%------------------------------------------------------------------------------
% 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/0006_Formal_Polynomial_Derivatives/prob_00118_003395__18241294_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1507 ( 131 unt; 230 typ;   0 def)
%            Number of atoms       : 5018 (1265 equ;   0 cnn)
%            Maximal formula atoms :   17 (   3 avg)
%            Number of connectives : 21749 ( 262   ~;  46   |; 113   &;17908   @)
%                                         (   0 <=>;3420  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   9 avg)
%            Number of types       :   20 (  19 usr)
%            Number of type conns  :  453 ( 453   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  212 ( 211 usr;  10 con; 0-4 aty)
%            Number of variables   : 3606 (  19   ^;3489   !;  98   ?;3606   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 13:21:54.288
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (211)
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thf(sy_c_Embedded__Algebras_Oring_Oindependent_001tf__a_001tf__b,type,
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thf(sy_c_Embedded__Algebras_Osubalgebra_001tf__a_001tf__b,type,
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thf(sy_c_Formal__Polynomial__Derivatives_Opderiv_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Formal__Polynomial__Derivatives_Opderiv_001tf__a_001tf__b,type,
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thf(sy_c_Ring_Ocring_001tf__a_001tf__b,type,
    cring_a_b: partia2175431115845679010xt_a_b > $o ).

thf(sy_c_Ring_Odomain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    domain7810152921033798211t_unit: partia2956882679547061052t_unit > $o ).

thf(sy_c_Ring_Odomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    domain6553523120543210313t_unit: partia2670972154091845814t_unit > $o ).

thf(sy_c_Ring_Odomain_001tf__a_001tf__b,type,
    domain_a_b: partia2175431115845679010xt_a_b > $o ).

thf(sy_c_Ring_Ofield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    field_6388047844668329575t_unit: partia2670972154091845814t_unit > $o ).

thf(sy_c_Ring_Ofield_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    field_26233345952514695t_unit: partia7496981018696276118t_unit > $o ).

thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
    field_a_b: partia2175431115845679010xt_a_b > $o ).

thf(sy_c_Ring_Oring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_l1939023646219158831t_unit: partia2956882679547061052t_unit > $o ).

thf(sy_c_Ring_Oring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_l6212528067271185461t_unit: partia2670972154091845814t_unit > $o ).

thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
    ring_a_b: partia2175431115845679010xt_a_b > $o ).

thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    add_li174743652000525320t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).

thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).

thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
    add_a_b: partia2175431115845679010xt_a_b > a > a > a ).

thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).

thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).

thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
    zero_a_b: partia2175431115845679010xt_a_b > a ).

thf(sy_c_Ring_Osemiring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    semiri2265994252334843677t_unit: partia2956882679547061052t_unit > $o ).

thf(sy_c_Ring_Osemiring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    semiri2871908745932252451t_unit: partia2670972154091845814t_unit > $o ).

thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
    semiring_a_b: partia2175431115845679010xt_a_b > $o ).

thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).

thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).

thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_list_a > $o ).

thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).

thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
    ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).

thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_r5437400583859147359t_unit: partia2956882679547061052t_unit > list_list_a > $o ).

thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).

thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
    ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).

thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
    collect_list_a: ( list_a > $o ) > set_list_a ).

thf(sy_c_Set_OCollect_001tf__a,type,
    collect_a: ( a > $o ) > set_a ).

thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
    insert_list_a: list_a > set_list_a > set_list_a ).

thf(sy_c_Set_Oinsert_001tf__a,type,
    insert_a: a > set_a > set_a ).

thf(sy_c_Subrings_Osubcring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    subcri8676831449680469861t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).

thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    subdom561091866123308472t_unit: set_list_list_a > partia2956882679547061052t_unit > $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_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_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_Itf__a_J_001t__Product____Type__Ounit,type,
    subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_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_001tf__a_001tf__b,type,
    subring_a_b: set_a > partia2175431115845679010xt_a_b > $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_Itf__a_J_J,type,
    member_list_list_a: list_list_a > set_list_list_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_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 ).

thf(sy_v_g,type,
    g: list_a ).

% Relevant facts (1276)
thf(fact_0_domain__axioms,axiom,
    domain_a_b @ r ).

% domain_axioms
thf(fact_1_assms_I1_J,axiom,
    subring_a_b @ k @ r ).

% assms(1)
thf(fact_2_assms_I3_J,axiom,
    member_list_a @ g @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% assms(3)
thf(fact_3_assms_I2_J,axiom,
    member_list_a @ f @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% assms(2)
thf(fact_4_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_5_local_Osemiring__axioms,axiom,
    semiring_a_b @ r ).

% local.semiring_axioms
thf(fact_6_abelian__monoid__axioms,axiom,
    abelian_monoid_a_b @ r ).

% abelian_monoid_axioms
thf(fact_7_local_Oring__axioms,axiom,
    ring_a_b @ r ).

% local.ring_axioms
thf(fact_8_is__abelian__group,axiom,
    abelian_group_a_b @ r ).

% is_abelian_group
thf(fact_9_associated__sym,axiom,
    ! [A: a,B: a] :
      ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
     => ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).

% associated_sym
thf(fact_10_comm__monoid__axioms,axiom,
    comm_m952295370001973751xt_a_b @ r ).

% comm_monoid_axioms
thf(fact_11_monoid__axioms,axiom,
    monoid8385113658579753027xt_a_b @ r ).

% monoid_axioms
thf(fact_12_is__cring,axiom,
    cring_a_b @ r ).

% is_cring
thf(fact_13_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_14_pderiv__carr,axiom,
    ! [K: set_a,F: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_list_a @ ( formal4452980811800949548iv_a_b @ r @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ).

% pderiv_carr
thf(fact_15_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_16_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_17_domain_Opderiv__carr,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( member_list_list_a @ ( formal6075833236969493044t_unit @ R @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ).

% domain.pderiv_carr
thf(fact_18_domain_Opderiv__carr,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,F: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( member_list_a @ ( formal4452980811800949548iv_a_b @ R @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ).

% domain.pderiv_carr
thf(fact_19_domain_Opderiv__var,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( formal6075833236969493044t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) )
        = ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.pderiv_var
thf(fact_20_domain_Opderiv__var,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( formal4452980811800949548iv_a_b @ R @ ( var_a_b @ R ) )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.pderiv_var
thf(fact_21_domain_Opderiv__zero,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( formal6075833236969493044t_unit @ R @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
        = ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.pderiv_zero
thf(fact_22_domain_Opderiv__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( formal4452980811800949548iv_a_b @ R @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) ) )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.pderiv_zero
thf(fact_23_long__division__add_I1_J,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_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ A @ Q ) @ ( polynomial_pdiv_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).

% long_division_add(1)
thf(fact_24_monoid_Oone__closed,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) ) ) ).

% monoid.one_closed
thf(fact_25_monoid_Oone__closed,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).

% monoid.one_closed
thf(fact_26_monoid_Oone__closed,axiom,
    ! [G: partia2956882679547061052t_unit] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( member_list_list_a @ ( one_li8234411390022467901t_unit @ G ) @ ( partia2464479390973590831t_unit @ G ) ) ) ).

% monoid.one_closed
thf(fact_27_long__division__closed_I1_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( 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 @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% long_division_closed(1)
thf(fact_28_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_29_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_30_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_31_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_32_long__division__add__iff,axiom,
    ! [K: set_a,A: list_a,B: list_a,C: 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 @ C @ ( 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 ) )
                = ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ C ) @ Q )
                  = ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).

% long_division_add_iff
thf(fact_33_long__division__add_I2_J,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 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ A @ Q ) @ ( polynomial_pmod_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).

% long_division_add(2)
thf(fact_34_ring_Ocarrier__polynomial__shell,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ).

% ring.carrier_polynomial_shell
thf(fact_35_ring_Ocarrier__polynomial__shell,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( ring_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).

% ring.carrier_polynomial_shell
thf(fact_36_ring_Ocarrier__polynomial__shell,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).

% ring.carrier_polynomial_shell
thf(fact_37_subdomainI_H,axiom,
    ! [H: set_a] :
      ( ( subring_a_b @ H @ r )
     => ( subdomain_a_b @ H @ r ) ) ).

% subdomainI'
thf(fact_38_semiring_Osemiring__simprules_I11_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
          = X ) ) ) ).

% semiring.semiring_simprules(11)
thf(fact_39_semiring_Osemiring__simprules_I11_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
          = X ) ) ) ).

% semiring.semiring_simprules(11)
thf(fact_40_semiring_Osemiring__simprules_I11_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
          = X ) ) ) ).

% semiring.semiring_simprules(11)
thf(fact_41_semiring_Osemiring__simprules_I6_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(6)
thf(fact_42_semiring_Osemiring__simprules_I6_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(6)
thf(fact_43_semiring_Osemiring__simprules_I6_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(6)
thf(fact_44_abelian__monoidE_I4_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia226231641709521465t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = X ) ) ) ).

% abelian_monoidE(4)
thf(fact_45_abelian__monoidE_I4_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_monoid_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
          = X ) ) ) ).

% abelian_monoidE(4)
thf(fact_46_abelian__monoidE_I4_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = X ) ) ) ).

% abelian_monoidE(4)
thf(fact_47_add_Oinv__comm,axiom,
    ! [X: a,Y: a] :
      ( ( ( add_a_b @ r @ X @ Y )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( add_a_b @ r @ Y @ X )
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.inv_comm
thf(fact_48_add_Ol__inv__ex,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ? [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( add_a_b @ r @ X2 @ X )
            = ( zero_a_b @ r ) ) ) ) ).

