )
              | ( Q2 @ X5 ) ) ) )
      = ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P3 ) )
        & ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_183_finite__Collect__disjI,axiom,
    ! [P3: list_set_list_a > $o,Q2: list_set_list_a > $o] :
      ( ( finite3202133031812794515list_a
        @ ( collec5381118732811369429list_a
          @ ^ [X5: list_set_list_a] :
              ( ( P3 @ X5 )
              | ( Q2 @ X5 ) ) ) )
      = ( ( finite3202133031812794515list_a @ ( collec5381118732811369429list_a @ P3 ) )
        & ( finite3202133031812794515list_a @ ( collec5381118732811369429list_a @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_184_finite__Collect__disjI,axiom,
    ! [P3: list_a > $o,Q2: list_a > $o] :
      ( ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X5: list_a] :
              ( ( P3 @ X5 )
              | ( Q2 @ X5 ) ) ) )
      = ( ( finite_finite_list_a @ ( collect_list_a @ P3 ) )
        & ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_185_finite__Collect__disjI,axiom,
    ! [P3: set_list_a > $o,Q2: set_list_a > $o] :
      ( ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [X5: set_list_a] :
              ( ( P3 @ X5 )
              | ( Q2 @ X5 ) ) ) )
      = ( ( finite5282473924520328461list_a @ ( collect_set_list_a @ P3 ) )
        & ( finite5282473924520328461list_a @ ( collect_set_list_a @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_186_finite__Collect__disjI,axiom,
    ! [P3: nat > $o,Q2: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( P3 @ X5 )
              | ( Q2 @ X5 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
        & ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_187_finite__Collect__disjI,axiom,
    ! [P3: a > $o,Q2: a > $o] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [X5: a] :
              ( ( P3 @ X5 )
              | ( Q2 @ X5 ) ) ) )
      = ( ( finite_finite_a @ ( collect_a @ P3 ) )
        & ( finite_finite_a @ ( collect_a @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_188_finite__Collect__conjI,axiom,
    ! [P3: list_list_a > $o,Q2: list_list_a > $o] :
      ( ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P3 ) )
        | ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q2 ) ) )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [X5: list_list_a] :
              ( ( P3 @ X5 )
              & ( Q2 @ X5 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_189_finite__Collect__conjI,axiom,
    ! [P3: list_set_list_a > $o,Q2: list_set_list_a > $o] :
      ( ( ( finite3202133031812794515list_a @ ( collec5381118732811369429list_a @ P3 ) )
        | ( finite3202133031812794515list_a @ ( collec5381118732811369429list_a @ Q2 ) ) )
     => ( finite3202133031812794515list_a
        @ ( collec5381118732811369429list_a
          @ ^ [X5: list_set_list_a] :
              ( ( P3 @ X5 )
              & ( Q2 @ X5 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_190_finite__Collect__conjI,axiom,
    ! [P3: list_a > $o,Q2: list_a > $o] :
      ( ( ( finite_finite_list_a @ ( collect_list_a @ P3 ) )
        | ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X5: list_a] :
              ( ( P3 @ X5 )
              & ( Q2 @ X5 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_191_finite__Collect__conjI,axiom,
    ! [P3: set_list_a > $o,Q2: set_list_a > $o] :
      ( ( ( finite5282473924520328461list_a @ ( collect_set_list_a @ P3 ) )
        | ( finite5282473924520328461list_a @ ( collect_set_list_a @ Q2 ) ) )
     => ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [X5: set_list_a] :
              ( ( P3 @ X5 )
              & ( Q2 @ X5 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_192_finite__Collect__conjI,axiom,
    ! [P3: nat > $o,Q2: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
        | ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X5: nat] :
              ( ( P3 @ X5 )
              & ( Q2 @ X5 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_193_finite__Collect__conjI,axiom,
    ! [P3: a > $o,Q2: a > $o] :
      ( ( ( finite_finite_a @ ( collect_a @ P3 ) )
        | ( finite_finite_a @ ( collect_a @ Q2 ) ) )
     => ( finite_finite_a
        @ ( collect_a
          @ ^ [X5: a] :
              ( ( P3 @ X5 )
              & ( Q2 @ X5 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_194_finite__Collect__less__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_nat @ N3 @ K2 ) ) ) ).

% finite_Collect_less_nat
thf(fact_195_finite__Collect__le__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K2 ) ) ) ).

% finite_Collect_le_nat
thf(fact_196_long__division__zero_I2_J,axiom,
    ! [K: set_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( polynomial_pmod_a_b @ r @ nil_a @ Q )
          = nil_a ) ) ) ).

% long_division_zero(2)
thf(fact_197_long__division__zero_I1_J,axiom,
    ! [K: set_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q )
          = nil_a ) ) ) ).

% long_division_zero(1)
thf(fact_198_p_Olong__division__closed_I1_J,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( member_list_list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ) ) ).

% p.long_division_closed(1)
thf(fact_199_rupture__surj__composed__with__pmod,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q )
            = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ ( polynomial_pmod_a_b @ r @ Q @ P2 ) ) ) ) ) ) ).

% rupture_surj_composed_with_pmod
thf(fact_200_finite__imageI,axiom,
    ! [F: set_nat,H: nat > nat] :
      ( ( finite_finite_nat @ F )
     => ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_201_finite__imageI,axiom,
    ! [F: set_nat,H: nat > a] :
      ( ( finite_finite_nat @ F )
     => ( finite_finite_a @ ( image_nat_a @ H @ F ) ) ) ).

% finite_imageI
thf(fact_202_finite__imageI,axiom,
    ! [F: set_a,H: a > nat] :
      ( ( finite_finite_a @ F )
     => ( finite_finite_nat @ ( image_a_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_203_finite__imageI,axiom,
    ! [F: set_a,H: a > a] :
      ( ( finite_finite_a @ F )
     => ( finite_finite_a @ ( image_a_a @ H @ F ) ) ) ).

% finite_imageI
thf(fact_204_finite__imageI,axiom,
    ! [F: set_list_a,H: list_a > nat] :
      ( ( finite_finite_list_a @ F )
     => ( finite_finite_nat @ ( image_list_a_nat @ H @ F ) ) ) ).

% finite_imageI
thf(fact_205_finite__imageI,axiom,
    ! [F: set_list_a,H: list_a > a] :
      ( ( finite_finite_list_a @ F )
     => ( finite_finite_a @ ( image_list_a_a @ H @ F ) ) ) ).

% finite_imageI
thf(fact_206_finite__imageI,axiom,
    ! [F: set_nat,H: nat > list_a] :
      ( ( finite_finite_nat @ F )
     => ( finite_finite_list_a @ ( image_nat_list_a @ H @ F ) ) ) ).

% finite_imageI
thf(fact_207_finite__imageI,axiom,
    ! [F: set_a,H: a > set_a] :
      ( ( finite_finite_a @ F )
     => ( finite_finite_set_a @ ( image_a_set_a @ H @ F ) ) ) ).

% finite_imageI
thf(fact_208_finite__imageI,axiom,
    ! [F: set_a,H: a > list_a] :
      ( ( finite_finite_a @ F )
     => ( finite_finite_list_a @ ( image_a_list_a @ H @ F ) ) ) ).

% finite_imageI
thf(fact_209_finite__imageI,axiom,
    ! [F: set_list_a,H: list_a > list_a] :
      ( ( finite_finite_list_a @ F )
     => ( finite_finite_list_a @ ( image_list_a_list_a @ H @ F ) ) ) ).

% finite_imageI
thf(fact_210_finite__Union,axiom,
    ! [A3: set_se895765194286668842list_a] :
      ( ( finite191489321941288691list_a @ A3 )
     => ( ! [M2: set_list_set_list_a] :
            ( ( member2758200387949059059list_a @ M2 @ A3 )
           => ( finite3202133031812794515list_a @ M2 ) )
       => ( finite3202133031812794515list_a @ ( comple4047202532333093431list_a @ A3 ) ) ) ) ).

% finite_Union
thf(fact_211_finite__Union,axiom,
    ! [A3: set_set_set_list_a] :
      ( ( finite6594153429226962157list_a @ A3 )
     => ( ! [M2: set_set_list_a] :
            ( ( member8857465052274545133list_a @ M2 @ A3 )
           => ( finite5282473924520328461list_a @ M2 ) )
       => ( finite5282473924520328461list_a @ ( comple4119837387707116081list_a @ A3 ) ) ) ) ).

% finite_Union
thf(fact_212_finite__Union,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ! [M2: set_nat] :
            ( ( member_set_nat @ M2 @ A3 )
           => ( finite_finite_nat @ M2 ) )
       => ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A3 ) ) ) ) ).

% finite_Union
thf(fact_213_finite__Union,axiom,
    ! [A3: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ( ! [M2: set_list_a] :
            ( ( member_set_list_a @ M2 @ A3 )
           => ( finite_finite_list_a @ M2 ) )
       => ( finite_finite_list_a @ ( comple6928918032620976721list_a @ A3 ) ) ) ) ).

% finite_Union
thf(fact_214_finite__Union,axiom,
    ! [A3: set_set_a] :
      ( ( finite_finite_set_a @ A3 )
     => ( ! [M2: set_a] :
            ( ( member_set_a @ M2 @ A3 )
           => ( finite_finite_a @ M2 ) )
       => ( finite_finite_a @ ( comple2307003609928055243_set_a @ A3 ) ) ) ) ).

% finite_Union
thf(fact_215_finite__Collect__subsets,axiom,
    ! [A3: set_list_set_list_a] :
      ( ( finite3202133031812794515list_a @ A3 )
     => ( finite191489321941288691list_a
        @ ( collec3034293182684082741list_a
          @ ^ [B3: set_list_set_list_a] : ( ord_le1961058059034850666list_a @ B3 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_216_finite__Collect__subsets,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_217_finite__Collect__subsets,axiom,
    ! [A3: set_list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [B3: set_list_a] : ( ord_le8861187494160871172list_a @ B3 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_218_finite__Collect__subsets,axiom,
    ! [A3: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ( finite6594153429226962157list_a
        @ ( collec3809296942973202735list_a
          @ ^ [B3: set_set_list_a] : ( ord_le8877086941679407844list_a @ B3 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_219_finite__Collect__subsets,axiom,
    ! [A3: set_a] :
      ( ( finite_finite_a @ A3 )
     => ( finite_finite_set_a
        @ ( collect_set_a
          @ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_220_finite__Collect__subsets,axiom,
    ! [A3: set_set_a] :
      ( ( finite_finite_set_a @ A3 )
     => ( finite7209287970140883943_set_a
        @ ( collect_set_set_a
          @ ^ [B3: set_set_a] : ( ord_le3724670747650509150_set_a @ B3 @ A3 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_221_pmod__const_I2_J,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
           => ( ( polynomial_pmod_a_b @ r @ P2 @ Q )
              = P2 ) ) ) ) ) ).

% pmod_const(2)
thf(fact_222_pmod__degree,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ( ( ( polynomial_pmod_a_b @ r @ P2 @ Q )
                = nil_a )
              | ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).

% pmod_degree
thf(fact_223_univ__poly__zero__closed,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ).

% univ_poly_zero_closed
thf(fact_224_univ__poly__zero__closed,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a] : ( member5524387281408368019list_a @ nil_set_list_a @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) ) ).

% univ_poly_zero_closed
thf(fact_225_univ__poly__zero__closed,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ).

% univ_poly_zero_closed
thf(fact_226_rupture__surj__inj__on,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) )
          @ ( image_list_a_list_a
            @ ^ [Q3: list_a] : ( polynomial_pmod_a_b @ r @ Q3 @ P2 )
            @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% rupture_surj_inj_on
thf(fact_227_finite__UN,axiom,
    ! [A3: set_nat,B2: nat > set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B2 @ A3 ) ) )
        = ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
             => ( finite_finite_nat @ ( B2 @ X5 ) ) ) ) ) ) ).

% finite_UN
thf(fact_228_finite__UN,axiom,
    ! [A3: set_a,B2: a > set_nat] :
      ( ( finite_finite_a @ A3 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_a_set_nat @ B2 @ A3 ) ) )
        = ( ! [X5: a] :
              ( ( member_a @ X5 @ A3 )
             => ( finite_finite_nat @ ( B2 @ X5 ) ) ) ) ) ) ).

% finite_UN
thf(fact_229_finite__UN,axiom,
    ! [A3: set_nat,B2: nat > set_a] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_nat_set_a @ B2 @ A3 ) ) )
        = ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
             => ( finite_finite_a @ ( B2 @ X5 ) ) ) ) ) ) ).

% finite_UN
thf(fact_230_finite__UN,axiom,
    ! [A3: set_a,B2: a > set_a] :
      ( ( finite_finite_a @ A3 )
     => ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B2 @ A3 ) ) )
        = ( ! [X5: a] :
              ( ( member_a @ X5 @ A3 )
             => ( finite_finite_a @ ( B2 @ X5 ) ) ) ) ) ) ).

% finite_UN
thf(fact_231_finite__UN,axiom,
    ! [A3: set_list_a,B2: list_a > set_nat] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_list_a_set_nat @ B2 @ A3 ) ) )
        = ( ! [X5: list_a] :
              ( ( member_list_a @ X5 @ A3 )
             => ( finite_finite_nat @ ( B2 @ X5 ) ) ) ) ) ) ).

% finite_UN
thf(fact_232_finite__UN,axiom,
    ! [A3: set_nat,B2: nat > set_list_a] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_nat_set_list_a @ B2 @ A3 ) ) )
        = ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
             => ( finite_finite_list_a @ ( B2 @ X5 ) ) ) ) ) ) ).

% finite_UN
thf(fact_233_finite__UN,axiom,
    ! [A3: set_a,B2: a > set_list_a] :
      ( ( finite_finite_a @ A3 )
     => ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ ( image_a_set_list_a @ B2 @ A3 ) ) )
        = ( ! [X5: a] :
              ( ( member_a @ X5 @ A3 )
             => ( finite_finite_list_a @ ( B2 @ X5 ) ) ) ) ) ) ).

% finite_UN
thf(fact_234_finite__UN,axiom,
    ! [A3: set_list_a,B2: list_a > set_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ B2 @ A3 ) ) )
        = ( ! [X5: list_a] :
              ( ( member_list_a @ X5 @ A3 )
             => ( finite_finite_a @ ( B2 @ X5 ) ) ) ) ) ) ).

% finite_UN
thf(fact_235_finite__UN,axiom,
    ! [A3: set_set_list_a,B2: set_list_a > set_nat] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ ( image_7346808305986241189et_nat @ B2 @ A3 ) ) )
        = ( ! [X5: set_list_a] :
              ( ( member_set_list_a @ X5 @ A3 )
             => ( finite_finite_nat @ ( B2 @ X5 ) ) ) ) ) ) ).

% finite_UN
thf(fact_236_finite__UN,axiom,
    ! [A3: set_nat,B2: nat > set_set_list_a] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite5282473924520328461list_a @ ( comple4119837387707116081list_a @ ( image_6263066288311102637list_a @ B2 @ A3 ) ) )
        = ( ! [X5: nat] :
              ( ( member_nat @ X5 @ A3 )
             => ( finite5282473924520328461list_a @ ( B2 @ X5 ) ) ) ) ) ) ).

% finite_UN
thf(fact_237_pmod__image__characterization,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( P2 != nil_a )
         => ( ( image_list_a_list_a
              @ ^ [Q3: list_a] : ( polynomial_pmod_a_b @ r @ Q3 @ P2 )
              @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
            = ( collect_list_a
              @ ^ [Q3: list_a] :
                  ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                  & ( ord_less_eq_nat @ ( size_size_list_a @ Q3 ) @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% pmod_image_characterization
thf(fact_238_p_Orupture__dimension,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) )
         => ( embedd7670639971858547848t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( image_551801017575455717list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) ) @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) ) @ ( partia3317168157747563407t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 ) ) ) ) ) ) ).

% p.rupture_dimension
thf(fact_239_ring_Opmod_Ocong,axiom,
    polynomial_pmod_a_b = polynomial_pmod_a_b ).

% ring.pmod.cong
thf(fact_240_ring_Opmod_Ocong,axiom,
    polyno1727750685288865234t_unit = polyno1727750685288865234t_unit ).

% ring.pmod.cong
thf(fact_241_ring_Ois__root_Ocong,axiom,
    polyno6951661231331188332t_unit = polyno6951661231331188332t_unit ).

% ring.is_root.cong
thf(fact_242_ring_Ois__root_Ocong,axiom,
    polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).

% ring.is_root.cong
thf(fact_243_ring_Ois__root_Ocong,axiom,
    polyno4320237611291262604t_unit = polyno4320237611291262604t_unit ).

% ring.is_root.cong
thf(fact_244_ring_Osplitted_Ocong,axiom,
    polyno6259083269128200473t_unit = polyno6259083269128200473t_unit ).

% ring.splitted.cong
thf(fact_245_ring_Osplitted_Ocong,axiom,
    polyno8329700637149614481ed_a_b = polyno8329700637149614481ed_a_b ).

% ring.splitted.cong
thf(fact_246_ring_Osplitted_Ocong,axiom,
    polyno7858167711734664505t_unit = polyno7858167711734664505t_unit ).

% ring.splitted.cong
thf(fact_247_finite__psubset__induct,axiom,
    ! [A3: set_list_set_list_a,P3: set_list_set_list_a > $o] :
      ( ( finite3202133031812794515list_a @ A3 )
     => ( ! [A5: set_list_set_list_a] :
            ( ( finite3202133031812794515list_a @ A5 )
           => ( ! [B4: set_list_set_list_a] :
                  ( ( ord_le6726094527366843510list_a @ B4 @ A5 )
                 => ( P3 @ B4 ) )
             => ( P3 @ A5 ) ) )
       => ( P3 @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_248_finite__psubset__induct,axiom,
    ! [A3: set_list_a,P3: set_list_a > $o] :
      ( ( finite_finite_list_a @ A3 )
     => ( ! [A5: set_list_a] :
            ( ( finite_finite_list_a @ A5 )
           => ( ! [B4: set_list_a] :
                  ( ( ord_less_set_list_a @ B4 @ A5 )
                 => ( P3 @ B4 ) )
             => ( P3 @ A5 ) ) )
       => ( P3 @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_249_finite__psubset__induct,axiom,
    ! [A3: set_set_list_a,P3: set_set_list_a > $o] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ( ! [A5: set_set_list_a] :
            ( ( finite5282473924520328461list_a @ A5 )
           => ( ! [B4: set_set_list_a] :
                  ( ( ord_le1620366983259561968list_a @ B4 @ A5 )
                 => ( P3 @ B4 ) )
             => ( P3 @ A5 ) ) )
       => ( P3 @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_250_finite__psubset__induct,axiom,
    ! [A3: set_nat,P3: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [A5: set_nat] :
            ( ( finite_finite_nat @ A5 )
           => ( ! [B4: set_nat] :
                  ( ( ord_less_set_nat @ B4 @ A5 )
                 => ( P3 @ B4 ) )
             => ( P3 @ A5 ) ) )
       => ( P3 @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_251_finite__psubset__induct,axiom,
    ! [A3: set_a,P3: set_a > $o] :
      ( ( finite_finite_a @ A3 )
     => ( ! [A5: set_a] :
            ( ( finite_finite_a @ A5 )
           => ( ! [B4: set_a] :
                  ( ( ord_less_set_a @ B4 @ A5 )
                 => ( P3 @ B4 ) )
             => ( P3 @ A5 ) ) )
       => ( P3 @ A3 ) ) ) ).

% finite_psubset_induct
thf(fact_252_domain_Olong__division__zero_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( polyno2750255454319713356t_unit @ R @ nil_list_list_a @ Q )
            = nil_list_list_a ) ) ) ) ).

% domain.long_division_zero(2)
thf(fact_253_domain_Olong__division__zero_I2_J,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( polyno427657113825607788t_unit @ R @ nil_list_set_list_a @ Q )
            = nil_list_set_list_a ) ) ) ) ).

% domain.long_division_zero(2)
thf(fact_254_domain_Olong__division__zero_I2_J,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( polyno8414006224137659890t_unit @ R @ nil_set_list_a @ Q )
            = nil_set_list_a ) ) ) ) ).

% domain.long_division_zero(2)
thf(fact_255_domain_Olong__division__zero_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( polynomial_pmod_a_b @ R @ nil_a @ Q )
            = nil_a ) ) ) ) ).

% domain.long_division_zero(2)
thf(fact_256_domain_Olong__division__zero_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( polyno1727750685288865234t_unit @ R @ nil_list_a @ Q )
            = nil_list_a ) ) ) ) ).

% domain.long_division_zero(2)
thf(fact_257_Diff__infinite__finite,axiom,
    ! [T: set_list_set_list_a,S: set_list_set_list_a] :
      ( ( finite3202133031812794515list_a @ T )
     => ( ~ ( finite3202133031812794515list_a @ S )
       => ~ ( finite3202133031812794515list_a @ ( minus_4926175736359902257list_a @ S @ T ) ) ) ) ).

% Diff_infinite_finite
thf(fact_258_Diff__infinite__finite,axiom,
    ! [T: set_list_a,S: set_list_a] :
      ( ( finite_finite_list_a @ T )
     => ( ~ ( finite_finite_list_a @ S )
       => ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T ) ) ) ) ).

% Diff_infinite_finite
thf(fact_259_Diff__infinite__finite,axiom,
    ! [T: set_set_list_a,S: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ T )
     => ( ~ ( finite5282473924520328461list_a @ S )
       => ~ ( finite5282473924520328461list_a @ ( minus_4782336368215558443list_a @ S @ T ) ) ) ) ).

% Diff_infinite_finite
thf(fact_260_Diff__infinite__finite,axiom,
    ! [T: set_nat,S: set_nat] :
      ( ( finite_finite_nat @ T )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T ) ) ) ) ).

% Diff_infinite_finite
thf(fact_261_Diff__infinite__finite,axiom,
    ! [T: set_a,S: set_a] :
      ( ( finite_finite_a @ T )
     => ( ~ ( finite_finite_a @ S )
       => ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T ) ) ) ) ).

% Diff_infinite_finite
thf(fact_262_domain_Opmod__image__characterization,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( P2 != nil_list_list_a )
           => ( ( image_865451852121373573list_a
                @ ^ [Q3: list_list_list_a] : ( polyno2750255454319713356t_unit @ R @ Q3 @ P2 )
                @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
              = ( collec1292721268053437947list_a
                @ ^ [Q3: list_list_list_a] :
                    ( ( member5342144027231129785list_a @ Q3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
                    & ( ord_less_eq_nat @ ( size_s2403821588304063868list_a @ Q3 ) @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).

% domain.pmod_image_characterization
thf(fact_263_domain_Opmod__image__characterization,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( P2 != nil_list_set_list_a )
           => ( ( image_1936032423783470981list_a
                @ ^ [Q3: list_list_set_list_a] : ( polyno427657113825607788t_unit @ R @ Q3 @ P2 )
                @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
              = ( collec4413339279609262555list_a
                @ ^ [Q3: list_list_set_list_a] :
                    ( ( member352051402189872281list_a @ Q3 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
                    & ( ord_less_eq_nat @ ( size_s4069122225494125916list_a @ Q3 ) @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).

% domain.pmod_image_characterization
thf(fact_264_domain_Opmod__image__characterization,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( P2 != nil_set_list_a )
           => ( ( image_341778791481906565list_a
                @ ^ [Q3: list_set_list_a] : ( polyno8414006224137659890t_unit @ R @ Q3 @ P2 )
                @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
              = ( collec5381118732811369429list_a
                @ ^ [Q3: list_set_list_a] :
                    ( ( member5524387281408368019list_a @ Q3 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
                    & ( ord_less_eq_nat @ ( size_s1991367317912710102list_a @ Q3 ) @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).

% domain.pmod_image_characterization
thf(fact_265_domain_Opmod__image__characterization,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( P2 != nil_a )
           => ( ( image_list_a_list_a
                @ ^ [Q3: list_a] : ( polynomial_pmod_a_b @ R @ Q3 @ P2 )
                @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
              = ( collect_list_a
                @ ^ [Q3: list_a] :
                    ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                    & ( ord_less_eq_nat @ ( size_size_list_a @ Q3 ) @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).

% domain.pmod_image_characterization
thf(fact_266_domain_Opmod__image__characterization,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( P2 != nil_list_a )
           => ( ( image_5446147394702115205list_a
                @ ^ [Q3: list_list_a] : ( polyno1727750685288865234t_unit @ R @ Q3 @ P2 )
                @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
              = ( collect_list_list_a
                @ ^ [Q3: list_list_a] :
                    ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                    & ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Q3 ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ) ) ) ).

% domain.pmod_image_characterization
thf(fact_267_domain_Opmod__degree,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_list_a )
             => ( ( ( polyno2750255454319713356t_unit @ R @ P2 @ Q )
                  = nil_list_list_a )
                | ( ord_less_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( polyno2750255454319713356t_unit @ R @ P2 @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% domain.pmod_degree
thf(fact_268_domain_Opmod__degree,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_set_list_a )
             => ( ( ( polyno427657113825607788t_unit @ R @ P2 @ Q )
                  = nil_list_set_list_a )
                | ( ord_less_nat @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ ( polyno427657113825607788t_unit @ R @ P2 @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% domain.pmod_degree
thf(fact_269_domain_Opmod__degree,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
           => ( ( Q != nil_set_list_a )
             => ( ( ( polyno8414006224137659890t_unit @ R @ P2 @ Q )
                  = nil_set_list_a )
                | ( ord_less_nat @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ ( polyno8414006224137659890t_unit @ R @ P2 @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% domain.pmod_degree
thf(fact_270_domain_Opmod__degree,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( Q != nil_a )
             => ( ( ( polynomial_pmod_a_b @ R @ P2 @ Q )
                  = nil_a )
                | ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ R @ P2 @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% domain.pmod_degree
thf(fact_271_domain_Opmod__degree,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_a )
             => ( ( ( polyno1727750685288865234t_unit @ R @ P2 @ Q )
                  = nil_list_a )
                | ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% domain.pmod_degree
thf(fact_272_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B2: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A6: nat] :
                        ( ( member_nat @ A6 @ A3 )
                        & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_273_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B2: set_a,R: nat > a > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_a @ B2 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A3 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B2 )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: a] :
              ( ( member_a @ X2 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A6: nat] :
                        ( ( member_nat @ A6 @ A3 )
                        & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_274_pigeonhole__infinite__rel,axiom,
    ! [A3: set_a,B2: set_nat,R: a > nat > $o] :
      ( ~ ( finite_finite_a @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B2 )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A6: a] :
                        ( ( member_a @ A6 @ A3 )
                        & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_275_pigeonhole__infinite__rel,axiom,
    ! [A3: set_a,B2: set_a,R: a > a > $o] :
      ( ~ ( finite_finite_a @ A3 )
     => ( ( finite_finite_a @ B2 )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ A3 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B2 )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: a] :
              ( ( member_a @ X2 @ B2 )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A6: a] :
                        ( ( member_a @ A6 @ A3 )
                        & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_276_pigeonhole__infinite__rel,axiom,
    ! [A3: set_list_a,B2: set_nat,R: list_a > nat > $o] :
      ( ~ ( finite_finite_list_a @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X2: list_a] :
              ( ( member_list_a @ X2 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B2 )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A6: list_a] :
                        ( ( member_list_a @ A6 @ A3 )
                        & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_277_pigeonhole__infinite__rel,axiom,
    ! [A3: set_list_a,B2: set_a,R: list_a > a > $o] :
      ( ~ ( finite_finite_list_a @ A3 )
     => ( ( finite_finite_a @ B2 )
       => ( ! [X2: list_a] :
              ( ( member_list_a @ X2 @ A3 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B2 )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: a] :
              ( ( member_a @ X2 @ B2 )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A6: list_a] :
                        ( ( member_list_a @ A6 @ A3 )
                        & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_278_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat,B2: set_list_a,R: nat > list_a > $o] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_list_a @ B2 )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A3 )
             => ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ B2 )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: list_a] :
              ( ( member_list_a @ X2 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A6: nat] :
                        ( ( member_nat @ A6 @ A3 )
                        & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_279_pigeonhole__infinite__rel,axiom,
    ! [A3: set_a,B2: set_list_a,R: a > list_a > $o] :
      ( ~ ( finite_finite_a @ A3 )
     => ( ( finite_finite_list_a @ B2 )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ A3 )
             => ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ B2 )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: list_a] :
              ( ( member_list_a @ X2 @ B2 )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A6: a] :
                        ( ( member_a @ A6 @ A3 )
                        & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_280_pigeonhole__infinite__rel,axiom,
    ! [A3: set_nat_a,B2: set_nat,R: ( nat > a ) > nat > $o] :
      ( ~ ( finite_finite_nat_a @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X2: nat > a] :
              ( ( member_nat_a @ X2 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B2 )
              & ~ ( finite_finite_nat_a
                  @ ( collect_nat_a
                    @ ^ [A6: nat > a] :
                        ( ( member_nat_a @ A6 @ A3 )
                        & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_281_pigeonhole__infinite__rel,axiom,
    ! [A3: set_list_list_a,B2: set_nat,R: list_list_a > nat > $o] :
      ( ~ ( finite1660835950917165235list_a @ A3 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X2: list_list_a] :
              ( ( member_list_list_a @ X2 @ A3 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B2 )
              & ~ ( finite1660835950917165235list_a
                  @ ( collect_list_list_a
                    @ ^ [A6: list_list_a] :
                        ( ( member_list_list_a @ A6 @ A3 )
                        & ( R @ A6 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_282_not__finite__existsD,axiom,
    ! [P3: list_list_a > $o] :
      ( ~ ( finite1660835950917165235list_a @ ( collect_list_list_a @ P3 ) )
     => ? [X_1: list_list_a] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_283_not__finite__existsD,axiom,
    ! [P3: list_set_list_a > $o] :
      ( ~ ( finite3202133031812794515list_a @ ( collec5381118732811369429list_a @ P3 ) )
     => ? [X_1: list_set_list_a] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_284_not__finite__existsD,axiom,
    ! [P3: list_a > $o] :
      ( ~ ( finite_finite_list_a @ ( collect_list_a @ P3 ) )
     => ? [X_1: list_a] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_285_not__finite__existsD,axiom,
    ! [P3: set_list_a > $o] :
      ( ~ ( finite5282473924520328461list_a @ ( collect_set_list_a @ P3 ) )
     => ? [X_1: set_list_a] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_286_not__finite__existsD,axiom,
    ! [P3: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P3 ) )
     => ? [X_1: nat] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_287_not__finite__existsD,axiom,
    ! [P3: a > $o] :
      ( ~ ( finite_finite_a @ ( collect_a @ P3 ) )
     => ? [X_1: a] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_288_domain_Olong__division__closed_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
           => ( member5342144027231129785list_a @ ( polyno2750255454319713356t_unit @ R @ P2 @ Q ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(2)
thf(fact_289_domain_Olong__division__closed_I2_J,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
           => ( member352051402189872281list_a @ ( polyno427657113825607788t_unit @ R @ P2 @ Q ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(2)
thf(fact_290_domain_Olong__division__closed_I2_J,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
           => ( member5524387281408368019list_a @ ( polyno8414006224137659890t_unit @ R @ P2 @ Q ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(2)
thf(fact_291_domain_Olong__division__closed_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( member_list_a @ ( polynomial_pmod_a_b @ R @ P2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(2)
thf(fact_292_domain_Olong__division__closed_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( member_list_list_a @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(2)
thf(fact_293_domain_Olong__division__zero_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( polyno4115915122720352731t_unit @ R @ nil_list_list_a @ Q )
            = nil_list_list_a ) ) ) ) ).

% domain.long_division_zero(1)
thf(fact_294_domain_Olong__division__zero_I1_J,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( polyno5933693564459536891t_unit @ R @ nil_list_set_list_a @ Q )
            = nil_list_set_list_a ) ) ) ) ).

% domain.long_division_zero(1)
thf(fact_295_domain_Olong__division__zero_I1_J,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( polyno6856256803263433473t_unit @ R @ nil_set_list_a @ Q )
            = nil_set_list_a ) ) ) ) ).

% domain.long_division_zero(1)
thf(fact_296_domain_Olong__division__zero_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( polynomial_pdiv_a_b @ R @ nil_a @ Q )
            = nil_a ) ) ) ) ).

% domain.long_division_zero(1)
thf(fact_297_domain_Olong__division__zero_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( polyno5893782122288709345t_unit @ R @ nil_list_a @ Q )
            = nil_list_a ) ) ) ) ).