% add.l_inv_ex
thf(fact_49_add_Oone__unique,axiom,
    ! [U: a] :
      ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ! [X2: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( add_a_b @ r @ U @ X2 )
              = X2 ) )
       => ( U
          = ( zero_a_b @ r ) ) ) ) ).

% add.one_unique
thf(fact_50_add_Or__inv__ex,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ? [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( add_a_b @ r @ X @ X2 )
            = ( zero_a_b @ r ) ) ) ) ).

% add.r_inv_ex
thf(fact_51_local_Ominus__unique,axiom,
    ! [Y: a,X: a,Y2: a] :
      ( ( ( add_a_b @ r @ Y @ X )
        = ( zero_a_b @ r ) )
     => ( ( ( add_a_b @ r @ X @ Y2 )
          = ( zero_a_b @ r ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
             => ( Y = Y2 ) ) ) ) ) ) ).

% local.minus_unique
thf(fact_52_zero__not__one,axiom,
    ( ( zero_a_b @ r )
   != ( one_a_ring_ext_a_b @ r ) ) ).

% zero_not_one
thf(fact_53_add_Ol__cancel,axiom,
    ! [C: a,A: a,B: a] :
      ( ( ( add_a_b @ r @ C @ A )
        = ( add_a_b @ r @ C @ B ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
           => ( A = B ) ) ) ) ) ).

% add.l_cancel
thf(fact_54_add_Or__cancel,axiom,
    ! [A: a,C: a,B: a] :
      ( ( ( add_a_b @ r @ A @ C )
        = ( add_a_b @ r @ B @ C ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
           => ( A = B ) ) ) ) ) ).

% add.r_cancel
thf(fact_55_a__assoc,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
            = ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).

% a_assoc
thf(fact_56_a__comm,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( add_a_b @ r @ X @ Y )
          = ( add_a_b @ r @ Y @ X ) ) ) ) ).

% a_comm
thf(fact_57_a__lcomm,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
            = ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).

% a_lcomm
thf(fact_58_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_59_subring__props_I7_J,axiom,
    ! [K: set_a,H1: a,H2: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ H1 @ K )
       => ( ( member_a @ H2 @ K )
         => ( member_a @ ( add_a_b @ r @ H1 @ H2 ) @ K ) ) ) ) ).

% subring_props(7)
thf(fact_60_mem__Collect__eq,axiom,
    ! [A: list_a,P2: list_a > $o] :
      ( ( member_list_a @ A @ ( collect_list_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
    ! [A: a,P2: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
    ! [A: list_list_a,P2: list_list_a > $o] :
      ( ( member_list_list_a @ A @ ( collect_list_list_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_63_Collect__mem__eq,axiom,
    ! [A2: set_list_a] :
      ( ( collect_list_a
        @ ^ [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_64_Collect__mem__eq,axiom,
    ! [A2: set_a] :
      ( ( collect_a
        @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
    ! [A2: set_list_list_a] :
      ( ( collect_list_list_a
        @ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_66_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_67_carrier__is__subring,axiom,
    subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% carrier_is_subring
thf(fact_68_assoc__subst,axiom,
    ! [A: a,B: a,F: a > a] :
      ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
     => ( ! [A3: a,B2: a] :
            ( ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
              & ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
              & ( associ5860276527279195403xt_a_b @ r @ A3 @ B2 ) )
           => ( ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) )
              & ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) )
              & ( associ5860276527279195403xt_a_b @ r @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
           => ( associ5860276527279195403xt_a_b @ r @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).

% assoc_subst
thf(fact_69_associated__trans,axiom,
    ! [A: a,B: a,C: a] :
      ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
     => ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
           => ( associ5860276527279195403xt_a_b @ r @ A @ C ) ) ) ) ) ).

% associated_trans
thf(fact_70_univ__poly__is__monoid,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( monoid5589397312508706001t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_is_monoid
thf(fact_71_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_72_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_73_univ__poly__is__abelian__monoid,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_is_abelian_monoid
thf(fact_74_long__division__closed_I2_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( 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 @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% long_division_closed(2)
thf(fact_75_add_Ol__cancel__one,axiom,
    ! [X: a,A: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ X @ A )
            = X )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.l_cancel_one
thf(fact_76_add_Ol__cancel__one_H,axiom,
    ! [X: a,A: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( X
            = ( add_a_b @ r @ X @ A ) )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.l_cancel_one'
thf(fact_77_add_Or__cancel__one,axiom,
    ! [X: a,A: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ A @ X )
            = X )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.r_cancel_one
thf(fact_78_add_Or__cancel__one_H,axiom,
    ! [X: a,A: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( X
            = ( add_a_b @ r @ A @ X ) )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.r_cancel_one'
thf(fact_79_l__zero,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
        = X ) ) ).

% l_zero
thf(fact_80_r__zero,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
        = X ) ) ).

% r_zero
thf(fact_81_one__closed,axiom,
    member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% one_closed
thf(fact_82_local_Oadd_Oright__cancel,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ Y @ X )
              = ( add_a_b @ r @ Z @ X ) )
            = ( Y = Z ) ) ) ) ) ).

% local.add.right_cancel
thf(fact_83_a__closed,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% a_closed
thf(fact_84_zero__closed,axiom,
    member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% zero_closed
thf(fact_85_associated__refl,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).

% associated_refl
thf(fact_86_carrier__polynomial__shell,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% carrier_polynomial_shell
thf(fact_87_ring_Opmod_Ocong,axiom,
    polynomial_pmod_a_b = polynomial_pmod_a_b ).

% ring.pmod.cong
thf(fact_88_ring_Opdiv_Ocong,axiom,
    polynomial_pdiv_a_b = polynomial_pdiv_a_b ).

% ring.pdiv.cong
thf(fact_89_domain_Ouniv__poly__is__monoid,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( monoid5589397312508706001t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.univ_poly_is_monoid
thf(fact_90_domain_Ouniv__poly__is__monoid,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( monoid5729698748631984209t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_monoid
thf(fact_91_domain_Ouniv__poly__is__ring,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.univ_poly_is_ring
thf(fact_92_domain_Ouniv__poly__is__ring,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ring_l1939023646219158831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_ring
thf(fact_93_domain_Ouniv__poly__is__abelian__monoid,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.univ_poly_is_abelian_monoid
thf(fact_94_domain_Ouniv__poly__is__abelian__monoid,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( abelia3641329199688042803t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_abelian_monoid
thf(fact_95_domain_Olong__division__closed_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( 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 @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(2)
thf(fact_96_domain_Olong__division__closed_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( 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 @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(2)
thf(fact_97_cring_Ois__cring,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( cring_a_b @ R )
     => ( cring_a_b @ R ) ) ).

% cring.is_cring
thf(fact_98_cring_Ois__cring,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( cring_3148771470849435808t_unit @ R ) ) ).

% cring.is_cring
thf(fact_99_domain_Olong__division__closed_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( 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 @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(1)
thf(fact_100_domain_Olong__division__closed_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( 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 @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).

% domain.long_division_closed(1)
thf(fact_101_domain_Olong__division__add__iff,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
             => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
               => ( ( ( polyno1727750685288865234t_unit @ R @ A @ Q )
                    = ( polyno1727750685288865234t_unit @ R @ B @ Q ) )
                  = ( ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ C ) @ Q )
                    = ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ) ).

% domain.long_division_add_iff
thf(fact_102_domain_Olong__division__add__iff,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( 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 @ C @ ( 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 ) )
                  = ( ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ C ) @ Q )
                    = ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ) ).

% domain.long_division_add_iff
thf(fact_103_domain_Olong__division__add_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
             => ( ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) @ Q )
                = ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno1727750685288865234t_unit @ R @ A @ Q ) @ ( polyno1727750685288865234t_unit @ R @ B @ Q ) ) ) ) ) ) ) ) ).

% domain.long_division_add(2)
thf(fact_104_domain_Olong__division__add_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( 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 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ B ) @ Q )
                = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pmod_a_b @ R @ A @ Q ) @ ( polynomial_pmod_a_b @ R @ B @ Q ) ) ) ) ) ) ) ) ).

% domain.long_division_add(2)
thf(fact_105_domain_Olong__division__add_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
             => ( ( polyno5893782122288709345t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) @ Q )
                = ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno5893782122288709345t_unit @ R @ A @ Q ) @ ( polyno5893782122288709345t_unit @ R @ B @ Q ) ) ) ) ) ) ) ) ).

% domain.long_division_add(1)
thf(fact_106_domain_Olong__division__add_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( 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_pdiv_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ B ) @ Q )
                = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pdiv_a_b @ R @ A @ Q ) @ ( polynomial_pdiv_a_b @ R @ B @ Q ) ) ) ) ) ) ) ) ).

% domain.long_division_add(1)
thf(fact_107_cring_Oaxioms_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( cring_a_b @ R )
     => ( ring_a_b @ R ) ) ).

% cring.axioms(1)
thf(fact_108_cring_Oaxioms_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ring_l6212528067271185461t_unit @ R ) ) ).

% cring.axioms(1)
thf(fact_109_domain_Oaxioms_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( domain_a_b @ R )
     => ( cring_a_b @ R ) ) ).

% domain.axioms(1)
thf(fact_110_domain_Oaxioms_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( cring_3148771470849435808t_unit @ R ) ) ).

% domain.axioms(1)
thf(fact_111_ring_Ois__abelian__group,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R )
     => ( abelian_group_a_b @ R ) ) ).

% ring.is_abelian_group
thf(fact_112_ring_Ois__abelian__group,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( abelia3891852623213500406t_unit @ R ) ) ).

% ring.is_abelian_group
thf(fact_113_Group_Ocomm__monoid_Oaxioms_I1_J,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( comm_m1219397618491936389t_unit @ G )
     => ( monoid5589397312508706001t_unit @ G ) ) ).