% domain.long_division_zero(1)
thf(fact_298_finite__has__minimal2,axiom,
    ! [A3: set_nat_set_list_a,A: nat > set_list_a] :
      ( ( finite1571261242143456884list_a @ A3 )
     => ( ( member491565700723299188list_a @ A @ A3 )
       => ? [X2: nat > set_list_a] :
            ( ( member491565700723299188list_a @ X2 @ A3 )
            & ( ord_le3697158753621733867list_a @ X2 @ A )
            & ! [Xa: nat > set_list_a] :
                ( ( member491565700723299188list_a @ Xa @ A3 )
               => ( ( ord_le3697158753621733867list_a @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_299_finite__has__minimal2,axiom,
    ! [A3: set_a_set_list_a,A: a > set_list_a] :
      ( ( finite9041730431398133218list_a @ A3 )
     => ( ( member_a_set_list_a @ A @ A3 )
       => ? [X2: a > set_list_a] :
            ( ( member_a_set_list_a @ X2 @ A3 )
            & ( ord_le7804183552102971947list_a @ X2 @ A )
            & ! [Xa: a > set_list_a] :
                ( ( member_a_set_list_a @ Xa @ A3 )
               => ( ( ord_le7804183552102971947list_a @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_300_finite__has__minimal2,axiom,
    ! [A3: set_set_list_a,A: set_list_a] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ( ( member_set_list_a @ A @ A3 )
       => ? [X2: set_list_a] :
            ( ( member_set_list_a @ X2 @ A3 )
            & ( ord_le8861187494160871172list_a @ X2 @ A )
            & ! [Xa: set_list_a] :
                ( ( member_set_list_a @ Xa @ A3 )
               => ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_301_finite__has__minimal2,axiom,
    ! [A3: set_set_set_list_a,A: set_set_list_a] :
      ( ( finite6594153429226962157list_a @ A3 )
     => ( ( member8857465052274545133list_a @ A @ A3 )
       => ? [X2: set_set_list_a] :
            ( ( member8857465052274545133list_a @ X2 @ A3 )
            & ( ord_le8877086941679407844list_a @ X2 @ A )
            & ! [Xa: set_set_list_a] :
                ( ( member8857465052274545133list_a @ Xa @ A3 )
               => ( ( ord_le8877086941679407844list_a @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_302_finite__has__minimal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ( ord_less_eq_nat @ X2 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_303_finite__has__minimal2,axiom,
    ! [A3: set_set_a,A: set_a] :
      ( ( finite_finite_set_a @ A3 )
     => ( ( member_set_a @ A @ A3 )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A3 )
            & ( ord_less_eq_set_a @ X2 @ A )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A3 )
               => ( ( ord_less_eq_set_a @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_304_finite__has__minimal2,axiom,
    ! [A3: set_set_set_a,A: set_set_a] :
      ( ( finite7209287970140883943_set_a @ A3 )
     => ( ( member_set_set_a @ A @ A3 )
       => ? [X2: set_set_a] :
            ( ( member_set_set_a @ X2 @ A3 )
            & ( ord_le3724670747650509150_set_a @ X2 @ A )
            & ! [Xa: set_set_a] :
                ( ( member_set_set_a @ Xa @ A3 )
               => ( ( ord_le3724670747650509150_set_a @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_305_finite__has__maximal2,axiom,
    ! [A3: set_nat_set_list_a,A: nat > set_list_a] :
      ( ( finite1571261242143456884list_a @ A3 )
     => ( ( member491565700723299188list_a @ A @ A3 )
       => ? [X2: nat > set_list_a] :
            ( ( member491565700723299188list_a @ X2 @ A3 )
            & ( ord_le3697158753621733867list_a @ A @ X2 )
            & ! [Xa: nat > set_list_a] :
                ( ( member491565700723299188list_a @ Xa @ A3 )
               => ( ( ord_le3697158753621733867list_a @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_306_finite__has__maximal2,axiom,
    ! [A3: set_a_set_list_a,A: a > set_list_a] :
      ( ( finite9041730431398133218list_a @ A3 )
     => ( ( member_a_set_list_a @ A @ A3 )
       => ? [X2: a > set_list_a] :
            ( ( member_a_set_list_a @ X2 @ A3 )
            & ( ord_le7804183552102971947list_a @ A @ X2 )
            & ! [Xa: a > set_list_a] :
                ( ( member_a_set_list_a @ Xa @ A3 )
               => ( ( ord_le7804183552102971947list_a @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_307_finite__has__maximal2,axiom,
    ! [A3: set_set_list_a,A: set_list_a] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ( ( member_set_list_a @ A @ A3 )
       => ? [X2: set_list_a] :
            ( ( member_set_list_a @ X2 @ A3 )
            & ( ord_le8861187494160871172list_a @ A @ X2 )
            & ! [Xa: set_list_a] :
                ( ( member_set_list_a @ Xa @ A3 )
               => ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_308_finite__has__maximal2,axiom,
    ! [A3: set_set_set_list_a,A: set_set_list_a] :
      ( ( finite6594153429226962157list_a @ A3 )
     => ( ( member8857465052274545133list_a @ A @ A3 )
       => ? [X2: set_set_list_a] :
            ( ( member8857465052274545133list_a @ X2 @ A3 )
            & ( ord_le8877086941679407844list_a @ A @ X2 )
            & ! [Xa: set_set_list_a] :
                ( ( member8857465052274545133list_a @ Xa @ A3 )
               => ( ( ord_le8877086941679407844list_a @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_309_finite__has__maximal2,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat @ A @ A3 )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ( ord_less_eq_nat @ A @ X2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A3 )
               => ( ( ord_less_eq_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_310_finite__has__maximal2,axiom,
    ! [A3: set_set_a,A: set_a] :
      ( ( finite_finite_set_a @ A3 )
     => ( ( member_set_a @ A @ A3 )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A3 )
            & ( ord_less_eq_set_a @ A @ X2 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A3 )
               => ( ( ord_less_eq_set_a @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_311_finite__has__maximal2,axiom,
    ! [A3: set_set_set_a,A: set_set_a] :
      ( ( finite7209287970140883943_set_a @ A3 )
     => ( ( member_set_set_a @ A @ A3 )
       => ? [X2: set_set_a] :
            ( ( member_set_set_a @ X2 @ A3 )
            & ( ord_le3724670747650509150_set_a @ A @ X2 )
            & ! [Xa: set_set_a] :
                ( ( member_set_set_a @ Xa @ A3 )
               => ( ( ord_le3724670747650509150_set_a @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_312_all__subset__image,axiom,
    ! [F2: a > a,A3: set_a,P3: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F2 @ A3 ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ A3 )
           => ( P3 @ ( image_a_a @ F2 @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_313_all__subset__image,axiom,
    ! [F2: a > list_a,A3: set_a,P3: set_list_a > $o] :
      ( ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ ( image_a_list_a @ F2 @ A3 ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ A3 )
           => ( P3 @ ( image_a_list_a @ F2 @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_314_all__subset__image,axiom,
    ! [F2: list_a > a,A3: set_list_a,P3: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ ( image_list_a_a @ F2 @ A3 ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ A3 )
           => ( P3 @ ( image_list_a_a @ F2 @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_315_all__subset__image,axiom,
    ! [F2: set_a > a,A3: set_set_a,P3: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ ( image_set_a_a @ F2 @ A3 ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
           => ( P3 @ ( image_set_a_a @ F2 @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_316_all__subset__image,axiom,
    ! [F2: a > set_a,A3: set_a,P3: set_set_a > $o] :
      ( ( ! [B3: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F2 @ A3 ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ A3 )
           => ( P3 @ ( image_a_set_a @ F2 @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_317_all__subset__image,axiom,
    ! [F2: list_a > list_a,A3: set_list_a,P3: set_list_a > $o] :
      ( ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ ( image_list_a_list_a @ F2 @ A3 ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ A3 )
           => ( P3 @ ( image_list_a_list_a @ F2 @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_318_all__subset__image,axiom,
    ! [F2: set_a > list_a,A3: set_set_a,P3: set_list_a > $o] :
      ( ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ ( image_set_a_list_a @ F2 @ A3 ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ B3 @ A3 )
           => ( P3 @ ( image_set_a_list_a @ F2 @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_319_all__subset__image,axiom,
    ! [F2: a > set_list_a,A3: set_a,P3: set_set_list_a > $o] :
      ( ( ! [B3: set_set_list_a] :
            ( ( ord_le8877086941679407844list_a @ B3 @ ( image_a_set_list_a @ F2 @ A3 ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ A3 )
           => ( P3 @ ( image_a_set_list_a @ F2 @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_320_all__subset__image,axiom,
    ! [F2: set_list_a > a,A3: set_set_list_a,P3: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ord_less_eq_set_a @ B3 @ ( image_set_list_a_a @ F2 @ A3 ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_set_list_a] :
            ( ( ord_le8877086941679407844list_a @ B3 @ A3 )
           => ( P3 @ ( image_set_list_a_a @ F2 @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_321_all__subset__image,axiom,
    ! [F2: list_a > set_a,A3: set_list_a,P3: set_set_a > $o] :
      ( ( ! [B3: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_list_a_set_a @ F2 @ A3 ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ B3 @ A3 )
           => ( P3 @ ( image_list_a_set_a @ F2 @ B3 ) ) ) ) ) ).

% all_subset_image
thf(fact_322_rev__finite__subset,axiom,
    ! [B2: set_list_set_list_a,A3: set_list_set_list_a] :
      ( ( finite3202133031812794515list_a @ B2 )
     => ( ( ord_le1961058059034850666list_a @ A3 @ B2 )
       => ( finite3202133031812794515list_a @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_323_rev__finite__subset,axiom,
    ! [B2: set_nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A3 @ B2 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_324_rev__finite__subset,axiom,
    ! [B2: set_list_a,A3: set_list_a] :
      ( ( finite_finite_list_a @ B2 )
     => ( ( ord_le8861187494160871172list_a @ A3 @ B2 )
       => ( finite_finite_list_a @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_325_rev__finite__subset,axiom,
    ! [B2: set_set_list_a,A3: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ B2 )
     => ( ( ord_le8877086941679407844list_a @ A3 @ B2 )
       => ( finite5282473924520328461list_a @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_326_rev__finite__subset,axiom,
    ! [B2: set_a,A3: set_a] :
      ( ( finite_finite_a @ B2 )
     => ( ( ord_less_eq_set_a @ A3 @ B2 )
       => ( finite_finite_a @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_327_rev__finite__subset,axiom,
    ! [B2: set_set_a,A3: set_set_a] :
      ( ( finite_finite_set_a @ B2 )
     => ( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
       => ( finite_finite_set_a @ A3 ) ) ) ).

% rev_finite_subset
thf(fact_328_infinite__super,axiom,
    ! [S: set_list_set_list_a,T: set_list_set_list_a] :
      ( ( ord_le1961058059034850666list_a @ S @ T )
     => ( ~ ( finite3202133031812794515list_a @ S )
       => ~ ( finite3202133031812794515list_a @ T ) ) ) ).

% infinite_super
thf(fact_329_infinite__super,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_super
thf(fact_330_infinite__super,axiom,
    ! [S: set_list_a,T: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ S @ T )
     => ( ~ ( finite_finite_list_a @ S )
       => ~ ( finite_finite_list_a @ T ) ) ) ).

% infinite_super
thf(fact_331_infinite__super,axiom,
    ! [S: set_set_list_a,T: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ S @ T )
     => ( ~ ( finite5282473924520328461list_a @ S )
       => ~ ( finite5282473924520328461list_a @ T ) ) ) ).

% infinite_super
thf(fact_332_infinite__super,axiom,
    ! [S: set_a,T: set_a] :
      ( ( ord_less_eq_set_a @ S @ T )
     => ( ~ ( finite_finite_a @ S )
       => ~ ( finite_finite_a @ T ) ) ) ).

% infinite_super
thf(fact_333_infinite__super,axiom,
    ! [S: set_set_a,T: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ S @ T )
     => ( ~ ( finite_finite_set_a @ S )
       => ~ ( finite_finite_set_a @ T ) ) ) ).

% infinite_super
thf(fact_334_finite__subset,axiom,
    ! [A3: set_list_set_list_a,B2: set_list_set_list_a] :
      ( ( ord_le1961058059034850666list_a @ A3 @ B2 )
     => ( ( finite3202133031812794515list_a @ B2 )
       => ( finite3202133031812794515list_a @ A3 ) ) ) ).

% finite_subset
thf(fact_335_finite__subset,axiom,
    ! [A3: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B2 )
     => ( ( finite_finite_nat @ B2 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% finite_subset
thf(fact_336_finite__subset,axiom,
    ! [A3: set_list_a,B2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A3 @ B2 )
     => ( ( finite_finite_list_a @ B2 )
       => ( finite_finite_list_a @ A3 ) ) ) ).

% finite_subset
thf(fact_337_finite__subset,axiom,
    ! [A3: set_set_list_a,B2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A3 @ B2 )
     => ( ( finite5282473924520328461list_a @ B2 )
       => ( finite5282473924520328461list_a @ A3 ) ) ) ).

% finite_subset
thf(fact_338_finite__subset,axiom,
    ! [A3: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ B2 )
     => ( ( finite_finite_a @ B2 )
       => ( finite_finite_a @ A3 ) ) ) ).

% finite_subset
thf(fact_339_finite__subset,axiom,
    ! [A3: set_set_a,B2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A3 @ B2 )
     => ( ( finite_finite_set_a @ B2 )
       => ( finite_finite_set_a @ A3 ) ) ) ).

% finite_subset
thf(fact_340_finite__UnionD,axiom,
    ! [A3: set_se895765194286668842list_a] :
      ( ( finite3202133031812794515list_a @ ( comple4047202532333093431list_a @ A3 ) )
     => ( finite191489321941288691list_a @ A3 ) ) ).

% finite_UnionD
thf(fact_341_finite__UnionD,axiom,
    ! [A3: set_set_set_list_a] :
      ( ( finite5282473924520328461list_a @ ( comple4119837387707116081list_a @ A3 ) )
     => ( finite6594153429226962157list_a @ A3 ) ) ).

% finite_UnionD
thf(fact_342_finite__UnionD,axiom,
    ! [A3: set_set_nat] :
      ( ( finite_finite_nat @ ( comple7399068483239264473et_nat @ A3 ) )
     => ( finite1152437895449049373et_nat @ A3 ) ) ).

% finite_UnionD
thf(fact_343_finite__UnionD,axiom,
    ! [A3: set_set_list_a] :
      ( ( finite_finite_list_a @ ( comple6928918032620976721list_a @ A3 ) )
     => ( finite5282473924520328461list_a @ A3 ) ) ).

% finite_UnionD
thf(fact_344_finite__UnionD,axiom,
    ! [A3: set_set_a] :
      ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ A3 ) )
     => ( finite_finite_set_a @ A3 ) ) ).

% finite_UnionD
thf(fact_345_domain_Orupture__surj__composed__with__pmod,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
           => ( ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 ) @ Q )
              = ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 ) @ ( polyno2750255454319713356t_unit @ R @ Q @ P2 ) ) ) ) ) ) ) ).

% domain.rupture_surj_composed_with_pmod
thf(fact_346_domain_Orupture__surj__composed__with__pmod,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
           => ( ( a_r_co6659872811071116064t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ ( cgenid3857712956745501987t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ P2 ) @ Q )
              = ( a_r_co6659872811071116064t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ ( cgenid3857712956745501987t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ P2 ) @ ( polyno427657113825607788t_unit @ R @ Q @ P2 ) ) ) ) ) ) ) ).

% domain.rupture_surj_composed_with_pmod
thf(fact_347_domain_Orupture__surj__composed__with__pmod,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
           => ( ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 ) @ Q )
              = ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 ) @ ( polyno8414006224137659890t_unit @ R @ Q @ P2 ) ) ) ) ) ) ) ).

% domain.rupture_surj_composed_with_pmod
thf(fact_348_domain_Orupture__surj__composed__with__pmod,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) @ Q )
              = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) @ ( polynomial_pmod_a_b @ R @ Q @ P2 ) ) ) ) ) ) ) ).

% domain.rupture_surj_composed_with_pmod
thf(fact_349_domain_Orupture__surj__composed__with__pmod,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) @ Q )
              = ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) @ ( polyno1727750685288865234t_unit @ R @ Q @ P2 ) ) ) ) ) ) ) ).

% domain.rupture_surj_composed_with_pmod
thf(fact_350_domain_Ofinite__number__of__roots,axiom,
    ! [R: partia2956882679547061052t_unit,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( finite1660835950917165235list_a @ ( collect_list_list_a @ ( polyno5142720416380192742t_unit @ R @ P2 ) ) ) ) ) ).

% domain.finite_number_of_roots
thf(fact_351_domain_Ofinite__number__of__roots,axiom,
    ! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R )
     => ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
       => ( finite7235877257443484435list_a @ ( collec191490921587283541list_a @ ( polyno875383481011669382t_unit @ R @ P2 ) ) ) ) ) ).

% domain.finite_number_of_roots
thf(fact_352_domain_Ofinite__number__of__roots,axiom,
    ! [R: partia6043505979758434576t_unit,P2: list_set_a] :
      ( ( domain4236798911309298543t_unit @ R )
     => ( ( member_list_set_a @ P2 @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
       => ( finite_finite_set_a @ ( collect_set_a @ ( polyno4890645956962836498t_unit @ R @ P2 ) ) ) ) ) ).

% domain.finite_number_of_roots
thf(fact_353_domain_Ofinite__number__of__roots,axiom,
    ! [R: partia2670972154091845814t_unit,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( finite_finite_list_a @ ( collect_list_a @ ( polyno6951661231331188332t_unit @ R @ P2 ) ) ) ) ) ).

% domain.finite_number_of_roots
thf(fact_354_domain_Ofinite__number__of__roots,axiom,
    ! [R: partia2175431115845679010xt_a_b,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ R @ P2 ) ) ) ) ) ).

% domain.finite_number_of_roots
thf(fact_355_domain_Ofinite__number__of__roots,axiom,
    ! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
       => ( finite5282473924520328461list_a @ ( collect_set_list_a @ ( polyno4320237611291262604t_unit @ R @ P2 ) ) ) ) ) ).

% domain.finite_number_of_roots
thf(fact_356_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F2: nat > nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A3 ) )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A6: nat] :
                      ( ( member_nat @ A6 @ A3 )
                      & ( ( F2 @ A6 )
                        = ( F2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_357_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F2: nat > a] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_a @ ( image_nat_a @ F2 @ A3 ) )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A6: nat] :
                      ( ( member_nat @ A6 @ A3 )
                      & ( ( F2 @ A6 )
                        = ( F2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_358_pigeonhole__infinite,axiom,
    ! [A3: set_a,F2: a > nat] :
      ( ~ ( finite_finite_a @ A3 )
     => ( ( finite_finite_nat @ ( image_a_nat @ F2 @ A3 ) )
       => ? [X2: a] :
            ( ( member_a @ X2 @ A3 )
            & ~ ( finite_finite_a
                @ ( collect_a
                  @ ^ [A6: a] :
                      ( ( member_a @ A6 @ A3 )
                      & ( ( F2 @ A6 )
                        = ( F2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_359_pigeonhole__infinite,axiom,
    ! [A3: set_a,F2: a > a] :
      ( ~ ( finite_finite_a @ A3 )
     => ( ( finite_finite_a @ ( image_a_a @ F2 @ A3 ) )
       => ? [X2: a] :
            ( ( member_a @ X2 @ A3 )
            & ~ ( finite_finite_a
                @ ( collect_a
                  @ ^ [A6: a] :
                      ( ( member_a @ A6 @ A3 )
                      & ( ( F2 @ A6 )
                        = ( F2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_360_pigeonhole__infinite,axiom,
    ! [A3: set_list_a,F2: list_a > nat] :
      ( ~ ( finite_finite_list_a @ A3 )
     => ( ( finite_finite_nat @ ( image_list_a_nat @ F2 @ A3 ) )
       => ? [X2: list_a] :
            ( ( member_list_a @ X2 @ A3 )
            & ~ ( finite_finite_list_a
                @ ( collect_list_a
                  @ ^ [A6: list_a] :
                      ( ( member_list_a @ A6 @ A3 )
                      & ( ( F2 @ A6 )
                        = ( F2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_361_pigeonhole__infinite,axiom,
    ! [A3: set_list_a,F2: list_a > a] :
      ( ~ ( finite_finite_list_a @ A3 )
     => ( ( finite_finite_a @ ( image_list_a_a @ F2 @ A3 ) )
       => ? [X2: list_a] :
            ( ( member_list_a @ X2 @ A3 )
            & ~ ( finite_finite_list_a
                @ ( collect_list_a
                  @ ^ [A6: list_a] :
                      ( ( member_list_a @ A6 @ A3 )
                      & ( ( F2 @ A6 )
                        = ( F2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_362_pigeonhole__infinite,axiom,
    ! [A3: set_nat,F2: nat > list_a] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( finite_finite_list_a @ ( image_nat_list_a @ F2 @ A3 ) )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A3 )
            & ~ ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [A6: nat] :
                      ( ( member_nat @ A6 @ A3 )
                      & ( ( F2 @ A6 )
                        = ( F2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_363_pigeonhole__infinite,axiom,
    ! [A3: set_a,F2: a > set_a] :
      ( ~ ( finite_finite_a @ A3 )
     => ( ( finite_finite_set_a @ ( image_a_set_a @ F2 @ A3 ) )
       => ? [X2: a] :
            ( ( member_a @ X2 @ A3 )
            & ~ ( finite_finite_a
                @ ( collect_a
                  @ ^ [A6: a] :
                      ( ( member_a @ A6 @ A3 )
                      & ( ( F2 @ A6 )
                        = ( F2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_364_pigeonhole__infinite,axiom,
    ! [A3: set_a,F2: a > list_a] :
      ( ~ ( finite_finite_a @ A3 )
     => ( ( finite_finite_list_a @ ( image_a_list_a @ F2 @ A3 ) )
       => ? [X2: a] :
            ( ( member_a @ X2 @ A3 )
            & ~ ( finite_finite_a
                @ ( collect_a
                  @ ^ [A6: a] :
                      ( ( member_a @ A6 @ A3 )
                      & ( ( F2 @ A6 )
                        = ( F2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_365_pigeonhole__infinite,axiom,
    ! [A3: set_nat_a,F2: ( nat > a ) > nat] :
      ( ~ ( finite_finite_nat_a @ A3 )
     => ( ( finite_finite_nat @ ( image_nat_a_nat @ F2 @ A3 ) )
       => ? [X2: nat > a] :
            ( ( member_nat_a @ X2 @ A3 )
            & ~ ( finite_finite_nat_a
                @ ( collect_nat_a
                  @ ^ [A6: nat > a] :
                      ( ( member_nat_a @ A6 @ A3 )
                      & ( ( F2 @ A6 )
                        = ( F2 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite
thf(fact_366_domain_Odegree__one__imp__pirreducible,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat )
              = one_one_nat )
           => ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 ) ) ) ) ) ).

% domain.degree_one_imp_pirreducible
thf(fact_367_domain_Odegree__one__imp__pirreducible,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P2 ) @ one_one_nat )
              = one_one_nat )
           => ( ring_r7962978046438709730t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ P2 ) ) ) ) ) ).

% domain.degree_one_imp_pirreducible
thf(fact_368_domain_Odegree__one__imp__pirreducible,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat )
              = one_one_nat )
           => ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 ) ) ) ) ) ).

% domain.degree_one_imp_pirreducible
thf(fact_369_domain_Odegree__one__imp__pirreducible,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
              = one_one_nat )
           => ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) ) ) ) ) ).

% domain.degree_one_imp_pirreducible
thf(fact_370_domain_Odegree__one__imp__pirreducible,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
              = one_one_nat )
           => ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) ) ) ) ).

% domain.degree_one_imp_pirreducible
thf(fact_371_finite__inverse__image__gen,axiom,
    ! [A3: set_nat,F2: nat > nat,D: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on_nat_nat @ F2 @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_nat @ ( F2 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_372_finite__inverse__image__gen,axiom,
    ! [A3: set_nat,F2: a > nat,D: set_a] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on_a_nat @ F2 @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_nat @ ( F2 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_373_finite__inverse__image__gen,axiom,
    ! [A3: set_a,F2: nat > a,D: set_nat] :
      ( ( finite_finite_a @ A3 )
     => ( ( inj_on_nat_a @ F2 @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_a @ ( F2 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_374_finite__inverse__image__gen,axiom,
    ! [A3: set_a,F2: a > a,D: set_a] :
      ( ( finite_finite_a @ A3 )
     => ( ( inj_on_a_a @ F2 @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_a @ ( F2 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_375_finite__inverse__image__gen,axiom,
    ! [A3: set_list_a,F2: nat > list_a,D: set_nat] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( inj_on_nat_list_a @ F2 @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_list_a @ ( F2 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_376_finite__inverse__image__gen,axiom,
    ! [A3: set_list_a,F2: a > list_a,D: set_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( inj_on_a_list_a @ F2 @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_list_a @ ( F2 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_377_finite__inverse__image__gen,axiom,
    ! [A3: set_nat,F2: list_a > nat,D: set_list_a] :
      ( ( finite_finite_nat @ A3 )
     => ( ( inj_on_list_a_nat @ F2 @ D )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [J: list_a] :
                ( ( member_list_a @ J @ D )
                & ( member_nat @ ( F2 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_378_finite__inverse__image__gen,axiom,
    ! [A3: set_a,F2: list_a > a,D: set_list_a] :
      ( ( finite_finite_a @ A3 )
     => ( ( inj_on_list_a_a @ F2 @ D )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [J: list_a] :
                ( ( member_list_a @ J @ D )
                & ( member_a @ ( F2 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_379_finite__inverse__image__gen,axiom,
    ! [A3: set_nat_a,F2: nat > nat > a,D: set_nat] :
      ( ( finite_finite_nat_a @ A3 )
     => ( ( inj_on_nat_nat_a @ F2 @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_nat_a @ ( F2 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_380_finite__inverse__image__gen,axiom,
    ! [A3: set_nat_a,F2: a > nat > a,D: set_a] :
      ( ( finite_finite_nat_a @ A3 )
     => ( ( inj_on_a_nat_a @ F2 @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_nat_a @ ( F2 @ J ) @ A3 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_381_domain_Opmod__const_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
           => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Q ) @ one_one_nat ) )
             => ( ( polyno4115915122720352731t_unit @ R @ P2 @ Q )
                = nil_list_list_a ) ) ) ) ) ) ).

% domain.pmod_const(1)
thf(fact_382_domain_Opmod__const_I1_J,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
           => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ Q ) @ one_one_nat ) )
             => ( ( polyno5933693564459536891t_unit @ R @ P2 @ Q )
                = nil_list_set_list_a ) ) ) ) ) ) ).

% domain.pmod_const(1)
thf(fact_383_domain_Opmod__const_I1_J,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
           => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ Q ) @ one_one_nat ) )
             => ( ( polyno6856256803263433473t_unit @ R @ P2 @ Q )
                = nil_set_list_a ) ) ) ) ) ) ).

% domain.pmod_const(1)
thf(fact_384_domain_Opmod__const_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
             => ( ( polynomial_pdiv_a_b @ R @ P2 @ Q )
                = nil_a ) ) ) ) ) ) ).

% domain.pmod_const(1)
thf(fact_385_domain_Opmod__const_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) )
             => ( ( polyno5893782122288709345t_unit @ R @ P2 @ Q )
                = nil_list_a ) ) ) ) ) ) ).

% domain.pmod_const(1)
thf(fact_386_domain_Opmod__const_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
           => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Q ) @ one_one_nat ) )
             => ( ( polyno2750255454319713356t_unit @ R @ P2 @ Q )
                = P2 ) ) ) ) ) ) ).

% domain.pmod_const(2)
thf(fact_387_domain_Opmod__const_I2_J,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
           => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ Q ) @ one_one_nat ) )
             => ( ( polyno427657113825607788t_unit @ R @ P2 @ Q )
                = P2 ) ) ) ) ) ) ).

% domain.pmod_const(2)
thf(fact_388_domain_Opmod__const_I2_J,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
           => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ Q ) @ one_one_nat ) )
             => ( ( polyno8414006224137659890t_unit @ R @ P2 @ Q )
                = P2 ) ) ) ) ) ) ).

% domain.pmod_const(2)
thf(fact_389_domain_Opmod__const_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
             => ( ( polynomial_pmod_a_b @ R @ P2 @ Q )
                = P2 ) ) ) ) ) ) ).

% domain.pmod_const(2)
thf(fact_390_domain_Opmod__const_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) )
             => ( ( polyno1727750685288865234t_unit @ R @ P2 @ Q )
                = P2 ) ) ) ) ) ) ).

% domain.pmod_const(2)
thf(fact_391_all__finite__subset__image,axiom,
    ! [F2: nat > nat,A3: set_nat,P3: set_nat > $o] :
      ( ( ! [B3: set_nat] :
            ( ( ( finite_finite_nat @ B3 )
              & ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A3 ) ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_nat] :
            ( ( ( finite_finite_nat @ B3 )
              & ( ord_less_eq_set_nat @ B3 @ A3 ) )
           => ( P3 @ ( image_nat_nat @ F2 @ B3 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_392_all__finite__subset__image,axiom,
    ! [F2: a > nat,A3: set_a,P3: set_nat > $o] :
      ( ( ! [B3: set_nat] :
            ( ( ( finite_finite_nat @ B3 )
              & ( ord_less_eq_set_nat @ B3 @ ( image_a_nat @ F2 @ A3 ) ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ( finite_finite_a @ B3 )
              & ( ord_less_eq_set_a @ B3 @ A3 ) )
           => ( P3 @ ( image_a_nat @ F2 @ B3 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_393_all__finite__subset__image,axiom,
    ! [F2: nat > a,A3: set_nat,P3: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ( finite_finite_a @ B3 )
              & ( ord_less_eq_set_a @ B3 @ ( image_nat_a @ F2 @ A3 ) ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_nat] :
            ( ( ( finite_finite_nat @ B3 )
              & ( ord_less_eq_set_nat @ B3 @ A3 ) )
           => ( P3 @ ( image_nat_a @ F2 @ B3 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_394_all__finite__subset__image,axiom,
    ! [F2: a > a,A3: set_a,P3: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ( finite_finite_a @ B3 )
              & ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F2 @ A3 ) ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ( finite_finite_a @ B3 )
              & ( ord_less_eq_set_a @ B3 @ A3 ) )
           => ( P3 @ ( image_a_a @ F2 @ B3 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_395_all__finite__subset__image,axiom,
    ! [F2: list_a > nat,A3: set_list_a,P3: set_nat > $o] :
      ( ( ! [B3: set_nat] :
            ( ( ( finite_finite_nat @ B3 )
              & ( ord_less_eq_set_nat @ B3 @ ( image_list_a_nat @ F2 @ A3 ) ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_list_a] :
            ( ( ( finite_finite_list_a @ B3 )
              & ( ord_le8861187494160871172list_a @ B3 @ A3 ) )
           => ( P3 @ ( image_list_a_nat @ F2 @ B3 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_396_all__finite__subset__image,axiom,
    ! [F2: set_a > nat,A3: set_set_a,P3: set_nat > $o] :
      ( ( ! [B3: set_nat] :
            ( ( ( finite_finite_nat @ B3 )
              & ( ord_less_eq_set_nat @ B3 @ ( image_set_a_nat @ F2 @ A3 ) ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_set_a] :
            ( ( ( finite_finite_set_a @ B3 )
              & ( ord_le3724670747650509150_set_a @ B3 @ A3 ) )
           => ( P3 @ ( image_set_a_nat @ F2 @ B3 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_397_all__finite__subset__image,axiom,
    ! [F2: nat > list_a,A3: set_nat,P3: set_list_a > $o] :
      ( ( ! [B3: set_list_a] :
            ( ( ( finite_finite_list_a @ B3 )
              & ( ord_le8861187494160871172list_a @ B3 @ ( image_nat_list_a @ F2 @ A3 ) ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_nat] :
            ( ( ( finite_finite_nat @ B3 )
              & ( ord_less_eq_set_nat @ B3 @ A3 ) )
           => ( P3 @ ( image_nat_list_a @ F2 @ B3 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_398_all__finite__subset__image,axiom,
    ! [F2: a > list_a,A3: set_a,P3: set_list_a > $o] :
      ( ( ! [B3: set_list_a] :
            ( ( ( finite_finite_list_a @ B3 )
              & ( ord_le8861187494160871172list_a @ B3 @ ( image_a_list_a @ F2 @ A3 ) ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_a] :
            ( ( ( finite_finite_a @ B3 )
              & ( ord_less_eq_set_a @ B3 @ A3 ) )
           => ( P3 @ ( image_a_list_a @ F2 @ B3 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_399_all__finite__subset__image,axiom,
    ! [F2: list_a > a,A3: set_list_a,P3: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ( finite_finite_a @ B3 )
              & ( ord_less_eq_set_a @ B3 @ ( image_list_a_a @ F2 @ A3 ) ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_list_a] :
            ( ( ( finite_finite_list_a @ B3 )
              & ( ord_le8861187494160871172list_a @ B3 @ A3 ) )
           => ( P3 @ ( image_list_a_a @ F2 @ B3 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_400_all__finite__subset__image,axiom,
    ! [F2: set_a > a,A3: set_set_a,P3: set_a > $o] :
      ( ( ! [B3: set_a] :
            ( ( ( finite_finite_a @ B3 )
              & ( ord_less_eq_set_a @ B3 @ ( image_set_a_a @ F2 @ A3 ) ) )
           => ( P3 @ B3 ) ) )
      = ( ! [B3: set_set_a] :
            ( ( ( finite_finite_set_a @ B3 )
              & ( ord_le3724670747650509150_set_a @ B3 @ A3 ) )
           => ( P3 @ ( image_set_a_a @ F2 @ B3 ) ) ) ) ) ).

% all_finite_subset_image
thf(fact_401_ex__finite__subset__image,axiom,
    ! [F2: nat > nat,A3: set_nat,P3: set_nat > $o] :
      ( ( ? [B3: set_nat] :
            ( ( finite_finite_nat @ B3 )
            & ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F2 @ A3 ) )
            & ( P3 @ B3 ) ) )
      = ( ? [B3: set_nat] :
            ( ( finite_finite_nat @ B3 )
            & ( ord_less_eq_set_nat @ B3 @ A3 )
            & ( P3 @ ( image_nat_nat @ F2 @ B3 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_402_ex__finite__subset__image,axiom,
    ! [F2: a > nat,A3: set_a,P3: set_nat > $o] :
      ( ( ? [B3: set_nat] :
            ( ( finite_finite_nat @ B3 )
            & ( ord_less_eq_set_nat @ B3 @ ( image_a_nat @ F2 @ A3 ) )
            & ( P3 @ B3 ) ) )
      = ( ? [B3: set_a] :
            ( ( finite_finite_a @ B3 )
            & ( ord_less_eq_set_a @ B3 @ A3 )
            & ( P3 @ ( image_a_nat @ F2 @ B3 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_403_ex__finite__subset__image,axiom,
    ! [F2: nat > a,A3: set_nat,P3: set_a > $o] :
      ( ( ? [B3: set_a] :
            ( ( finite_finite_a @ B3 )
            & ( ord_less_eq_set_a @ B3 @ ( image_nat_a @ F2 @ A3 ) )
            & ( P3 @ B3 ) ) )
      = ( ? [B3: set_nat] :
            ( ( finite_finite_nat @ B3 )
            & ( ord_less_eq_set_nat @ B3 @ A3 )
            & ( P3 @ ( image_nat_a @ F2 @ B3 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_404_ex__finite__subset__image,axiom,
    ! [F2: a > a,A3: set_a,P3: set_a > $o] :
      ( ( ? [B3: set_a] :
            ( ( finite_finite_a @ B3 )
            & ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F2 @ A3 ) )
            & ( P3 @ B3 ) ) )
      = ( ? [B3: set_a] :
            ( ( finite_finite_a @ B3 )
            & ( ord_less_eq_set_a @ B3 @ A3 )
            & ( P3 @ ( image_a_a @ F2 @ B3 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_405_ex__finite__subset__image,axiom,
    ! [F2: list_a > nat,A3: set_list_a,P3: set_nat > $o] :
      ( ( ? [B3: set_nat] :
            ( ( finite_finite_nat @ B3 )
            & ( ord_less_eq_set_nat @ B3 @ ( image_list_a_nat @ F2 @ A3 ) )
            & ( P3 @ B3 ) ) )
      = ( ? [B3: set_list_a] :
            ( ( finite_finite_list_a @ B3 )
            & ( ord_le8861187494160871172list_a @ B3 @ A3 )
            & ( P3 @ ( image_list_a_nat @ F2 @ B3 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_406_ex__finite__subset__image,axiom,
    ! [F2: set_a > nat,A3: set_set_a,P3: set_nat > $o] :
      ( ( ? [B3: set_nat] :
            ( ( finite_finite_nat @ B3 )
            & ( ord_less_eq_set_nat @ B3 @ ( image_set_a_nat @ F2 @ A3 ) )
            & ( P3 @ B3 ) ) )
      = ( ? [B3: set_set_a] :
            ( ( finite_finite_set_a @ B3 )
            & ( ord_le3724670747650509150_set_a @ B3 @ A3 )
            & ( P3 @ ( image_set_a_nat @ F2 @ B3 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_407_ex__finite__subset__image,axiom,
    ! [F2: nat > list_a,A3: set_nat,P3: set_list_a > $o] :
      ( ( ? [B3: set_list_a] :
            ( ( finite_finite_list_a @ B3 )
            & ( ord_le8861187494160871172list_a @ B3 @ ( image_nat_list_a @ F2 @ A3 ) )
            & ( P3 @ B3 ) ) )
      = ( ? [B3: set_nat] :
            ( ( finite_finite_nat @ B3 )
            & ( ord_less_eq_set_nat @ B3 @ A3 )
            & ( P3 @ ( image_nat_list_a @ F2 @ B3 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_408_ex__finite__subset__image,axiom,
    ! [F2: a > list_a,A3: set_a,P3: set_list_a > $o] :
      ( ( ? [B3: set_list_a] :
            ( ( finite_finite_list_a @ B3 )
            & ( ord_le8861187494160871172list_a @ B3 @ ( image_a_list_a @ F2 @ A3 ) )
            & ( P3 @ B3 ) ) )
      = ( ? [B3: set_a] :
            ( ( finite_finite_a @ B3 )
            & ( ord_less_eq_set_a @ B3 @ A3 )
            & ( P3 @ ( image_a_list_a @ F2 @ B3 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_409_ex__finite__subset__image,axiom,
    ! [F2: list_a > a,A3: set_list_a,P3: set_a > $o] :
      ( ( ? [B3: set_a] :
            ( ( finite_finite_a @ B3 )
            & ( ord_less_eq_set_a @ B3 @ ( image_list_a_a @ F2 @ A3 ) )
            & ( P3 @ B3 ) ) )
      = ( ? [B3: set_list_a] :
            ( ( finite_finite_list_a @ B3 )
            & ( ord_le8861187494160871172list_a @ B3 @ A3 )
            & ( P3 @ ( image_list_a_a @ F2 @ B3 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_410_ex__finite__subset__image,axiom,
    ! [F2: set_a > a,A3: set_set_a,P3: set_a > $o] :
      ( ( ? [B3: set_a] :
            ( ( finite_finite_a @ B3 )
            & ( ord_less_eq_set_a @ B3 @ ( image_set_a_a @ F2 @ A3 ) )
            & ( P3 @ B3 ) ) )
      = ( ? [B3: set_set_a] :
            ( ( finite_finite_set_a @ B3 )
            & ( ord_le3724670747650509150_set_a @ B3 @ A3 )
            & ( P3 @ ( image_set_a_a @ F2 @ B3 ) ) ) ) ) ).

% ex_finite_subset_image
thf(fact_411_finite__subset__image,axiom,
    ! [B2: set_nat,F2: nat > nat,A3: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A3 ) )
       => ? [C2: set_nat] :
            ( ( ord_less_eq_set_nat @ C2 @ A3 )
            & ( finite_finite_nat @ C2 )
            & ( B2
              = ( image_nat_nat @ F2 @ C2 ) ) ) ) ) ).

% finite_subset_image
thf(fact_412_finite__subset__image,axiom,
    ! [B2: set_nat,F2: a > nat,A3: set_a] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F2 @ A3 ) )
       => ? [C2: set_a] :
            ( ( ord_less_eq_set_a @ C2 @ A3 )
            & ( finite_finite_a @ C2 )
            & ( B2
              = ( image_a_nat @ F2 @ C2 ) ) ) ) ) ).

% finite_subset_image
thf(fact_413_finite__subset__image,axiom,
    ! [B2: set_a,F2: nat > a,A3: set_nat] :
      ( ( finite_finite_a @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F2 @ A3 ) )
       => ? [C2: set_nat] :
            ( ( ord_less_eq_set_nat @ C2 @ A3 )
            & ( finite_finite_nat @ C2 )
            & ( B2
              = ( image_nat_a @ F2 @ C2 ) ) ) ) ) ).

% finite_subset_image
thf(fact_414_finite__subset__image,axiom,
    ! [B2: set_a,F2: a > a,A3: set_a] :
      ( ( finite_finite_a @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A3 ) )
       => ? [C2: set_a] :
            ( ( ord_less_eq_set_a @ C2 @ A3 )
            & ( finite_finite_a @ C2 )
            & ( B2
              = ( image_a_a @ F2 @ C2 ) ) ) ) ) ).

% finite_subset_image
thf(fact_415_finite__subset__image,axiom,
    ! [B2: set_nat,F2: list_a > nat,A3: set_list_a] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( image_list_a_nat @ F2 @ A3 ) )
       => ? [C2: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C2 @ A3 )
            & ( finite_finite_list_a @ C2 )
            & ( B2
              = ( image_list_a_nat @ F2 @ C2 ) ) ) ) ) ).

% finite_subset_image
thf(fact_416_finite__subset__image,axiom,
    ! [B2: set_nat,F2: set_a > nat,A3: set_set_a] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( image_set_a_nat @ F2 @ A3 ) )
       => ? [C2: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ C2 @ A3 )
            & ( finite_finite_set_a @ C2 )
            & ( B2
              = ( image_set_a_nat @ F2 @ C2 ) ) ) ) ) ).

% finite_subset_image
thf(fact_417_finite__subset__image,axiom,
    ! [B2: set_list_a,F2: nat > list_a,A3: set_nat] :
      ( ( finite_finite_list_a @ B2 )
     => ( ( ord_le8861187494160871172list_a @ B2 @ ( image_nat_list_a @ F2 @ A3 ) )
       => ? [C2: set_nat] :
            ( ( ord_less_eq_set_nat @ C2 @ A3 )
            & ( finite_finite_nat @ C2 )
            & ( B2
              = ( image_nat_list_a @ F2 @ C2 ) ) ) ) ) ).

% finite_subset_image
thf(fact_418_finite__subset__image,axiom,
    ! [B2: set_list_a,F2: a > list_a,A3: set_a] :
      ( ( finite_finite_list_a @ B2 )
     => ( ( ord_le8861187494160871172list_a @ B2 @ ( image_a_list_a @ F2 @ A3 ) )
       => ? [C2: set_a] :
            ( ( ord_less_eq_set_a @ C2 @ A3 )
            & ( finite_finite_a @ C2 )
            & ( B2
              = ( image_a_list_a @ F2 @ C2 ) ) ) ) ) ).

% finite_subset_image
thf(fact_419_finite__subset__image,axiom,
    ! [B2: set_a,F2: list_a > a,A3: set_list_a] :
      ( ( finite_finite_a @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ ( image_list_a_a @ F2 @ A3 ) )
       => ? [C2: set_list_a] :
            ( ( ord_le8861187494160871172list_a @ C2 @ A3 )
            & ( finite_finite_list_a @ C2 )
            & ( B2
              = ( image_list_a_a @ F2 @ C2 ) ) ) ) ) ).

% finite_subset_image
thf(fact_420_finite__subset__image,axiom,
    ! [B2: set_a,F2: set_a > a,A3: set_set_a] :
      ( ( finite_finite_a @ B2 )
     => ( ( ord_less_eq_set_a @ B2 @ ( image_set_a_a @ F2 @ A3 ) )
       => ? [C2: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ C2 @ A3 )
            & ( finite_finite_set_a @ C2 )
            & ( B2
              = ( image_set_a_a @ F2 @ C2 ) ) ) ) ) ).

% finite_subset_image
thf(fact_421_finite__surj,axiom,
    ! [A3: set_nat,B2: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( image_nat_nat @ F2 @ A3 ) )
       => ( finite_finite_nat @ B2 ) ) ) ).

% finite_surj
thf(fact_422_finite__surj,axiom,
    ! [A3: set_a,B2: set_nat,F2: a > nat] :
      ( ( finite_finite_a @ A3 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( image_a_nat @ F2 @ A3 ) )
       => ( finite_finite_nat @ B2 ) ) ) ).

% finite_surj
thf(fact_423_finite__surj,axiom,
    ! [A3: set_nat,B2: set_a,F2: nat > a] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_a @ B2 @ ( image_nat_a @ F2 @ A3 ) )
       => ( finite_finite_a @ B2 ) ) ) ).

% finite_surj
thf(fact_424_finite__surj,axiom,
    ! [A3: set_a,B2: set_a,F2: a > a] :
      ( ( finite_finite_a @ A3 )
     => ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F2 @ A3 ) )
       => ( finite_finite_a @ B2 ) ) ) ).

% finite_surj
thf(fact_425_finite__surj,axiom,
    ! [A3: set_list_a,B2: set_nat,F2: list_a > nat] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( ord_less_eq_set_nat @ B2 @ ( image_list_a_nat @ F2 @ A3 ) )
       => ( finite_finite_nat @ B2 ) ) ) ).

% finite_surj
thf(fact_426_finite__surj,axiom,
    ! [A3: set_nat,B2: set_list_a,F2: nat > list_a] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le8861187494160871172list_a @ B2 @ ( image_nat_list_a @ F2 @ A3 ) )
       => ( finite_finite_list_a @ B2 ) ) ) ).

% finite_surj
thf(fact_427_finite__surj,axiom,
    ! [A3: set_a,B2: set_list_a,F2: a > list_a] :
      ( ( finite_finite_a @ A3 )
     => ( ( ord_le8861187494160871172list_a @ B2 @ ( image_a_list_a @ F2 @ A3 ) )
       => ( finite_finite_list_a @ B2 ) ) ) ).

% finite_surj
thf(fact_428_finite__surj,axiom,
    ! [A3: set_list_a,B2: set_a,F2: list_a > a] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( ord_less_eq_set_a @ B2 @ ( image_list_a_a @ F2 @ A3 ) )
       => ( finite_finite_a @ B2 ) ) ) ).

% finite_surj
thf(fact_429_finite__surj,axiom,
    ! [A3: set_nat,B2: set_set_a,F2: nat > set_a] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_le3724670747650509150_set_a @ B2 @ ( image_nat_set_a @ F2 @ A3 ) )
       => ( finite_finite_set_a @ B2 ) ) ) ).

% finite_surj
thf(fact_430_finite__surj,axiom,
    ! [A3: set_a,B2: set_set_a,F2: a > set_a] :
      ( ( finite_finite_a @ A3 )
     => ( ( ord_le3724670747650509150_set_a @ B2 @ ( image_a_set_a @ F2 @ A3 ) )
       => ( finite_finite_set_a @ B2 ) ) ) ).

% finite_surj
thf(fact_431_domain_Orupture__surj__inj__on,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( inj_on7192984512851773393list_a @ ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 ) )
            @ ( image_865451852121373573list_a
              @ ^ [Q3: list_list_list_a] : ( polyno2750255454319713356t_unit @ R @ Q3 @ P2 )
              @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.rupture_surj_inj_on
thf(fact_432_domain_Orupture__surj__inj__on,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( inj_on5158904088876548049list_a @ ( a_r_co6659872811071116064t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ ( cgenid3857712956745501987t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ P2 ) )
            @ ( image_1936032423783470981list_a
              @ ^ [Q3: list_list_set_list_a] : ( polyno427657113825607788t_unit @ R @ Q3 @ P2 )
              @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.rupture_surj_inj_on
thf(fact_433_domain_Orupture__surj__inj__on,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( inj_on3583008398376989521list_a @ ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 ) )
            @ ( image_341778791481906565list_a
              @ ^ [Q3: list_set_list_a] : ( polyno8414006224137659890t_unit @ R @ Q3 @ P2 )
              @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.rupture_surj_inj_on
thf(fact_434_domain_Orupture__surj__inj__on,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) )
            @ ( image_list_a_list_a
              @ ^ [Q3: list_a] : ( polynomial_pmod_a_b @ R @ Q3 @ P2 )
              @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).

% domain.rupture_surj_inj_on
thf(fact_435_domain_Orupture__surj__inj__on,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( inj_on3733059618185315153list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) )
            @ ( image_5446147394702115205list_a
              @ ^ [Q3: list_list_a] : ( polyno1727750685288865234t_unit @ R @ Q3 @ P2 )
              @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.rupture_surj_inj_on
thf(fact_436_finite__image__iff,axiom,
    ! [F2: nat > nat,A3: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A3 )
     => ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A3 ) )
        = ( finite_finite_nat @ A3 ) ) ) ).

% finite_image_iff
thf(fact_437_finite__image__iff,axiom,
    ! [F2: a > nat,A3: set_a] :
      ( ( inj_on_a_nat @ F2 @ A3 )
     => ( ( finite_finite_nat @ ( image_a_nat @ F2 @ A3 ) )
        = ( finite_finite_a @ A3 ) ) ) ).

% finite_image_iff
thf(fact_438_finite__image__iff,axiom,
    ! [F2: nat > a,A3: set_nat] :
      ( ( inj_on_nat_a @ F2 @ A3 )
     => ( ( finite_finite_a @ ( image_nat_a @ F2 @ A3 ) )
        = ( finite_finite_nat @ A3 ) ) ) ).

% finite_image_iff
thf(fact_439_finite__image__iff,axiom,
    ! [F2: a > a,A3: set_a] :
      ( ( inj_on_a_a @ F2 @ A3 )
     => ( ( finite_finite_a @ ( image_a_a @ F2 @ A3 ) )
        = ( finite_finite_a @ A3 ) ) ) ).

% finite_image_iff
thf(fact_440_finite__image__iff,axiom,
    ! [F2: a > set_a,A3: set_a] :
      ( ( inj_on_a_set_a @ F2 @ A3 )
     => ( ( finite_finite_set_a @ ( image_a_set_a @ F2 @ A3 ) )
        = ( finite_finite_a @ A3 ) ) ) ).

% finite_image_iff
thf(fact_441_finite__image__iff,axiom,
    ! [F2: nat > list_a,A3: set_nat] :
      ( ( inj_on_nat_list_a @ F2 @ A3 )
     => ( ( finite_finite_list_a @ ( image_nat_list_a @ F2 @ A3 ) )
        = ( finite_finite_nat @ A3 ) ) ) ).

% finite_image_iff
thf(fact_442_finite__image__iff,axiom,
    ! [F2: a > list_a,A3: set_a] :
      ( ( inj_on_a_list_a @ F2 @ A3 )
     => ( ( finite_finite_list_a @ ( image_a_list_a @ F2 @ A3 ) )
        = ( finite_finite_a @ A3 ) ) ) ).

% finite_image_iff
thf(fact_443_finite__image__iff,axiom,
    ! [F2: list_a > nat,A3: set_list_a] :
      ( ( inj_on_list_a_nat @ F2 @ A3 )
     => ( ( finite_finite_nat @ ( image_list_a_nat @ F2 @ A3 ) )
        = ( finite_finite_list_a @ A3 ) ) ) ).

% finite_image_iff
thf(fact_444_finite__image__iff,axiom,
    ! [F2: list_a > a,A3: set_list_a] :
      ( ( inj_on_list_a_a @ F2 @ A3 )
     => ( ( finite_finite_a @ ( image_list_a_a @ F2 @ A3 ) )
        = ( finite_finite_list_a @ A3 ) ) ) ).

% finite_image_iff
thf(fact_445_finite__image__iff,axiom,
    ! [F2: list_a > list_a,A3: set_list_a] :
      ( ( inj_on_list_a_list_a @ F2 @ A3 )
     => ( ( finite_finite_list_a @ ( image_list_a_list_a @ F2 @ A3 ) )
        = ( finite_finite_list_a @ A3 ) ) ) ).

% finite_image_iff
thf(fact_446_finite__imageD,axiom,
    ! [F2: nat > nat,A3: set_nat] :
      ( ( finite_finite_nat @ ( image_nat_nat @ F2 @ A3 ) )
     => ( ( inj_on_nat_nat @ F2 @ A3 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% finite_imageD
thf(fact_447_finite__imageD,axiom,
    ! [F2: a > nat,A3: set_a] :
      ( ( finite_finite_nat @ ( image_a_nat @ F2 @ A3 ) )
     => ( ( inj_on_a_nat @ F2 @ A3 )
       => ( finite_finite_a @ A3 ) ) ) ).

% finite_imageD
thf(fact_448_finite__imageD,axiom,
    ! [F2: nat > a,A3: set_nat] :
      ( ( finite_finite_a @ ( image_nat_a @ F2 @ A3 ) )
     => ( ( inj_on_nat_a @ F2 @ A3 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% finite_imageD
thf(fact_449_finite__imageD,axiom,
    ! [F2: a > a,A3: set_a] :
      ( ( finite_finite_a @ ( image_a_a @ F2 @ A3 ) )
     => ( ( inj_on_a_a @ F2 @ A3 )
       => ( finite_finite_a @ A3 ) ) ) ).

% finite_imageD
thf(fact_450_finite__imageD,axiom,
    ! [F2: a > set_a,A3: set_a] :
      ( ( finite_finite_set_a @ ( image_a_set_a @ F2 @ A3 ) )
     => ( ( inj_on_a_set_a @ F2 @ A3 )
       => ( finite_finite_a @ A3 ) ) ) ).

% finite_imageD
thf(fact_451_finite__imageD,axiom,
    ! [F2: nat > list_a,A3: set_nat] :
      ( ( finite_finite_list_a @ ( image_nat_list_a @ F2 @ A3 ) )
     => ( ( inj_on_nat_list_a @ F2 @ A3 )
       => ( finite_finite_nat @ A3 ) ) ) ).

% finite_imageD
thf(fact_452_finite__imageD,axiom,
    ! [F2: a > list_a,A3: set_a] :
      ( ( finite_finite_list_a @ ( image_a_list_a @ F2 @ A3 ) )
     => ( ( inj_on_a_list_a @ F2 @ A3 )
       => ( finite_finite_a @ A3 ) ) ) ).

% finite_imageD
thf(fact_453_finite__imageD,axiom,
    ! [F2: list_a > nat,A3: set_list_a] :
      ( ( finite_finite_nat @ ( image_list_a_nat @ F2 @ A3 ) )
     => ( ( inj_on_list_a_nat @ F2 @ A3 )
       => ( finite_finite_list_a @ A3 ) ) ) ).

% finite_imageD
thf(fact_454_finite__imageD,axiom,
    ! [F2: list_a > a,A3: set_list_a] :
      ( ( finite_finite_a @ ( image_list_a_a @ F2 @ A3 ) )
     => ( ( inj_on_list_a_a @ F2 @ A3 )
       => ( finite_finite_list_a @ A3 ) ) ) ).

% finite_imageD
thf(fact_455_finite__imageD,axiom,
    ! [F2: list_a > list_a,A3: set_list_a] :
      ( ( finite_finite_list_a @ ( image_list_a_list_a @ F2 @ A3 ) )
     => ( ( inj_on_list_a_list_a @ F2 @ A3 )
       => ( finite_finite_list_a @ A3 ) ) ) ).

% finite_imageD
thf(fact_456_domain_Odegree__zero__imp__not__is__root,axiom,
    ! [R: partia2956882679547061052t_unit,P2: list_list_list_a,X: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ~ ( polyno5142720416380192742t_unit @ R @ P2 @ X ) ) ) ) ).

% domain.degree_zero_imp_not_is_root
thf(fact_457_domain_Odegree__zero__imp__not__is__root,axiom,
    ! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a,X: set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R )
     => ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ~ ( polyno875383481011669382t_unit @ R @ P2 @ X ) ) ) ) ).

% domain.degree_zero_imp_not_is_root
thf(fact_458_domain_Odegree__zero__imp__not__is__root,axiom,
    ! [R: partia6043505979758434576t_unit,P2: list_set_a,X: set_a] :
      ( ( domain4236798911309298543t_unit @ R )
     => ( ( member_list_set_a @ P2 @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_size_list_set_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ~ ( polyno4890645956962836498t_unit @ R @ P2 @ X ) ) ) ) ).

% domain.degree_zero_imp_not_is_root
thf(fact_459_domain_Odegree__zero__imp__not__is__root,axiom,
    ! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ~ ( polyno6951661231331188332t_unit @ R @ P2 @ X ) ) ) ) ).

% domain.degree_zero_imp_not_is_root
thf(fact_460_domain_Odegree__zero__imp__not__is__root,axiom,
    ! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ~ ( polyno4133073214067823460ot_a_b @ R @ P2 @ X ) ) ) ) ).

% domain.degree_zero_imp_not_is_root
thf(fact_461_domain_Odegree__zero__imp__not__is__root,axiom,
    ! [R: partia7496981018696276118t_unit,P2: list_set_list_a,X: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ~ ( polyno4320237611291262604t_unit @ R @ P2 @ X ) ) ) ) ).

% domain.degree_zero_imp_not_is_root
thf(fact_462_domain_Orupture__carrier__as__pmod__image,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( image_5096927308667620453list_a @ ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 ) )
              @ ( image_865451852121373573list_a
                @ ^ [Q3: list_list_list_a] : ( polyno2750255454319713356t_unit @ R @ Q3 @ P2 )
                @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) ) )
            = ( partia7210328371471923989t_unit @ ( polyno8931960069169623149t_unit @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_carrier_as_pmod_image
thf(fact_463_domain_Orupture__carrier__as__pmod__image,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( image_3381647565828645989list_a @ ( a_r_co6659872811071116064t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ ( cgenid3857712956745501987t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ P2 ) )
              @ ( image_1936032423783470981list_a
                @ ^ [Q3: list_list_set_list_a] : ( polyno427657113825607788t_unit @ R @ Q3 @ P2 )
                @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) ) )
            = ( partia5834437165032953589t_unit @ ( polyno1128427740506820749t_unit @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_carrier_as_pmod_image
thf(fact_464_domain_Orupture__carrier__as__pmod__image,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( image_1309896914751589349list_a @ ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 ) )
              @ ( image_341778791481906565list_a
                @ ^ [Q3: list_set_list_a] : ( polyno8414006224137659890t_unit @ R @ Q3 @ P2 )
                @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) ) )
            = ( partia5585283898957912943t_unit @ ( polyno7054224608333822611t_unit @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_carrier_as_pmod_image
thf(fact_465_domain_Orupture__carrier__as__pmod__image,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) )
              @ ( image_list_a_list_a
                @ ^ [Q3: list_a] : ( polynomial_pmod_a_b @ R @ Q3 @ P2 )
                @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) )
            = ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_carrier_as_pmod_image
thf(fact_466_domain_Orupture__carrier__as__pmod__image,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( image_551801017575455717list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) )
              @ ( image_5446147394702115205list_a
                @ ^ [Q3: list_list_a] : ( polyno1727750685288865234t_unit @ R @ Q3 @ P2 )
                @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) )
            = ( partia3317168157747563407t_unit @ ( polyno859807163042199155t_unit @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_carrier_as_pmod_image
thf(fact_467_finite__surj__inj,axiom,
    ! [A3: set_list_set_list_a,F2: list_set_list_a > list_set_list_a] :
      ( ( finite3202133031812794515list_a @ A3 )
     => ( ( ord_le1961058059034850666list_a @ A3 @ ( image_341778791481906565list_a @ F2 @ A3 ) )
       => ( inj_on320113931233832177list_a @ F2 @ A3 ) ) ) ).

% finite_surj_inj
thf(fact_468_finite__surj__inj,axiom,
    ! [A3: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ A3 @ ( image_nat_nat @ F2 @ A3 ) )
       => ( inj_on_nat_nat @ F2 @ A3 ) ) ) ).

% finite_surj_inj
thf(fact_469_finite__surj__inj,axiom,
    ! [A3: set_list_a,F2: list_a > list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( ord_le8861187494160871172list_a @ A3 @ ( image_list_a_list_a @ F2 @ A3 ) )
       => ( inj_on_list_a_list_a @ F2 @ A3 ) ) ) ).

% finite_surj_inj
thf(fact_470_finite__surj__inj,axiom,
    ! [A3: set_set_list_a,F2: set_list_a > set_list_a] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ( ( ord_le8877086941679407844list_a @ A3 @ ( image_5749939591322298757list_a @ F2 @ A3 ) )
       => ( inj_on727569427812895985list_a @ F2 @ A3 ) ) ) ).

% finite_surj_inj
thf(fact_471_finite__surj__inj,axiom,
    ! [A3: set_a,F2: a > a] :
      ( ( finite_finite_a @ A3 )
     => ( ( ord_less_eq_set_a @ A3 @ ( image_a_a @ F2 @ A3 ) )
       => ( inj_on_a_a @ F2 @ A3 ) ) ) ).

% finite_surj_inj
thf(fact_472_finite__surj__inj,axiom,
    ! [A3: set_set_a,F2: set_a > set_a] :
      ( ( finite_finite_set_a @ A3 )
     => ( ( ord_le3724670747650509150_set_a @ A3 @ ( image_set_a_set_a @ F2 @ A3 ) )
       => ( inj_on_set_a_set_a @ F2 @ A3 ) ) ) ).

% finite_surj_inj
thf(fact_473_inj__on__finite,axiom,
    ! [F2: nat > nat,A3: set_nat,B2: set_nat] :
      ( ( inj_on_nat_nat @ F2 @ A3 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A3 ) @ B2 )
       => ( ( finite_finite_nat @ B2 )
         => ( finite_finite_nat @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_474_inj__on__finite,axiom,
    ! [F2: a > nat,A3: set_a,B2: set_nat] :
      ( ( inj_on_a_nat @ F2 @ A3 )
     => ( ( ord_less_eq_set_nat @ ( image_a_nat @ F2 @ A3 ) @ B2 )
       => ( ( finite_finite_nat @ B2 )
         => ( finite_finite_a @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_475_inj__on__finite,axiom,
    ! [F2: nat > a,A3: set_nat,B2: set_a] :
      ( ( inj_on_nat_a @ F2 @ A3 )
     => ( ( ord_less_eq_set_a @ ( image_nat_a @ F2 @ A3 ) @ B2 )
       => ( ( finite_finite_a @ B2 )
         => ( finite_finite_nat @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_476_inj__on__finite,axiom,
    ! [F2: a > a,A3: set_a,B2: set_a] :
      ( ( inj_on_a_a @ F2 @ A3 )
     => ( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A3 ) @ B2 )
       => ( ( finite_finite_a @ B2 )
         => ( finite_finite_a @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_477_inj__on__finite,axiom,
    ! [F2: list_a > nat,A3: set_list_a,B2: set_nat] :
      ( ( inj_on_list_a_nat @ F2 @ A3 )
     => ( ( ord_less_eq_set_nat @ ( image_list_a_nat @ F2 @ A3 ) @ B2 )
       => ( ( finite_finite_nat @ B2 )
         => ( finite_finite_list_a @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_478_inj__on__finite,axiom,
    ! [F2: nat > list_a,A3: set_nat,B2: set_list_a] :
      ( ( inj_on_nat_list_a @ F2 @ A3 )
     => ( ( ord_le8861187494160871172list_a @ ( image_nat_list_a @ F2 @ A3 ) @ B2 )
       => ( ( finite_finite_list_a @ B2 )
         => ( finite_finite_nat @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_479_inj__on__finite,axiom,
    ! [F2: a > list_a,A3: set_a,B2: set_list_a] :
      ( ( inj_on_a_list_a @ F2 @ A3 )
     => ( ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F2 @ A3 ) @ B2 )
       => ( ( finite_finite_list_a @ B2 )
         => ( finite_finite_a @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_480_inj__on__finite,axiom,
    ! [F2: list_a > a,A3: set_list_a,B2: set_a] :
      ( ( inj_on_list_a_a @ F2 @ A3 )
     => ( ( ord_less_eq_set_a @ ( image_list_a_a @ F2 @ A3 ) @ B2 )
       => ( ( finite_finite_a @ B2 )
         => ( finite_finite_list_a @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_481_inj__on__finite,axiom,
    ! [F2: nat > set_a,A3: set_nat,B2: set_set_a] :
      ( ( inj_on_nat_set_a @ F2 @ A3 )
     => ( ( ord_le3724670747650509150_set_a @ ( image_nat_set_a @ F2 @ A3 ) @ B2 )
       => ( ( finite_finite_set_a @ B2 )
         => ( finite_finite_nat @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_482_inj__on__finite,axiom,
    ! [F2: a > set_a,A3: set_a,B2: set_set_a] :
      ( ( inj_on_a_set_a @ F2 @ A3 )
     => ( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F2 @ A3 ) @ B2 )
       => ( ( finite_finite_set_a @ B2 )
         => ( finite_finite_a @ A3 ) ) ) ) ).

% inj_on_finite
thf(fact_483_endo__inj__surj,axiom,
    ! [A3: set_list_set_list_a,F2: list_set_list_a > list_set_list_a] :
      ( ( finite3202133031812794515list_a @ A3 )
     => ( ( ord_le1961058059034850666list_a @ ( image_341778791481906565list_a @ F2 @ A3 ) @ A3 )
       => ( ( inj_on320113931233832177list_a @ F2 @ A3 )
         => ( ( image_341778791481906565list_a @ F2 @ A3 )
            = A3 ) ) ) ) ).

% endo_inj_surj
thf(fact_484_endo__inj__surj,axiom,
    ! [A3: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A3 ) @ A3 )
       => ( ( inj_on_nat_nat @ F2 @ A3 )
         => ( ( image_nat_nat @ F2 @ A3 )
            = A3 ) ) ) ) ).

% endo_inj_surj
thf(fact_485_endo__inj__surj,axiom,
    ! [A3: set_list_a,F2: list_a > list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ( ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F2 @ A3 ) @ A3 )
       => ( ( inj_on_list_a_list_a @ F2 @ A3 )
         => ( ( image_list_a_list_a @ F2 @ A3 )
            = A3 ) ) ) ) ).

% endo_inj_surj
thf(fact_486_endo__inj__surj,axiom,
    ! [A3: set_set_list_a,F2: set_list_a > set_list_a] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ( ( ord_le8877086941679407844list_a @ ( image_5749939591322298757list_a @ F2 @ A3 ) @ A3 )
       => ( ( inj_on727569427812895985list_a @ F2 @ A3 )
         => ( ( image_5749939591322298757list_a @ F2 @ A3 )
            = A3 ) ) ) ) ).

% endo_inj_surj
thf(fact_487_endo__inj__surj,axiom,
    ! [A3: set_a,F2: a > a] :
      ( ( finite_finite_a @ A3 )
     => ( ( ord_less_eq_set_a @ ( image_a_a @ F2 @ A3 ) @ A3 )
       => ( ( inj_on_a_a @ F2 @ A3 )
         => ( ( image_a_a @ F2 @ A3 )
            = A3 ) ) ) ) ).

% endo_inj_surj
thf(fact_488_endo__inj__surj,axiom,
    ! [A3: set_set_a,F2: set_a > set_a] :
      ( ( finite_finite_set_a @ A3 )
     => ( ( ord_le3724670747650509150_set_a @ ( image_set_a_set_a @ F2 @ A3 ) @ A3 )
       => ( ( inj_on_set_a_set_a @ F2 @ A3 )
         => ( ( image_set_a_set_a @ F2 @ A3 )
            = A3 ) ) ) ) ).

% endo_inj_surj
thf(fact_489_domain_Odegree__zero__imp__splitted,axiom,
    ! [R: partia2956882679547061052t_unit,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( polyno5970451904377802771t_unit @ R @ P2 ) ) ) ) ).

% domain.degree_zero_imp_splitted
thf(fact_490_domain_Odegree__zero__imp__splitted,axiom,
    ! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R )
     => ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( polyno3744827648284794291t_unit @ R @ P2 ) ) ) ) ).

% domain.degree_zero_imp_splitted
thf(fact_491_domain_Odegree__zero__imp__splitted,axiom,
    ! [R: partia6043505979758434576t_unit,P2: list_set_a] :
      ( ( domain4236798911309298543t_unit @ R )
     => ( ( member_list_set_a @ P2 @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_size_list_set_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( polyno8339612155817273663t_unit @ R @ P2 ) ) ) ) ).

% domain.degree_zero_imp_splitted
thf(fact_492_domain_Odegree__zero__imp__splitted,axiom,
    ! [R: partia2670972154091845814t_unit,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( polyno6259083269128200473t_unit @ R @ P2 ) ) ) ) ).

% domain.degree_zero_imp_splitted
thf(fact_493_domain_Odegree__zero__imp__splitted,axiom,
    ! [R: partia2175431115845679010xt_a_b,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( polyno8329700637149614481ed_a_b @ R @ P2 ) ) ) ) ).

% domain.degree_zero_imp_splitted
thf(fact_494_domain_Odegree__zero__imp__splitted,axiom,
    ! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( polyno7858167711734664505t_unit @ R @ P2 ) ) ) ) ).

% domain.degree_zero_imp_splitted
thf(fact_495_d_Oorder__gt__0__iff__finite,axiom,
    ( ( ord_less_nat @ zero_zero_nat @ ( order_1351569949434154782t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
    = ( finite5282473924520328461list_a @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.order_gt_0_iff_finite
thf(fact_496_p_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [P2: list_list_a,X: list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X ) )
        = ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X ) ) ) ).

% p.alg_mult_gt_zero_iff_is_root
thf(fact_497_exists__unique__long__division,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ? [X2: produc9164743771328383783list_a] :
                ( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ X2 )
                & ! [Y2: produc9164743771328383783list_a] :
                    ( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ Y2 )
                   => ( Y2 = X2 ) ) ) ) ) ) ) ).

% exists_unique_long_division
thf(fact_498_length__greater__0__conv,axiom,
    ! [Xs: list_list_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) )
      = ( Xs != nil_list_a ) ) ).

% length_greater_0_conv
thf(fact_499_length__greater__0__conv,axiom,
    ! [Xs: list_set_list_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s1991367317912710102list_a @ Xs ) )
      = ( Xs != nil_set_list_a ) ) ).

% length_greater_0_conv
thf(fact_500_length__greater__0__conv,axiom,
    ! [Xs: list_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
      = ( Xs != nil_a ) ) ).

% length_greater_0_conv
thf(fact_501_d_Ofinite__carr__imp__char__ge__0,axiom,
    ( ( finite5282473924520328461list_a @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
   => ( ord_less_nat @ zero_zero_nat @ ( ring_c6053888738502451990t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.finite_carr_imp_char_ge_0
thf(fact_502_p_Obounded__degree__dimension,axiom,
    ! [K: set_list_a,N2: nat] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( embedd2305571234642070248t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ N2 @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K )
        @ ( collect_list_list_a
          @ ^ [Q3: list_list_a] :
              ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
              & ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Q3 ) @ N2 ) ) ) ) ) ).

% p.bounded_degree_dimension
thf(fact_503_p_Odegree__zero__imp__empty__roots,axiom,
    ! [P2: list_list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
          = zero_zero_nat )
       => ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
          = zero_z4454100511807792257list_a ) ) ) ).

% p.degree_zero_imp_empty_roots
thf(fact_504_p_Opirreducible__degree,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).

% p.pirreducible_degree
thf(fact_505_p_Opmod__const_I1_J,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) )
           => ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q )
              = nil_list_a ) ) ) ) ) ).

% p.pmod_const(1)
thf(fact_506_subring__props_I1_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% subring_props(1)
thf(fact_507_finite__number__of__roots,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ r @ P2 ) ) ) ) ).

% finite_number_of_roots
thf(fact_508__092_060open_062inj__on_Apoly__of__const_AK_092_060close_062,axiom,
    inj_on_a_list_a @ ( poly_of_const_a_b @ r ) @ k ).

% \<open>inj_on poly_of_const K\<close>
thf(fact_509_h_Osubring__props_I1_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ord_less_eq_set_a @ K
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.subring_props(1)
thf(fact_510_p_Olong__division__closed_I2_J,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( member_list_list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ) ) ).

% p.long_division_closed(2)
thf(fact_511_degree__zero__imp__not__is__root,axiom,
    ! [P2: list_a,X: a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
          = zero_zero_nat )
       => ~ ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).

% degree_zero_imp_not_is_root
thf(fact_512_degree__zero__imp__splitted,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
          = zero_zero_nat )
       => ( polyno8329700637149614481ed_a_b @ r @ P2 ) ) ) ).

% degree_zero_imp_splitted
thf(fact_513_degree__one__imp__pirreducible,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
            = one_one_nat )
         => ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).

% degree_one_imp_pirreducible
thf(fact_514_p_Olong__division__zero_I2_J,axiom,
    ! [K: set_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a @ Q )
          = nil_list_a ) ) ) ).

% p.long_division_zero(2)
thf(fact_515_pirreducible__degree,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).

% pirreducible_degree
thf(fact_516_p_Olong__division__zero_I1_J,axiom,
    ! [K: set_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a @ Q )
          = nil_list_a ) ) ) ).

% p.long_division_zero(1)
thf(fact_517_p_Orupture__surj__composed__with__pmod,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) @ Q )
            = ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ Q @ P2 ) ) ) ) ) ) ).

% p.rupture_surj_composed_with_pmod
thf(fact_518_p_Orupture__surj__inj__on,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( inj_on3733059618185315153list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) )
          @ ( image_5446147394702115205list_a
            @ ^ [Q3: list_list_a] : ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ Q3 @ P2 )
            @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ) ) ).

% p.rupture_surj_inj_on
thf(fact_519_p_Opmod__const_I2_J,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) )
           => ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q )
              = P2 ) ) ) ) ) ).

% p.pmod_const(2)
thf(fact_520_h_Opirreducible__degree,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( member_list_a @ P2
          @ ( partia5361259788508890537t_unit
            @ ( univ_poly_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ K ) ) )
       => ( ( ring_r932985474545269838t_unit
            @ ( univ_poly_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ K )
            @ P2 )
         => ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).