% Group.comm_monoid.axioms(1)
thf(fact_114_Group_Ocomm__monoid_Oaxioms_I1_J,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( comm_m952295370001973751xt_a_b @ G )
     => ( monoid8385113658579753027xt_a_b @ G ) ) ).

% Group.comm_monoid.axioms(1)
thf(fact_115_abelian__group_Oaxioms_I1_J,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( abelian_group_a_b @ G )
     => ( abelian_monoid_a_b @ G ) ) ).

% abelian_group.axioms(1)
thf(fact_116_abelian__group_Oaxioms_I1_J,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( abelia226231641709521465t_unit @ G ) ) ).

% abelian_group.axioms(1)
thf(fact_117_ring_Ois__monoid,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R )
     => ( monoid8385113658579753027xt_a_b @ R ) ) ).

% ring.is_monoid
thf(fact_118_ring_Ois__monoid,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( monoid5589397312508706001t_unit @ R ) ) ).

% ring.is_monoid
thf(fact_119_semiring_Oaxioms_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( semiring_a_b @ R )
     => ( abelian_monoid_a_b @ R ) ) ).

% semiring.axioms(1)
thf(fact_120_semiring_Oaxioms_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( abelia226231641709521465t_unit @ R ) ) ).

% semiring.axioms(1)
thf(fact_121_cring_Oaxioms_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( comm_m1219397618491936389t_unit @ R ) ) ).

% cring.axioms(2)
thf(fact_122_cring_Oaxioms_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( cring_a_b @ R )
     => ( comm_m952295370001973751xt_a_b @ R ) ) ).

% cring.axioms(2)
thf(fact_123_semiring_Oaxioms_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( monoid5589397312508706001t_unit @ R ) ) ).

% semiring.axioms(2)
thf(fact_124_semiring_Oaxioms_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( semiring_a_b @ R )
     => ( monoid8385113658579753027xt_a_b @ R ) ) ).

% semiring.axioms(2)
thf(fact_125_ring_Oring__simprules_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% ring.ring_simprules(2)
thf(fact_126_ring_Oring__simprules_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R )
     => ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% ring.ring_simprules(2)
thf(fact_127_ring_Oring__simprules_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% ring.ring_simprules(2)
thf(fact_128_ring_Oring__simprules_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% ring.ring_simprules(1)
thf(fact_129_ring_Oring__simprules_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% ring.ring_simprules(1)
thf(fact_130_ring_Oring__simprules_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% ring.ring_simprules(1)
thf(fact_131_ring_Oring__simprules_I7_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(7)
thf(fact_132_ring_Oring__simprules_I7_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(7)
thf(fact_133_ring_Oring__simprules_I7_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(7)
thf(fact_134_ring_Oring__simprules_I10_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
            = ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).

% ring.ring_simprules(10)
thf(fact_135_ring_Oring__simprules_I10_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ X @ Y )
            = ( add_a_b @ R @ Y @ X ) ) ) ) ) ).

% ring.ring_simprules(10)
thf(fact_136_ring_Oring__simprules_I10_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ X @ Y )
            = ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).

% ring.ring_simprules(10)
thf(fact_137_ring_Oring__simprules_I22_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
              = ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(22)
thf(fact_138_ring_Oring__simprules_I22_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
              = ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(22)
thf(fact_139_ring_Oring__simprules_I22_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) )
              = ( add_li174743652000525320t_unit @ R @ Y @ ( add_li174743652000525320t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(22)
thf(fact_140_ring_Oring__simprules_I6_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% ring.ring_simprules(6)
thf(fact_141_ring_Oring__simprules_I6_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R )
     => ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% ring.ring_simprules(6)
thf(fact_142_ring_Oring__simprules_I6_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% ring.ring_simprules(6)
thf(fact_143_cring_Ocring__simprules_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% cring.cring_simprules(2)
thf(fact_144_cring_Ocring__simprules_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( cring_a_b @ R )
     => ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% cring.cring_simprules(2)
thf(fact_145_cring_Ocring__simprules_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% cring.cring_simprules(2)
thf(fact_146_cring_Ocring__simprules_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% cring.cring_simprules(1)
thf(fact_147_cring_Ocring__simprules_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% cring.cring_simprules(1)
thf(fact_148_cring_Ocring__simprules_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% cring.cring_simprules(1)
thf(fact_149_cring_Ocring__simprules_I7_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(7)
thf(fact_150_cring_Ocring__simprules_I7_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(7)
thf(fact_151_cring_Ocring__simprules_I7_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(7)
thf(fact_152_cring_Ocring__simprules_I10_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
            = ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).

% cring.cring_simprules(10)
thf(fact_153_cring_Ocring__simprules_I10_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ X @ Y )
            = ( add_a_b @ R @ Y @ X ) ) ) ) ) ).

% cring.cring_simprules(10)
thf(fact_154_cring_Ocring__simprules_I10_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ X @ Y )
            = ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).

% cring.cring_simprules(10)
thf(fact_155_cring_Ocring__simprules_I23_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
              = ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(23)
thf(fact_156_cring_Ocring__simprules_I23_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
              = ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(23)
thf(fact_157_cring_Ocring__simprules_I23_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) )
              = ( add_li174743652000525320t_unit @ R @ Y @ ( add_li174743652000525320t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(23)
thf(fact_158_domain_Ozero__not__one,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( zero_l4142658623432671053t_unit @ R )
       != ( one_li8328186300101108157t_unit @ R ) ) ) ).

% domain.zero_not_one
thf(fact_159_domain_Ozero__not__one,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( domain_a_b @ R )
     => ( ( zero_a_b @ R )
       != ( one_a_ring_ext_a_b @ R ) ) ) ).

% domain.zero_not_one
thf(fact_160_domain_Oone__not__zero,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( one_li8328186300101108157t_unit @ R )
       != ( zero_l4142658623432671053t_unit @ R ) ) ) ).

% domain.one_not_zero
thf(fact_161_domain_Oone__not__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( domain_a_b @ R )
     => ( ( one_a_ring_ext_a_b @ R )
       != ( zero_a_b @ R ) ) ) ).

% domain.one_not_zero
thf(fact_162_abelian__groupE_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( abelia3891852623213500406t_unit @ R )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% abelian_groupE(2)
thf(fact_163_abelian__groupE_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( abelian_group_a_b @ R )
     => ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% abelian_groupE(2)
thf(fact_164_abelian__groupE_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( abelia2778853791629620336t_unit @ R )
     => ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% abelian_groupE(2)
thf(fact_165_abelian__groupE_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia3891852623213500406t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% abelian_groupE(1)
thf(fact_166_abelian__groupE_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_group_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% abelian_groupE(1)
thf(fact_167_abelian__groupE_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% abelian_groupE(1)
thf(fact_168_abelian__groupE_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( abelia3891852623213500406t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_groupE(3)
thf(fact_169_abelian__groupE_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( abelian_group_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_groupE(3)
thf(fact_170_abelian__groupE_I3_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_groupE(3)
thf(fact_171_abelian__groupE_I4_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia3891852623213500406t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
            = ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).

% abelian_groupE(4)
thf(fact_172_abelian__groupE_I4_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_group_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ X @ Y )
            = ( add_a_b @ R @ Y @ X ) ) ) ) ) ).

% abelian_groupE(4)
thf(fact_173_abelian__groupE_I4_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ X @ Y )
            = ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).

% abelian_groupE(4)
thf(fact_174_cring_Ocring__simprules_I6_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% cring.cring_simprules(6)
thf(fact_175_cring_Ocring__simprules_I6_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( cring_a_b @ R )
     => ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% cring.cring_simprules(6)
thf(fact_176_cring_Ocring__simprules_I6_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% cring.cring_simprules(6)
thf(fact_177_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_178_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_179_abelian__monoid_Ozero__closed,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( abelia226231641709521465t_unit @ G )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) ) ) ).

% abelian_monoid.zero_closed
thf(fact_180_abelian__monoid_Ozero__closed,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( abelian_monoid_a_b @ G )
     => ( member_a @ ( zero_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).

% abelian_monoid.zero_closed
thf(fact_181_abelian__monoid_Ozero__closed,axiom,
    ! [G: partia2956882679547061052t_unit] :
      ( ( abelia3641329199688042803t_unit @ G )
     => ( member_list_list_a @ ( zero_l347298301471573063t_unit @ G ) @ ( partia2464479390973590831t_unit @ G ) ) ) ).

% abelian_monoid.zero_closed
thf(fact_182_abelian__monoidE_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( abelia226231641709521465t_unit @ R )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% abelian_monoidE(2)
thf(fact_183_abelian__monoidE_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( abelian_monoid_a_b @ R )
     => ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% abelian_monoidE(2)
thf(fact_184_abelian__monoidE_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( abelia3641329199688042803t_unit @ R )
     => ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% abelian_monoidE(2)
thf(fact_185_abelian__monoid_Oa__closed,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia226231641709521465t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).

% abelian_monoid.a_closed
thf(fact_186_abelian__monoid_Oa__closed,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_monoid_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( member_a @ ( add_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).

% abelian_monoid.a_closed
thf(fact_187_abelian__monoid_Oa__closed,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( member_list_list_a @ ( add_li174743652000525320t_unit @ G @ X @ Y ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ).