% h.pirreducible_degree
thf(fact_521_p_Odegree__one__imp__pirreducible,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
            = one_one_nat )
         => ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) ) ) ) ).

% p.degree_one_imp_pirreducible
thf(fact_522_length__0__conv,axiom,
    ! [Xs: list_list_a] :
      ( ( ( size_s349497388124573686list_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_list_a ) ) ).

% length_0_conv
thf(fact_523_length__0__conv,axiom,
    ! [Xs: list_set_list_a] :
      ( ( ( size_s1991367317912710102list_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_set_list_a ) ) ).

% length_0_conv
thf(fact_524_length__0__conv,axiom,
    ! [Xs: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_a ) ) ).

% length_0_conv
thf(fact_525_p_Opmod__degree,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( Q != nil_list_a )
           => ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q )
                = nil_list_a )
              | ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).

% p.pmod_degree
thf(fact_526_p_Opmod__image__characterization,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( P2 != nil_list_a )
         => ( ( image_5446147394702115205list_a
              @ ^ [Q3: list_list_a] : ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ Q3 @ P2 )
              @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
            = ( collect_list_list_a
              @ ^ [Q3: list_list_a] :
                  ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
                  & ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Q3 ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% p.pmod_image_characterization
thf(fact_527_p_Opirreducible__roots,axiom,
    ! [P2: list_list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P2 )
       => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
           != one_one_nat )
         => ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
            = zero_z4454100511807792257list_a ) ) ) ) ).

% p.pirreducible_roots
thf(fact_528_p_Orupture__carrier__as__pmod__image,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( image_551801017575455717list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) )
            @ ( image_5446147394702115205list_a
              @ ^ [Q3: list_list_a] : ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ Q3 @ P2 )
              @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) )
          = ( partia3317168157747563407t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 ) ) ) ) ) ).

% p.rupture_carrier_as_pmod_image
thf(fact_529_ring_Olong__divides_Ocong,axiom,
    polyno2806191415236617128es_a_b = polyno2806191415236617128es_a_b ).

% ring.long_divides.cong
thf(fact_530_ring_Olong__divides_Ocong,axiom,
    polyno6947042923167803568t_unit = polyno6947042923167803568t_unit ).

% ring.long_divides.cong
thf(fact_531_ring_Oalg__mult_Ocong,axiom,
    polyno4259638811958763678t_unit = polyno4259638811958763678t_unit ).

% ring.alg_mult.cong
thf(fact_532_ring_Oalg__mult_Ocong,axiom,
    polyno4422430861927485590lt_a_b = polyno4422430861927485590lt_a_b ).

% ring.alg_mult.cong
thf(fact_533_ring_Oalg__mult_Ocong,axiom,
    polyno1088517687229135038t_unit = polyno1088517687229135038t_unit ).

% ring.alg_mult.cong
thf(fact_534_ring_Oroots_Ocong,axiom,
    polyno7858422826990252003t_unit = polyno7858422826990252003t_unit ).

% ring.roots.cong
thf(fact_535_ring_Oroots_Ocong,axiom,
    polynomial_roots_a_b = polynomial_roots_a_b ).

% ring.roots.cong
thf(fact_536_ring_Oroots_Ocong,axiom,
    polyno4169377219242390531t_unit = polyno4169377219242390531t_unit ).

% ring.roots.cong
thf(fact_537_finite__imp__inj__to__nat__seg,axiom,
    ! [A3: set_list_set_list_a] :
      ( ( finite3202133031812794515list_a @ A3 )
     => ? [F3: list_set_list_a > nat,N: nat] :
          ( ( ( image_5823034270878669301_a_nat @ F3 @ A3 )
            = ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
          & ( inj_on6368251998611587977_a_nat @ F3 @ A3 ) ) ) ).

% finite_imp_inj_to_nat_seg
thf(fact_538_finite__imp__inj__to__nat__seg,axiom,
    ! [A3: set_list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ? [F3: list_a > nat,N: nat] :
          ( ( ( image_list_a_nat @ F3 @ A3 )
            = ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
          & ( inj_on_list_a_nat @ F3 @ A3 ) ) ) ).

% finite_imp_inj_to_nat_seg
thf(fact_539_finite__imp__inj__to__nat__seg,axiom,
    ! [A3: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ? [F3: set_list_a > nat,N: nat] :
          ( ( ( image_set_list_a_nat @ F3 @ A3 )
            = ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
          & ( inj_on6672046521841440387_a_nat @ F3 @ A3 ) ) ) ).

% finite_imp_inj_to_nat_seg
thf(fact_540_finite__imp__inj__to__nat__seg,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [F3: nat > nat,N: nat] :
          ( ( ( image_nat_nat @ F3 @ A3 )
            = ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
          & ( inj_on_nat_nat @ F3 @ A3 ) ) ) ).

% finite_imp_inj_to_nat_seg
thf(fact_541_finite__imp__inj__to__nat__seg,axiom,
    ! [A3: set_a] :
      ( ( finite_finite_a @ A3 )
     => ? [F3: a > nat,N: nat] :
          ( ( ( image_a_nat @ F3 @ A3 )
            = ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) )
          & ( inj_on_a_nat @ F3 @ A3 ) ) ) ).

% finite_imp_inj_to_nat_seg
thf(fact_542_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A3: set_list_set_list_a] :
      ( ( finite3202133031812794515list_a @ A3 )
     => ? [N: nat,F3: nat > list_set_list_a] :
          ( ( A3
            = ( image_6038183786030946387list_a @ F3
              @ ( collect_nat
                @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
          & ( inj_on6583401513763865063list_a @ F3
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) ) ) ).

% finite_imp_nat_seg_image_inj_on
thf(fact_543_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A3: set_list_a] :
      ( ( finite_finite_list_a @ A3 )
     => ? [N: nat,F3: nat > list_a] :
          ( ( A3
            = ( image_nat_list_a @ F3
              @ ( collect_nat
                @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
          & ( inj_on_nat_list_a @ F3
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) ) ) ).

% finite_imp_nat_seg_image_inj_on
thf(fact_544_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A3: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A3 )
     => ? [N: nat,F3: nat > set_list_a] :
          ( ( A3
            = ( image_nat_set_list_a @ F3
              @ ( collect_nat
                @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
          & ( inj_on977100749562638433list_a @ F3
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) ) ) ).

% finite_imp_nat_seg_image_inj_on
thf(fact_545_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ? [N: nat,F3: nat > nat] :
          ( ( A3
            = ( image_nat_nat @ F3
              @ ( collect_nat
                @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
          & ( inj_on_nat_nat @ F3
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) ) ) ).

% finite_imp_nat_seg_image_inj_on
thf(fact_546_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A3: set_a] :
      ( ( finite_finite_a @ A3 )
     => ? [N: nat,F3: nat > a] :
          ( ( A3
            = ( image_nat_a @ F3
              @ ( collect_nat
                @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) )
          & ( inj_on_nat_a @ F3
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N ) ) ) ) ) ).

% finite_imp_nat_seg_image_inj_on
thf(fact_547_nat__seg__image__imp__finite,axiom,
    ! [A3: set_list_set_list_a,F2: nat > list_set_list_a,N2: nat] :
      ( ( A3
        = ( image_6038183786030946387list_a @ F2
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) ) )
     => ( finite3202133031812794515list_a @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_548_nat__seg__image__imp__finite,axiom,
    ! [A3: set_list_a,F2: nat > list_a,N2: nat] :
      ( ( A3
        = ( image_nat_list_a @ F2
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) ) )
     => ( finite_finite_list_a @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_549_nat__seg__image__imp__finite,axiom,
    ! [A3: set_set_list_a,F2: nat > set_list_a,N2: nat] :
      ( ( A3
        = ( image_nat_set_list_a @ F2
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) ) )
     => ( finite5282473924520328461list_a @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_550_nat__seg__image__imp__finite,axiom,
    ! [A3: set_nat,F2: nat > nat,N2: nat] :
      ( ( A3
        = ( image_nat_nat @ F2
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) ) )
     => ( finite_finite_nat @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_551_nat__seg__image__imp__finite,axiom,
    ! [A3: set_a,F2: nat > a,N2: nat] :
      ( ( A3
        = ( image_nat_a @ F2
          @ ( collect_nat
            @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) ) )
     => ( finite_finite_a @ A3 ) ) ).

% nat_seg_image_imp_finite
thf(fact_552_finite__conv__nat__seg__image,axiom,
    ( finite3202133031812794515list_a
    = ( ^ [A7: set_list_set_list_a] :
        ? [N3: nat,F4: nat > list_set_list_a] :
          ( A7
          = ( image_6038183786030946387list_a @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N3 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_553_finite__conv__nat__seg__image,axiom,
    ( finite_finite_list_a
    = ( ^ [A7: set_list_a] :
        ? [N3: nat,F4: nat > list_a] :
          ( A7
          = ( image_nat_list_a @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N3 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_554_finite__conv__nat__seg__image,axiom,
    ( finite5282473924520328461list_a
    = ( ^ [A7: set_set_list_a] :
        ? [N3: nat,F4: nat > set_list_a] :
          ( A7
          = ( image_nat_set_list_a @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N3 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_555_finite__conv__nat__seg__image,axiom,
    ( finite_finite_nat
    = ( ^ [A7: set_nat] :
        ? [N3: nat,F4: nat > nat] :
          ( A7
          = ( image_nat_nat @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N3 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_556_finite__conv__nat__seg__image,axiom,
    ( finite_finite_a
    = ( ^ [A7: set_a] :
        ? [N3: nat,F4: nat > a] :
          ( A7
          = ( image_nat_a @ F4
            @ ( collect_nat
              @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N3 ) ) ) ) ) ) ).

% finite_conv_nat_seg_image
thf(fact_557_neq__if__length__neq,axiom,
    ! [Xs: list_list_a,Ys: list_list_a] :
      ( ( ( size_s349497388124573686list_a @ Xs )
       != ( size_s349497388124573686list_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_558_neq__if__length__neq,axiom,
    ! [Xs: list_set_list_a,Ys: list_set_list_a] :
      ( ( ( size_s1991367317912710102list_a @ Xs )
       != ( size_s1991367317912710102list_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_559_neq__if__length__neq,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs )
       != ( size_size_list_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_560_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_list_a] :
      ( ( size_s349497388124573686list_a @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_561_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_set_list_a] :
      ( ( size_s1991367317912710102list_a @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_562_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_a] :
      ( ( size_size_list_a @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_563_length__induct,axiom,
    ! [P3: list_list_a > $o,Xs: list_list_a] :
      ( ! [Xs2: list_list_a] :
          ( ! [Ys2: list_list_a] :
              ( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ys2 ) @ ( size_s349497388124573686list_a @ Xs2 ) )
             => ( P3 @ Ys2 ) )
         => ( P3 @ Xs2 ) )
     => ( P3 @ Xs ) ) ).

% length_induct
thf(fact_564_length__induct,axiom,
    ! [P3: list_set_list_a > $o,Xs: list_set_list_a] :
      ( ! [Xs2: list_set_list_a] :
          ( ! [Ys2: list_set_list_a] :
              ( ( ord_less_nat @ ( size_s1991367317912710102list_a @ Ys2 ) @ ( size_s1991367317912710102list_a @ Xs2 ) )
             => ( P3 @ Ys2 ) )
         => ( P3 @ Xs2 ) )
     => ( P3 @ Xs ) ) ).

% length_induct
thf(fact_565_length__induct,axiom,
    ! [P3: list_a > $o,Xs: list_a] :
      ( ! [Xs2: list_a] :
          ( ! [Ys2: list_a] :
              ( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
             => ( P3 @ Ys2 ) )
         => ( P3 @ Xs2 ) )
     => ( P3 @ Xs ) ) ).

% length_induct
thf(fact_566_domain_Obounded__degree__dimension,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,N2: nat] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( embedd1837541058045081570t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ N2 @ ( image_1156962946714028939list_a @ ( poly_o1617770581224298896t_unit @ R ) @ K )
          @ ( collec1292721268053437947list_a
            @ ^ [Q3: list_list_list_a] :
                ( ( member5342144027231129785list_a @ Q3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
                & ( ord_less_eq_nat @ ( size_s2403821588304063868list_a @ Q3 ) @ N2 ) ) ) ) ) ) ).

% domain.bounded_degree_dimension
thf(fact_567_domain_Obounded__degree__dimension,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,N2: nat] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( embedd5477863166044162050t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ N2 @ ( image_3300606519373034379list_a @ ( poly_o1450351294650543536t_unit @ R ) @ K )
          @ ( collec4413339279609262555list_a
            @ ^ [Q3: list_list_set_list_a] :
                ( ( member352051402189872281list_a @ Q3 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
                & ( ord_less_eq_nat @ ( size_s4069122225494125916list_a @ Q3 ) @ N2 ) ) ) ) ) ) ).

% domain.bounded_degree_dimension
thf(fact_568_domain_Obounded__degree__dimension,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,N2: nat] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( embedd3148316222609325832t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ N2 @ ( image_3509949574358669579list_a @ ( poly_o3535013565302768950t_unit @ R ) @ K )
          @ ( collec5381118732811369429list_a
            @ ^ [Q3: list_set_list_a] :
                ( ( member5524387281408368019list_a @ Q3 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
                & ( ord_less_eq_nat @ ( size_s1991367317912710102list_a @ Q3 ) @ N2 ) ) ) ) ) ) ).

% domain.bounded_degree_dimension
thf(fact_569_domain_Obounded__degree__dimension,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,N2: nat] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( embedd2305571234642070248t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ N2 @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ R ) @ K )
          @ ( collect_list_list_a
            @ ^ [Q3: list_list_a] :
                ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                & ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Q3 ) @ N2 ) ) ) ) ) ) ).

% domain.bounded_degree_dimension
thf(fact_570_domain_Obounded__degree__dimension,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,N2: nat] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ R @ K ) @ N2 @ ( image_a_list_a @ ( poly_of_const_a_b @ R ) @ K )
          @ ( collect_list_a
            @ ^ [Q3: list_a] :
                ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                & ( ord_less_eq_nat @ ( size_size_list_a @ Q3 ) @ N2 ) ) ) ) ) ) ).

% domain.bounded_degree_dimension
thf(fact_571_domain_Oexists__unique__long__division,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_list_a )
             => ? [X2: produc3789376428941379879list_a] :
                  ( ( polyno8316157972930027562t_unit @ R @ P2 @ Q @ X2 )
                  & ! [Y2: produc3789376428941379879list_a] :
                      ( ( polyno8316157972930027562t_unit @ R @ P2 @ Q @ Y2 )
                     => ( Y2 = X2 ) ) ) ) ) ) ) ) ).

% domain.exists_unique_long_division
thf(fact_572_domain_Oexists__unique__long__division,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_set_list_a )
             => ? [X2: produc655077077081492775list_a] :
                  ( ( polyno1752669682718493770t_unit @ R @ P2 @ Q @ X2 )
                  & ! [Y2: produc655077077081492775list_a] :
                      ( ( polyno1752669682718493770t_unit @ R @ P2 @ Q @ Y2 )
                     => ( Y2 = X2 ) ) ) ) ) ) ) ) ).

% domain.exists_unique_long_division
thf(fact_573_domain_Oexists__unique__long__division,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
           => ( ( Q != nil_set_list_a )
             => ? [X2: produc76396112498735911list_a] :
                  ( ( polyno8465598152942450896t_unit @ R @ P2 @ Q @ X2 )
                  & ! [Y2: produc76396112498735911list_a] :
                      ( ( polyno8465598152942450896t_unit @ R @ P2 @ Q @ Y2 )
                     => ( Y2 = X2 ) ) ) ) ) ) ) ) ).

% domain.exists_unique_long_division
thf(fact_574_domain_Oexists__unique__long__division,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( Q != nil_a )
             => ? [X2: produc9164743771328383783list_a] :
                  ( ( polyno2806191415236617128es_a_b @ R @ P2 @ Q @ X2 )
                  & ! [Y2: produc9164743771328383783list_a] :
                      ( ( polyno2806191415236617128es_a_b @ R @ P2 @ Q @ Y2 )
                     => ( Y2 = X2 ) ) ) ) ) ) ) ) ).

% domain.exists_unique_long_division
thf(fact_575_domain_Oexists__unique__long__division,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_a )
             => ? [X2: produc7709606177366032167list_a] :
                  ( ( polyno6947042923167803568t_unit @ R @ P2 @ Q @ X2 )
                  & ! [Y2: produc7709606177366032167list_a] :
                      ( ( polyno6947042923167803568t_unit @ R @ P2 @ Q @ Y2 )
                     => ( Y2 = X2 ) ) ) ) ) ) ) ) ).

% domain.exists_unique_long_division
thf(fact_576_domain_Odegree__zero__imp__empty__roots,axiom,
    ! [R: partia2956882679547061052t_unit,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( ( polyno3707469075594375645t_unit @ R @ P2 )
            = zero_z1542645121299710087list_a ) ) ) ) ).

% domain.degree_zero_imp_empty_roots
thf(fact_577_domain_Odegree__zero__imp__empty__roots,axiom,
    ! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R )
     => ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( ( polyno2127442156181624701t_unit @ R @ P2 )
            = zero_z6145066983645916903list_a ) ) ) ) ).

% domain.degree_zero_imp_empty_roots
thf(fact_578_domain_Odegree__zero__imp__empty__roots,axiom,
    ! [R: partia6043505979758434576t_unit,P2: list_set_a] :
      ( ( domain4236798911309298543t_unit @ R )
     => ( ( member_list_set_a @ P2 @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_size_list_set_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( ( polyno8580752123319221513t_unit @ R @ P2 )
            = zero_z5079479921072680283_set_a ) ) ) ) ).

% domain.degree_zero_imp_empty_roots
thf(fact_579_domain_Odegree__zero__imp__empty__roots,axiom,
    ! [R: partia2670972154091845814t_unit,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( ( polyno7858422826990252003t_unit @ R @ P2 )
            = zero_z4454100511807792257list_a ) ) ) ) ).

% domain.degree_zero_imp_empty_roots
thf(fact_580_domain_Odegree__zero__imp__empty__roots,axiom,
    ! [R: partia2175431115845679010xt_a_b,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( ( polynomial_roots_a_b @ R @ P2 )
            = zero_zero_multiset_a ) ) ) ) ).

% domain.degree_zero_imp_empty_roots
thf(fact_581_domain_Odegree__zero__imp__empty__roots,axiom,
    ! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
       => ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat )
            = zero_zero_nat )
         => ( ( polyno4169377219242390531t_unit @ R @ P2 )
            = zero_z7061913751530476641list_a ) ) ) ) ).

% domain.degree_zero_imp_empty_roots
thf(fact_582_domain_Opirreducible__roots,axiom,
    ! [R: partia2956882679547061052t_unit,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P2 )
         => ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat )
             != one_one_nat )
           => ( ( polyno3707469075594375645t_unit @ R @ P2 )
              = zero_z1542645121299710087list_a ) ) ) ) ) ).

% domain.pirreducible_roots
thf(fact_583_domain_Opirreducible__roots,axiom,
    ! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R )
     => ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
       => ( ( ring_r97889109428395874t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ P2 )
         => ( ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ P2 ) @ one_one_nat )
             != one_one_nat )
           => ( ( polyno2127442156181624701t_unit @ R @ P2 )
              = zero_z6145066983645916903list_a ) ) ) ) ) ).

% domain.pirreducible_roots
thf(fact_584_domain_Opirreducible__roots,axiom,
    ! [R: partia6043505979758434576t_unit,P2: list_set_a] :
      ( ( domain4236798911309298543t_unit @ R )
     => ( ( member_list_set_a @ P2 @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
       => ( ( ring_r8677422918745460462t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) @ P2 )
         => ( ( ( minus_minus_nat @ ( size_size_list_set_a @ P2 ) @ one_one_nat )
             != one_one_nat )
           => ( ( polyno8580752123319221513t_unit @ R @ P2 )
              = zero_z5079479921072680283_set_a ) ) ) ) ) ).

% domain.pirreducible_roots
thf(fact_585_domain_Opirreducible__roots,axiom,
    ! [R: partia7496981018696276118t_unit,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
       => ( ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ P2 )
         => ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat )
             != one_one_nat )
           => ( ( polyno4169377219242390531t_unit @ R @ P2 )
              = zero_z7061913751530476641list_a ) ) ) ) ) ).

% domain.pirreducible_roots
thf(fact_586_domain_Opirreducible__roots,axiom,
    ! [R: partia2670972154091845814t_unit,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 )
         => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
             != one_one_nat )
           => ( ( polyno7858422826990252003t_unit @ R @ P2 )
              = zero_z4454100511807792257list_a ) ) ) ) ) ).

% domain.pirreducible_roots
thf(fact_587_domain_Opirreducible__roots,axiom,
    ! [R: partia2175431115845679010xt_a_b,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 )
         => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
             != one_one_nat )
           => ( ( polynomial_roots_a_b @ R @ P2 )
              = zero_zero_multiset_a ) ) ) ) ) ).

% domain.pirreducible_roots
thf(fact_588_list_Osize_I3_J,axiom,
    ( ( size_s349497388124573686list_a @ nil_list_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_589_list_Osize_I3_J,axiom,
    ( ( size_s1991367317912710102list_a @ nil_set_list_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_590_list_Osize_I3_J,axiom,
    ( ( size_size_list_a @ nil_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_591_finite__maxlen,axiom,
    ! [M3: set_list_list_a] :
      ( ( finite1660835950917165235list_a @ M3 )
     => ? [N: nat] :
        ! [X3: list_list_a] :
          ( ( member_list_list_a @ X3 @ M3 )
         => ( ord_less_nat @ ( size_s349497388124573686list_a @ X3 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_592_finite__maxlen,axiom,
    ! [M3: set_list_set_list_a] :
      ( ( finite3202133031812794515list_a @ M3 )
     => ? [N: nat] :
        ! [X3: list_set_list_a] :
          ( ( member5524387281408368019list_a @ X3 @ M3 )
         => ( ord_less_nat @ ( size_s1991367317912710102list_a @ X3 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_593_finite__maxlen,axiom,
    ! [M3: set_list_a] :
      ( ( finite_finite_list_a @ M3 )
     => ? [N: nat] :
        ! [X3: list_a] :
          ( ( member_list_a @ X3 @ M3 )
         => ( ord_less_nat @ ( size_size_list_a @ X3 ) @ N ) ) ) ).

% finite_maxlen
thf(fact_594_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [R: partia2956882679547061052t_unit,P2: list_list_list_a,X: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( polyno1672195411705137432t_unit @ R @ P2 @ X ) )
          = ( polyno5142720416380192742t_unit @ R @ P2 @ X ) ) ) ) ).

% domain.alg_mult_gt_zero_iff_is_root
thf(fact_595_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a,X: set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R )
     => ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( polyno5417740675737174712t_unit @ R @ P2 @ X ) )
          = ( polyno875383481011669382t_unit @ R @ P2 @ X ) ) ) ) ).

% domain.alg_mult_gt_zero_iff_is_root
thf(fact_596_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [R: partia6043505979758434576t_unit,P2: list_set_a,X: set_a] :
      ( ( domain4236798911309298543t_unit @ R )
     => ( ( member_list_set_a @ P2 @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( polyno197644361506364228t_unit @ R @ P2 @ X ) )
          = ( polyno4890645956962836498t_unit @ R @ P2 @ X ) ) ) ) ).

% domain.alg_mult_gt_zero_iff_is_root
thf(fact_597_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( polyno4259638811958763678t_unit @ R @ P2 @ X ) )
          = ( polyno6951661231331188332t_unit @ R @ P2 @ X ) ) ) ) ).

% domain.alg_mult_gt_zero_iff_is_root
thf(fact_598_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ R @ P2 @ X ) )
          = ( polyno4133073214067823460ot_a_b @ R @ P2 @ X ) ) ) ) ).

% domain.alg_mult_gt_zero_iff_is_root
thf(fact_599_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [R: partia7496981018696276118t_unit,P2: list_set_list_a,X: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( polyno1088517687229135038t_unit @ R @ P2 @ X ) )
          = ( polyno4320237611291262604t_unit @ R @ P2 @ X ) ) ) ) ).

% domain.alg_mult_gt_zero_iff_is_root
thf(fact_600_p_Oexists__unique__long__division,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( Q != nil_list_a )
           => ? [X2: produc7709606177366032167list_a] :
                ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q @ X2 )
                & ! [Y2: produc7709606177366032167list_a] :
                    ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q @ Y2 )
                   => ( Y2 = X2 ) ) ) ) ) ) ) ).

% p.exists_unique_long_division
thf(fact_601_p_Opprime__iff__pirreducible,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 )
          = ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) ) ) ) ).

% p.pprime_iff_pirreducible
thf(fact_602_p_OpprimeE_I1_J,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 )
         => ( P2 != nil_list_a ) ) ) ) ).

% p.pprimeE(1)
thf(fact_603_h_Oorder__gt__0__iff__finite,axiom,
    ( ( ord_less_nat @ zero_zero_nat
      @ ( order_a_ring_ext_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) )
    = ( finite_finite_a
      @ ( partia707051561876973205xt_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) ) ) ).

% h.order_gt_0_iff_finite
thf(fact_604_p_Oirreducible__imp__maximalideal,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
       => ( maxima6585700282301356660t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.irreducible_imp_maximalideal
thf(fact_605_h_Oonepideal,axiom,
    ( principalideal_a_b
    @ ( partia707051561876973205xt_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) )
    @ ( partia8674076737563717228xt_a_b
      @ ^ [Uu: set_a] : k
      @ r ) ) ).

% h.onepideal
thf(fact_606_alg__mult__gt__zero__iff__is__root,axiom,
    ! [P2: list_a,X: a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
        = ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).

% alg_mult_gt_zero_iff_is_root
thf(fact_607_order__gt__0__iff__finite,axiom,
    ( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
    = ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% order_gt_0_iff_finite
thf(fact_608_onepideal,axiom,
    principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% onepideal
thf(fact_609_p_Odimension__is__inj,axiom,
    ! [K: set_list_a,N2: nat,E: set_list_a,M: nat] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N2 @ K @ E )
       => ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ K @ E )
         => ( N2 = M ) ) ) ) ).

% p.dimension_is_inj
thf(fact_610_p_Ofinite__dimensionE_H,axiom,
    ! [K: set_list_a,E: set_list_a] :
      ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
     => ~ ! [N: nat] :
            ~ ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N @ K @ E ) ) ).

% p.finite_dimensionE'
thf(fact_611_p_Ofinite__dimensionI,axiom,
    ! [N2: nat,K: set_list_a,E: set_list_a] :
      ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N2 @ K @ E )
     => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E ) ) ).

% p.finite_dimensionI
thf(fact_612_p_Ofinite__dimension__def,axiom,
    ! [K: set_list_a,E: set_list_a] :
      ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
      = ( ? [N3: nat] : ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N3 @ K @ E ) ) ) ).

% p.finite_dimension_def
thf(fact_613_p_Otelescopic__base__aux,axiom,
    ! [K: set_list_a,F: set_list_a,N2: nat,E: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( subfie1779122896746047282t_unit @ F @ ( univ_poly_a_b @ r @ k ) )
       => ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N2 @ K @ F )
         => ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ one_one_nat @ F @ E )
           => ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N2 @ K @ E ) ) ) ) ) ).

% p.telescopic_base_aux
thf(fact_614_p_Ounique__dimension,axiom,
    ! [K: set_list_a,E: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
       => ? [X2: nat] :
            ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ K @ E )
            & ! [Y2: nat] :
                ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ Y2 @ K @ E )
               => ( Y2 = X2 ) ) ) ) ) ).

% p.unique_dimension
thf(fact_615_p_Ospace__subgroup__props_I1_J,axiom,
    ! [K: set_list_a,N2: nat,E: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N2 @ K @ E )
       => ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.space_subgroup_props(1)
thf(fact_616_degree__zero__imp__empty__roots,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
          = zero_zero_nat )
       => ( ( polynomial_roots_a_b @ r @ P2 )
          = zero_zero_multiset_a ) ) ) ).

% degree_zero_imp_empty_roots
thf(fact_617_pirreducible__roots,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
       => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
           != one_one_nat )
         => ( ( polynomial_roots_a_b @ r @ P2 )
            = zero_zero_multiset_a ) ) ) ) ).

% pirreducible_roots
thf(fact_618_bounded__degree__dimension,axiom,
    ! [K: set_a,N2: nat] :
      ( ( subfield_a_b @ K @ r )
     => ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ K ) @ N2 @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K )
        @ ( collect_list_a
          @ ^ [Q3: list_a] :
              ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
              & ( ord_less_eq_nat @ ( size_size_list_a @ Q3 ) @ N2 ) ) ) ) ) ).

% bounded_degree_dimension
thf(fact_619_p_Odimension__one,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ one_one_nat @ K @ K ) ) ).

% p.dimension_one
thf(fact_620_domain_OpprimeE_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( ring_r346321679897941977t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 )
           => ( P2 != nil_list_list_a ) ) ) ) ) ).

% domain.pprimeE(1)
thf(fact_621_domain_OpprimeE_I1_J,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( ring_r8041554798613056505t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ P2 )
           => ( P2 != nil_list_set_list_a ) ) ) ) ) ).

% domain.pprimeE(1)
thf(fact_622_domain_OpprimeE_I1_J,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( ring_r294914423999638143t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 )
           => ( P2 != nil_set_list_a ) ) ) ) ) ).

% domain.pprimeE(1)
thf(fact_623_domain_OpprimeE_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
           => ( P2 != nil_list_a ) ) ) ) ) ).

% domain.pprimeE(1)
thf(fact_624_domain_OpprimeE_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
           => ( P2 != nil_a ) ) ) ) ) ).

% domain.pprimeE(1)
thf(fact_625_domain_Opprime__iff__pirreducible,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( ring_r346321679897941977t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 )
            = ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 ) ) ) ) ) ).

% domain.pprime_iff_pirreducible
thf(fact_626_domain_Opprime__iff__pirreducible,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( ring_r8041554798613056505t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ P2 )
            = ( ring_r7962978046438709730t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) @ P2 ) ) ) ) ) ).

% domain.pprime_iff_pirreducible
thf(fact_627_domain_Opprime__iff__pirreducible,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( ring_r294914423999638143t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 )
            = ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 ) ) ) ) ) ).

% domain.pprime_iff_pirreducible
thf(fact_628_domain_Opprime__iff__pirreducible,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
            = ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) ) ) ) ) ).

% domain.pprime_iff_pirreducible
thf(fact_629_domain_Opprime__iff__pirreducible,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
            = ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) ) ) ) ).

% domain.pprime_iff_pirreducible
thf(fact_630_h_Osubdomain__iff,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( subdomain_a_b @ H2
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
        = ( domain_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.subdomain_iff
thf(fact_631_p_Olong__divisionI,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a,B: list_list_a,R2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( Q != nil_list_a )
           => ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q @ ( produc8696003437204565271list_a @ B @ R2 ) )
             => ( ( produc8696003437204565271list_a @ B @ R2 )
                = ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q ) ) ) ) ) ) ) ) ).

% p.long_divisionI
thf(fact_632_p_Olong__divisionE,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( Q != nil_list_a )
           => ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q ) ) ) ) ) ) ) ).

% p.long_divisionE
thf(fact_633_d_Odegree__var,axiom,
    ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ ( var_se6008125447796440765t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) @ one_one_nat )
    = one_one_nat ) ).

% d.degree_var
thf(fact_634_p_Osubalbegra__incl__imp__finite__dimension,axiom,
    ! [K: set_list_a,E: set_list_a,V: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
       => ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ k ) )
         => ( ( ord_le8861187494160871172list_a @ V @ E )
           => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ V ) ) ) ) ) ).

% p.subalbegra_incl_imp_finite_dimension
thf(fact_635_d_Ofinite__poly_I2_J,axiom,
    ! [K: set_set_list_a,N2: nat] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( finite5282473924520328461list_a @ K )
       => ( finite3202133031812794515list_a
          @ ( collec5381118732811369429list_a
            @ ^ [F4: list_set_list_a] :
                ( ( member5524387281408368019list_a @ F4 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K ) ) )
                & ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ F4 ) @ one_one_nat ) @ N2 ) ) ) ) ) ) ).

% d.finite_poly(2)
thf(fact_636_p_Oprimeness__condition,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
        = ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) ) ).

% p.primeness_condition
thf(fact_637_p_Opoly__add_Ocases,axiom,
    ! [X: produc7709606177366032167list_a] :
      ~ ! [P1: list_list_a,P22: list_list_a] :
          ( X
         != ( produc8696003437204565271list_a @ P1 @ P22 ) ) ).

% p.poly_add.cases
thf(fact_638_subdomain__is__domain,axiom,
    ! [H2: set_a] :
      ( ( subdomain_a_b @ H2 @ r )
     => ( domain_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ r ) ) ) ).

% subdomain_is_domain
thf(fact_639_pprimeE_I1_J,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
         => ( P2 != nil_a ) ) ) ) ).

% pprimeE(1)
thf(fact_640_pprime__iff__pirreducible,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
          = ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).

% pprime_iff_pirreducible
thf(fact_641_h_Osubdomain__is__domain,axiom,
    ! [H2: set_a] :
      ( ( subdomain_a_b @ H2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( domain_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.subdomain_is_domain
thf(fact_642_p_Osubalgebra__in__carrier,axiom,
    ! [K: set_list_a,V: set_list_a] :
      ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ k ) )
     => ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.subalgebra_in_carrier
thf(fact_643_p_Ocarrier__is__subalgebra,axiom,
    ! [K: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( embedd1768981623711841426t_unit @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ) ) ).

% p.carrier_is_subalgebra
thf(fact_644_d_Ocarrier__is__subring,axiom,
    subrin5643252653130547402t_unit @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ).

% d.carrier_is_subring
thf(fact_645_p_Ofinite__dimension__imp__subalgebra,axiom,
    ! [K: set_list_a,E: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E )
       => ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.finite_dimension_imp_subalgebra
thf(fact_646_subdomain__iff,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( subdomain_a_b @ H2 @ r )
        = ( domain_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ r ) ) ) ) ).

% subdomain_iff
thf(fact_647_p_Oexists__long__division,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( Q != nil_list_a )
           => ~ ! [B5: list_list_a] :
                  ( ( member_list_list_a @ B5 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
                 => ! [R3: list_list_a] :
                      ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
                     => ~ ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q @ ( produc8696003437204565271list_a @ B5 @ R3 ) ) ) ) ) ) ) ) ).

% p.exists_long_division
thf(fact_648_d_Ofinite__poly_I1_J,axiom,
    ! [K: set_set_list_a,N2: nat] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( finite5282473924520328461list_a @ K )
       => ( finite3202133031812794515list_a
          @ ( collec5381118732811369429list_a
            @ ^ [F4: list_set_list_a] :
                ( ( member5524387281408368019list_a @ F4 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K ) ) )
                & ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ F4 ) @ one_one_nat )
                  = N2 ) ) ) ) ) ) ).

% d.finite_poly(1)
thf(fact_649_d_Ocarrier__polynomial__shell,axiom,
    ! [K: set_set_list_a,P2: list_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K ) ) )
       => ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ) ) ).

% d.carrier_polynomial_shell
thf(fact_650_domain_Ovar__closed_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( member5342144027231129785list_a @ ( var_li3532061862469730199t_unit @ R ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) ) ) ) ).

% domain.var_closed(1)
thf(fact_651_domain_Ovar__closed_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( member_list_list_a @ ( var_li8453953174693405341t_unit @ R ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ).

% domain.var_closed(1)
thf(fact_652_domain_Ovar__closed_I1_J,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( member5524387281408368019list_a @ ( var_se6008125447796440765t_unit @ R ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) ) ) ) ).