% abelian_monoid.a_closed
thf(fact_188_abelian__monoid_Oa__lcomm,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( abelia226231641709521465t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( add_li7652885771158616974t_unit @ G @ X @ ( add_li7652885771158616974t_unit @ G @ Y @ Z ) )
              = ( add_li7652885771158616974t_unit @ G @ Y @ ( add_li7652885771158616974t_unit @ G @ X @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_lcomm
thf(fact_189_abelian__monoid_Oa__lcomm,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( abelian_monoid_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( add_a_b @ G @ X @ ( add_a_b @ G @ Y @ Z ) )
              = ( add_a_b @ G @ Y @ ( add_a_b @ G @ X @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_lcomm
thf(fact_190_abelian__monoid_Oa__lcomm,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( add_li174743652000525320t_unit @ G @ X @ ( add_li174743652000525320t_unit @ G @ Y @ Z ) )
              = ( add_li174743652000525320t_unit @ G @ Y @ ( add_li174743652000525320t_unit @ G @ X @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_lcomm
thf(fact_191_abelian__monoid_Oa__assoc,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( abelia226231641709521465t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( add_li7652885771158616974t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ G @ X @ ( add_li7652885771158616974t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_assoc
thf(fact_192_abelian__monoid_Oa__assoc,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( abelian_monoid_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( add_a_b @ G @ ( add_a_b @ G @ X @ Y ) @ Z )
              = ( add_a_b @ G @ X @ ( add_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_assoc
thf(fact_193_abelian__monoid_Oa__assoc,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( add_li174743652000525320t_unit @ G @ ( add_li174743652000525320t_unit @ G @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ G @ X @ ( add_li174743652000525320t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_assoc
thf(fact_194_abelian__monoid_Oa__comm,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia226231641709521465t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( add_li7652885771158616974t_unit @ G @ X @ Y )
            = ( add_li7652885771158616974t_unit @ G @ Y @ X ) ) ) ) ) ).

% abelian_monoid.a_comm
thf(fact_195_abelian__monoid_Oa__comm,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_monoid_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( add_a_b @ G @ X @ Y )
            = ( add_a_b @ G @ Y @ X ) ) ) ) ) ).

% abelian_monoid.a_comm
thf(fact_196_abelian__monoid_Oa__comm,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( add_li174743652000525320t_unit @ G @ X @ Y )
            = ( add_li174743652000525320t_unit @ G @ Y @ X ) ) ) ) ) ).

% abelian_monoid.a_comm
thf(fact_197_abelian__monoidE_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia226231641709521465t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% abelian_monoidE(1)
thf(fact_198_abelian__monoidE_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_monoid_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% abelian_monoidE(1)
thf(fact_199_abelian__monoidE_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% abelian_monoidE(1)
thf(fact_200_abelian__monoidE_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( abelia226231641709521465t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoidE(3)
thf(fact_201_abelian__monoidE_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( abelian_monoid_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoidE(3)
thf(fact_202_abelian__monoidE_I3_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoidE(3)
thf(fact_203_abelian__monoidE_I5_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia226231641709521465t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
            = ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).

% abelian_monoidE(5)
thf(fact_204_abelian__monoidE_I5_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_monoid_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ X @ Y )
            = ( add_a_b @ R @ Y @ X ) ) ) ) ) ).

% abelian_monoidE(5)
thf(fact_205_abelian__monoidE_I5_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ X @ Y )
            = ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).

% abelian_monoidE(5)
thf(fact_206_semiring_Osemiring__simprules_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% semiring.semiring_simprules(2)
thf(fact_207_semiring_Osemiring__simprules_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( semiring_a_b @ R )
     => ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% semiring.semiring_simprules(2)
thf(fact_208_semiring_Osemiring__simprules_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% semiring.semiring_simprules(2)
thf(fact_209_semiring_Osemiring__simprules_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% semiring.semiring_simprules(1)
thf(fact_210_semiring_Osemiring__simprules_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% semiring.semiring_simprules(1)
thf(fact_211_semiring_Osemiring__simprules_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% semiring.semiring_simprules(1)
thf(fact_212_semiring_Osemiring__simprules_I5_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(5)
thf(fact_213_semiring_Osemiring__simprules_I5_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(5)
thf(fact_214_semiring_Osemiring__simprules_I5_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(5)
thf(fact_215_semiring_Osemiring__simprules_I7_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
            = ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).

% semiring.semiring_simprules(7)
thf(fact_216_semiring_Osemiring__simprules_I7_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ X @ Y )
            = ( add_a_b @ R @ Y @ X ) ) ) ) ) ).

% semiring.semiring_simprules(7)
thf(fact_217_semiring_Osemiring__simprules_I7_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ X @ Y )
            = ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).

% semiring.semiring_simprules(7)
thf(fact_218_semiring_Osemiring__simprules_I12_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
              = ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(12)
thf(fact_219_semiring_Osemiring__simprules_I12_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
              = ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(12)
thf(fact_220_semiring_Osemiring__simprules_I12_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) )
              = ( add_li174743652000525320t_unit @ R @ Y @ ( add_li174743652000525320t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(12)
thf(fact_221_semiring_Osemiring__simprules_I4_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% semiring.semiring_simprules(4)
thf(fact_222_semiring_Osemiring__simprules_I4_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( semiring_a_b @ R )
     => ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% semiring.semiring_simprules(4)
thf(fact_223_semiring_Osemiring__simprules_I4_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% semiring.semiring_simprules(4)
thf(fact_224_cring__def,axiom,
    ( cring_3148771470849435808t_unit
    = ( ^ [R2: partia2670972154091845814t_unit] :
          ( ( ring_l6212528067271185461t_unit @ R2 )
          & ( comm_m1219397618491936389t_unit @ R2 ) ) ) ) ).

% cring_def
thf(fact_225_cring__def,axiom,
    ( cring_a_b
    = ( ^ [R2: partia2175431115845679010xt_a_b] :
          ( ( ring_a_b @ R2 )
          & ( comm_m952295370001973751xt_a_b @ R2 ) ) ) ) ).

% cring_def
thf(fact_226_cring_Ointro,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( comm_m1219397618491936389t_unit @ R )
       => ( cring_3148771470849435808t_unit @ R ) ) ) ).

% cring.intro
thf(fact_227_cring_Ointro,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R )
     => ( ( comm_m952295370001973751xt_a_b @ R )
       => ( cring_a_b @ R ) ) ) ).

% cring.intro
thf(fact_228_ring_Oring__simprules_I8_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = X ) ) ) ).

% ring.ring_simprules(8)
thf(fact_229_ring_Oring__simprules_I8_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
          = X ) ) ) ).

% ring.ring_simprules(8)
thf(fact_230_ring_Oring__simprules_I8_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = X ) ) ) ).

% ring.ring_simprules(8)
thf(fact_231_ring_Oring__simprules_I15_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
          = X ) ) ) ).

% ring.ring_simprules(15)
thf(fact_232_ring_Oring__simprules_I15_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
          = X ) ) ) ).

% ring.ring_simprules(15)
thf(fact_233_ring_Oring__simprules_I15_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
          = X ) ) ) ).

% ring.ring_simprules(15)
thf(fact_234_cring_Ocring__simprules_I8_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = X ) ) ) ).

% cring.cring_simprules(8)
thf(fact_235_cring_Ocring__simprules_I8_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
          = X ) ) ) ).

% cring.cring_simprules(8)
thf(fact_236_cring_Ocring__simprules_I8_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = X ) ) ) ).

% cring.cring_simprules(8)
thf(fact_237_cring_Ocring__simprules_I16_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
          = X ) ) ) ).

% cring.cring_simprules(16)
thf(fact_238_cring_Ocring__simprules_I16_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
          = X ) ) ) ).

% cring.cring_simprules(16)
thf(fact_239_cring_Ocring__simprules_I16_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
          = X ) ) ) ).

% cring.cring_simprules(16)
thf(fact_240_abelian__groupI,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ! [X2: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
         => ! [Y3: list_a] :
              ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
             => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
     => ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
       => ( ! [X2: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
             => ! [Y3: list_a] :
                  ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
                 => ! [Z2: list_a] :
                      ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
                     => ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) @ Z2 )
                        = ( add_li7652885771158616974t_unit @ R @ X2 @ ( add_li7652885771158616974t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
               => ! [Y3: list_a] :
                    ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
                   => ( ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 )
                      = ( add_li7652885771158616974t_unit @ R @ Y3 @ X2 ) ) ) )
           => ( ! [X2: list_a] :
                  ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
                 => ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X2 )
                    = X2 ) )
             => ( ! [X2: list_a] :
                    ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
                   => ? [Xa: list_a] :
                        ( ( member_list_a @ Xa @ ( partia5361259788508890537t_unit @ R ) )
                        & ( ( add_li7652885771158616974t_unit @ R @ Xa @ X2 )
                          = ( zero_l4142658623432671053t_unit @ R ) ) ) )
               => ( abelia3891852623213500406t_unit @ R ) ) ) ) ) ) ) ).

% abelian_groupI
thf(fact_241_abelian__groupI,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
         => ! [Y3: a] :
              ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( member_a @ ( add_a_b @ R @ X2 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
     => ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
             => ! [Y3: a] :
                  ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
                 => ! [Z2: a] :
                      ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
                     => ( ( add_a_b @ R @ ( add_a_b @ R @ X2 @ Y3 ) @ Z2 )
                        = ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
               => ! [Y3: a] :
                    ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
                   => ( ( add_a_b @ R @ X2 @ Y3 )
                      = ( add_a_b @ R @ Y3 @ X2 ) ) ) )
           => ( ! [X2: a] :
                  ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
                 => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X2 )
                    = X2 ) )
             => ( ! [X2: a] :
                    ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
                   => ? [Xa: a] :
                        ( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ R ) )
                        & ( ( add_a_b @ R @ Xa @ X2 )
                          = ( zero_a_b @ R ) ) ) )
               => ( abelian_group_a_b @ R ) ) ) ) ) ) ) ).