% domain.var_closed(1)
thf(fact_653_domain_Ovar__closed_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( member_list_a @ ( var_a_b @ R ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ).

% domain.var_closed(1)
thf(fact_654_domain_Ouniv__poly__is__domain,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( domain9211287710782191037t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_domain
thf(fact_655_domain_Ouniv__poly__is__domain,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.univ_poly_is_domain
thf(fact_656_domain_Ouniv__poly__is__domain,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_domain
thf(fact_657_domain_Ouniv__poly__is__domain,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( domain2898972329295444579t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_domain
thf(fact_658_domain_OpirreducibleE_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 )
           => ( P2 != nil_list_list_a ) ) ) ) ) ).

% domain.pirreducibleE(1)
thf(fact_659_domain_OpirreducibleE_I1_J,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 )
           => ( P2 != nil_set_list_a ) ) ) ) ) ).

% domain.pirreducibleE(1)
thf(fact_660_domain_OpirreducibleE_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
           => ( P2 != nil_list_a ) ) ) ) ) ).

% domain.pirreducibleE(1)
thf(fact_661_domain_OpirreducibleE_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
           => ( P2 != nil_a ) ) ) ) ) ).

% domain.pirreducibleE(1)
thf(fact_662_domain_Oexists__long__division,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_list_a )
             => ~ ! [B5: list_list_list_a] :
                    ( ( member5342144027231129785list_a @ B5 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
                   => ! [R3: list_list_list_a] :
                        ( ( member5342144027231129785list_a @ R3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
                       => ~ ( polyno8316157972930027562t_unit @ R @ P2 @ Q @ ( produc1091363791885468951list_a @ B5 @ R3 ) ) ) ) ) ) ) ) ) ).

% domain.exists_long_division
thf(fact_663_domain_Oexists__long__division,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_set_list_a )
             => ~ ! [B5: list_list_set_list_a] :
                    ( ( member352051402189872281list_a @ B5 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
                   => ! [R3: list_list_set_list_a] :
                        ( ( member352051402189872281list_a @ R3 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
                       => ~ ( polyno1752669682718493770t_unit @ R @ P2 @ Q @ ( produc8939343616531927319list_a @ B5 @ R3 ) ) ) ) ) ) ) ) ) ).

% domain.exists_long_division
thf(fact_664_domain_Oexists__long__division,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
           => ( ( Q != nil_set_list_a )
             => ~ ! [B5: list_set_list_a] :
                    ( ( member5524387281408368019list_a @ B5 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
                   => ! [R3: list_set_list_a] :
                        ( ( member5524387281408368019list_a @ R3 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
                       => ~ ( polyno8465598152942450896t_unit @ R @ P2 @ Q @ ( produc8430489356965245207list_a @ B5 @ R3 ) ) ) ) ) ) ) ) ) ).

% domain.exists_long_division
thf(fact_665_domain_Oexists__long__division,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( Q != nil_a )
             => ~ ! [B5: list_a] :
                    ( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                   => ! [R3: list_a] :
                        ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                       => ~ ( polyno2806191415236617128es_a_b @ R @ P2 @ Q @ ( produc6837034575241423639list_a @ B5 @ R3 ) ) ) ) ) ) ) ) ) ).

% domain.exists_long_division
thf(fact_666_domain_Oexists__long__division,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_a )
             => ~ ! [B5: list_list_a] :
                    ( ( member_list_list_a @ B5 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                   => ! [R3: list_list_a] :
                        ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                       => ~ ( polyno6947042923167803568t_unit @ R @ P2 @ Q @ ( produc8696003437204565271list_a @ B5 @ R3 ) ) ) ) ) ) ) ) ) ).

% domain.exists_long_division
thf(fact_667_domain_Olong__divisionE,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a,Q: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_list_a )
             => ( polyno8316157972930027562t_unit @ R @ P2 @ Q @ ( produc1091363791885468951list_a @ ( polyno4115915122720352731t_unit @ R @ P2 @ Q ) @ ( polyno2750255454319713356t_unit @ R @ P2 @ Q ) ) ) ) ) ) ) ) ).

% domain.long_divisionE
thf(fact_668_domain_Olong__divisionE,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a,Q: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_set_list_a )
             => ( polyno1752669682718493770t_unit @ R @ P2 @ Q @ ( produc8939343616531927319list_a @ ( polyno5933693564459536891t_unit @ R @ P2 @ Q ) @ ( polyno427657113825607788t_unit @ R @ P2 @ Q ) ) ) ) ) ) ) ) ).

% domain.long_divisionE
thf(fact_669_domain_Olong__divisionE,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
           => ( ( Q != nil_set_list_a )
             => ( polyno8465598152942450896t_unit @ R @ P2 @ Q @ ( produc8430489356965245207list_a @ ( polyno6856256803263433473t_unit @ R @ P2 @ Q ) @ ( polyno8414006224137659890t_unit @ R @ P2 @ Q ) ) ) ) ) ) ) ) ).

% domain.long_divisionE
thf(fact_670_domain_Olong__divisionE,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( Q != nil_a )
             => ( polyno2806191415236617128es_a_b @ R @ P2 @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P2 @ Q ) @ ( polynomial_pmod_a_b @ R @ P2 @ Q ) ) ) ) ) ) ) ) ).

% domain.long_divisionE
thf(fact_671_domain_Olong__divisionE,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_a )
             => ( polyno6947042923167803568t_unit @ R @ P2 @ Q @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q ) @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q ) ) ) ) ) ) ) ) ).

% domain.long_divisionE
thf(fact_672_domain_Olong__divisionI,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a,Q: list_list_list_a,B: list_list_list_a,R2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_list_a )
             => ( ( polyno8316157972930027562t_unit @ R @ P2 @ Q @ ( produc1091363791885468951list_a @ B @ R2 ) )
               => ( ( produc1091363791885468951list_a @ B @ R2 )
                  = ( produc1091363791885468951list_a @ ( polyno4115915122720352731t_unit @ R @ P2 @ Q ) @ ( polyno2750255454319713356t_unit @ R @ P2 @ Q ) ) ) ) ) ) ) ) ) ).

% domain.long_divisionI
thf(fact_673_domain_Olong__divisionI,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a,P2: list_list_set_list_a,Q: list_list_set_list_a,B: list_list_set_list_a,R2: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( ( member352051402189872281list_a @ P2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
         => ( ( member352051402189872281list_a @ Q @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_set_list_a )
             => ( ( polyno1752669682718493770t_unit @ R @ P2 @ Q @ ( produc8939343616531927319list_a @ B @ R2 ) )
               => ( ( produc8939343616531927319list_a @ B @ R2 )
                  = ( produc8939343616531927319list_a @ ( polyno5933693564459536891t_unit @ R @ P2 @ Q ) @ ( polyno427657113825607788t_unit @ R @ P2 @ Q ) ) ) ) ) ) ) ) ) ).

% domain.long_divisionI
thf(fact_674_domain_Olong__divisionI,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q: list_set_list_a,B: list_set_list_a,R2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
           => ( ( Q != nil_set_list_a )
             => ( ( polyno8465598152942450896t_unit @ R @ P2 @ Q @ ( produc8430489356965245207list_a @ B @ R2 ) )
               => ( ( produc8430489356965245207list_a @ B @ R2 )
                  = ( produc8430489356965245207list_a @ ( polyno6856256803263433473t_unit @ R @ P2 @ Q ) @ ( polyno8414006224137659890t_unit @ R @ P2 @ Q ) ) ) ) ) ) ) ) ) ).

% domain.long_divisionI
thf(fact_675_domain_Olong__divisionI,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q: list_a,B: list_a,R2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( Q != nil_a )
             => ( ( polyno2806191415236617128es_a_b @ R @ P2 @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) )
               => ( ( produc6837034575241423639list_a @ B @ R2 )
                  = ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P2 @ Q ) @ ( polynomial_pmod_a_b @ R @ P2 @ Q ) ) ) ) ) ) ) ) ) ).

% domain.long_divisionI
thf(fact_676_domain_Olong__divisionI,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q: list_list_a,B: list_list_a,R2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_a )
             => ( ( polyno6947042923167803568t_unit @ R @ P2 @ Q @ ( produc8696003437204565271list_a @ B @ R2 ) )
               => ( ( produc8696003437204565271list_a @ B @ R2 )
                  = ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q ) @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q ) ) ) ) ) ) ) ) ) ).

% domain.long_divisionI
thf(fact_677_p_Oring__primeE_I3_J,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
       => ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) ) ).

% p.ring_primeE(3)
thf(fact_678_h_Ofinite__poly_I2_J,axiom,
    ! [K: set_a,N2: nat] :
      ( ( subring_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( finite_finite_a @ K )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [F4: list_a] :
                ( ( member_list_a @ F4
                  @ ( partia5361259788508890537t_unit
                    @ ( univ_poly_a_b
                      @ ( partia8674076737563717228xt_a_b
                        @ ^ [Uu: set_a] : k
                        @ r )
                      @ K ) ) )
                & ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ F4 ) @ one_one_nat ) @ N2 ) ) ) ) ) ) ).

% h.finite_poly(2)
thf(fact_679_h_Oinj__on__domain,axiom,
    ( ( inj_on_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
      @ ( partia707051561876973205xt_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) )
   => ( ( domain1617769409708967785t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( domain_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) ) ) ).

% h.inj_on_domain
thf(fact_680_h_Ohom__closed,axiom,
    ! [X: a] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( member_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ X ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% h.hom_closed
thf(fact_681_h_Ofinite__poly_I1_J,axiom,
    ! [K: set_a,N2: nat] :
      ( ( subring_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( finite_finite_a @ K )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [F4: list_a] :
                ( ( member_list_a @ F4
                  @ ( partia5361259788508890537t_unit
                    @ ( univ_poly_a_b
                      @ ( partia8674076737563717228xt_a_b
                        @ ^ [Uu: set_a] : k
                        @ r )
                      @ K ) ) )
                & ( ( minus_minus_nat @ ( size_size_list_a @ F4 ) @ one_one_nat )
                  = N2 ) ) ) ) ) ) ).

% h.finite_poly(1)
thf(fact_682_p_Ofinite__poly_I2_J,axiom,
    ! [K: set_list_a,N2: nat] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( finite_finite_list_a @ K )
       => ( finite1660835950917165235list_a
          @ ( collect_list_list_a
            @ ^ [F4: list_list_a] :
                ( ( member_list_list_a @ F4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
                & ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F4 ) @ one_one_nat ) @ N2 ) ) ) ) ) ) ).

% p.finite_poly(2)
thf(fact_683_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% diff_is_0_eq
thf(fact_684_carrier__is__subring,axiom,
    subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% carrier_is_subring
thf(fact_685_e,axiom,
    subring_a_b @ k @ r ).

% e
thf(fact_686_subdomainI_H,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( subdomain_a_b @ H2 @ r ) ) ).

% subdomainI'
thf(fact_687_var__closed_I1_J,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).

% var_closed(1)
thf(fact_688_degree__var,axiom,
    ( ( minus_minus_nat @ ( size_size_list_a @ ( var_a_b @ r ) ) @ one_one_nat )
    = one_one_nat ) ).

% degree_var
thf(fact_689_univ__poly__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( univ_poly_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r ) )
        = ( univ_poly_a_b @ r ) ) ) ).

% univ_poly_consistent
thf(fact_690_poly__of__const__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( poly_of_const_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r ) )
        = ( poly_of_const_a_b @ r ) ) ) ).

% poly_of_const_consistent
thf(fact_691_univ__poly__is__domain,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_is_domain
thf(fact_692_subring__is__domain,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( domain_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ r ) ) ) ).

% subring_is_domain
thf(fact_693_p_Ocarrier__is__subring,axiom,
    subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).

% p.carrier_is_subring
thf(fact_694_p_Ouniv__poly__is__domain,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% p.univ_poly_is_domain
thf(fact_695_h_Ouniv__poly__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( univ_poly_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
        = ( univ_poly_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.univ_poly_consistent
thf(fact_696_h_Ocarrier__is__subring,axiom,
    ( subring_a_b
    @ ( partia707051561876973205xt_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) )
    @ ( partia8674076737563717228xt_a_b
      @ ^ [Uu: set_a] : k
      @ r ) ) ).

% h.carrier_is_subring
thf(fact_697_h_Opoly__of__const__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( poly_of_const_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
        = ( poly_of_const_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.poly_of_const_consistent
thf(fact_698_pirreducibleE_I1_J,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
         => ( P2 != nil_a ) ) ) ) ).

% pirreducibleE(1)
thf(fact_699_p_Odegree__var,axiom,
    ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ one_one_nat )
    = one_one_nat ) ).

% p.degree_var
thf(fact_700_exists__long__division,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ~ ! [B5: list_a] :
                  ( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                 => ! [R3: list_a] :
                      ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                     => ~ ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ ( produc6837034575241423639list_a @ B5 @ R3 ) ) ) ) ) ) ) ) ).

% exists_long_division
thf(fact_701_p_Ovar__closed_I1_J,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( member_list_list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ).

% p.var_closed(1)
thf(fact_702_h_Odegree__var,axiom,
    ( ( minus_minus_nat
      @ ( size_size_list_a
        @ ( var_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
      @ one_one_nat )
    = one_one_nat ) ).

% h.degree_var
thf(fact_703_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_704_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_705_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_706_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_707_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_708_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_709_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_710_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_711_p_OpirreducibleE_I1_J,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 )
         => ( P2 != nil_list_a ) ) ) ) ).

% p.pirreducibleE(1)
thf(fact_712_long__divisionI,axiom,
    ! [K: set_a,P2: list_a,Q: list_a,B: list_a,R2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) )
             => ( ( produc6837034575241423639list_a @ B @ R2 )
                = ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ) ) ).

% long_divisionI
thf(fact_713_long__divisionE,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ) ).

% long_divisionE
thf(fact_714_finite__poly_I1_J,axiom,
    ! [K: set_a,N2: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( finite_finite_a @ K )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [F4: list_a] :
                ( ( member_list_a @ F4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                & ( ( minus_minus_nat @ ( size_size_list_a @ F4 ) @ one_one_nat )
                  = N2 ) ) ) ) ) ) ).

% finite_poly(1)
thf(fact_715_finite__poly_I2_J,axiom,
    ! [K: set_a,N2: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( finite_finite_a @ K )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [F4: list_a] :
                ( ( member_list_a @ F4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                & ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ F4 ) @ one_one_nat ) @ N2 ) ) ) ) ) ) ).

% finite_poly(2)
thf(fact_716_d,axiom,
    inj_on_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ k ).

% d
thf(fact_717_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% zero_less_diff
thf(fact_718_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_719_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_720_p_Ofinite__poly_I1_J,axiom,
    ! [K: set_list_a,N2: nat] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( finite_finite_list_a @ K )
       => ( finite1660835950917165235list_a
          @ ( collect_list_list_a
            @ ^ [F4: list_list_a] :
                ( ( member_list_list_a @ F4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
                & ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ F4 ) @ one_one_nat )
                  = N2 ) ) ) ) ) ) ).

% p.finite_poly(1)
thf(fact_721_h_Oimg__is__subring,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( subrin5643252653130547402t_unit @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ K ) @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% h.img_is_subring
thf(fact_722_carrier__polynomial__shell,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% carrier_polynomial_shell
thf(fact_723_p_Ocarrier__polynomial__shell,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).

% p.carrier_polynomial_shell
thf(fact_724_h_Ocarrier__polynomial__shell,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( member_list_a @ P2
          @ ( partia5361259788508890537t_unit
            @ ( univ_poly_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ K ) ) )
       => ( member_list_a @ P2
          @ ( partia5361259788508890537t_unit
            @ ( univ_poly_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( partia707051561876973205xt_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r ) ) ) ) ) ) ) ).

% h.carrier_polynomial_shell
thf(fact_725_size__neq__size__imp__neq,axiom,
    ! [X: multiset_set_list_a,Y: multiset_set_list_a] :
      ( ( ( size_s1226348209404258454list_a @ X )
       != ( size_s1226348209404258454list_a @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_726_size__neq__size__imp__neq,axiom,
    ! [X: multiset_a,Y: multiset_a] :
      ( ( ( size_size_multiset_a @ X )
       != ( size_size_multiset_a @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_727_size__neq__size__imp__neq,axiom,
    ! [X: list_list_a,Y: list_list_a] :
      ( ( ( size_s349497388124573686list_a @ X )
       != ( size_s349497388124573686list_a @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_728_size__neq__size__imp__neq,axiom,
    ! [X: list_set_list_a,Y: list_set_list_a] :
      ( ( ( size_s1991367317912710102list_a @ X )
       != ( size_s1991367317912710102list_a @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_729_size__neq__size__imp__neq,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( ( size_size_list_a @ X )
       != ( size_size_list_a @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_730_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_731_infinite__descent,axiom,
    ! [P3: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ~ ( P3 @ N )
         => ? [M4: nat] :
              ( ( ord_less_nat @ M4 @ N )
              & ~ ( P3 @ M4 ) ) )
     => ( P3 @ N2 ) ) ).

% infinite_descent
thf(fact_732_nat__less__induct,axiom,
    ! [P3: nat > $o,N2: nat] :
      ( ! [N: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N )
             => ( P3 @ M4 ) )
         => ( P3 @ N ) )
     => ( P3 @ N2 ) ) ).

% nat_less_induct
thf(fact_733_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_734_less__not__refl3,axiom,
    ! [S2: nat,T2: nat] :
      ( ( ord_less_nat @ S2 @ T2 )
     => ( S2 != T2 ) ) ).

% less_not_refl3
thf(fact_735_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_736_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_737_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
      = ( ( ord_less_nat @ M @ N2 )
        | ( ord_less_nat @ N2 @ M ) ) ) ).

% nat_neq_iff
thf(fact_738_diff__commute,axiom,
    ! [I: nat,J2: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J2 ) @ K2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J2 ) ) ).

% diff_commute
thf(fact_739_Nat_Oex__has__greatest__nat,axiom,
    ! [P3: nat > $o,K2: nat,B: nat] :
      ( ( P3 @ K2 )
     => ( ! [Y3: nat] :
            ( ( P3 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X2: nat] :
            ( ( P3 @ X2 )
            & ! [Y2: nat] :
                ( ( P3 @ Y2 )
               => ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_740_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
      | ( ord_less_eq_nat @ N2 @ M ) ) ).

% nat_le_linear
thf(fact_741_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% le_antisym
thf(fact_742_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_743_le__trans,axiom,
    ! [I: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K2 )
       => ( ord_less_eq_nat @ I @ K2 ) ) ) ).

% le_trans
thf(fact_744_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_745_infinite__descent0,axiom,
    ! [P3: nat > $o,N2: nat] :
      ( ( P3 @ zero_zero_nat )
     => ( ! [N: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( ~ ( P3 @ N )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N )
                  & ~ ( P3 @ M4 ) ) ) )
       => ( P3 @ N2 ) ) ) ).

% infinite_descent0
thf(fact_746_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_747_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_748_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_749_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_750_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_751_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_752_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_753_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M )
          = zero_zero_nat )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_754_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_755_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_756_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_757_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_758_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_759_less__imp__diff__less,axiom,
    ! [J2: nat,K2: nat,N2: nat] :
      ( ( ord_less_nat @ J2 @ K2 )
     => ( ord_less_nat @ ( minus_minus_nat @ J2 @ N2 ) @ K2 ) ) ).

% less_imp_diff_less
thf(fact_760_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
          & ( M5 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_761_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_imp_le_nat
thf(fact_762_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N3: nat] :
          ( ( ord_less_nat @ M5 @ N3 )
          | ( M5 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_763_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_764_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( M != N2 )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_765_less__mono__imp__le__mono,axiom,
    ! [F2: nat > nat,I: nat,J2: nat] :
      ( ! [I4: nat,J3: nat] :
          ( ( ord_less_nat @ I4 @ J3 )
         => ( ord_less_nat @ ( F2 @ I4 ) @ ( F2 @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I @ J2 )
       => ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_766_eq__diff__iff,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N2 )
       => ( ( ( minus_minus_nat @ M @ K2 )
            = ( minus_minus_nat @ N2 @ K2 ) )
          = ( M = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_767_le__diff__iff,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
          = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_768_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
          = ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_769_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_770_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_771_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_772_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_773_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K2: nat] :
      ( ! [N: nat] :
          ( ( member_nat @ N @ N4 )
         => ( ord_less_eq_nat @ K2 @ N ) )
     => ( inj_on_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K2 )
        @ N4 ) ) ).

% inj_on_diff_nat
thf(fact_774_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).

% diff_less
thf(fact_775_ex__least__nat__le,axiom,
    ! [P3: nat > $o,N2: nat] :
      ( ( P3 @ N2 )
     => ( ~ ( P3 @ zero_zero_nat )
       => ? [K3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N2 )
            & ! [I5: nat] :
                ( ( ord_less_nat @ I5 @ K3 )
               => ~ ( P3 @ I5 ) )
            & ( P3 @ K3 ) ) ) ) ).

% ex_least_nat_le
thf(fact_776_less__diff__iff,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K2 @ M )
     => ( ( ord_less_eq_nat @ K2 @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
          = ( ord_less_nat @ M @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_777_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_778_h_Ois__abelian__group__hom,axiom,
    ( abelia7016813173424565629t_unit
    @ ( partia8674076737563717228xt_a_b
      @ ^ [Uu: set_a] : k
      @ r )
    @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
    @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) ).

% h.is_abelian_group_hom
thf(fact_779_p_Ouniv__poly__a__minus__consistent,axiom,
    ! [K: set_list_a,Q: list_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 @ Q )
          = ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P2 @ Q ) ) ) ) ).

% p.univ_poly_a_minus_consistent
thf(fact_780_h_Oimg__is__subalgebra,axiom,
    ! [K: set_a,V: set_a] :
      ( ( ord_less_eq_set_a @ K
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( embedd9027525575939734154ra_a_b @ K @ V
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
       => ( embedd3586486045337765042t_unit @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ K ) @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ V ) @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% h.img_is_subalgebra
thf(fact_781_local_Oh_Ochar__consistent,axiom,
    ( ( inj_on_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
      @ ( partia707051561876973205xt_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) )
   => ( ( ring_char_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
      = ( ring_c6053888738502451990t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% local.h.char_consistent
thf(fact_782_h_Oline__extension__hom,axiom,
    ! [K: set_a,A: a,E: set_a] :
      ( ( ord_less_eq_set_a @ K
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( member_a @ A
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( ( ord_less_eq_set_a @ E
            @ ( partia707051561876973205xt_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r ) ) )
         => ( ( embedd5951720228509767443t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ K ) @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ A ) @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ E ) )
            = ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
              @ ( embedd971793762689825387on_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ K
                @ A
                @ E ) ) ) ) ) ) ).

% h.line_extension_hom
thf(fact_783_h_Oabelian__group_Ozero__closed,axiom,
    ( member_set_list_a
    @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r )
      @ ( zero_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) )
    @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% h.abelian_group.zero_closed
thf(fact_784_poly__add_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ~ ! [P1: list_a,P22: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).

% poly_add.cases
thf(fact_785_subring__props_I2_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( member_a @ ( zero_a_b @ r ) @ K ) ) ).

% subring_props(2)
thf(fact_786_line__extension__in__carrier,axiom,
    ! [K: set_a,A: a,E: set_a] :
      ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% line_extension_in_carrier
thf(fact_787_subalgebra__in__carrier,axiom,
    ! [K: set_a,V: set_a] :
      ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
     => ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% subalgebra_in_carrier
thf(fact_788_carrier__is__subalgebra,axiom,
    ! [K: set_a] :
      ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
     => ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).

% carrier_is_subalgebra
thf(fact_789_line__extension__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( embedd971793762689825387on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r ) )
        = ( embedd971793762689825387on_a_b @ r ) ) ) ).

% line_extension_consistent
thf(fact_790_char__consistent,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( ( ring_char_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ r ) )
        = ( ring_char_a_b @ r ) ) ) ).

% char_consistent
thf(fact_791_h_Osubring__props_I2_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( member_a
        @ ( zero_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
        @ K ) ) ).

% h.subring_props(2)
thf(fact_792_h_Oline__extension__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( embedd971793762689825387on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
        = ( embedd971793762689825387on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.line_extension_consistent
thf(fact_793_h_OR_Ochar__consistent,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ring_char_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
        = ( ring_char_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.R.char_consistent
thf(fact_794_finite__carr__imp__char__ge__0,axiom,
    ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
   => ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) ) ) ).

% finite_carr_imp_char_ge_0
thf(fact_795_h_Oline__extension__in__carrier,axiom,
    ! [K: set_a,A: a,E: set_a] :
      ( ( ord_less_eq_set_a @ K
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( member_a @ A
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( ( ord_less_eq_set_a @ E
            @ ( partia707051561876973205xt_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r ) ) )
         => ( ord_less_eq_set_a
            @ ( embedd971793762689825387on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ K
              @ A
              @ E )
            @ ( partia707051561876973205xt_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r ) ) ) ) ) ) ).

% h.line_extension_in_carrier
thf(fact_796_h_Osubalgebra__in__carrier,axiom,
    ! [K: set_a,V: set_a] :
      ( ( embedd9027525575939734154ra_a_b @ K @ V
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ord_less_eq_set_a @ V
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.subalgebra_in_carrier
thf(fact_797_h_Ocarrier__is__subalgebra,axiom,
    ! [K: set_a] :
      ( ( ord_less_eq_set_a @ K
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( embedd9027525575939734154ra_a_b @ K
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) ) ).

% h.carrier_is_subalgebra
thf(fact_798_splitted__def,axiom,
    ! [P2: list_a] :
      ( ( polyno8329700637149614481ed_a_b @ r @ P2 )
      = ( ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P2 ) )
        = ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).

% splitted_def
thf(fact_799_d_Osplitted__def,axiom,
    ! [P2: list_set_list_a] :
      ( ( polyno7858167711734664505t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ P2 )
      = ( ( size_s1226348209404258454list_a @ ( polyno4169377219242390531t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ P2 ) )
        = ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat ) ) ) ).

% d.splitted_def
thf(fact_800_h_Ofinite__carr__imp__char__ge__0,axiom,
    ( ( finite_finite_a
      @ ( partia707051561876973205xt_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) )
   => ( ord_less_nat @ zero_zero_nat
      @ ( ring_char_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) ) ) ).

% h.finite_carr_imp_char_ge_0
thf(fact_801_p_Osplitted__def,axiom,
    ! [P2: list_list_a] :
      ( ( polyno6259083269128200473t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
      = ( ( size_s2335926164413107382list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) )
        = ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ).

% p.splitted_def
thf(fact_802_h_Osplitted__def,axiom,
    ! [P2: list_a] :
      ( ( polyno8329700637149614481ed_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ P2 )
      = ( ( size_size_multiset_a
          @ ( polynomial_roots_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ P2 ) )
        = ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).

% h.splitted_def
thf(fact_803_zero__closed,axiom,
    member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% zero_closed
thf(fact_804_h_OR_Ozero__closed,axiom,
    ( member_a
    @ ( zero_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) )
    @ ( partia707051561876973205xt_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) ) ) ).

% h.R.zero_closed
thf(fact_805_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 @ Q )
            = ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P2 @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_806_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ Q )
            = ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_807_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia4960592913263135132t_unit,K: set_set_list_list_a,Q: list_set_list_list_a,P2: list_set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R )
     => ( ( subrin3011427358523593924t_unit @ K @ R )
       => ( ( member6124916891863447321list_a @ Q @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ K ) ) )
         => ( ( a_minu1178922365601208552t_unit @ ( univ_p7077926387201515752t_unit @ R @ K ) @ P2 @ Q )
            = ( a_minu1178922365601208552t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ P2 @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_808_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,Q: list_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( a_minu6874796375791416686t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 @ Q )
            = ( a_minu6874796375791416686t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ P2 @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_809_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia6043505979758434576t_unit,K: set_set_a,Q: list_set_a,P2: list_set_a] :
      ( ( domain4236798911309298543t_unit @ R )
     => ( ( subrin1511138061850335568t_unit @ K @ R )
       => ( ( member_list_set_a @ Q @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
         => ( ( a_minu6204024409777241716t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) @ P2 @ Q )
            = ( a_minu6204024409777241716t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) @ P2 @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_810_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 @ Q )
            = ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_811_h_OboundD__carrier,axiom,
    ! [N2: nat,F2: nat > a,M: nat] :
      ( ( bound_a
        @ ( zero_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
        @ N2
        @ F2 )
     => ( ( ord_less_nat @ N2 @ M )
       => ( member_a @ ( F2 @ M )
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.boundD_carrier
thf(fact_812_ring__primeE_I1_J,axiom,
    ! [P2: a] :
      ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_ring_prime_a_b @ r @ P2 )
       => ( P2
         != ( zero_a_b @ r ) ) ) ) ).

% ring_primeE(1)
thf(fact_813_ring__irreducibleE_I1_J,axiom,
    ! [R2: a] :
      ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_r999134135267193926le_a_b @ r @ R2 )
       => ( R2
         != ( zero_a_b @ r ) ) ) ) ).

% ring_irreducibleE(1)
thf(fact_814_p_Orupture__surj__hom_I1_J,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( member2776101604912167772list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) ) @ ( ring_h8245347068061950880t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 ) ) ) ) ) ).

% p.rupture_surj_hom(1)
thf(fact_815_d_Osplitted__on__def,axiom,
    ! [K: set_set_list_a,P2: list_set_list_a] :
      ( ( polyno3250933163413791416t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ P2 )
      = ( ( size_s1226348209404258454list_a @ ( polyno3381972410027130754t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K @ P2 ) )
        = ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat ) ) ) ).

% d.splitted_on_def
thf(fact_816_univ__poly__a__minus__consistent,axiom,
    ! [K: set_a,Q: list_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q )
          = ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) ) ) ) ).

% univ_poly_a_minus_consistent
thf(fact_817_boundD__carrier,axiom,
    ! [N2: nat,F2: nat > a,M: nat] :
      ( ( bound_a @ ( zero_a_b @ r ) @ N2 @ F2 )
     => ( ( ord_less_nat @ N2 @ M )
       => ( member_a @ ( F2 @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% boundD_carrier
thf(fact_818_p_Ominus__closed,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.minus_closed
thf(fact_819_ring_Osplitted__on_Ocong,axiom,
    polyno3250933163413791416t_unit = polyno3250933163413791416t_unit ).

% ring.splitted_on.cong
thf(fact_820_ring_Osplitted__on_Ocong,axiom,
    polyno1986131841096413848t_unit = polyno1986131841096413848t_unit ).

% ring.splitted_on.cong
thf(fact_821_ring_Osplitted__on_Ocong,axiom,
    polyno2453258491555121552on_a_b = polyno2453258491555121552on_a_b ).

% ring.splitted_on.cong
thf(fact_822_ring_Oroots__on_Ocong,axiom,
    polyno3381972410027130754t_unit = polyno3381972410027130754t_unit ).

% ring.roots_on.cong
thf(fact_823_ring_Oroots__on_Ocong,axiom,
    polyno5990348478334826338t_unit = polyno5990348478334826338t_unit ).

% ring.roots_on.cong
thf(fact_824_ring_Oroots__on_Ocong,axiom,
    polyno5714441830345289050on_a_b = polyno5714441830345289050on_a_b ).

% ring.roots_on.cong
thf(fact_825_domain_Orupture__surj__hom_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( member6369349398066642652list_a @ ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 ) ) @ ( ring_h5921505907431827092t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ ( polyno8931960069169623149t_unit @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_surj_hom(1)
thf(fact_826_domain_Orupture__surj__hom_I1_J,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( member2273002913129892700list_a @ ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 ) ) @ ( ring_h45727557117266912t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( polyno7054224608333822611t_unit @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_surj_hom(1)
thf(fact_827_domain_Orupture__surj__hom_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( member2776101604912167772list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) ) @ ( ring_h8245347068061950880t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno859807163042199155t_unit @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_surj_hom(1)
thf(fact_828_domain_Orupture__surj__hom_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( member4263473470251683292list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) @ ( ring_h6188449271506562988t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polyno5459750281392823787re_a_b @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_surj_hom(1)
thf(fact_829_domain_Orupture__surj__norm__is__hom,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( member3350673534532848354list_a @ ( comp_l4894990565352033927list_a @ ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P2 ) ) @ ( poly_o1617770581224298896t_unit @ R ) )
            @ ( ring_h168092891735524890t_unit
              @ ( partia5725451877363000582t_unit
                @ ^ [Uu: set_list_list_a] : K
                @ R )
              @ ( polyno8931960069169623149t_unit @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_surj_norm_is_hom
thf(fact_830_domain_Orupture__surj__norm__is__hom,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( member_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) @ ( poly_of_const_a_b @ R ) )
            @ ( ring_h6109298854714515236t_unit
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : K
                @ R )
              @ ( polyno5459750281392823787re_a_b @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_surj_norm_is_hom
thf(fact_831_domain_Orupture__surj__norm__is__hom,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( member5222132204073479906list_a @ ( comp_l92744468816104577list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) ) @ ( poly_o8716471131768098070t_unit @ R ) )
            @ ( ring_h5771512479963622054t_unit
              @ ( partia9041243232023819264t_unit
                @ ^ [Uu: set_list_a] : K
                @ R )
              @ ( polyno859807163042199155t_unit @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_surj_norm_is_hom
thf(fact_832_domain_Orupture__surj__norm__is__hom,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( member3288041530688867042list_a @ ( comp_l2462527268875606113list_a @ ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 ) ) @ ( poly_o3535013565302768950t_unit @ R ) )
            @ ( ring_h8230865552078659302t_unit
              @ ( partia2314006361284536288t_unit
                @ ^ [Uu: set_set_list_a] : K
                @ R )
              @ ( polyno7054224608333822611t_unit @ R @ K @ P2 ) ) ) ) ) ) ).

% domain.rupture_surj_norm_is_hom
thf(fact_833_h_Oring__primeI,axiom,
    ! [P2: a] :
      ( ( P2
       != ( zero_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( prime_a_ring_ext_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ P2 )
       => ( ring_ring_prime_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ P2 ) ) ) ).

% h.ring_primeI
thf(fact_834_p_Osplitted__on__def,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( polyno1986131841096413848t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 )
      = ( ( size_s2335926164413107382list_a @ ( polyno5990348478334826338t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 ) )
        = ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ).

% p.splitted_on_def
thf(fact_835_h_Ohomh,axiom,
    ( member_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
    @ ( ring_h6109298854714515236t_unit
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r )
      @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% h.homh
thf(fact_836_rupture__surj__norm__is__hom,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) @ ( poly_of_const_a_b @ r ) )
          @ ( ring_h6109298854714515236t_unit
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : K
              @ r )
            @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) ) ) ) ) ).

% rupture_surj_norm_is_hom
thf(fact_837_h_Oabelian__group_Oa__inv__closed,axiom,
    ! [X: a] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( member_set_list_a
        @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r )
          @ ( a_inv_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ X ) )
        @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% h.abelian_group.a_inv_closed
thf(fact_838_a__inv__inj,axiom,
    inj_on_a_a @ ( a_inv_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% a_inv_inj
thf(fact_839_subring__props_I5_J,axiom,
    ! [K: set_a,H: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ H @ K )
       => ( member_a @ ( a_inv_a_b @ r @ H ) @ K ) ) ) ).

% subring_props(5)
thf(fact_840_ring__hom__restrict,axiom,
    ! [F2: a > list_a,S: partia2670972154091845814t_unit,G: a > list_a] :
      ( ( member_a_list_a @ F2 @ ( ring_h405018892823518980t_unit @ r @ S ) )
     => ( ! [R3: a] :
            ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( F2 @ R3 )
              = ( G @ R3 ) ) )
       => ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ r @ S ) ) ) ) ).