% abelian_groupI
thf(fact_242_abelian__groupI,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ! [X2: list_list_a] :
          ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
         => ! [Y3: list_list_a] :
              ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
             => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X2 @ Y3 ) @ ( partia2464479390973590831t_unit @ R ) ) ) )
     => ( ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) )
       => ( ! [X2: list_list_a] :
              ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
             => ! [Y3: list_list_a] :
                  ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
                 => ! [Z2: list_list_a] :
                      ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
                     => ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X2 @ Y3 ) @ Z2 )
                        = ( add_li174743652000525320t_unit @ R @ X2 @ ( add_li174743652000525320t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: list_list_a] :
                ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
               => ! [Y3: list_list_a] :
                    ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
                   => ( ( add_li174743652000525320t_unit @ R @ X2 @ Y3 )
                      = ( add_li174743652000525320t_unit @ R @ Y3 @ X2 ) ) ) )
           => ( ! [X2: list_list_a] :
                  ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
                 => ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X2 )
                    = X2 ) )
             => ( ! [X2: list_list_a] :
                    ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
                   => ? [Xa: list_list_a] :
                        ( ( member_list_list_a @ Xa @ ( partia2464479390973590831t_unit @ R ) )
                        & ( ( add_li174743652000525320t_unit @ R @ Xa @ X2 )
                          = ( zero_l347298301471573063t_unit @ R ) ) ) )
               => ( abelia2778853791629620336t_unit @ R ) ) ) ) ) ) ) ).

% abelian_groupI
thf(fact_243_abelian__groupE_I5_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia3891852623213500406t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = X ) ) ) ).

% abelian_groupE(5)
thf(fact_244_abelian__groupE_I5_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_group_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
          = X ) ) ) ).

% abelian_groupE(5)
thf(fact_245_abelian__groupE_I5_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = X ) ) ) ).

% abelian_groupE(5)
thf(fact_246_abelian__groupE_I6_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia3891852623213500406t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ? [X2: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
            & ( ( add_li7652885771158616974t_unit @ R @ X2 @ X )
              = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).

% abelian_groupE(6)
thf(fact_247_abelian__groupE_I6_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_group_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ? [X2: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
            & ( ( add_a_b @ R @ X2 @ X )
              = ( zero_a_b @ R ) ) ) ) ) ).

% abelian_groupE(6)
thf(fact_248_abelian__groupE_I6_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ? [X2: list_list_a] :
            ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
            & ( ( add_li174743652000525320t_unit @ R @ X2 @ X )
              = ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).

% abelian_groupE(6)
thf(fact_249_abelian__monoidI,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ! [X2: list_a,Y3: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
           => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
     => ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
       => ( ! [X2: list_a,Y3: list_a,Z2: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
                 => ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) @ Z2 )
                    = ( add_li7652885771158616974t_unit @ R @ X2 @ ( add_li7652885771158616974t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X2 )
                  = X2 ) )
           => ( ! [X2: list_a,Y3: list_a] :
                  ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
                 => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
                   => ( ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 )
                      = ( add_li7652885771158616974t_unit @ R @ Y3 @ X2 ) ) ) )
             => ( abelia226231641709521465t_unit @ R ) ) ) ) ) ) ).

% abelian_monoidI
thf(fact_250_abelian__monoidI,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ! [X2: a,Y3: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
           => ( member_a @ ( add_a_b @ R @ X2 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
     => ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ! [X2: a,Y3: a,Z2: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
                 => ( ( add_a_b @ R @ ( add_a_b @ R @ X2 @ Y3 ) @ Z2 )
                    = ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X2 )
                  = X2 ) )
           => ( ! [X2: a,Y3: a] :
                  ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
                 => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
                   => ( ( add_a_b @ R @ X2 @ Y3 )
                      = ( add_a_b @ R @ Y3 @ X2 ) ) ) )
             => ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).

% abelian_monoidI
thf(fact_251_abelian__monoidI,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ! [X2: list_list_a,Y3: list_list_a] :
          ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
           => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X2 @ Y3 ) @ ( partia2464479390973590831t_unit @ R ) ) ) )
     => ( ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) )
       => ( ! [X2: list_list_a,Y3: list_list_a,Z2: list_list_a] :
              ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
               => ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
                 => ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X2 @ Y3 ) @ Z2 )
                    = ( add_li174743652000525320t_unit @ R @ X2 @ ( add_li174743652000525320t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: list_list_a] :
                ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
               => ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X2 )
                  = X2 ) )
           => ( ! [X2: list_list_a,Y3: list_list_a] :
                  ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
                 => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
                   => ( ( add_li174743652000525320t_unit @ R @ X2 @ Y3 )
                      = ( add_li174743652000525320t_unit @ R @ Y3 @ X2 ) ) ) )
             => ( abelia3641329199688042803t_unit @ R ) ) ) ) ) ) ).

% abelian_monoidI
thf(fact_252_abelian__monoid_Ominus__unique,axiom,
    ! [G: partia2670972154091845814t_unit,Y: list_a,X: list_a,Y2: list_a] :
      ( ( abelia226231641709521465t_unit @ G )
     => ( ( ( add_li7652885771158616974t_unit @ G @ Y @ X )
          = ( zero_l4142658623432671053t_unit @ G ) )
       => ( ( ( add_li7652885771158616974t_unit @ G @ X @ Y2 )
            = ( zero_l4142658623432671053t_unit @ G ) )
         => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
             => ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
               => ( Y = Y2 ) ) ) ) ) ) ) ).

% abelian_monoid.minus_unique
thf(fact_253_abelian__monoid_Ominus__unique,axiom,
    ! [G: partia2175431115845679010xt_a_b,Y: a,X: a,Y2: a] :
      ( ( abelian_monoid_a_b @ G )
     => ( ( ( add_a_b @ G @ Y @ X )
          = ( zero_a_b @ G ) )
       => ( ( ( add_a_b @ G @ X @ Y2 )
            = ( zero_a_b @ G ) )
         => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
             => ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
               => ( Y = Y2 ) ) ) ) ) ) ) ).

% abelian_monoid.minus_unique
thf(fact_254_abelian__monoid_Ominus__unique,axiom,
    ! [G: partia2956882679547061052t_unit,Y: list_list_a,X: list_list_a,Y2: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G )
     => ( ( ( add_li174743652000525320t_unit @ G @ Y @ X )
          = ( zero_l347298301471573063t_unit @ G ) )
       => ( ( ( add_li174743652000525320t_unit @ G @ X @ Y2 )
            = ( zero_l347298301471573063t_unit @ G ) )
         => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
             => ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
               => ( Y = Y2 ) ) ) ) ) ) ) ).

% abelian_monoid.minus_unique
thf(fact_255_abelian__monoid_Or__zero,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia226231641709521465t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( add_li7652885771158616974t_unit @ G @ X @ ( zero_l4142658623432671053t_unit @ G ) )
          = X ) ) ) ).

% abelian_monoid.r_zero
thf(fact_256_abelian__monoid_Or__zero,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_monoid_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( add_a_b @ G @ X @ ( zero_a_b @ G ) )
          = X ) ) ) ).

% abelian_monoid.r_zero
thf(fact_257_abelian__monoid_Or__zero,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( add_li174743652000525320t_unit @ G @ X @ ( zero_l347298301471573063t_unit @ G ) )
          = X ) ) ) ).

% abelian_monoid.r_zero
thf(fact_258_abelian__monoid_Ol__zero,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia226231641709521465t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( add_li7652885771158616974t_unit @ G @ ( zero_l4142658623432671053t_unit @ G ) @ X )
          = X ) ) ) ).

% abelian_monoid.l_zero
thf(fact_259_abelian__monoid_Ol__zero,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_monoid_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( add_a_b @ G @ ( zero_a_b @ G ) @ X )
          = X ) ) ) ).

% abelian_monoid.l_zero
thf(fact_260_abelian__monoid_Ol__zero,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( add_li174743652000525320t_unit @ G @ ( zero_l347298301471573063t_unit @ G ) @ X )
          = X ) ) ) ).

% abelian_monoid.l_zero
thf(fact_261_onepideal,axiom,
    principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% onepideal
thf(fact_262_univ__poly__a__minus__consistent,axiom,
    ! [K: set_a,Q: list_a,P: 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 ) @ P @ Q )
          = ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ).

% univ_poly_a_minus_consistent
thf(fact_263_ring__primeE_I1_J,axiom,
    ! [P: a] :
      ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_ring_prime_a_b @ r @ P )
       => ( P
         != ( zero_a_b @ r ) ) ) ) ).

% ring_primeE(1)
thf(fact_264_pdiv__pmod,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( P
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).

% pdiv_pmod
thf(fact_265_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_266_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_267_const__term__simprules__shell_I3_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
            = ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).

% const_term_simprules_shell(3)
thf(fact_268_ring__irreducibleE_I1_J,axiom,
    ! [R3: a] :
      ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_r999134135267193926le_a_b @ r @ R3 )
       => ( R3
         != ( zero_a_b @ r ) ) ) ) ).

% ring_irreducibleE(1)
thf(fact_269_const__term__simprules__shell_I1_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_a @ ( const_term_a_b @ r @ P ) @ K ) ) ) ).

% const_term_simprules_shell(1)
thf(fact_270_var__pow__closed,axiom,
    ! [K: set_a,N: nat] :
      ( ( subring_a_b @ K @ r )
     => ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).

% var_pow_closed
thf(fact_271_long__division__a__inv_I1_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( 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 ) @ P ) @ Q )
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) ) ) ) ) ).

% long_division_a_inv(1)
thf(fact_272_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_273_univ__poly__is__abelian__group,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( abelia3891852623213500406t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_is_abelian_group
thf(fact_274_const__term__not__zero,axiom,
    ! [P: list_a] :
      ( ( ( const_term_a_b @ r @ P )
       != ( zero_a_b @ r ) )
     => ( P != nil_a ) ) ).

% const_term_not_zero
thf(fact_275_polynomial__pow__not__zero,axiom,
    ! [P: list_a,N: nat] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( P != nil_a )
       => ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N )
         != nil_a ) ) ) ).

% polynomial_pow_not_zero
thf(fact_276_subring__polynomial__pow__not__zero,axiom,
    ! [K: set_a,P: list_a,N: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( P != nil_a )
         => ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N )
           != nil_a ) ) ) ) ).

% subring_polynomial_pow_not_zero
thf(fact_277_univ__poly__a__inv__consistent,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ).

% univ_poly_a_inv_consistent
thf(fact_278_long__division__a__inv_I2_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( 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 ) @ P ) @ Q )
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).

% long_division_a_inv(2)
thf(fact_279_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_280_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_281_monoid_Ol__one,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X )
          = X ) ) ) ).

% monoid.l_one
thf(fact_282_monoid_Ol__one,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X )
          = X ) ) ) ).

% monoid.l_one
thf(fact_283_monoid_Ol__one,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( mult_l4853965630390486993t_unit @ G @ ( one_li8234411390022467901t_unit @ G ) @ X )
          = X ) ) ) ).

% monoid.l_one
thf(fact_284_monoid_Or__one,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( mult_l7073676228092353617t_unit @ G @ X @ ( one_li8328186300101108157t_unit @ G ) )
          = X ) ) ) ).

% monoid.r_one
thf(fact_285_monoid_Or__one,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( mult_a_ring_ext_a_b @ G @ X @ ( one_a_ring_ext_a_b @ G ) )
          = X ) ) ) ).

% monoid.r_one
thf(fact_286_monoid_Or__one,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( mult_l4853965630390486993t_unit @ G @ X @ ( one_li8234411390022467901t_unit @ G ) )
          = X ) ) ) ).

% monoid.r_one
thf(fact_287_ring_Oconst__term_Ocong,axiom,
    const_term_a_b = const_term_a_b ).

% ring.const_term.cong
thf(fact_288_a__minus__def,axiom,
    ( a_minu3984020753470702548t_unit
    = ( ^ [R2: partia2670972154091845814t_unit,X3: list_a,Y4: list_a] : ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( a_inv_8944721093294617173t_unit @ R2 @ Y4 ) ) ) ) ).

% a_minus_def
thf(fact_289_a__minus__def,axiom,
    ( a_minus_a_b
    = ( ^ [R2: partia2175431115845679010xt_a_b,X3: a,Y4: a] : ( add_a_b @ R2 @ X3 @ ( a_inv_a_b @ R2 @ Y4 ) ) ) ) ).

% a_minus_def
thf(fact_290_ring_Ol__minus,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( mult_l7073676228092353617t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y )
            = ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).

% ring.l_minus
thf(fact_291_ring_Ol__minus,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( mult_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y )
            = ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).

% ring.l_minus
thf(fact_292_ring_Ol__minus,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( mult_l4853965630390486993t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ Y )
            = ( a_inv_7033018035630854991t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) ) ) ) ) ) ).

% ring.l_minus
thf(fact_293_ring_Or__minus,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( mult_l7073676228092353617t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) )
            = ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).

% ring.r_minus
thf(fact_294_ring_Or__minus,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( mult_a_ring_ext_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) )
            = ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).

% ring.r_minus
thf(fact_295_ring_Or__minus,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( mult_l4853965630390486993t_unit @ R @ X @ ( a_inv_7033018035630854991t_unit @ R @ Y ) )
            = ( a_inv_7033018035630854991t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) ) ) ) ) ) ).

% ring.r_minus
thf(fact_296_cring_Ocring__simprules_I29_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( mult_l7073676228092353617t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) )
            = ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).

% cring.cring_simprules(29)
thf(fact_297_cring_Ocring__simprules_I29_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( mult_a_ring_ext_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) )
            = ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).

% cring.cring_simprules(29)
thf(fact_298_cring_Ocring__simprules_I29_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( mult_l4853965630390486993t_unit @ R @ X @ ( a_inv_7033018035630854991t_unit @ R @ Y ) )
            = ( a_inv_7033018035630854991t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) ) ) ) ) ) ).

% cring.cring_simprules(29)
thf(fact_299_cring_Ocring__simprules_I28_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( mult_l7073676228092353617t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y )
            = ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).

% cring.cring_simprules(28)
thf(fact_300_cring_Ocring__simprules_I28_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( mult_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y )
            = ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).

% cring.cring_simprules(28)
thf(fact_301_cring_Ocring__simprules_I28_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( mult_l4853965630390486993t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ Y )
            = ( a_inv_7033018035630854991t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) ) ) ) ) ) ).

% cring.cring_simprules(28)
thf(fact_302_monoid_Onat__pow__comm,axiom,
    ! [G: partia8223610829204095565t_unit,X: a,N: nat,M: nat] :
      ( ( monoid2746444814143937472t_unit @ G )
     => ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
       => ( ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ ( pow_a_1875594501834816709it_nat @ G @ X @ M ) )
          = ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ M ) @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) ) ) ) ) ).

% monoid.nat_pow_comm
thf(fact_303_monoid_Onat__pow__comm,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,N: nat,M: nat] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( mult_l7073676228092353617t_unit @ G @ ( pow_li1142815632869257134it_nat @ G @ X @ N ) @ ( pow_li1142815632869257134it_nat @ G @ X @ M ) )
          = ( mult_l7073676228092353617t_unit @ G @ ( pow_li1142815632869257134it_nat @ G @ X @ M ) @ ( pow_li1142815632869257134it_nat @ G @ X @ N ) ) ) ) ) ).

% monoid.nat_pow_comm
thf(fact_304_monoid_Onat__pow__comm,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,N: nat,M: nat] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ G @ X @ M ) )
          = ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X @ M ) @ ( pow_a_1026414303147256608_b_nat @ G @ X @ N ) ) ) ) ) ).

% monoid.nat_pow_comm
thf(fact_305_monoid_Onat__pow__comm,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,N: nat,M: nat] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( mult_l4853965630390486993t_unit @ G @ ( pow_li488931774710091566it_nat @ G @ X @ N ) @ ( pow_li488931774710091566it_nat @ G @ X @ M ) )
          = ( mult_l4853965630390486993t_unit @ G @ ( pow_li488931774710091566it_nat @ G @ X @ M ) @ ( pow_li488931774710091566it_nat @ G @ X @ N ) ) ) ) ) ).

% monoid.nat_pow_comm
thf(fact_306_monoid_Opow__mult__distrib,axiom,
    ! [G: partia8223610829204095565t_unit,X: a,Y: a,N: nat] :
      ( ( monoid2746444814143937472t_unit @ G )
     => ( ( ( mult_a_Product_unit @ G @ X @ Y )
          = ( mult_a_Product_unit @ G @ Y @ X ) )
       => ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
         => ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
           => ( ( pow_a_1875594501834816709it_nat @ G @ ( mult_a_Product_unit @ G @ X @ Y ) @ N )
              = ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ ( pow_a_1875594501834816709it_nat @ G @ Y @ N ) ) ) ) ) ) ) ).

% monoid.pow_mult_distrib
thf(fact_307_monoid_Opow__mult__distrib,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,N: nat] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y )
          = ( mult_l7073676228092353617t_unit @ G @ Y @ X ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( pow_li1142815632869257134it_nat @ G @ ( mult_l7073676228092353617t_unit @ G @ X @ Y ) @ N )
              = ( mult_l7073676228092353617t_unit @ G @ ( pow_li1142815632869257134it_nat @ G @ X @ N ) @ ( pow_li1142815632869257134it_nat @ G @ Y @ N ) ) ) ) ) ) ) ).

% monoid.pow_mult_distrib
thf(fact_308_monoid_Opow__mult__distrib,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a,N: nat] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
          = ( mult_a_ring_ext_a_b @ G @ Y @ X ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( pow_a_1026414303147256608_b_nat @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ N )
              = ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ G @ Y @ N ) ) ) ) ) ) ) ).

% monoid.pow_mult_distrib
thf(fact_309_monoid_Opow__mult__distrib,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,N: nat] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( ( mult_l4853965630390486993t_unit @ G @ X @ Y )
          = ( mult_l4853965630390486993t_unit @ G @ Y @ X ) )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( pow_li488931774710091566it_nat @ G @ ( mult_l4853965630390486993t_unit @ G @ X @ Y ) @ N )
              = ( mult_l4853965630390486993t_unit @ G @ ( pow_li488931774710091566it_nat @ G @ X @ N ) @ ( pow_li488931774710091566it_nat @ G @ Y @ N ) ) ) ) ) ) ) ).

% monoid.pow_mult_distrib
thf(fact_310_monoid_Ogroup__commutes__pow,axiom,
    ! [G: partia8223610829204095565t_unit,X: a,Y: a,N: nat] :
      ( ( monoid2746444814143937472t_unit @ G )
     => ( ( ( mult_a_Product_unit @ G @ X @ Y )
          = ( mult_a_Product_unit @ G @ Y @ X ) )
       => ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
         => ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
           => ( ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ Y )
              = ( mult_a_Product_unit @ G @ Y @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) ) ) ) ) ) ) ).

% monoid.group_commutes_pow
thf(fact_311_monoid_Ogroup__commutes__pow,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,N: nat] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y )
          = ( mult_l7073676228092353617t_unit @ G @ Y @ X ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( mult_l7073676228092353617t_unit @ G @ ( pow_li1142815632869257134it_nat @ G @ X @ N ) @ Y )
              = ( mult_l7073676228092353617t_unit @ G @ Y @ ( pow_li1142815632869257134it_nat @ G @ X @ N ) ) ) ) ) ) ) ).