% ring_hom_restrict
thf(fact_841_ring__hom__restrict,axiom,
    ! [F2: a > set_list_a,S: partia7496981018696276118t_unit,G: a > set_list_a] :
      ( ( member_a_set_list_a @ F2 @ ( ring_h6109298854714515236t_unit @ r @ S ) )
     => ( ! [R3: a] :
            ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( F2 @ R3 )
              = ( G @ R3 ) ) )
       => ( member_a_set_list_a @ G @ ( ring_h6109298854714515236t_unit @ r @ S ) ) ) ) ).

% ring_hom_restrict
thf(fact_842_zero__is__prime_I1_J,axiom,
    prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).

% zero_is_prime(1)
thf(fact_843_ring__primeE_I3_J,axiom,
    ! [P2: a] :
      ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_ring_prime_a_b @ r @ P2 )
       => ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).

% ring_primeE(3)
thf(fact_844_ring__primeI,axiom,
    ! [P2: a] :
      ( ( P2
       != ( zero_a_b @ r ) )
     => ( ( prime_a_ring_ext_a_b @ r @ P2 )
       => ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).

% ring_primeI
thf(fact_845_splitted__on__def,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( polyno2453258491555121552on_a_b @ r @ K @ P2 )
      = ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ r @ K @ P2 ) )
        = ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).

% splitted_on_def
thf(fact_846_h_Oadd_Oinv__inj,axiom,
    ( inj_on_a_a
    @ ( a_inv_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) )
    @ ( partia707051561876973205xt_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) ) ) ).

% h.add.inv_inj
thf(fact_847_p_Ouniv__poly__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( univ_p7953238456130426574t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) ) )
        = ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.univ_poly_consistent
thf(fact_848_h_Osubring__props_I5_J,axiom,
    ! [K: set_a,H: a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( member_a @ H @ K )
       => ( member_a
          @ ( a_inv_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ H )
          @ K ) ) ) ).

% h.subring_props(5)
thf(fact_849_h_Oring__hom__restrict,axiom,
    ! [F2: a > list_a,S: partia2670972154091845814t_unit,G: a > list_a] :
      ( ( member_a_list_a @ F2
        @ ( ring_h405018892823518980t_unit
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ S ) )
     => ( ! [R3: a] :
            ( ( member_a @ R3
              @ ( partia707051561876973205xt_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r ) ) )
           => ( ( F2 @ R3 )
              = ( G @ R3 ) ) )
       => ( member_a_list_a @ G
          @ ( ring_h405018892823518980t_unit
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ S ) ) ) ) ).

% h.ring_hom_restrict
thf(fact_850_h_Oring__hom__restrict,axiom,
    ! [F2: a > set_list_a,S: partia7496981018696276118t_unit,G: a > set_list_a] :
      ( ( member_a_set_list_a @ F2
        @ ( ring_h6109298854714515236t_unit
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ S ) )
     => ( ! [R3: a] :
            ( ( member_a @ R3
              @ ( partia707051561876973205xt_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r ) ) )
           => ( ( F2 @ R3 )
              = ( G @ R3 ) ) )
       => ( member_a_set_list_a @ G
          @ ( ring_h6109298854714515236t_unit
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ S ) ) ) ) ).

% h.ring_hom_restrict
thf(fact_851_p_Opoly__of__const__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( poly_o8716471131768098070t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) ) )
        = ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.poly_of_const_consistent
thf(fact_852_p_Osubring__is__domain,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( domain6553523120543210313t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : H2
          @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.subring_is_domain
thf(fact_853_rupture__surj__hom_I1_J,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member4263473470251683292list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) @ ( ring_h6188449271506562988t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) ) ) ) ) ).

% rupture_surj_hom(1)
thf(fact_854_d_Ouniv__poly__consistent,axiom,
    ! [K: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( univ_p863672496597069550t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : K
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
        = ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.univ_poly_consistent
thf(fact_855_d_Oline__extension__consistent,axiom,
    ! [K: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( embedd5951720228509767443t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : K
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
        = ( embedd5951720228509767443t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.line_extension_consistent
thf(fact_856_d_Ochar__consistent,axiom,
    ! [H2: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( ring_c6053888738502451990t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : H2
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
        = ( ring_c6053888738502451990t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.char_consistent
thf(fact_857_d_Opoly__of__const__consistent,axiom,
    ! [K: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( poly_o3535013565302768950t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : K
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
        = ( poly_o3535013565302768950t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.poly_of_const_consistent
thf(fact_858_h_Osplitted__on__def,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( polyno2453258491555121552on_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ K
        @ P2 )
      = ( ( size_size_multiset_a
          @ ( polyno5714441830345289050on_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ K
            @ P2 ) )
        = ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).

% h.splitted_on_def
thf(fact_859_p_Orupture__surj__norm__is__hom,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( member5222132204073479906list_a @ ( comp_l92744468816104577list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          @ ( ring_h5771512479963622054t_unit
            @ ( partia9041243232023819264t_unit
              @ ^ [Uu: set_list_a] : K
              @ ( univ_poly_a_b @ r @ k ) )
            @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 ) ) ) ) ) ).

% p.rupture_surj_norm_is_hom
thf(fact_860_local_Ominus__minus,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
        = X ) ) ).

% local.minus_minus
thf(fact_861_a__inv__closed,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% a_inv_closed
thf(fact_862_local_Ominus__zero,axiom,
    ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
    = ( zero_a_b @ r ) ) ).

% local.minus_zero
thf(fact_863_h_Oimg__is__domain,axiom,
    ( ( domain1617769409708967785t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
   => ( domain1617769409708967785t_unit
      @ ( partia2314006361284536288t_unit
        @ ^ [Uu: set_set_list_a] :
            ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
            @ ( partia707051561876973205xt_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uv: set_a] : k
                @ r ) ) )
        @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% h.img_is_domain
thf(fact_864_add_Oinv__eq__1__iff,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ( a_inv_a_b @ r @ X )
          = ( zero_a_b @ r ) )
        = ( X
          = ( zero_a_b @ r ) ) ) ) ).

% add.inv_eq_1_iff
thf(fact_865_h_Ominus__minus,axiom,
    ! [X: a] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( a_inv_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ ( a_inv_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ X ) )
        = X ) ) ).

% h.minus_minus
thf(fact_866_h_Oadd_Oinv__closed,axiom,
    ! [X: a] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( member_a
        @ ( a_inv_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ X )
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.add.inv_closed
thf(fact_867_h_Ominus__zero,axiom,
    ( ( a_inv_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r )
      @ ( zero_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) )
    = ( zero_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) ) ) ).

% h.minus_zero
thf(fact_868_h_Oadd_Oinv__eq__1__iff,axiom,
    ! [X: a] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( ( a_inv_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ X )
          = ( zero_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
        = ( X
          = ( zero_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.add.inv_eq_1_iff
thf(fact_869_d_Osubdomain__iff,axiom,
    ! [H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( subdom3220114454046903646t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
        = ( domain1617769409708967785t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : H2
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ).

% d.subdomain_iff
thf(fact_870_d_Osubdomain__is__domain,axiom,
    ! [H2: set_set_list_a] :
      ( ( subdom3220114454046903646t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( domain1617769409708967785t_unit
        @ ( partia2314006361284536288t_unit
          @ ^ [Uu: set_set_list_a] : H2
          @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.subdomain_is_domain
thf(fact_871_p_Osubdomain__iff,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
        = ( domain6553523120543210313t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : H2
            @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.subdomain_iff
thf(fact_872_h_Oimg__is__ring,axiom,
    ( ring_s8247141995668492373t_unit
    @ ( partia2314006361284536288t_unit
      @ ^ [Uu: set_set_list_a] :
          ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uv: set_a] : k
              @ r ) ) )
      @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% h.img_is_ring
thf(fact_873_p_Osubdomain__is__domain,axiom,
    ! [H2: set_list_a] :
      ( ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( domain6553523120543210313t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : H2
          @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.subdomain_is_domain
thf(fact_874_canonical__embedding__is__hom,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( member_a_list_a @ ( poly_of_const_a_b @ r )
        @ ( ring_h405018892823518980t_unit
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r )
          @ ( univ_poly_a_b @ r @ K ) ) ) ) ).

% canonical_embedding_is_hom
thf(fact_875_p_Oring__hom__restrict,axiom,
    ! [F2: list_a > set_list_a,S: partia7496981018696276118t_unit,G: list_a > set_list_a] :
      ( ( member4263473470251683292list_a @ F2 @ ( ring_h6188449271506562988t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) )
     => ( ! [R3: list_a] :
            ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( F2 @ R3 )
              = ( G @ R3 ) ) )
       => ( member4263473470251683292list_a @ G @ ( ring_h6188449271506562988t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) ) ) ) ).

% p.ring_hom_restrict
thf(fact_876_p_OsubdomainI_H,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) ) ) ).

% p.subdomainI'
thf(fact_877_d_Oring__axioms,axiom,
    ring_s8247141995668492373t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ).

% d.ring_axioms
thf(fact_878_d_Osubring__is__ring,axiom,
    ! [H2: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ring_s8247141995668492373t_unit
        @ ( partia2314006361284536288t_unit
          @ ^ [Uu: set_set_list_a] : H2
          @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.subring_is_ring
thf(fact_879_d_Oring__incl__imp__subring,axiom,
    ! [H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( ring_s8247141995668492373t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : H2
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( subrin5643252653130547402t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.ring_incl_imp_subring
thf(fact_880_d_Osubring__iff,axiom,
    ! [H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( subrin5643252653130547402t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
        = ( ring_s8247141995668492373t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : H2
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ).

% d.subring_iff
thf(fact_881_ring_Opoly__add_Ocases,axiom,
    ! [R: partia2956882679547061052t_unit,X: produc3789376428941379879list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ~ ! [P1: list_list_list_a,P22: list_list_list_a] :
            ( X
           != ( produc1091363791885468951list_a @ P1 @ P22 ) ) ) ).

% ring.poly_add.cases
thf(fact_882_ring_Opoly__add_Ocases,axiom,
    ! [R: partia2670972154091845814t_unit,X: produc7709606177366032167list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ~ ! [P1: list_list_a,P22: list_list_a] :
            ( X
           != ( produc8696003437204565271list_a @ P1 @ P22 ) ) ) ).

% ring.poly_add.cases
thf(fact_883_ring_Opoly__add_Ocases,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: produc9164743771328383783list_a] :
      ( ( ring_a_b @ R )
     => ~ ! [P1: list_a,P22: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ P1 @ P22 ) ) ) ).

% ring.poly_add.cases
thf(fact_884_ring_Opoly__add_Ocases,axiom,
    ! [R: partia7496981018696276118t_unit,X: produc76396112498735911list_a] :
      ( ( ring_s8247141995668492373t_unit @ R )
     => ~ ! [P1: list_set_list_a,P22: list_set_list_a] :
            ( X
           != ( produc8430489356965245207list_a @ P1 @ P22 ) ) ) ).

% ring.poly_add.cases
thf(fact_885_ring_Ouniv__poly__consistent,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( univ_p2250591967980070728t_unit
            @ ( partia5725451877363000582t_unit
              @ ^ [Uu: set_list_list_a] : K
              @ R ) )
          = ( univ_p2250591967980070728t_unit @ R ) ) ) ) ).

% ring.univ_poly_consistent
thf(fact_886_ring_Ouniv__poly__consistent,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( ring_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( univ_poly_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : K
              @ R ) )
          = ( univ_poly_a_b @ R ) ) ) ) ).

% ring.univ_poly_consistent
thf(fact_887_ring_Ouniv__poly__consistent,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( univ_p7953238456130426574t_unit
            @ ( partia9041243232023819264t_unit
              @ ^ [Uu: set_list_a] : K
              @ R ) )
          = ( univ_p7953238456130426574t_unit @ R ) ) ) ) ).

% ring.univ_poly_consistent
thf(fact_888_ring_Ouniv__poly__consistent,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( ring_s8247141995668492373t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( ( univ_p863672496597069550t_unit
            @ ( partia2314006361284536288t_unit
              @ ^ [Uu: set_set_list_a] : K
              @ R ) )
          = ( univ_p863672496597069550t_unit @ R ) ) ) ) ).

% ring.univ_poly_consistent
thf(fact_889_ring_Opoly__of__const__consistent,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( poly_o1617770581224298896t_unit
            @ ( partia5725451877363000582t_unit
              @ ^ [Uu: set_list_list_a] : K
              @ R ) )
          = ( poly_o1617770581224298896t_unit @ R ) ) ) ) ).

% ring.poly_of_const_consistent
thf(fact_890_ring_Opoly__of__const__consistent,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( ring_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( poly_of_const_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : K
              @ R ) )
          = ( poly_of_const_a_b @ R ) ) ) ) ).

% ring.poly_of_const_consistent
thf(fact_891_ring_Opoly__of__const__consistent,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( poly_o8716471131768098070t_unit
            @ ( partia9041243232023819264t_unit
              @ ^ [Uu: set_list_a] : K
              @ R ) )
          = ( poly_o8716471131768098070t_unit @ R ) ) ) ) ).

% ring.poly_of_const_consistent
thf(fact_892_ring_Opoly__of__const__consistent,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( ring_s8247141995668492373t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( ( poly_o3535013565302768950t_unit
            @ ( partia2314006361284536288t_unit
              @ ^ [Uu: set_set_list_a] : K
              @ R ) )
          = ( poly_o3535013565302768950t_unit @ R ) ) ) ) ).

% ring.poly_of_const_consistent
thf(fact_893_ring_Ocarrier__polynomial__shell,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P2: list_list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ).

% ring.carrier_polynomial_shell
thf(fact_894_ring_Ocarrier__polynomial__shell,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).

% ring.carrier_polynomial_shell
thf(fact_895_ring_Ocarrier__polynomial__shell,axiom,
    ! [R: partia4960592913263135132t_unit,K: set_set_list_list_a,P2: list_set_list_list_a] :
      ( ( ring_s5628540378388669135t_unit @ R )
     => ( ( subrin3011427358523593924t_unit @ K @ R )
       => ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ K ) ) )
         => ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) ) ) ) ).

% ring.carrier_polynomial_shell
thf(fact_896_ring_Ocarrier__polynomial__shell,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
      ( ( ring_s8247141995668492373t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ) ) ).

% ring.carrier_polynomial_shell
thf(fact_897_ring_Ocarrier__polynomial__shell,axiom,
    ! [R: partia6043505979758434576t_unit,K: set_set_a,P2: list_set_a] :
      ( ( ring_s5549108798478166619t_unit @ R )
     => ( ( subrin1511138061850335568t_unit @ K @ R )
       => ( ( member_list_set_a @ P2 @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
         => ( member_list_set_a @ P2 @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ) ).

% ring.carrier_polynomial_shell
thf(fact_898_ring_Ocarrier__polynomial__shell,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
      ( ( ring_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).

% ring.carrier_polynomial_shell
thf(fact_899_ring_Ouniv__poly__subfield__of__consts,axiom,
    ! [R: partia4556295656693239580t_unit,K: set_list_set_list_a] :
      ( ( ring_l1106216629139447119t_unit @ R )
     => ( ( subfie868023609828841932t_unit @ K @ R )
       => ( subfie2054853380319850694t_unit @ ( image_3300606519373034379list_a @ ( poly_o1450351294650543536t_unit @ R ) @ K ) @ ( univ_p2555602637952293736t_unit @ R @ K ) ) ) ) ).

% ring.univ_poly_subfield_of_consts
thf(fact_900_ring_Ouniv__poly__subfield__of__consts,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( ring_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ R ) @ K ) @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% ring.univ_poly_subfield_of_consts
thf(fact_901_ring_Ouniv__poly__subfield__of__consts,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ R ) @ K ) @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% ring.univ_poly_subfield_of_consts
thf(fact_902_ring_Ouniv__poly__subfield__of__consts,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( ring_s8247141995668492373t_unit @ R )
     => ( ( subfie4339374749748326226t_unit @ K @ R )
       => ( subfie868023609828841932t_unit @ ( image_3509949574358669579list_a @ ( poly_o3535013565302768950t_unit @ R ) @ K ) @ ( univ_p863672496597069550t_unit @ R @ K ) ) ) ) ).

% ring.univ_poly_subfield_of_consts
thf(fact_903_h_Oimg__is__cring,axiom,
    ( ( cring_3470013030684506304t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
   => ( cring_3470013030684506304t_unit
      @ ( partia2314006361284536288t_unit
        @ ^ [Uu: set_set_list_a] :
            ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
            @ ( partia707051561876973205xt_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uv: set_a] : k
                @ r ) ) )
        @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% h.img_is_cring
thf(fact_904_local_Oring__axioms,axiom,
    ring_a_b @ r ).

% local.ring_axioms
thf(fact_905_univ__poly__is__ring,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_is_ring
thf(fact_906_p_Oring__axioms,axiom,
    ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ k ) ).

% p.ring_axioms
thf(fact_907_subring__is__ring,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( ring_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ r ) ) ) ).

% subring_is_ring
thf(fact_908_h_Oring__axioms,axiom,
    ( ring_a_b
    @ ( partia8674076737563717228xt_a_b
      @ ^ [Uu: set_a] : k
      @ r ) ) ).

% h.ring_axioms
thf(fact_909_h_Osubring__is__ring,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ring_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.subring_is_ring
thf(fact_910_ring__incl__imp__subring,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ r ) )
       => ( subring_a_b @ H2 @ r ) ) ) ).

% ring_incl_imp_subring
thf(fact_911_subring__iff,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( subring_a_b @ H2 @ r )
        = ( ring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ r ) ) ) ) ).

% subring_iff
thf(fact_912_p_Ouniv__poly__is__ring,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ring_l1939023646219158831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% p.univ_poly_is_ring
thf(fact_913_p_Osubring__is__ring,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( ring_l6212528067271185461t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : H2
          @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.subring_is_ring
thf(fact_914_h_Osubring__iff,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( subring_a_b @ H2
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
        = ( ring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.subring_iff
thf(fact_915_h_Oring__incl__imp__subring,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( ring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( subring_a_b @ H2
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.ring_incl_imp_subring
thf(fact_916_p_Osubring__iff,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
        = ( ring_l6212528067271185461t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : H2
            @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.subring_iff
thf(fact_917_p_Oring__incl__imp__subring,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ring_l6212528067271185461t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : H2
            @ ( univ_poly_a_b @ r @ k ) ) )
       => ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.ring_incl_imp_subring
thf(fact_918_p_Ocanonical__embedding__is__hom,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( member6714375691612171394list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) )
        @ ( ring_h8002040739877300486t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) )
          @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ).

% p.canonical_embedding_is_hom
thf(fact_919_I_Oa__rcos__module__minus,axiom,
    ! [X: list_a,X6: list_a] :
      ( ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ X6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( member_list_a @ X6 @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ X ) )
            = ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ X6 @ X ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) ) ) ) ) ).

% I.a_rcos_module_minus
thf(fact_920_d_Osubcring__iff,axiom,
    ! [H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( subcri7783154434480317835t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
        = ( cring_3470013030684506304t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : H2
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ).

% d.subcring_iff
thf(fact_921_h_Odegree__oneE,axiom,
    ! [P2: list_a,K: set_a] :
      ( ( member_list_a @ P2
        @ ( partia5361259788508890537t_unit
          @ ( univ_poly_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ K ) ) )
     => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
          = one_one_nat )
       => ~ ! [A4: a] :
              ( ( member_a @ A4 @ K )
             => ( ( A4
                 != ( zero_a_b
                    @ ( partia8674076737563717228xt_a_b
                      @ ^ [Uu: set_a] : k
                      @ r ) ) )
               => ! [B5: a] :
                    ( ( member_a @ B5 @ K )
                   => ( P2
                     != ( cons_a @ A4 @ ( cons_a @ B5 @ nil_a ) ) ) ) ) ) ) ) ).

% h.degree_oneE
thf(fact_922_p_Opdivides__imp__degree__le,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( Q != nil_list_a )
           => ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q )
             => ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).

% p.pdivides_imp_degree_le
thf(fact_923_normalize_Ocases,axiom,
    ! [X: list_a] :
      ( ( X != nil_a )
     => ~ ! [V2: a,Va: list_a] :
            ( X
           != ( cons_a @ V2 @ Va ) ) ) ).

% normalize.cases
thf(fact_924_h_Ocombine_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [K3: a,Ks: list_a,U: a,Us: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ ( cons_a @ K3 @ Ks ) @ ( cons_a @ U @ Us ) ) )
     => ( ! [Us: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ nil_a @ Us ) )
       => ~ ! [Ks: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).

% h.combine.cases
thf(fact_925_poly__mult_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [P22: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ nil_a @ P22 ) )
     => ~ ! [V2: a,Va: list_a,P22: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ ( cons_a @ V2 @ Va ) @ P22 ) ) ) ).

% poly_mult.cases
thf(fact_926_p_Osubring__props_I2_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) ) ).

% p.subring_props(2)
thf(fact_927_p_Ozero__pdivides,axiom,
    ! [P2: list_list_a] :
      ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a @ P2 )
      = ( P2 = nil_list_a ) ) ).

% p.zero_pdivides
thf(fact_928_p_Ozero__pdivides__zero,axiom,
    polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a @ nil_list_a ).

% p.zero_pdivides_zero
thf(fact_929_p_Ozero__is__prime_I1_J,axiom,
    prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% p.zero_is_prime(1)
thf(fact_930_p_Oring__irreducibleE_I1_J,axiom,
    ! [R2: list_a] :
      ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ k ) @ R2 )
       => ( R2
         != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.ring_irreducibleE(1)
thf(fact_931_p_Ospace__subgroup__props_I2_J,axiom,
    ! [K: set_list_a,N2: nat,E: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N2 @ K @ E )
       => ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ E ) ) ) ).

% p.space_subgroup_props(2)
thf(fact_932_p_Oring__primeE_I1_J,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
       => ( P2
         != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.ring_primeE(1)
thf(fact_933_I_Ozero__closed,axiom,
    member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ).

% I.zero_closed
thf(fact_934_p_Oring__primeI,axiom,
    ! [P2: list_a] :
      ( ( P2
       != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
       => ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) ) ).

% p.ring_primeI
thf(fact_935_p_Oa__lcos__mult__one,axiom,
    ! [M3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ M3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ M3 )
        = M3 ) ) ).

% p.a_lcos_mult_one
thf(fact_936_p_Opdivides__zero,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ nil_list_a ) ) ) ).

% p.pdivides_zero
thf(fact_937_degree__oneE,axiom,
    ! [P2: list_a,K: set_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
     => ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
          = one_one_nat )
       => ~ ! [A4: a] :
              ( ( member_a @ A4 @ K )
             => ( ( A4
                 != ( zero_a_b @ r ) )
               => ! [B5: a] :
                    ( ( member_a @ B5 @ K )
                   => ( P2
                     != ( cons_a @ A4 @ ( cons_a @ B5 @ nil_a ) ) ) ) ) ) ) ) ).

% degree_oneE
thf(fact_938_p_Opmod__zero__iff__pdivides,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q )
              = nil_list_a )
            = ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ Q @ P2 ) ) ) ) ) ).

% p.pmod_zero_iff_pdivides
thf(fact_939_p_Osame__pmod__iff__pdivides,axiom,
    ! [K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
           => ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ Q )
                = ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ Q ) )
              = ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ Q @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ A @ B ) ) ) ) ) ) ) ).

% p.same_pmod_iff_pdivides
thf(fact_940_p_Ozero__closed,axiom,
    member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% p.zero_closed
thf(fact_941_p_Or__right__minus__eq,axiom,
    ! [A: list_a,B: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          = ( A = B ) ) ) ) ).

% p.r_right_minus_eq
thf(fact_942_p_Oa__coset__add__zero,axiom,
    ! [M3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ M3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ M3 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        = M3 ) ) ).

% p.a_coset_add_zero
thf(fact_943_p_OboundD__carrier,axiom,
    ! [N2: nat,F2: nat > list_a,M: nat] :
      ( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N2 @ F2 )
     => ( ( ord_less_nat @ N2 @ M )
       => ( member_list_a @ ( F2 @ M ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.boundD_carrier
thf(fact_944_p_Orupture__eq__0__iff,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) @ Q )
              = ( zero_s2920163772466840039t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 ) ) )
            = ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q ) ) ) ) ) ).

% p.rupture_eq_0_iff
thf(fact_945_p_Onormalize_Ocases,axiom,
    ! [X: list_list_a] :
      ( ( X != nil_list_a )
     => ~ ! [V2: list_a,Va: list_list_a] :
            ( X
           != ( cons_list_a @ V2 @ Va ) ) ) ).

% p.normalize.cases
thf(fact_946_is__cring,axiom,
    cring_a_b @ r ).

% is_cring
thf(fact_947_p_Opoly__mult_Ocases,axiom,
    ! [X: produc7709606177366032167list_a] :
      ( ! [P22: list_list_a] :
          ( X
         != ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
     => ~ ! [V2: list_a,Va: list_list_a,P22: list_list_a] :
            ( X
           != ( produc8696003437204565271list_a @ ( cons_list_a @ V2 @ Va ) @ P22 ) ) ) ).

% p.poly_mult.cases
thf(fact_948_p_Ocombine_Ocases,axiom,
    ! [X: produc7709606177366032167list_a] :
      ( ! [K3: list_a,Ks: list_list_a,U: list_a,Us: list_list_a] :
          ( X
         != ( produc8696003437204565271list_a @ ( cons_list_a @ K3 @ Ks ) @ ( cons_list_a @ U @ Us ) ) )
     => ( ! [Us: list_list_a] :
            ( X
           != ( produc8696003437204565271list_a @ nil_list_a @ Us ) )
       => ~ ! [Ks: list_list_a] :
              ( X
             != ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).

% p.combine.cases
thf(fact_949_zero__pdivides__zero,axiom,
    polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).

% zero_pdivides_zero
thf(fact_950_zero__pdivides,axiom,
    ! [P2: list_a] :
      ( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P2 )
      = ( P2 = nil_a ) ) ).

% zero_pdivides
thf(fact_951_univ__poly__is__cring,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_is_cring
thf(fact_952_p_Ois__cring,axiom,
    cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ k ) ).

% p.is_cring
thf(fact_953_pdivides__zero,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( polyno5814909790663948098es_a_b @ r @ P2 @ nil_a ) ) ) ).

% pdivides_zero
thf(fact_954_p_Ouniv__poly__is__cring,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( cring_5991999922451032090t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% p.univ_poly_is_cring
thf(fact_955_d_Osubring__props_I2_J,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ K ) ) ).

% d.subring_props(2)
thf(fact_956_pmod__zero__iff__pdivides,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( ( polynomial_pmod_a_b @ r @ P2 @ Q )
              = nil_a )
            = ( polyno5814909790663948098es_a_b @ r @ Q @ P2 ) ) ) ) ) ).

% pmod_zero_iff_pdivides
thf(fact_957_same__pmod__iff__pdivides,axiom,
    ! [K: set_a,A: list_a,B: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
                = ( polynomial_pmod_a_b @ r @ B @ Q ) )
              = ( polyno5814909790663948098es_a_b @ r @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) ) ) ) ) ) ) ).

% same_pmod_iff_pdivides
thf(fact_958_rupture__eq__0__iff,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q )
              = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) ) )
            = ( polyno5814909790663948098es_a_b @ r @ P2 @ Q ) ) ) ) ) ).

% rupture_eq_0_iff
thf(fact_959_d_Ospace__subgroup__props_I2_J,axiom,
    ! [K: set_set_list_a,N2: nat,E: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N2 @ K @ E )
       => ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ E ) ) ) ).

% d.space_subgroup_props(2)
thf(fact_960_d_Oa__lcos__mult__one,axiom,
    ! [M3: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ M3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( a_l_co4135707798524667186t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ M3 )
        = M3 ) ) ).

% d.a_lcos_mult_one
thf(fact_961_pdivides__imp__degree__le,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
             => ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).

% pdivides_imp_degree_le
thf(fact_962_d_Odegree__oneE,axiom,
    ! [P2: list_set_list_a,K: set_set_list_a] :
      ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K ) ) )
     => ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P2 ) @ one_one_nat )
          = one_one_nat )
       => ~ ! [A4: set_list_a] :
              ( ( member_set_list_a @ A4 @ K )
             => ( ( A4
                 != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
               => ! [B5: set_list_a] :
                    ( ( member_set_list_a @ B5 @ K )
                   => ( P2
                     != ( cons_set_list_a @ A4 @ ( cons_set_list_a @ B5 @ nil_set_list_a ) ) ) ) ) ) ) ) ).

% d.degree_oneE
thf(fact_963_p_Odegree__oneE,axiom,
    ! [P2: list_list_a,K: set_list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
     => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
          = one_one_nat )
       => ~ ! [A4: list_a] :
              ( ( member_list_a @ A4 @ K )
             => ( ( A4
                 != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
               => ! [B5: list_a] :
                    ( ( member_list_a @ B5 @ K )
                   => ( P2
                     != ( cons_list_a @ A4 @ ( cons_list_a @ B5 @ nil_list_a ) ) ) ) ) ) ) ) ).

% p.degree_oneE
thf(fact_964_d_Ozero__closed,axiom,
    member_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ).

% d.zero_closed
thf(fact_965_h_Ohom__zero,axiom,
    ( ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r )
      @ ( zero_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) )
    = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% h.hom_zero
thf(fact_966_d_OboundD__carrier,axiom,
    ! [N2: nat,F2: nat > set_list_a,M: nat] :
      ( ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ N2 @ F2 )
     => ( ( ord_less_nat @ N2 @ M )
       => ( member_set_list_a @ ( F2 @ M ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ).

% d.boundD_carrier
thf(fact_967_h_Oimg__is__subfield_I2_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
         != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( subfie4339374749748326226t_unit @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ K ) @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% h.img_is_subfield(2)
thf(fact_968_p_Orupture__one__not__zero,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) )
         => ( ( one_se2489417650821308733t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 ) )
           != ( zero_s2920163772466840039t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P2 ) ) ) ) ) ) ).

% p.rupture_one_not_zero
thf(fact_969_d_Ocombine_Ocases,axiom,
    ! [X: produc76396112498735911list_a] :
      ( ! [K3: set_list_a,Ks: list_set_list_a,U: set_list_a,Us: list_set_list_a] :
          ( X
         != ( produc8430489356965245207list_a @ ( cons_set_list_a @ K3 @ Ks ) @ ( cons_set_list_a @ U @ Us ) ) )
     => ( ! [Us: list_set_list_a] :
            ( X
           != ( produc8430489356965245207list_a @ nil_set_list_a @ Us ) )
       => ~ ! [Ks: list_set_list_a] :
              ( X
             != ( produc8430489356965245207list_a @ Ks @ nil_set_list_a ) ) ) ) ).

% d.combine.cases
thf(fact_970_d_Opoly__mult_Ocases,axiom,
    ! [X: produc76396112498735911list_a] :
      ( ! [P22: list_set_list_a] :
          ( X
         != ( produc8430489356965245207list_a @ nil_set_list_a @ P22 ) )
     => ~ ! [V2: set_list_a,Va: list_set_list_a,P22: list_set_list_a] :
            ( X
           != ( produc8430489356965245207list_a @ ( cons_set_list_a @ V2 @ Va ) @ P22 ) ) ) ).

% d.poly_mult.cases
thf(fact_971_d_Onormalize_Ocases,axiom,
    ! [X: list_set_list_a] :
      ( ( X != nil_set_list_a )
     => ~ ! [V2: set_list_a,Va: list_set_list_a] :
            ( X
           != ( cons_set_list_a @ V2 @ Va ) ) ) ).

% d.normalize.cases
thf(fact_972_d_Osubring__props_I3_J,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( member_set_list_a @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ K ) ) ).

% d.subring_props(3)
thf(fact_973_rupture__one__not__zero,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) )
         => ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) )
           != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) ) ) ) ) ) ).

% rupture_one_not_zero
thf(fact_974_h_Oimg__is__subfield_I1_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
         != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( inj_on_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ K ) ) ) ).

% h.img_is_subfield(1)
thf(fact_975_d_Oone__closed,axiom,
    member_set_list_a @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ).

% d.one_closed
thf(fact_976_h_Oinfinite__dimension__hom,axiom,
    ! [K: set_a,E: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
         != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( ( inj_on_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ E )
         => ( ( embedd9027525575939734154ra_a_b @ K @ E
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r ) )
           => ( ~ ( embedd8708762675212832759on_a_b
                  @ ( partia8674076737563717228xt_a_b
                    @ ^ [Uu: set_a] : k
                    @ r )
                  @ K
                  @ E )
             => ~ ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ K ) @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ E ) ) ) ) ) ) ) ).

% h.infinite_dimension_hom
thf(fact_977_h_Oinj__hom__dimension,axiom,
    ! [K: set_a,E: set_a,N2: nat] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
         != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( ( inj_on_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ E )
         => ( ( embedd2795209813406577254on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ N2
              @ K
              @ E )
           => ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N2 @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ K ) @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ E ) ) ) ) ) ) ).

% h.inj_hom_dimension
thf(fact_978_d_Opoly__add_Ocases,axiom,
    ! [X: produc76396112498735911list_a] :
      ~ ! [P1: list_set_list_a,P22: list_set_list_a] :
          ( X
         != ( produc8430489356965245207list_a @ P1 @ P22 ) ) ).

% d.poly_add.cases
thf(fact_979_zero__not__one,axiom,
    ( ( zero_a_b @ r )
   != ( one_a_ring_ext_a_b @ r ) ) ).

% zero_not_one
thf(fact_980_subring__props_I3_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K ) ) ).

% subring_props(3)
thf(fact_981_dimension__is__inj,axiom,
    ! [K: set_a,N2: nat,E: set_a,M: nat] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
       => ( ( embedd2795209813406577254on_a_b @ r @ M @ K @ E )
         => ( N2 = M ) ) ) ) ).

% dimension_is_inj
thf(fact_982_telescopic__base__dim_I1_J,axiom,
    ! [K: set_a,F: set_a,E: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( subfield_a_b @ F @ r )
       => ( ( embedd8708762675212832759on_a_b @ r @ K @ F )
         => ( ( embedd8708762675212832759on_a_b @ r @ F @ E )
           => ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ) ) ) ).

% telescopic_base_dim(1)
thf(fact_983_finite__dimensionE_H,axiom,
    ! [K: set_a,E: set_a] :
      ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
     => ~ ! [N: nat] :
            ~ ( embedd2795209813406577254on_a_b @ r @ N @ K @ E ) ) ).

% finite_dimensionE'
thf(fact_984_finite__dimensionI,axiom,
    ! [N2: nat,K: set_a,E: set_a] :
      ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
     => ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ).

% finite_dimensionI
thf(fact_985_finite__dimension__def,axiom,
    ! [K: set_a,E: set_a] :
      ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
      = ( ? [N3: nat] : ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E ) ) ) ).

% finite_dimension_def
thf(fact_986_space__subgroup__props_I2_J,axiom,
    ! [K: set_a,N2: nat,E: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
       => ( member_a @ ( zero_a_b @ r ) @ E ) ) ) ).

% space_subgroup_props(2)
thf(fact_987_telescopic__base__aux,axiom,
    ! [K: set_a,F: set_a,N2: nat,E: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( subfield_a_b @ F @ r )
       => ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ F )
         => ( ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ F @ E )
           => ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E ) ) ) ) ) ).

% telescopic_base_aux
thf(fact_988_space__subgroup__props_I4_J,axiom,
    ! [K: set_a,N2: nat,E: set_a,V3: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
       => ( ( member_a @ V3 @ E )
         => ( member_a @ ( a_inv_a_b @ r @ V3 ) @ E ) ) ) ) ).