% monoid.group_commutes_pow
thf(fact_312_monoid_Ogroup__commutes__pow,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a,N: nat] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
          = ( mult_a_ring_ext_a_b @ G @ Y @ X ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X @ N ) @ Y )
              = ( mult_a_ring_ext_a_b @ G @ Y @ ( pow_a_1026414303147256608_b_nat @ G @ X @ N ) ) ) ) ) ) ) ).

% monoid.group_commutes_pow
thf(fact_313_monoid_Ogroup__commutes__pow,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,N: nat] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( ( mult_l4853965630390486993t_unit @ G @ X @ Y )
          = ( mult_l4853965630390486993t_unit @ G @ Y @ X ) )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( mult_l4853965630390486993t_unit @ G @ ( pow_li488931774710091566it_nat @ G @ X @ N ) @ Y )
              = ( mult_l4853965630390486993t_unit @ G @ Y @ ( pow_li488931774710091566it_nat @ G @ X @ N ) ) ) ) ) ) ) ).

% monoid.group_commutes_pow
thf(fact_314_comm__monoid_Onat__pow__distrib,axiom,
    ! [G: partia8223610829204095565t_unit,X: a,Y: a,N: nat] :
      ( ( comm_m7681468956318391052t_unit @ G )
     => ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
       => ( ( member_a @ Y @ ( partia6735698275553448452t_unit @ G ) )
         => ( ( pow_a_1875594501834816709it_nat @ G @ ( mult_a_Product_unit @ G @ X @ Y ) @ N )
            = ( mult_a_Product_unit @ G @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ ( pow_a_1875594501834816709it_nat @ G @ Y @ N ) ) ) ) ) ) ).

% comm_monoid.nat_pow_distrib
thf(fact_315_comm__monoid_Onat__pow__distrib,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,N: nat] :
      ( ( comm_m1219397618491936389t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( pow_li1142815632869257134it_nat @ G @ ( mult_l7073676228092353617t_unit @ G @ X @ Y ) @ N )
            = ( mult_l7073676228092353617t_unit @ G @ ( pow_li1142815632869257134it_nat @ G @ X @ N ) @ ( pow_li1142815632869257134it_nat @ G @ Y @ N ) ) ) ) ) ) ).

% comm_monoid.nat_pow_distrib
thf(fact_316_comm__monoid_Onat__pow__distrib,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a,N: nat] :
      ( ( comm_m952295370001973751xt_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( pow_a_1026414303147256608_b_nat @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ N )
            = ( mult_a_ring_ext_a_b @ G @ ( pow_a_1026414303147256608_b_nat @ G @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ G @ Y @ N ) ) ) ) ) ) ).

% comm_monoid.nat_pow_distrib
thf(fact_317_comm__monoid_Onat__pow__distrib,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,N: nat] :
      ( ( comm_m2156883222399968773t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( pow_li488931774710091566it_nat @ G @ ( mult_l4853965630390486993t_unit @ G @ X @ Y ) @ N )
            = ( mult_l4853965630390486993t_unit @ G @ ( pow_li488931774710091566it_nat @ G @ X @ N ) @ ( pow_li488931774710091566it_nat @ G @ Y @ N ) ) ) ) ) ) ).

% comm_monoid.nat_pow_distrib
thf(fact_318_ring_Oring__simprules_I14_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( a_minu3984020753470702548t_unit @ R @ X @ Y )
        = ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ).

% ring.ring_simprules(14)
thf(fact_319_ring_Oring__simprules_I14_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_a_b @ R )
     => ( ( a_minus_a_b @ R @ X @ Y )
        = ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) ) ) ) ).

% ring.ring_simprules(14)
thf(fact_320_ring_Oconst__term__not__zero,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( ( const_6738166269504826821t_unit @ R @ P )
         != ( zero_l4142658623432671053t_unit @ R ) )
       => ( P != nil_list_a ) ) ) ).

% ring.const_term_not_zero
thf(fact_321_ring_Oconst__term__not__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a] :
      ( ( ring_a_b @ R )
     => ( ( ( const_term_a_b @ R @ P )
         != ( zero_a_b @ R ) )
       => ( P != nil_a ) ) ) ).

% ring.const_term_not_zero
thf(fact_322_cring_Ocring__simprules_I15_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( a_minu3984020753470702548t_unit @ R @ X @ Y )
        = ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ).

% cring.cring_simprules(15)
thf(fact_323_cring_Ocring__simprules_I15_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( a_minus_a_b @ R @ X @ Y )
        = ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) ) ) ) ).

% cring.cring_simprules(15)
thf(fact_324_abelian__group_Ominus__eq,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_group_a_b @ G )
     => ( ( a_minus_a_b @ G @ X @ Y )
        = ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ Y ) ) ) ) ).

% abelian_group.minus_eq
thf(fact_325_abelian__group_Ominus__eq,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( ( a_minu3984020753470702548t_unit @ G @ X @ Y )
        = ( add_li7652885771158616974t_unit @ G @ X @ ( a_inv_8944721093294617173t_unit @ G @ Y ) ) ) ) ).

% abelian_group.minus_eq
thf(fact_326_domain_Oconst__term__simprules__shell_I4_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( const_term_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) )
            = ( a_inv_a_b @ R @ ( const_term_a_b @ R @ P ) ) ) ) ) ) ).

% domain.const_term_simprules_shell(4)
thf(fact_327_domain_Oconst__term__simprules__shell_I4_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( const_6738166269504826821t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) )
            = ( a_inv_8944721093294617173t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) ) ) ) ) ) ).

% domain.const_term_simprules_shell(4)
thf(fact_328_domain_Oconst__term__simprules__shell_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( const_term_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q ) )
              = ( mult_a_ring_ext_a_b @ R @ ( const_term_a_b @ R @ P ) @ ( const_term_a_b @ R @ Q ) ) ) ) ) ) ) ).

% domain.const_term_simprules_shell(2)
thf(fact_329_domain_Oconst__term__simprules__shell_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( const_6738166269504826821t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q ) )
              = ( mult_l7073676228092353617t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) @ ( const_6738166269504826821t_unit @ R @ Q ) ) ) ) ) ) ) ).

% domain.const_term_simprules_shell(2)
thf(fact_330_domain_Osquare__eq__one,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( ( mult_l7073676228092353617t_unit @ R @ X @ X )
            = ( one_li8328186300101108157t_unit @ R ) )
         => ( ( X
              = ( one_li8328186300101108157t_unit @ R ) )
            | ( X
              = ( a_inv_8944721093294617173t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) ) ) ) ) ) ) ).

% domain.square_eq_one
thf(fact_331_domain_Osquare__eq__one,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( domain_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( ( mult_a_ring_ext_a_b @ R @ X @ X )
            = ( one_a_ring_ext_a_b @ R ) )
         => ( ( X
              = ( one_a_ring_ext_a_b @ R ) )
            | ( X
              = ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ).

% domain.square_eq_one
thf(fact_332_domain_Osquare__eq__one,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( ( mult_l4853965630390486993t_unit @ R @ X @ X )
            = ( one_li8234411390022467901t_unit @ R ) )
         => ( ( X
              = ( one_li8234411390022467901t_unit @ R ) )
            | ( X
              = ( a_inv_7033018035630854991t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) ) ) ) ) ) ) ).

% domain.square_eq_one
thf(fact_333_ring_Oring__simprules_I20_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( a_inv_8944721093294617173t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
          = X ) ) ) ).

% ring.ring_simprules(20)
thf(fact_334_ring_Oring__simprules_I20_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X ) )
          = X ) ) ) ).

% ring.ring_simprules(20)
thf(fact_335_ring_Oring__simprules_I20_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( a_inv_7033018035630854991t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) )
          = X ) ) ) ).

% ring.ring_simprules(20)
thf(fact_336_ring_Oring__simprules_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).

% ring.ring_simprules(3)
thf(fact_337_ring_Oring__simprules_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( member_a @ ( a_inv_a_b @ R @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).

% ring.ring_simprules(3)
thf(fact_338_ring_Oring__simprules_I3_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( member_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).

% ring.ring_simprules(3)
thf(fact_339_ring_Ominus__zero,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( a_inv_8944721093294617173t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) )
        = ( zero_l4142658623432671053t_unit @ R ) ) ) ).

% ring.minus_zero
thf(fact_340_ring_Ominus__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R )
     => ( ( a_inv_a_b @ R @ ( zero_a_b @ R ) )
        = ( zero_a_b @ R ) ) ) ).

% ring.minus_zero
thf(fact_341_domain_Opolynomial__pow__not__zero,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,N: nat] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( P != nil_list_list_a )
         => ( ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ N )
           != nil_list_list_a ) ) ) ) ).

% domain.polynomial_pow_not_zero
thf(fact_342_domain_Opolynomial__pow__not__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( P != nil_a )
         => ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ N )
           != nil_a ) ) ) ) ).

% domain.polynomial_pow_not_zero
thf(fact_343_domain_Opolynomial__pow__not__zero,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( P != nil_list_a )
         => ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ N )
           != nil_list_a ) ) ) ) ).

% domain.polynomial_pow_not_zero
thf(fact_344_cring_Ocring__simprules_I21_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( a_inv_8944721093294617173t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
          = X ) ) ) ).

% cring.cring_simprules(21)
thf(fact_345_cring_Ocring__simprules_I21_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X ) )
          = X ) ) ) ).

% cring.cring_simprules(21)
thf(fact_346_cring_Ocring__simprules_I21_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( a_inv_7033018035630854991t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) )
          = X ) ) ) ).