% space_subgroup_props(4)
thf(fact_989_p_Oadd_Oinv__inj,axiom,
    inj_on_list_a_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% p.add.inv_inj
thf(fact_990_p_Ozero__not__one,axiom,
    ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) )
   != ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% p.zero_not_one
thf(fact_991_unique__dimension,axiom,
    ! [K: set_a,E: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
       => ? [X2: nat] :
            ( ( embedd2795209813406577254on_a_b @ r @ X2 @ K @ E )
            & ! [Y2: nat] :
                ( ( embedd2795209813406577254on_a_b @ r @ Y2 @ K @ E )
               => ( Y2 = X2 ) ) ) ) ) ).

% unique_dimension
thf(fact_992_p_Osubring__props_I5_J,axiom,
    ! [K: set_list_a,H: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_a @ H @ K )
       => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ H ) @ K ) ) ) ).

% p.subring_props(5)
thf(fact_993_p_Osubring__props_I3_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) ) ).

% p.subring_props(3)
thf(fact_994_finite__dimension__imp__subalgebra,axiom,
    ! [K: set_a,E: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
       => ( embedd9027525575939734154ra_a_b @ K @ E @ r ) ) ) ).

% finite_dimension_imp_subalgebra
thf(fact_995_h_Osubring__props_I3_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( member_a
        @ ( one_a_ring_ext_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
        @ K ) ) ).

% h.subring_props(3)
thf(fact_996_univ__poly__a__inv__consistent,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ) ).

% univ_poly_a_inv_consistent
thf(fact_997_univ__poly__a__inv__length,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) )
          = ( size_size_list_a @ P2 ) ) ) ) ).

% univ_poly_a_inv_length
thf(fact_998_space__subgroup__props_I1_J,axiom,
    ! [K: set_a,N2: nat,E: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E )
       => ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% space_subgroup_props(1)
thf(fact_999_long__division__a__inv_I2_J,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q )
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ).

% long_division_a_inv(2)
thf(fact_1000_long__division__a__inv_I1_J,axiom,
    ! [K: set_a,P2: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q )
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) ) ) ) ) ) ).

% long_division_a_inv(1)
thf(fact_1001_subalbegra__incl__imp__finite__dimension,axiom,
    ! [K: set_a,E: set_a,V: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
       => ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
         => ( ( ord_less_eq_set_a @ V @ E )
           => ( embedd8708762675212832759on_a_b @ r @ K @ V ) ) ) ) ) ).

% subalbegra_incl_imp_finite_dimension
thf(fact_1002_p_Ospace__subgroup__props_I4_J,axiom,
    ! [K: set_list_a,N2: nat,E: set_list_a,V3: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N2 @ K @ E )
       => ( ( member_list_a @ V3 @ E )
         => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ V3 ) @ E ) ) ) ) ).

% p.space_subgroup_props(4)
thf(fact_1003_d_Oadd_Oinv__inj,axiom,
    inj_on727569427812895985list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ).

% d.add.inv_inj
thf(fact_1004_d_Osubring__props_I5_J,axiom,
    ! [K: set_set_list_a,H: set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( member_set_list_a @ H @ K )
       => ( member_set_list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ H ) @ K ) ) ) ).

% d.subring_props(5)
thf(fact_1005_h_Odimension__is__inj,axiom,
    ! [K: set_a,N2: nat,E: set_a,M: nat] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( embedd2795209813406577254on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ N2
          @ K
          @ E )
       => ( ( embedd2795209813406577254on_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ M
            @ K
            @ E )
         => ( N2 = M ) ) ) ) ).

% h.dimension_is_inj
thf(fact_1006_degree__one,axiom,
    ! [K: set_a] :
      ( ( minus_minus_nat @ ( size_size_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ K ) ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% degree_one
thf(fact_1007_h_Otelescopic__base__dim_I1_J,axiom,
    ! [K: set_a,F: set_a,E: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( subfield_a_b @ F
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
       => ( ( embedd8708762675212832759on_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ K
            @ F )
         => ( ( embedd8708762675212832759on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ F
              @ E )
           => ( embedd8708762675212832759on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ K
              @ E ) ) ) ) ) ).

% h.telescopic_base_dim(1)
thf(fact_1008_h_Ofinite__dimension__def,axiom,
    ! [K: set_a,E: set_a] :
      ( ( embedd8708762675212832759on_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ K
        @ E )
      = ( ? [N3: nat] :
            ( embedd2795209813406577254on_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ N3
            @ K
            @ E ) ) ) ).

% h.finite_dimension_def
thf(fact_1009_h_Ofinite__dimensionI,axiom,
    ! [N2: nat,K: set_a,E: set_a] :
      ( ( embedd2795209813406577254on_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ N2
        @ K
        @ E )
     => ( embedd8708762675212832759on_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ K
        @ E ) ) ).

% h.finite_dimensionI
thf(fact_1010_h_Ofinite__dimensionE_H,axiom,
    ! [K: set_a,E: set_a] :
      ( ( embedd8708762675212832759on_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ K
        @ E )
     => ~ ! [N: nat] :
            ~ ( embedd2795209813406577254on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ N
              @ K
              @ E ) ) ).

% h.finite_dimensionE'
thf(fact_1011_monic__degree__one__root__condition,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( polyno4133073214067823460ot_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) @ B )
        = ( A = B ) ) ) ).

% monic_degree_one_root_condition
thf(fact_1012_I_Oone__imp__carrier,axiom,
    ( ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) )
   => ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f )
      = ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% I.one_imp_carrier
thf(fact_1013_h_Ospace__subgroup__props_I2_J,axiom,
    ! [K: set_a,N2: nat,E: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( embedd2795209813406577254on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ N2
          @ K
          @ E )
       => ( member_a
          @ ( zero_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) )
          @ E ) ) ) ).

% h.space_subgroup_props(2)
thf(fact_1014_h_Otelescopic__base__aux,axiom,
    ! [K: set_a,F: set_a,N2: nat,E: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( subfield_a_b @ F
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
       => ( ( embedd2795209813406577254on_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ N2
            @ K
            @ F )
         => ( ( embedd2795209813406577254on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ one_one_nat
              @ F
              @ E )
           => ( embedd2795209813406577254on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ N2
              @ K
              @ E ) ) ) ) ) ).

% h.telescopic_base_aux
thf(fact_1015_h_Ospace__subgroup__props_I4_J,axiom,
    ! [K: set_a,N2: nat,E: set_a,V3: a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( embedd2795209813406577254on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ N2
          @ K
          @ E )
       => ( ( member_a @ V3 @ E )
         => ( member_a
            @ ( a_inv_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ V3 )
            @ E ) ) ) ) ).

% h.space_subgroup_props(4)
thf(fact_1016_univ__poly__a__inv__degree,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( minus_minus_nat @ ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) @ one_one_nat )
          = ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ).

% univ_poly_a_inv_degree
thf(fact_1017_d_Ospace__subgroup__props_I4_J,axiom,
    ! [K: set_set_list_a,N2: nat,E: set_set_list_a,V3: set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N2 @ K @ E )
       => ( ( member_set_list_a @ V3 @ E )
         => ( member_set_list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ V3 ) @ E ) ) ) ) ).

% d.space_subgroup_props(4)
thf(fact_1018_h_Ounique__dimension,axiom,
    ! [K: set_a,E: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( embedd8708762675212832759on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ K
          @ E )
       => ? [X2: nat] :
            ( ( embedd2795209813406577254on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ X2
              @ K
              @ E )
            & ! [Y2: nat] :
                ( ( embedd2795209813406577254on_a_b
                  @ ( partia8674076737563717228xt_a_b
                    @ ^ [Uu: set_a] : k
                    @ r )
                  @ Y2
                  @ K
                  @ E )
               => ( Y2 = X2 ) ) ) ) ) ).

% h.unique_dimension
thf(fact_1019_h_Ofinite__dimension__imp__subalgebra,axiom,
    ! [K: set_a,E: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( embedd8708762675212832759on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ K
          @ E )
       => ( embedd9027525575939734154ra_a_b @ K @ E
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.finite_dimension_imp_subalgebra
thf(fact_1020_p_Ouniv__poly__a__inv__consistent,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 )
          = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P2 ) ) ) ) ).

% p.univ_poly_a_inv_consistent
thf(fact_1021_p_Ouniv__poly__a__inv__length,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) )
          = ( size_s349497388124573686list_a @ P2 ) ) ) ) ).

% p.univ_poly_a_inv_length
thf(fact_1022_pdivides__imp__is__root,axiom,
    ! [P2: list_a,X: a] :
      ( ( P2 != nil_a )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 )
         => ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ) ).

% pdivides_imp_is_root
thf(fact_1023_p_Olong__division__a__inv_I2_J,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) @ Q )
            = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q ) ) ) ) ) ) ).

% p.long_division_a_inv(2)
thf(fact_1024_p_Omonic__degree__one__root__condition,axiom,
    ! [A: list_a,B: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ A ) @ nil_list_a ) ) @ B )
        = ( A = B ) ) ) ).

% p.monic_degree_one_root_condition
thf(fact_1025_h_Ospace__subgroup__props_I1_J,axiom,
    ! [K: set_a,N2: nat,E: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( embedd2795209813406577254on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ N2
          @ K
          @ E )
       => ( ord_less_eq_set_a @ E
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.space_subgroup_props(1)
thf(fact_1026_p_Olong__division__a__inv_I1_J,axiom,
    ! [K: set_list_a,P2: list_list_a,Q: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) @ Q )
            = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q ) ) ) ) ) ) ).

% p.long_division_a_inv(1)
thf(fact_1027_h_Osubalbegra__incl__imp__finite__dimension,axiom,
    ! [K: set_a,E: set_a,V: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( embedd8708762675212832759on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ K
          @ E )
       => ( ( embedd9027525575939734154ra_a_b @ K @ V
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) )
         => ( ( ord_less_eq_set_a @ V @ E )
           => ( embedd8708762675212832759on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ K
              @ V ) ) ) ) ) ).

% h.subalbegra_incl_imp_finite_dimension
thf(fact_1028_p_Odegree__one,axiom,
    ! [K: set_list_a] :
      ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% p.degree_one
thf(fact_1029_h_Odegree__one,axiom,
    ! [K: set_a] :
      ( ( minus_minus_nat
        @ ( size_size_list_a
          @ ( one_li8328186300101108157t_unit
            @ ( univ_poly_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ K ) ) )
        @ one_one_nat )
      = zero_zero_nat ) ).

% h.degree_one
thf(fact_1030_p_Opdivides__imp__is__root,axiom,
    ! [P2: list_list_a,X: list_a] :
      ( ( P2 != nil_list_a )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ nil_list_a ) ) @ P2 )
         => ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X ) ) ) ) ).

% p.pdivides_imp_is_root
thf(fact_1031_d_Omonic__degree__one__root__condition,axiom,
    ! [A: set_list_a,B: set_list_a] :
      ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( polyno4320237611291262604t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ A ) @ nil_set_list_a ) ) @ B )
        = ( A = B ) ) ) ).

% d.monic_degree_one_root_condition
thf(fact_1032_h_Omonic__degree__one__root__condition,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( polyno4133073214067823460ot_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ ( cons_a
            @ ( one_a_ring_ext_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r ) )
            @ ( cons_a
              @ ( a_inv_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ A )
              @ nil_a ) )
          @ B )
        = ( A = B ) ) ) ).

% h.monic_degree_one_root_condition
thf(fact_1033_one__closed,axiom,
    member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% one_closed
thf(fact_1034_p_Ouniv__poly__a__inv__degree,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) ) @ one_one_nat )
          = ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ).

% p.univ_poly_a_inv_degree
thf(fact_1035_I_Oa__rcos__inv,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( a_SET_7640846956710366103t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ X ) )
        = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) ) ) ) ).

% I.a_rcos_inv
thf(fact_1036_is__root__imp__pdivides,axiom,
    ! [P2: list_a,X: a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
       => ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 ) ) ) ).

% is_root_imp_pdivides
thf(fact_1037_d_Odegree__one,axiom,
    ! [K: set_set_list_a] :
      ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ K ) ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% d.degree_one
thf(fact_1038_p_Ois__root__imp__pdivides,axiom,
    ! [P2: list_list_a,X: list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X )
       => ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ nil_list_a ) ) @ P2 ) ) ) ).

% p.is_root_imp_pdivides
thf(fact_1039_dimension__one,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ K @ K ) ) ).

% dimension_one
thf(fact_1040_p_Ominus__minus,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) )
        = X ) ) ).

% p.minus_minus
thf(fact_1041_p_Oadd_Oinv__closed,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.add.inv_closed
thf(fact_1042_p_Oone__closed,axiom,
    member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% p.one_closed
thf(fact_1043_p_Ominus__zero,axiom,
    ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
    = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% p.minus_zero
thf(fact_1044_h_Oone__closed,axiom,
    ( member_a
    @ ( one_a_ring_ext_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) )
    @ ( partia707051561876973205xt_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) ) ) ).

% h.one_closed
thf(fact_1045_p_Oadd_Oinv__eq__1__iff,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        = ( X
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.add.inv_eq_1_iff
thf(fact_1046_d_Ominus__minus,axiom,
    ! [X: set_list_a] :
      ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X ) )
        = X ) ) ).

% d.minus_minus
thf(fact_1047_d_Oadd_Oinv__closed,axiom,
    ! [X: set_list_a] :
      ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( member_set_list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.add.inv_closed
thf(fact_1048_d_Ominus__zero,axiom,
    ( ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
    = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% d.minus_zero
thf(fact_1049_I_Oa__inv__closed,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) )
     => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) ) ).

% I.a_inv_closed
thf(fact_1050_d_Oadd_Oinv__eq__1__iff,axiom,
    ! [X: set_list_a] :
      ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X )
          = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
        = ( X
          = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ).

% d.add.inv_eq_1_iff
thf(fact_1051_h_Odimension__one,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( embedd2795209813406577254on_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ one_one_nat
        @ K
        @ K ) ) ).

% h.dimension_one
thf(fact_1052_h_Ohom__one,axiom,
    ( ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r )
      @ ( one_a_ring_ext_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) )
    = ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% h.hom_one
thf(fact_1053_h_Ohom__a__inv,axiom,
    ! [X: a] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r )
          @ ( a_inv_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ X ) )
        = ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ X ) ) ) ) ).

% h.hom_a_inv
thf(fact_1054_h_Orank__nullity__theorem,axiom,
    ! [K: set_a,N2: nat,E: set_a,M: nat] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
         != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( ( embedd2795209813406577254on_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ N2
            @ K
            @ E )
         => ( ( embedd2795209813406577254on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ M
              @ K
              @ ( a_kern2981893169186190118t_unit
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : E
                  @ ( partia8674076737563717228xt_a_b
                    @ ^ [Uu: set_a] : k
                    @ r ) )
                @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
                @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) )
           => ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( minus_minus_nat @ N2 @ M ) @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ K ) @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ E ) ) ) ) ) ) ).

% h.rank_nullity_theorem
thf(fact_1055_p_Osubcring__iff,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
        = ( cring_3148771470849435808t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : H2
            @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.subcring_iff
thf(fact_1056_p_Ocarrier__is__subcring,axiom,
    subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).

% p.carrier_is_subcring
thf(fact_1057_p_OsubcringI_H,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) ) ) ).

% p.subcringI'
thf(fact_1058_h_Oadditive__subgroup__a__kernel,axiom,
    ( additi2834746164131130830up_a_b
    @ ( a_kern2981893169186190118t_unit
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r )
      @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
      @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) )
    @ ( partia8674076737563717228xt_a_b
      @ ^ [Uu: set_a] : k
      @ r ) ) ).

% h.additive_subgroup_a_kernel
thf(fact_1059_h_Oabelian__subgroup__a__kernel,axiom,
    ( abelian_subgroup_a_b
    @ ( a_kern2981893169186190118t_unit
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r )
      @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
      @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) )
    @ ( partia8674076737563717228xt_a_b
      @ ^ [Uu: set_a] : k
      @ r ) ) ).

% h.abelian_subgroup_a_kernel
thf(fact_1060_h_Oset__add__ker__hom_I2_J,axiom,
    ! [I2: set_a] :
      ( ( ord_less_eq_set_a @ I2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
          @ ( set_add_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ ( a_kern2981893169186190118t_unit
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
              @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) )
            @ I2 ) )
        = ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ I2 ) ) ) ).

% h.set_add_ker_hom(2)
thf(fact_1061_add__additive__subgroups,axiom,
    ! [H2: set_a,K: set_a] :
      ( ( additi2834746164131130830up_a_b @ H2 @ r )
     => ( ( additi2834746164131130830up_a_b @ K @ r )
       => ( additi2834746164131130830up_a_b @ ( set_add_a_b @ r @ H2 @ K ) @ r ) ) ) ).

% add_additive_subgroups
thf(fact_1062_setadd__subset__G,axiom,
    ! [H2: set_a,K: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H2 @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% setadd_subset_G
thf(fact_1063_set__add__comm,axiom,
    ! [I2: set_a,J4: set_a] :
      ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ J4 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( set_add_a_b @ r @ I2 @ J4 )
          = ( set_add_a_b @ r @ J4 @ I2 ) ) ) ) ).

% set_add_comm
thf(fact_1064_set__add__closed,axiom,
    ! [A3: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A3 @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% set_add_closed
thf(fact_1065_sum__space__dim_I1_J,axiom,
    ! [K: set_a,E: set_a,F: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
       => ( ( embedd8708762675212832759on_a_b @ r @ K @ F )
         => ( embedd8708762675212832759on_a_b @ r @ K @ ( set_add_a_b @ r @ E @ F ) ) ) ) ) ).

% sum_space_dim(1)
thf(fact_1066_h_Oadd__additive__subgroups,axiom,
    ! [H2: set_a,K: set_a] :
      ( ( additi2834746164131130830up_a_b @ H2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( additi2834746164131130830up_a_b @ K
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
       => ( additi2834746164131130830up_a_b
          @ ( set_add_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ H2
            @ K )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.add_additive_subgroups
thf(fact_1067_h_Osetadd__subset__G,axiom,
    ! [H2: set_a,K: set_a] :
      ( ( ord_less_eq_set_a @ H2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( ord_less_eq_set_a @ K
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( ord_less_eq_set_a
          @ ( set_add_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ H2
            @ K )
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.setadd_subset_G
thf(fact_1068_h_Oset__add__comm,axiom,
    ! [I2: set_a,J4: set_a] :
      ( ( ord_less_eq_set_a @ I2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( ord_less_eq_set_a @ J4
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( ( set_add_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ I2
            @ J4 )
          = ( set_add_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ J4
            @ I2 ) ) ) ) ).

% h.set_add_comm
thf(fact_1069_h_Oset__add__closed,axiom,
    ! [A3: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A3
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( ord_less_eq_set_a @ B2
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( ord_less_eq_set_a
          @ ( set_add_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ A3
            @ B2 )
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.set_add_closed
thf(fact_1070_h_Osum__space__dim_I1_J,axiom,
    ! [K: set_a,E: set_a,F: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( embedd8708762675212832759on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ K
          @ E )
       => ( ( embedd8708762675212832759on_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ K
            @ F )
         => ( embedd8708762675212832759on_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ K
            @ ( set_add_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ E
              @ F ) ) ) ) ) ).

% h.sum_space_dim(1)
thf(fact_1071_h_Oset__add__ker__hom_I1_J,axiom,
    ! [I2: set_a] :
      ( ( ord_less_eq_set_a @ I2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
          @ ( set_add_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ I2
            @ ( a_kern2981893169186190118t_unit
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
              @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) ) )
        = ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ I2 ) ) ) ).

% h.set_add_ker_hom(1)
thf(fact_1072_p_Onot__empty__rootsE,axiom,
    ! [P2: list_list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
         != zero_z4454100511807792257list_a )
       => ~ ! [A4: list_a] :
              ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
             => ( ( member_list_a @ A4 @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) )
               => ( ( member_list_list_a @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ A4 ) @ nil_list_a ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
                 => ~ ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ A4 ) @ nil_list_a ) ) @ P2 ) ) ) ) ) ) ).

% p.not_empty_rootsE
thf(fact_1073_p_Oalg__multE_I2_J,axiom,
    ! [X: list_a,P2: list_list_a,N2: nat] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
       => ( ( P2 != nil_list_a )
         => ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ nil_list_a ) ) @ N2 ) @ P2 )
           => ( ord_less_eq_nat @ N2 @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X ) ) ) ) ) ) ).

% p.alg_multE(2)
thf(fact_1074_p_Ole__alg__mult__imp__pdivides,axiom,
    ! [X: list_a,P2: list_list_a,N2: nat] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
       => ( ( ord_less_eq_nat @ N2 @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X ) )
         => ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ nil_list_a ) ) @ N2 ) @ P2 ) ) ) ) ).

% p.le_alg_mult_imp_pdivides
thf(fact_1075_p_Opolynomial__pow__not__zero,axiom,
    ! [P2: list_list_a,N2: nat] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( P2 != nil_list_a )
       => ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P2 @ N2 )
         != nil_list_a ) ) ) ).

% p.polynomial_pow_not_zero
thf(fact_1076_p_Osubring__polynomial__pow__not__zero,axiom,
    ! [K: set_list_a,P2: list_list_a,N2: nat] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( P2 != nil_list_a )
         => ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 @ N2 )
           != nil_list_a ) ) ) ) ).

% p.subring_polynomial_pow_not_zero
thf(fact_1077_p_Ovar__pow__closed,axiom,
    ! [K: set_list_a,N2: nat] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ).

% p.var_pow_closed
thf(fact_1078_p_Opolynomial__pow__division,axiom,
    ! [P2: list_list_a,N2: nat,M: nat] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P2 @ N2 ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P2 @ M ) ) ) ) ).

% p.polynomial_pow_division
thf(fact_1079_p_Oroots__mem__iff__is__root,axiom,
    ! [P2: list_list_a,X: list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( member_list_a @ X @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) )
        = ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X ) ) ) ).

% p.roots_mem_iff_is_root
thf(fact_1080_p_Ovar__pow__degree,axiom,
    ! [K: set_list_a,N2: nat] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N2 ) ) @ one_one_nat )
        = N2 ) ) ).

% p.var_pow_degree
thf(fact_1081_p_Oalg__multE_I1_J,axiom,
    ! [X: list_a,P2: list_list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
       => ( ( P2 != nil_list_a )
         => ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X ) ) @ P2 ) ) ) ) ).

% p.alg_multE(1)
thf(fact_1082_p_Oalg__mult__def,axiom,
    ! [P2: list_list_a,X: list_a] :
      ( ( ( P2 = nil_list_a )
       => ( ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X )
          = zero_zero_nat ) )
      & ( ( P2 != nil_list_a )
       => ( ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X )
              = ( order_Greatest_nat
                @ ^ [N3: nat] : ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X ) @ nil_list_a ) ) @ N3 ) @ P2 ) ) ) )
          & ( ~ ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ X )
              = zero_zero_nat ) ) ) ) ) ).

% p.alg_mult_def
thf(fact_1083_p_Oroots__inclI,axiom,
    ! [P2: list_list_a,Q: list_list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
       => ( ( Q != nil_list_a )
         => ( ! [A4: list_a] :
                ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
               => ( ( P2 != nil_list_a )
                 => ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ A4 ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ A4 ) ) @ Q ) ) )
           => ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ Q ) ) ) ) ) ) ).

% p.roots_inclI
thf(fact_1084_p_Ounitary__monom__eq__var__pow,axiom,
    ! [K: set_list_a,N2: nat] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N2 )
        = ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N2 ) ) ) ).

% p.unitary_monom_eq_var_pow
thf(fact_1085_nat__pow__consistent,axiom,
    ! [X: a,N2: nat,H2: set_a] :
      ( ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 )
      = ( pow_a_1026414303147256608_b_nat
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ r )
        @ X
        @ N2 ) ) ).

% nat_pow_consistent
thf(fact_1086_pow__non__zero,axiom,
    ! [X: a,N2: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( X
         != ( zero_a_b @ r ) )
       => ( ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 )
         != ( zero_a_b @ r ) ) ) ) ).

% pow_non_zero
thf(fact_1087_nat__pow__zero,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N2 )
        = ( zero_a_b @ r ) ) ) ).

% nat_pow_zero
thf(fact_1088_h_Onat__pow__consistent,axiom,
    ! [X: a,N2: nat,H2: set_a] :
      ( ( pow_a_1026414303147256608_b_nat
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ X
        @ N2 )
      = ( pow_a_1026414303147256608_b_nat
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
        @ X
        @ N2 ) ) ).

% h.nat_pow_consistent
thf(fact_1089_polynomial__pow__not__zero,axiom,
    ! [P2: list_a,N2: nat] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( P2 != nil_a )
       => ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ N2 )
         != nil_a ) ) ) ).

% polynomial_pow_not_zero
thf(fact_1090_p_Onat__pow__consistent,axiom,
    ! [X: list_a,N2: nat,H2: set_list_a] :
      ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N2 )
      = ( pow_li1142815632869257134it_nat
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : H2
          @ ( univ_poly_a_b @ r @ k ) )
        @ X
        @ N2 ) ) ).

% p.nat_pow_consistent
thf(fact_1091_subring__polynomial__pow__not__zero,axiom,
    ! [K: set_a,P2: list_a,N2: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( P2 != nil_a )
         => ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P2 @ N2 )
           != nil_a ) ) ) ) ).

% subring_polynomial_pow_not_zero
thf(fact_1092_var__pow__closed,axiom,
    ! [K: set_a,N2: nat] :
      ( ( subring_a_b @ K @ r )
     => ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).

% var_pow_closed
thf(fact_1093_p_Opow__non__zero,axiom,
    ! [X: list_a,N2: nat] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( X
         != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N2 )
         != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.pow_non_zero
thf(fact_1094_p_Onat__pow__zero,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.nat_pow_zero
thf(fact_1095_h_Onat__pow__zero,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( pow_a_1026414303147256608_b_nat
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ ( zero_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) )
          @ N2 )
        = ( zero_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.nat_pow_zero
thf(fact_1096_polynomial__pow__division,axiom,
    ! [P2: list_a,N2: nat,M: nat] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ N2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ M ) ) ) ) ).

% polynomial_pow_division
thf(fact_1097_d_Onat__pow__consistent,axiom,
    ! [X: set_list_a,N2: nat,H2: set_set_list_a] :
      ( ( pow_se8252319793075206062it_nat @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X @ N2 )
      = ( pow_se8252319793075206062it_nat
        @ ( partia2314006361284536288t_unit
          @ ^ [Uu: set_set_list_a] : H2
          @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
        @ X
        @ N2 ) ) ).

% d.nat_pow_consistent
thf(fact_1098_roots__mem__iff__is__root,axiom,
    ! [P2: list_a,X: a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_a @ X @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) ) )
        = ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).

% roots_mem_iff_is_root
thf(fact_1099_d_Onat__pow__zero,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( pow_se8252319793075206062it_nat @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ N2 )
        = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.nat_pow_zero
thf(fact_1100_var__pow__degree,axiom,
    ! [K: set_a,N2: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N2 ) ) @ one_one_nat )
        = N2 ) ) ).

% var_pow_degree
thf(fact_1101_nat__pow__closed,axiom,
    ! [X: a,N2: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% nat_pow_closed
thf(fact_1102_nat__pow__one,axiom,
    ! [N2: nat] :
      ( ( pow_a_1026414303147256608_b_nat @ r @ ( one_a_ring_ext_a_b @ r ) @ N2 )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% nat_pow_one
thf(fact_1103_alg__multE_I1_J,axiom,
    ! [X: a,P2: list_a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( P2 != nil_a )
         => ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) ) @ P2 ) ) ) ) ).

% alg_multE(1)
thf(fact_1104_le__alg__mult__imp__pdivides,axiom,
    ! [X: a,P2: list_a,N2: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ord_less_eq_nat @ N2 @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
         => ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N2 ) @ P2 ) ) ) ) ).

% le_alg_mult_imp_pdivides
thf(fact_1105_alg__multE_I2_J,axiom,
    ! [X: a,P2: list_a,N2: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( P2 != nil_a )
         => ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N2 ) @ P2 )
           => ( ord_less_eq_nat @ N2 @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) ) ) ) ) ) ).

% alg_multE(2)
thf(fact_1106_p_Opdivides__imp__roots__incl,axiom,
    ! [P2: list_list_a,Q: list_list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
       => ( ( Q != nil_list_a )
         => ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ Q )
           => ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ Q ) ) ) ) ) ) ).

% p.pdivides_imp_roots_incl
thf(fact_1107_not__empty__rootsE,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ( polynomial_roots_a_b @ r @ P2 )
         != zero_zero_multiset_a )
       => ~ ! [A4: a] :
              ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
             => ( ( member_a @ A4 @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) ) )
               => ( ( member_list_a @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A4 ) @ nil_a ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
                 => ~ ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A4 ) @ nil_a ) ) @ P2 ) ) ) ) ) ) ).

% not_empty_rootsE
thf(fact_1108_h_Ohom__nat__pow,axiom,
    ! [X: a,N2: nat] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r )
          @ ( pow_a_1026414303147256608_b_nat
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ X
            @ N2 ) )
        = ( pow_se8252319793075206062it_nat @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ X ) @ N2 ) ) ) ).

% h.hom_nat_pow
thf(fact_1109_alg__mult__def,axiom,
    ! [P2: list_a,X: a] :
      ( ( ( P2 = nil_a )
       => ( ( polyno4422430861927485590lt_a_b @ r @ P2 @ X )
          = zero_zero_nat ) )
      & ( ( P2 != nil_a )
       => ( ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( polyno4422430861927485590lt_a_b @ r @ P2 @ X )
              = ( order_Greatest_nat
                @ ^ [N3: nat] : ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N3 ) @ P2 ) ) ) )
          & ( ~ ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( polyno4422430861927485590lt_a_b @ r @ P2 @ X )
              = zero_zero_nat ) ) ) ) ) ).

% alg_mult_def
thf(fact_1110_nat__pow__eone,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( pow_a_1026414303147256608_b_nat @ r @ X @ one_one_nat )
        = X ) ) ).

% nat_pow_eone
thf(fact_1111_local_Onat__pow__0,axiom,
    ! [X: a] :
      ( ( pow_a_1026414303147256608_b_nat @ r @ X @ zero_zero_nat )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% local.nat_pow_0
thf(fact_1112_d_Oalg__mult__def,axiom,
    ! [P2: list_set_list_a,X: set_list_a] :
      ( ( ( P2 = nil_set_list_a )
       => ( ( polyno1088517687229135038t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ P2 @ X )
          = zero_zero_nat ) )
      & ( ( P2 != nil_set_list_a )
       => ( ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
           => ( ( polyno1088517687229135038t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ P2 @ X )
              = ( order_Greatest_nat
                @ ^ [N3: nat] : ( polyno9075941895896075626t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X ) @ nil_set_list_a ) ) @ N3 ) @ P2 ) ) ) )
          & ( ~ ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
           => ( ( polyno1088517687229135038t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ P2 @ X )
              = zero_zero_nat ) ) ) ) ) ).

% d.alg_mult_def
thf(fact_1113_h_Oalg__mult__def,axiom,
    ! [P2: list_a,X: a] :
      ( ( ( P2 = nil_a )
       => ( ( polyno4422430861927485590lt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ P2
            @ X )
          = zero_zero_nat ) )
      & ( ( P2 != nil_a )
       => ( ( ( member_a @ X
              @ ( partia707051561876973205xt_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r ) ) )
           => ( ( polyno4422430861927485590lt_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ P2
                @ X )
              = ( order_Greatest_nat
                @ ^ [N3: nat] :
                    ( polyno5814909790663948098es_a_b
                    @ ( partia8674076737563717228xt_a_b
                      @ ^ [Uu: set_a] : k
                      @ r )
                    @ ( pow_li1142815632869257134it_nat
                      @ ( univ_poly_a_b
                        @ ( partia8674076737563717228xt_a_b
                          @ ^ [Uu: set_a] : k
                          @ r )
                        @ ( partia707051561876973205xt_a_b
                          @ ( partia8674076737563717228xt_a_b
                            @ ^ [Uu: set_a] : k
                            @ r ) ) )
                      @ ( cons_a
                        @ ( one_a_ring_ext_a_b
                          @ ( partia8674076737563717228xt_a_b
                            @ ^ [Uu: set_a] : k
                            @ r ) )
                        @ ( cons_a
                          @ ( a_inv_a_b
                            @ ( partia8674076737563717228xt_a_b
                              @ ^ [Uu: set_a] : k
                              @ r )
                            @ X )
                          @ nil_a ) )
                      @ N3 )
                    @ P2 ) ) ) )
          & ( ~ ( member_a @ X
                @ ( partia707051561876973205xt_a_b
                  @ ( partia8674076737563717228xt_a_b
                    @ ^ [Uu: set_a] : k
                    @ r ) ) )
           => ( ( polyno4422430861927485590lt_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ P2
                @ X )
              = zero_zero_nat ) ) ) ) ) ).

% h.alg_mult_def
thf(fact_1114_p_Onat__pow__closed,axiom,
    ! [X: list_a,N2: nat] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.nat_pow_closed
thf(fact_1115_p_Onat__pow__one,axiom,
    ! [N2: nat] :
      ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N2 )
      = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% p.nat_pow_one
thf(fact_1116_h_Onat__pow__closed,axiom,
    ! [X: a,N2: nat] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( member_a
        @ ( pow_a_1026414303147256608_b_nat
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ X
          @ N2 )
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.nat_pow_closed
thf(fact_1117_h_Onat__pow__one,axiom,
    ! [N2: nat] :
      ( ( pow_a_1026414303147256608_b_nat
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ ( one_a_ring_ext_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
        @ N2 )
      = ( one_a_ring_ext_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) ) ).

% h.nat_pow_one
thf(fact_1118_p_Onat__pow__eone,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ one_one_nat )
        = X ) ) ).

% p.nat_pow_eone
thf(fact_1119_p_Onat__pow__0,axiom,
    ! [X: list_a] :
      ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ zero_zero_nat )
      = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% p.nat_pow_0
thf(fact_1120_h_Onat__pow__eone,axiom,
    ! [X: a] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( pow_a_1026414303147256608_b_nat
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ X
          @ one_one_nat )
        = X ) ) ).

% h.nat_pow_eone
thf(fact_1121_d_Onat__pow__closed,axiom,
    ! [X: set_list_a,N2: nat] :
      ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( member_set_list_a @ ( pow_se8252319793075206062it_nat @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X @ N2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.nat_pow_closed
thf(fact_1122_h_Onat__pow__0,axiom,
    ! [X: a] :
      ( ( pow_a_1026414303147256608_b_nat
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ X
        @ zero_zero_nat )
      = ( one_a_ring_ext_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) ) ).

% h.nat_pow_0
thf(fact_1123_d_Onat__pow__one,axiom,
    ! [N2: nat] :
      ( ( pow_se8252319793075206062it_nat @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ N2 )
      = ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% d.nat_pow_one
thf(fact_1124_d_Onat__pow__eone,axiom,
    ! [X: set_list_a] :
      ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( pow_se8252319793075206062it_nat @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X @ one_one_nat )
        = X ) ) ).

% d.nat_pow_eone
thf(fact_1125_d_Onat__pow__0,axiom,
    ! [X: set_list_a] :
      ( ( pow_se8252319793075206062it_nat @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X @ zero_zero_nat )
      = ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% d.nat_pow_0
thf(fact_1126_GreatestI__nat,axiom,
    ! [P3: nat > $o,K2: nat,B: nat] :
      ( ( P3 @ K2 )
     => ( ! [Y3: nat] :
            ( ( P3 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P3 @ ( order_Greatest_nat @ P3 ) ) ) ) ).

% GreatestI_nat
thf(fact_1127_Greatest__le__nat,axiom,
    ! [P3: nat > $o,K2: nat,B: nat] :
      ( ( P3 @ K2 )
     => ( ! [Y3: nat] :
            ( ( P3 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P3 ) ) ) ) ).

% Greatest_le_nat
thf(fact_1128_GreatestI__ex__nat,axiom,
    ! [P3: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P3 @ X_12 )
     => ( ! [Y3: nat] :
            ( ( P3 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P3 @ ( order_Greatest_nat @ P3 ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_1129_roots__inclI,axiom,
    ! [P2: list_a,Q: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( Q != nil_a )
         => ( ! [A4: a] :
                ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
               => ( ( P2 != nil_a )
                 => ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A4 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ A4 ) ) @ Q ) ) )
           => ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).

% roots_inclI
thf(fact_1130_d_Oconst__term__not__zero,axiom,
    ! [P2: list_set_list_a] :
      ( ( ( const_3308765751713425893t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ P2 )
       != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( P2 != nil_set_list_a ) ) ).

% d.const_term_not_zero
thf(fact_1131_unitary__monom__eq__var__pow,axiom,
    ! [K: set_a,N2: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( monom_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ N2 )
        = ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N2 ) ) ) ).

% unitary_monom_eq_var_pow
thf(fact_1132_pdivides__imp__roots__incl,axiom,
    ! [P2: list_a,Q: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( Q != nil_a )
         => ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
           => ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).

% pdivides_imp_roots_incl
thf(fact_1133_d_Oroots__def,axiom,
    ! [P2: list_set_list_a] :
      ( ( polyno4169377219242390531t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ P2 )
      = ( abs_mu2555677686620187222list_a @ ( polyno1088517687229135038t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ P2 ) ) ) ).

% d.roots_def
thf(fact_1134_d_Oup__one__closed,axiom,
    ( member491565700723299188list_a
    @ ^ [N3: nat] : ( if_set_list_a @ ( N3 = zero_zero_nat ) @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
    @ ( up_set529185716248919906t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% d.up_one_closed
thf(fact_1135_const__term__not__zero,axiom,
    ! [P2: list_a] :
      ( ( ( const_term_a_b @ r @ P2 )
       != ( zero_a_b @ r ) )
     => ( P2 != nil_a ) ) ).

% const_term_not_zero
thf(fact_1136_const__term__simprules__shell_I1_J,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_a @ ( const_term_a_b @ r @ P2 ) @ K ) ) ) ).

% const_term_simprules_shell(1)
thf(fact_1137_p_Oconst__term__not__zero,axiom,
    ! [P2: list_list_a] :
      ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
       != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( P2 != nil_list_a ) ) ).

% p.const_term_not_zero
thf(fact_1138_h_Oconst__term__not__zero,axiom,
    ! [P2: list_a] :
      ( ( ( const_term_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ P2 )
       != ( zero_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( P2 != nil_a ) ) ).

% h.const_term_not_zero
thf(fact_1139_const__term__simprules__shell_I4_J,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) )
          = ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P2 ) ) ) ) ) ).

% const_term_simprules_shell(4)
thf(fact_1140_p_Oconst__term__simprules__shell_I1_J,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) @ K ) ) ) ).

% p.const_term_simprules_shell(1)
thf(fact_1141_d_Oup__a__inv__closed,axiom,
    ! [P2: nat > set_list_a] :
      ( ( member491565700723299188list_a @ P2 @ ( up_set529185716248919906t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( member491565700723299188list_a
        @ ^ [I3: nat] : ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( P2 @ I3 ) )
        @ ( up_set529185716248919906t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% d.up_a_inv_closed
thf(fact_1142_d_Obound__upD,axiom,
    ! [F2: nat > set_list_a] :
      ( ( member491565700723299188list_a @ F2 @ ( up_set529185716248919906t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ? [N: nat] : ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) @ N @ F2 ) ) ).

% d.bound_upD
thf(fact_1143_p_Oconst__term__simprules__shell_I4_J,axiom,
    ! [K: set_list_a,P2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 ) )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) ) ) ) ).

% p.const_term_simprules_shell(4)
thf(fact_1144_d_Oup__minus__closed,axiom,
    ! [P2: nat > set_list_a,Q: nat > set_list_a] :
      ( ( member491565700723299188list_a @ P2 @ ( up_set529185716248919906t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( member491565700723299188list_a @ Q @ ( up_set529185716248919906t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( member491565700723299188list_a
          @ ^ [I3: nat] : ( a_minu2642007939804611572t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( P2 @ I3 ) @ ( Q @ I3 ) )
          @ ( up_set529185716248919906t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ).

% d.up_minus_closed
thf(fact_1145_up__a__inv__closed,axiom,
    ! [P2: nat > a] :
      ( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
     => ( member_nat_a
        @ ^ [I3: nat] : ( a_inv_a_b @ r @ ( P2 @ I3 ) )
        @ ( up_a_b @ r ) ) ) ).

% up_a_inv_closed
thf(fact_1146_bound__upD,axiom,
    ! [F2: nat > a] :
      ( ( member_nat_a @ F2 @ ( up_a_b @ r ) )
     => ? [N: nat] : ( bound_a @ ( zero_a_b @ r ) @ N @ F2 ) ) ).

% bound_upD
thf(fact_1147_roots__def,axiom,
    ! [P2: list_a] :
      ( ( polynomial_roots_a_b @ r @ P2 )
      = ( abs_multiset_a @ ( polyno4422430861927485590lt_a_b @ r @ P2 ) ) ) ).

% roots_def
thf(fact_1148_up__one__closed,axiom,
    ( member_nat_a
    @ ^ [N3: nat] : ( if_a @ ( N3 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ r ) @ ( zero_a_b @ r ) )
    @ ( up_a_b @ r ) ) ).

% up_one_closed
thf(fact_1149_h_Oup__a__inv__closed,axiom,
    ! [P2: nat > a] :
      ( ( member_nat_a @ P2
        @ ( up_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( member_nat_a
        @ ^ [I3: nat] :
            ( a_inv_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ ( P2 @ I3 ) )
        @ ( up_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.up_a_inv_closed
thf(fact_1150_p_Oup__a__inv__closed,axiom,
    ! [P2: nat > list_a] :
      ( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( member_nat_list_a
        @ ^ [I3: nat] : ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( P2 @ I3 ) )
        @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% p.up_a_inv_closed
thf(fact_1151_p_Oup__minus__closed,axiom,
    ! [P2: nat > list_a,Q: nat > list_a] :
      ( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_nat_list_a @ Q @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( member_nat_list_a
          @ ^ [I3: nat] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ ( P2 @ I3 ) @ ( Q @ I3 ) )
          @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% p.up_minus_closed
thf(fact_1152_p_Obound__upD,axiom,
    ! [F2: nat > list_a] :
      ( ( member_nat_list_a @ F2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ? [N: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ N @ F2 ) ) ).

% p.bound_upD
thf(fact_1153_p_Oroots__def,axiom,
    ! [P2: list_list_a] :
      ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
      = ( abs_multiset_list_a @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) ) ).

% p.roots_def
thf(fact_1154_h_Obound__upD,axiom,
    ! [F2: nat > a] :
      ( ( member_nat_a @ F2
        @ ( up_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ? [N: nat] :
          ( bound_a
          @ ( zero_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) )
          @ N
          @ F2 ) ) ).

% h.bound_upD
thf(fact_1155_h_Oroots__def,axiom,
    ! [P2: list_a] :
      ( ( polynomial_roots_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ P2 )
      = ( abs_multiset_a
        @ ( polyno4422430861927485590lt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ P2 ) ) ) ).

% h.roots_def
thf(fact_1156_h_Oup__one__closed,axiom,
    ( member_nat_a
    @ ^ [N3: nat] :
        ( if_a @ ( N3 = zero_zero_nat )
        @ ( one_a_ring_ext_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
        @ ( zero_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
    @ ( up_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) ) ) ).

% h.up_one_closed
thf(fact_1157_p_Oup__one__closed,axiom,
    ( member_nat_list_a
    @ ^ [N3: nat] : ( if_list_a @ ( N3 = zero_zero_nat ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
    @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% p.up_one_closed
thf(fact_1158_d_Ominus__closed,axiom,
    ! [X: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( member_set_list_a @ ( a_minu2642007939804611572t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X @ Y ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ) ).

% d.minus_closed
thf(fact_1159_d_Or__right__minus__eq,axiom,
    ! [A: set_list_a,B: set_list_a] :
      ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
       => ( ( ( a_minu2642007939804611572t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ A @ B )
            = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
          = ( A = B ) ) ) ) ).

% d.r_right_minus_eq
thf(fact_1160_p_Osubring__polynomial__pow__degree,axiom,
    ! [K: set_list_a,P2: list_list_a,N2: nat] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P2 @ N2 ) ) @ one_one_nat )
          = ( times_times_nat @ N2 @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).

% p.subring_polynomial_pow_degree
thf(fact_1161_p_Opolynomial__pow__degree,axiom,
    ! [P2: list_list_a,N2: nat] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P2 @ N2 ) ) @ one_one_nat )
        = ( times_times_nat @ N2 @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ).

% p.polynomial_pow_degree
thf(fact_1162_up__minus__closed,axiom,
    ! [P2: nat > a,Q: nat > a] :
      ( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
     => ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
       => ( member_nat_a
          @ ^ [I3: nat] : ( a_minus_a_b @ r @ ( P2 @ I3 ) @ ( Q @ I3 ) )
          @ ( up_a_b @ r ) ) ) ) ).

% up_minus_closed
thf(fact_1163_nat__pow__pow,axiom,
    ! [X: a,N2: nat,M: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( pow_a_1026414303147256608_b_nat @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N2 ) @ M )
        = ( pow_a_1026414303147256608_b_nat @ r @ X @ ( times_times_nat @ N2 @ M ) ) ) ) ).

% nat_pow_pow
thf(fact_1164_telescopic__base,axiom,
    ! [K: set_a,F: set_a,N2: nat,M: nat,E: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( subfield_a_b @ F @ r )
       => ( ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ F )
         => ( ( embedd2795209813406577254on_a_b @ r @ M @ F @ E )
           => ( embedd2795209813406577254on_a_b @ r @ ( times_times_nat @ N2 @ M ) @ K @ E ) ) ) ) ) ).

% telescopic_base
thf(fact_1165_h_Oup__minus__closed,axiom,
    ! [P2: nat > a,Q: nat > a] :
      ( ( member_nat_a @ P2
        @ ( up_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( member_nat_a @ Q
          @ ( up_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( member_nat_a
          @ ^ [I3: nat] :
              ( a_minus_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( P2 @ I3 )
              @ ( Q @ I3 ) )
          @ ( up_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.up_minus_closed
thf(fact_1166_p_Onat__pow__pow,axiom,
    ! [X: list_a,N2: nat,M: nat] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ N2 ) @ M )
        = ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ k ) @ X @ ( times_times_nat @ N2 @ M ) ) ) ) ).

% p.nat_pow_pow
thf(fact_1167_p_Otelescopic__base,axiom,
    ! [K: set_list_a,F: set_list_a,N2: nat,M: nat,E: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( subfie1779122896746047282t_unit @ F @ ( univ_poly_a_b @ r @ k ) )
       => ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ N2 @ K @ F )
         => ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ F @ E )
           => ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ k ) @ ( times_times_nat @ N2 @ M ) @ K @ E ) ) ) ) ) ).

% p.telescopic_base
thf(fact_1168_h_Onat__pow__pow,axiom,
    ! [X: a,N2: nat,M: nat] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( pow_a_1026414303147256608_b_nat
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ ( pow_a_1026414303147256608_b_nat
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ X
            @ N2 )
          @ M )
        = ( pow_a_1026414303147256608_b_nat
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ X
          @ ( times_times_nat @ N2 @ M ) ) ) ) ).

% h.nat_pow_pow
thf(fact_1169_h_Otelescopic__base,axiom,
    ! [K: set_a,F: set_a,N2: nat,M: nat,E: set_a] :
      ( ( subfield_a_b @ K
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( subfield_a_b @ F
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
       => ( ( embedd2795209813406577254on_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ N2
            @ K
            @ F )
         => ( ( embedd2795209813406577254on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ M
              @ F
              @ E )
           => ( embedd2795209813406577254on_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( times_times_nat @ N2 @ M )
              @ K
              @ E ) ) ) ) ) ).

% h.telescopic_base
thf(fact_1170_d_Onat__pow__pow,axiom,
    ! [X: set_list_a,N2: nat,M: nat] :
      ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
     => ( ( pow_se8252319793075206062it_nat @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( pow_se8252319793075206062it_nat @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X @ N2 ) @ M )
        = ( pow_se8252319793075206062it_nat @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ X @ ( times_times_nat @ N2 @ M ) ) ) ) ).

% d.nat_pow_pow
thf(fact_1171_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_1172_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_1173_mult__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K2 @ M )
        = ( times_times_nat @ K2 @ N2 ) )
      = ( ( M = N2 )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_1174_mult__cancel2,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ K2 )
        = ( times_times_nat @ N2 @ K2 ) )
      = ( ( M = N2 )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_1175_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_1176_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_1177_d_Otelescopic__base,axiom,
    ! [K: set_set_list_a,F: set_set_list_a,N2: nat,M: nat,E: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
     => ( ( subfie4339374749748326226t_unit @ F @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) )
       => ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ N2 @ K @ F )
         => ( ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ M @ F @ E )
           => ( embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( times_times_nat @ N2 @ M ) @ K @ E ) ) ) ) ) ).

% d.telescopic_base
thf(fact_1178_polynomial__pow__degree,axiom,
    ! [P2: list_a,N2: nat] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ N2 ) ) @ one_one_nat )
        = ( times_times_nat @ N2 @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ).

% polynomial_pow_degree
thf(fact_1179_subring__polynomial__pow__degree,axiom,
    ! [K: set_a,P2: list_a,N2: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P2 @ N2 ) ) @ one_one_nat )
          = ( times_times_nat @ N2 @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).

% subring_polynomial_pow_degree
thf(fact_1180_minus__closed,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% minus_closed
thf(fact_1181_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_1182_mult__less__cancel2,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N2 @ K2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_1183_r__right__minus__eq,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( a_minus_a_b @ r @ A @ B )
            = ( zero_a_b @ r ) )
          = ( A = B ) ) ) ) ).

% r_right_minus_eq
thf(fact_1184_mult__le__cancel2,axiom,
    ! [M: nat,K2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N2 @ K2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_1185_h_Ominus__closed,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( member_a @ Y
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( member_a
          @ ( a_minus_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ X
            @ Y )
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.minus_closed
thf(fact_1186_h_Or__right__minus__eq,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( member_a @ B
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( ( ( a_minus_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ A
              @ B )
            = ( zero_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r ) ) )
          = ( A = B ) ) ) ) ).

% h.r_right_minus_eq
thf(fact_1187_mult__less__mono1,axiom,
    ! [I: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ) ).

% mult_less_mono1
thf(fact_1188_mult__less__mono2,axiom,
    ! [I: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J2 ) ) ) ) ).

% mult_less_mono2
thf(fact_1189_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( times_times_nat @ M @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_1190_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_1191_diff__mult__distrib2,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) ) ) ).

% diff_mult_distrib2
thf(fact_1192_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K2 )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N2 @ K2 ) ) ) ).

% diff_mult_distrib
thf(fact_1193_mult__le__mono2,axiom,
    ! [I: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J2 ) ) ) ).

% mult_le_mono2
thf(fact_1194_mult__le__mono1,axiom,
    ! [I: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ).

% mult_le_mono1
thf(fact_1195_mult__le__mono,axiom,
    ! [I: nat,J2: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J2 )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).

% mult_le_mono
thf(fact_1196_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_1197_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_1198_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_1199_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_1200_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_1201_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_1202_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K2 @ M )
        = ( times_times_nat @ K2 @ N2 ) )
      = ( ( K2 = zero_zero_nat )
        | ( M = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_1203_nat__mult__less__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
        = ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_1204_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ( times_times_nat @ K2 @ M )
          = ( times_times_nat @ K2 @ N2 ) )
        = ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_1205_nat__mult__le__cancel1,axiom,
    ! [K2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_1206_h_Ohomeq__imp__rcos,axiom,
    ! [A: a,X: a] :
      ( ( member_a @ A
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( member_a @ X
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( ( ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ X )
            = ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ A ) )
         => ( member_a @ X
            @ ( a_r_coset_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( a_kern2981893169186190118t_unit
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
                @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) )
              @ A ) ) ) ) ) ).

% h.homeq_imp_rcos
thf(fact_1207_a__r__coset__subset__G,axiom,
    ! [H2: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% a_r_coset_subset_G
thf(fact_1208_a__setmult__rcos__assoc,axiom,
    ! [H2: set_a,K: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( set_add_a_b @ r @ H2 @ ( a_r_coset_a_b @ r @ K @ X ) )
            = ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H2 @ K ) @ X ) ) ) ) ) ).

% a_setmult_rcos_assoc
thf(fact_1209_h_Oa__r__coset__subset__G,axiom,
    ! [H2: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( member_a @ X
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( ord_less_eq_set_a
          @ ( a_r_coset_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ H2
            @ X )
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.a_r_coset_subset_G
thf(fact_1210_h_Oa__setmult__rcos__assoc,axiom,
    ! [H2: set_a,K: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( ord_less_eq_set_a @ K
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( ( member_a @ X
            @ ( partia707051561876973205xt_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r ) ) )
         => ( ( set_add_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ H2
              @ ( a_r_coset_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ K
                @ X ) )
            = ( a_r_coset_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( set_add_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ H2
                @ K )
              @ X ) ) ) ) ) ).

% h.a_setmult_rcos_assoc
thf(fact_1211_h_Orcos__imp__homeq,axiom,
    ! [A: a,X: a] :
      ( ( member_a @ A
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( member_a @ X
          @ ( a_r_coset_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ ( a_kern2981893169186190118t_unit
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
              @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) )
            @ A ) )
       => ( ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ X )
          = ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ A ) ) ) ) ).

% h.rcos_imp_homeq
thf(fact_1212_h_Orcos__eq__homeq,axiom,
    ! [A: a] :
      ( ( member_a @ A
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( a_r_coset_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ ( a_kern2981893169186190118t_unit
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
            @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) )
          @ A )
        = ( collect_a
          @ ^ [X5: a] :
              ( ( member_a @ X5
                @ ( partia707051561876973205xt_a_b
                  @ ( partia8674076737563717228xt_a_b
                    @ ^ [Uu: set_a] : k
                    @ r ) ) )
              & ( ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ X5 )
                = ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ A ) ) ) ) ) ) ).

% h.rcos_eq_homeq
thf(fact_1213_a__coset__add__zero,axiom,
    ! [M3: set_a] :
      ( ( ord_less_eq_set_a @ M3 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( a_r_coset_a_b @ r @ M3 @ ( zero_a_b @ r ) )
        = M3 ) ) ).

% a_coset_add_zero
thf(fact_1214_h_Oa__coset__add__zero,axiom,
    ! [M3: set_a] :
      ( ( ord_less_eq_set_a @ M3
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( a_r_coset_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ M3
          @ ( zero_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
        = M3 ) ) ).

% h.a_coset_add_zero
thf(fact_1215_h_Oquot__mem,axiom,
    ! [X4: set_a] :
      ( ( member_set_a @ X4
        @ ( partia5907974310037520643t_unit
          @ ( factRing_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ ( a_kern2981893169186190118t_unit
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
              @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) ) ) )
     => ? [X2: a] :
          ( ( member_a @ X2
            @ ( partia707051561876973205xt_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r ) ) )
          & ( X4
            = ( a_r_coset_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( a_kern2981893169186190118t_unit
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
                @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) )
              @ X2 ) ) ) ) ).

% h.quot_mem
thf(fact_1216_h_Oa__rcosetsI,axiom,
    ! [H2: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( member_a @ X
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( member_set_a
          @ ( a_r_coset_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ H2
            @ X )
          @ ( a_RCOSETS_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ H2 ) ) ) ) ).

% h.a_rcosetsI
thf(fact_1217_carrier__is__subcring,axiom,
    subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% carrier_is_subcring
thf(fact_1218_subcringI_H,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( subcring_a_b @ H2 @ r ) ) ).

% subcringI'
thf(fact_1219_a__rcosetsI,axiom,
    ! [H2: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_set_a @ ( a_r_coset_a_b @ r @ H2 @ X ) @ ( a_RCOSETS_a_b @ r @ H2 ) ) ) ) ).

% a_rcosetsI
thf(fact_1220_subcring__iff,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( subcring_a_b @ H2 @ r )
        = ( cring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ r ) ) ) ) ).

% subcring_iff
thf(fact_1221_h_Osubcring__iff,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( subcring_a_b @ H2
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
        = ( cring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.subcring_iff
thf(fact_1222_h_OFactRing__iso,axiom,
    ( ( ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
      = ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
   => ( is_rin4164780639300504926t_unit
      @ ( factRing_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ ( a_kern2981893169186190118t_unit
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
          @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) )
      @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% h.FactRing_iso
thf(fact_1223_h_Othe__elem__surj,axiom,
    ( ( image_1661627690217262859list_a
      @ ^ [X7: set_a] : ( the_elem_set_list_a @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ X7 ) )
      @ ( partia5907974310037520643t_unit
        @ ( factRing_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ ( a_kern2981893169186190118t_unit
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
            @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) ) ) )
    = ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
      @ ( partia707051561876973205xt_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) ) ) ) ).

% h.the_elem_surj
thf(fact_1224_h_Othe__elem__hom,axiom,
    ( member1188894495758700418list_a
    @ ^ [X7: set_a] : ( the_elem_set_list_a @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ X7 ) )
    @ ( ring_h715520217600262354t_unit
      @ ( factRing_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ ( a_kern2981893169186190118t_unit
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
          @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) )
      @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ).

% h.the_elem_hom
thf(fact_1225_h_Othe__elem__inj,axiom,
    ! [X4: set_a,Y4: set_a] :
      ( ( member_set_a @ X4
        @ ( partia5907974310037520643t_unit
          @ ( factRing_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ ( a_kern2981893169186190118t_unit
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
              @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) ) ) )
     => ( ( member_set_a @ Y4
          @ ( partia5907974310037520643t_unit
            @ ( factRing_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( a_kern2981893169186190118t_unit
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
                @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) ) ) )
       => ( ( ( the_elem_set_list_a @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ X4 ) )
            = ( the_elem_set_list_a @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ Y4 ) ) )
         => ( X4 = Y4 ) ) ) ) ).

% h.the_elem_inj
thf(fact_1226_h_Othe__elem__simp,axiom,
    ! [X: a] :
      ( ( member_a @ X
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( the_elem_set_list_a
          @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
            @ ( a_r_coset_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ ( a_kern2981893169186190118t_unit
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
                @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) )
              @ X ) ) )
        = ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) @ X ) ) ) ).

% h.the_elem_simp
thf(fact_1227_h_OFactRing__iso__set__aux,axiom,
    ( member1188894495758700418list_a
    @ ^ [X7: set_a] : ( the_elem_set_list_a @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ X7 ) )
    @ ( ring_i3023096418695644136t_unit
      @ ( factRing_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r )
        @ ( a_kern2981893169186190118t_unit
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
          @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) )
      @ ( partia2314006361284536288t_unit
        @ ^ [Uu: set_set_list_a] :
            ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
            @ ( partia707051561876973205xt_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uv: set_a] : k
                @ r ) ) )
        @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% h.FactRing_iso_set_aux
thf(fact_1228_h_OFactRing__iso__set,axiom,
    ( ( ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) )
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
      = ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) )
   => ( member1188894495758700418list_a
      @ ^ [X7: set_a] : ( the_elem_set_list_a @ ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) @ X7 ) )
      @ ( ring_i3023096418695644136t_unit
        @ ( factRing_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r )
          @ ( a_kern2981893169186190118t_unit
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
            @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) ) )
        @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ) ) ).

% h.FactRing_iso_set
thf(fact_1229_p_Opderiv__const,axiom,
    ! [X: list_list_a,K: set_list_a] :
      ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ X ) @ one_one_nat )
        = zero_zero_nat )
     => ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ k ) @ X )
        = ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ).

% p.pderiv_const
thf(fact_1230_p_Oroots__inclI_H,axiom,
    ! [P2: list_list_a,M: multiset_list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ! [A4: list_a] :
            ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( P2 != nil_list_a )
             => ( ord_less_eq_nat @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ A4 ) @ ( count_list_a @ M @ A4 ) ) ) )
       => ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) @ M ) ) ) ).

% p.roots_inclI'
thf(fact_1231_p_Opderiv__zero,axiom,
    ! [K: set_list_a] :
      ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
      = ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% p.pderiv_zero
thf(fact_1232_p_Opderiv__carr,axiom,
    ! [K: set_list_a,F2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ F2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( member_list_list_a @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ k ) @ F2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ) ).

% p.pderiv_carr
thf(fact_1233_p_Opderiv__var,axiom,
    ! [K: set_list_a] :
      ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ k ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) )
      = ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% p.pderiv_var
thf(fact_1234_p_Opderiv__inv,axiom,
    ! [K: set_list_a,F2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ F2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ F2 ) )
          = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ k ) @ F2 ) ) ) ) ) ).

% p.pderiv_inv
thf(fact_1235_p_Oalg__mult__eq__count__roots,axiom,
    ! [P2: list_list_a] :
      ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 )
        = ( count_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 ) ) ) ) ).

% p.alg_mult_eq_count_roots
thf(fact_1236_alg__mult__eq__count__roots,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( polyno4422430861927485590lt_a_b @ r @ P2 )
        = ( count_a @ ( polynomial_roots_a_b @ r @ P2 ) ) ) ) ).

% alg_mult_eq_count_roots
thf(fact_1237_roots__inclI_H,axiom,
    ! [P2: list_a,M: multiset_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ! [A4: a] :
            ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( P2 != nil_a )
             => ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ A4 ) @ ( count_a @ M @ A4 ) ) ) )
       => ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ M ) ) ) ).

% roots_inclI'
thf(fact_1238_h_Okernel__is__ideal,axiom,
    ( ideal_a_b
    @ ( a_kern2981893169186190118t_unit
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r )
      @ ( polyno5459750281392823787re_a_b @ r @ k @ f )
      @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( poly_of_const_a_b @ r ) ) )
    @ ( partia8674076737563717228xt_a_b
      @ ^ [Uu: set_a] : k
      @ r ) ) ).

% h.kernel_is_ideal
thf(fact_1239_pderiv__zero,axiom,
    ! [K: set_a] :
      ( ( formal4452980811800949548iv_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% pderiv_zero
thf(fact_1240_oneideal,axiom,
    ideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% oneideal
thf(fact_1241_a__rcos__zero,axiom,
    ! [I2: set_a,I: a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( member_a @ I @ I2 )
       => ( ( a_r_coset_a_b @ r @ I2 @ I )
          = I2 ) ) ) ).

% a_rcos_zero
thf(fact_1242_add__ideals,axiom,
    ! [I2: set_a,J4: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ideal_a_b @ J4 @ r )
       => ( ideal_a_b @ ( set_add_a_b @ r @ I2 @ J4 ) @ r ) ) ) ).

% add_ideals
thf(fact_1243_pderiv__carr,axiom,
    ! [K: set_a,F2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ F2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_list_a @ ( formal4452980811800949548iv_a_b @ r @ F2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ).

% pderiv_carr
thf(fact_1244_pderiv__var,axiom,
    ! [K: set_a] :
      ( ( formal4452980811800949548iv_a_b @ r @ ( var_a_b @ r ) )
      = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% pderiv_var
thf(fact_1245_ideal__incl__iff,axiom,
    ! [I2: set_a,J4: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ideal_a_b @ J4 @ r )
       => ( ( ord_less_eq_set_a @ I2 @ J4 )
          = ( J4
            = ( comple2307003609928055243_set_a @ ( image_a_set_a @ ( a_r_coset_a_b @ r @ I2 ) @ J4 ) ) ) ) ) ) ).

% ideal_incl_iff
thf(fact_1246_pderiv__inv,axiom,
    ! [K: set_a,F2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ F2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ F2 ) )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( formal4452980811800949548iv_a_b @ r @ F2 ) ) ) ) ) ).

% pderiv_inv
thf(fact_1247_ideal__is__subalgebra,axiom,
    ! [K: set_a,I2: set_a] :
      ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ideal_a_b @ I2 @ r )
       => ( embedd9027525575939734154ra_a_b @ K @ I2 @ r ) ) ) ).

% ideal_is_subalgebra
thf(fact_1248_quotient__eq__iff__same__a__r__cos,axiom,
    ! [I2: set_a,A: a,B: a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ ( a_minus_a_b @ r @ A @ B ) @ I2 )
            = ( ( a_r_coset_a_b @ r @ I2 @ A )
              = ( a_r_coset_a_b @ r @ I2 @ B ) ) ) ) ) ) ).

% quotient_eq_iff_same_a_r_cos
thf(fact_1249_h_Ooneideal,axiom,
    ( ideal_a_b
    @ ( partia707051561876973205xt_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] : k
        @ r ) )
    @ ( partia8674076737563717228xt_a_b
      @ ^ [Uu: set_a] : k
      @ r ) ) ).

% h.oneideal
thf(fact_1250_quot__quot__hom,axiom,
    ! [I2: set_a,J4: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ideal_a_b @ J4 @ r )
       => ( ( ord_less_eq_set_a @ I2 @ J4 )
         => ( member_set_a_set_a @ ( set_add_a_b @ r @ J4 ) @ ( ring_h4754363464466234712t_unit @ ( factRing_a_b @ r @ I2 ) @ ( factRing_a_b @ r @ J4 ) ) ) ) ) ) ).

% quot_quot_hom
thf(fact_1251_h_Oa__rcos__zero,axiom,
    ! [I2: set_a,I: a] :
      ( ( ideal_a_b @ I2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( member_a @ I @ I2 )
       => ( ( a_r_coset_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ I2
            @ I )
          = I2 ) ) ) ).

% h.a_rcos_zero
thf(fact_1252_h_Oadd__ideals,axiom,
    ! [I2: set_a,J4: set_a] :
      ( ( ideal_a_b @ I2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ideal_a_b @ J4
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
       => ( ideal_a_b
          @ ( set_add_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r )
            @ I2
            @ J4 )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.add_ideals
thf(fact_1253_quot__carr,axiom,
    ! [I2: set_a,Y: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
       => ( ord_less_eq_set_a @ Y @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% quot_carr
thf(fact_1254_canonical__proj__vimage__in__carrier,axiom,
    ! [I2: set_a,J4: set_set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ord_le3724670747650509150_set_a @ J4 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
       => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ J4 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% canonical_proj_vimage_in_carrier
thf(fact_1255_canonical__proj__vimage__mem__iff,axiom,
    ! [I2: set_a,J4: set_set_a,A: a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ord_le3724670747650509150_set_a @ J4 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ A @ ( comple2307003609928055243_set_a @ J4 ) )
            = ( member_set_a @ ( a_r_coset_a_b @ r @ I2 @ A ) @ J4 ) ) ) ) ) ).

% canonical_proj_vimage_mem_iff
thf(fact_1256_pderiv__const,axiom,
    ! [X: list_a,K: set_a] :
      ( ( ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat )
        = zero_zero_nat )
     => ( ( formal4452980811800949548iv_a_b @ r @ X )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).

% pderiv_const
thf(fact_1257_h_Oideal__incl__iff,axiom,
    ! [I2: set_a,J4: set_a] :
      ( ( ideal_a_b @ I2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ideal_a_b @ J4
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
       => ( ( ord_less_eq_set_a @ I2 @ J4 )
          = ( J4
            = ( comple2307003609928055243_set_a
              @ ( image_a_set_a
                @ ( a_r_coset_a_b
                  @ ( partia8674076737563717228xt_a_b
                    @ ^ [Uu: set_a] : k
                    @ r )
                  @ I2 )
                @ J4 ) ) ) ) ) ) ).

% h.ideal_incl_iff
thf(fact_1258_h_Oideal__is__subalgebra,axiom,
    ! [K: set_a,I2: set_a] :
      ( ( ord_less_eq_set_a @ K
        @ ( partia707051561876973205xt_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) )
     => ( ( ideal_a_b @ I2
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
       => ( embedd9027525575939734154ra_a_b @ K @ I2
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) ) ) ) ).

% h.ideal_is_subalgebra
thf(fact_1259_h_Oquot__quot__hom,axiom,
    ! [I2: set_a,J4: set_a] :
      ( ( ideal_a_b @ I2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ideal_a_b @ J4
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : k
            @ r ) )
       => ( ( ord_less_eq_set_a @ I2 @ J4 )
         => ( member_set_a_set_a
            @ ( set_add_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ J4 )
            @ ( ring_h4754363464466234712t_unit
              @ ( factRing_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ I2 )
              @ ( factRing_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ J4 ) ) ) ) ) ) ).

% h.quot_quot_hom
thf(fact_1260_h_Oquotient__eq__iff__same__a__r__cos,axiom,
    ! [I2: set_a,A: a,B: a] :
      ( ( ideal_a_b @ I2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( member_a @ A
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) )
       => ( ( member_a @ B
            @ ( partia707051561876973205xt_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r ) ) )
         => ( ( member_a
              @ ( a_minus_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ A
                @ B )
              @ I2 )
            = ( ( a_r_coset_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ I2
                @ A )
              = ( a_r_coset_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ I2
                @ B ) ) ) ) ) ) ).

% h.quotient_eq_iff_same_a_r_cos
thf(fact_1261_h_Oquot__carr,axiom,
    ! [I2: set_a,Y: set_a] :
      ( ( ideal_a_b @ I2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( member_set_a @ Y
          @ ( partia5907974310037520643t_unit
            @ ( factRing_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ I2 ) ) )
       => ( ord_less_eq_set_a @ Y
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.quot_carr
thf(fact_1262_h_Ocanonical__proj__vimage__in__carrier,axiom,
    ! [I2: set_a,J4: set_set_a] :
      ( ( ideal_a_b @ I2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ord_le3724670747650509150_set_a @ J4
          @ ( partia5907974310037520643t_unit
            @ ( factRing_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ I2 ) ) )
       => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ J4 )
          @ ( partia707051561876973205xt_a_b
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : k
              @ r ) ) ) ) ) ).

% h.canonical_proj_vimage_in_carrier
thf(fact_1263_h_Ocanonical__proj__vimage__mem__iff,axiom,
    ! [I2: set_a,J4: set_set_a,A: a] :
      ( ( ideal_a_b @ I2
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : k
          @ r ) )
     => ( ( ord_le3724670747650509150_set_a @ J4
          @ ( partia5907974310037520643t_unit
            @ ( factRing_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r )
              @ I2 ) ) )
       => ( ( member_a @ A
            @ ( partia707051561876973205xt_a_b
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : k
                @ r ) ) )
         => ( ( member_a @ A @ ( comple2307003609928055243_set_a @ J4 ) )
            = ( member_set_a
              @ ( a_r_coset_a_b
                @ ( partia8674076737563717228xt_a_b
                  @ ^ [Uu: set_a] : k
                  @ r )
                @ I2
                @ A )
              @ J4 ) ) ) ) ) ).

% h.canonical_proj_vimage_mem_iff

% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
    ! [X: a,Y: a] :
      ( ( if_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__a_T,axiom,
    ! [X: a,Y: a] :
      ( ( if_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( if_list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( if_list_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [P3: $o] :
      ( ( P3 = $true )
      | ( P3 = $false ) ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X: set_list_a,Y: set_list_a] :
      ( ( if_set_list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X: set_list_a,Y: set_list_a] :
      ( ( if_set_list_a @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    embedd114012573182435982t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) @ ( minus_minus_nat @ ( size_size_list_a @ f ) @ one_one_nat ) @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ f ) ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ k ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ f ) ) ).

%------------------------------------------------------------------------------