% cring.cring_simprules(21)
thf(fact_347_cring_Ocring__simprules_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).

% cring.cring_simprules(3)
thf(fact_348_cring_Ocring__simprules_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( member_a @ ( a_inv_a_b @ R @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).

% cring.cring_simprules(3)
thf(fact_349_cring_Ocring__simprules_I3_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( member_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).

% cring.cring_simprules(3)
thf(fact_350_domain_Osubring__polynomial__pow__not__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,N: nat] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( P != nil_a )
           => ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ P @ N )
             != nil_a ) ) ) ) ) ).

% domain.subring_polynomial_pow_not_zero
thf(fact_351_domain_Osubring__polynomial__pow__not__zero,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,N: nat] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( P != nil_list_a )
           => ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ N )
             != nil_list_a ) ) ) ) ) ).

% domain.subring_polynomial_pow_not_zero
thf(fact_352_cring_Ocring__simprules_I22_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( a_inv_8944721093294617173t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) )
        = ( zero_l4142658623432671053t_unit @ R ) ) ) ).

% cring.cring_simprules(22)
thf(fact_353_cring_Ocring__simprules_I22_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( cring_a_b @ R )
     => ( ( a_inv_a_b @ R @ ( zero_a_b @ R ) )
        = ( zero_a_b @ R ) ) ) ).

% cring.cring_simprules(22)
thf(fact_354_abelian__group_Ominus__minus,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( a_inv_8944721093294617173t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) )
          = X ) ) ) ).

% abelian_group.minus_minus
thf(fact_355_abelian__group_Ominus__minus,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_group_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( a_inv_a_b @ G @ ( a_inv_a_b @ G @ X ) )
          = X ) ) ) ).

% abelian_group.minus_minus
thf(fact_356_abelian__group_Ominus__minus,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( a_inv_7033018035630854991t_unit @ G @ ( a_inv_7033018035630854991t_unit @ G @ X ) )
          = X ) ) ) ).

% abelian_group.minus_minus
thf(fact_357_abelian__group_Oa__inv__closed,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).

% abelian_group.a_inv_closed
thf(fact_358_abelian__group_Oa__inv__closed,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_group_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( member_a @ ( a_inv_a_b @ G @ X ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).

% abelian_group.a_inv_closed
thf(fact_359_abelian__group_Oa__inv__closed,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( member_list_list_a @ ( a_inv_7033018035630854991t_unit @ G @ X ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ).

% abelian_group.a_inv_closed
thf(fact_360_monoid_Onat__pow__closed,axiom,
    ! [G: partia8223610829204095565t_unit,X: a,N: nat] :
      ( ( monoid2746444814143937472t_unit @ G )
     => ( ( member_a @ X @ ( partia6735698275553448452t_unit @ G ) )
       => ( member_a @ ( pow_a_1875594501834816709it_nat @ G @ X @ N ) @ ( partia6735698275553448452t_unit @ G ) ) ) ) ).

% monoid.nat_pow_closed
thf(fact_361_monoid_Onat__pow__closed,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,N: nat] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( member_list_a @ ( pow_li1142815632869257134it_nat @ G @ X @ N ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).

% monoid.nat_pow_closed
thf(fact_362_monoid_Onat__pow__closed,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,N: nat] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( member_a @ ( pow_a_1026414303147256608_b_nat @ G @ X @ N ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).

% monoid.nat_pow_closed
thf(fact_363_monoid_Onat__pow__closed,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,N: nat] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( member_list_list_a @ ( pow_li488931774710091566it_nat @ G @ X @ N ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ).

% monoid.nat_pow_closed
thf(fact_364_monoid_Onat__pow__one,axiom,
    ! [G: partia8223610829204095565t_unit,N: nat] :
      ( ( monoid2746444814143937472t_unit @ G )
     => ( ( pow_a_1875594501834816709it_nat @ G @ ( one_a_Product_unit @ G ) @ N )
        = ( one_a_Product_unit @ G ) ) ) ).

% monoid.nat_pow_one
thf(fact_365_monoid_Onat__pow__one,axiom,
    ! [G: partia2670972154091845814t_unit,N: nat] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( pow_li1142815632869257134it_nat @ G @ ( one_li8328186300101108157t_unit @ G ) @ N )
        = ( one_li8328186300101108157t_unit @ G ) ) ) ).

% monoid.nat_pow_one
thf(fact_366_monoid_Onat__pow__one,axiom,
    ! [G: partia2175431115845679010xt_a_b,N: nat] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( pow_a_1026414303147256608_b_nat @ G @ ( one_a_ring_ext_a_b @ G ) @ N )
        = ( one_a_ring_ext_a_b @ G ) ) ) ).

% monoid.nat_pow_one
thf(fact_367_monoid_Om__assoc,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X @ Y ) @ Z )
              = ( mult_l7073676228092353617t_unit @ G @ X @ ( mult_l7073676228092353617t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).

% monoid.m_assoc
thf(fact_368_monoid_Om__assoc,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ Z )
              = ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).

% monoid.m_assoc
thf(fact_369_monoid_Om__assoc,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( mult_l4853965630390486993t_unit @ G @ ( mult_l4853965630390486993t_unit @ G @ X @ Y ) @ Z )
              = ( mult_l4853965630390486993t_unit @ G @ X @ ( mult_l4853965630390486993t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).

% monoid.m_assoc
thf(fact_370_monoid_Om__closed,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X @ Y ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).

% monoid.m_closed
thf(fact_371_monoid_Om__closed,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).

% monoid.m_closed
thf(fact_372_monoid_Om__closed,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ G @ X @ Y ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ).

% monoid.m_closed
thf(fact_373_ring_Oring__simprules_I11_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z )
              = ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(11)
thf(fact_374_ring_Oring__simprules_I11_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
              = ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(11)
thf(fact_375_ring_Oring__simprules_I11_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ Z )
              = ( mult_l4853965630390486993t_unit @ R @ X @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(11)
thf(fact_376_ring_Oring__simprules_I5_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% ring.ring_simprules(5)
thf(fact_377_ring_Oring__simprules_I5_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% ring.ring_simprules(5)
thf(fact_378_ring_Oring__simprules_I5_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% ring.ring_simprules(5)
thf(fact_379_comm__monoid_Om__ac_I1_J,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( comm_m1219397618491936389t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X @ Y ) @ Z )
              = ( mult_l7073676228092353617t_unit @ G @ X @ ( mult_l7073676228092353617t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).

% comm_monoid.m_ac(1)
thf(fact_380_comm__monoid_Om__ac_I1_J,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( comm_m952295370001973751xt_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ Z )
              = ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) ) ) ) ) ) ) ).

% comm_monoid.m_ac(1)
thf(fact_381_comm__monoid_Om__ac_I1_J,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( comm_m2156883222399968773t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( mult_l4853965630390486993t_unit @ G @ ( mult_l4853965630390486993t_unit @ G @ X @ Y ) @ Z )
              = ( mult_l4853965630390486993t_unit @ G @ X @ ( mult_l4853965630390486993t_unit @ G @ Y @ Z ) ) ) ) ) ) ) ).

% comm_monoid.m_ac(1)
thf(fact_382_comm__monoid_Om__comm,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( comm_m1219397618491936389t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( mult_l7073676228092353617t_unit @ G @ X @ Y )
            = ( mult_l7073676228092353617t_unit @ G @ Y @ X ) ) ) ) ) ).

% comm_monoid.m_comm
thf(fact_383_comm__monoid_Om__comm,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( comm_m952295370001973751xt_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
            = ( mult_a_ring_ext_a_b @ G @ Y @ X ) ) ) ) ) ).

% comm_monoid.m_comm
thf(fact_384_comm__monoid_Om__comm,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( comm_m2156883222399968773t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( mult_l4853965630390486993t_unit @ G @ X @ Y )
            = ( mult_l4853965630390486993t_unit @ G @ Y @ X ) ) ) ) ) ).

% comm_monoid.m_comm
thf(fact_385_comm__monoid_Om__lcomm,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( comm_m1219397618491936389t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( mult_l7073676228092353617t_unit @ G @ X @ ( mult_l7073676228092353617t_unit @ G @ Y @ Z ) )
              = ( mult_l7073676228092353617t_unit @ G @ Y @ ( mult_l7073676228092353617t_unit @ G @ X @ Z ) ) ) ) ) ) ) ).

% comm_monoid.m_lcomm
thf(fact_386_comm__monoid_Om__lcomm,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( comm_m952295370001973751xt_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( mult_a_ring_ext_a_b @ G @ X @ ( mult_a_ring_ext_a_b @ G @ Y @ Z ) )
              = ( mult_a_ring_ext_a_b @ G @ Y @ ( mult_a_ring_ext_a_b @ G @ X @ Z ) ) ) ) ) ) ) ).

% comm_monoid.m_lcomm
thf(fact_387_comm__monoid_Om__lcomm,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( comm_m2156883222399968773t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( mult_l4853965630390486993t_unit @ G @ X @ ( mult_l4853965630390486993t_unit @ G @ Y @ Z ) )
              = ( mult_l4853965630390486993t_unit @ G @ Y @ ( mult_l4853965630390486993t_unit @ G @ X @ Z ) ) ) ) ) ) ) ).

% comm_monoid.m_lcomm
thf(fact_388_cring_Ocring__simprules_I24_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) )
              = ( mult_l7073676228092353617t_unit @ R @ Y @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(24)
thf(fact_389_cring_Ocring__simprules_I24_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) )
              = ( mult_a_ring_ext_a_b @ R @ Y @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(24)
thf(fact_390_cring_Ocring__simprules_I24_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ X @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) )
              = ( mult_l4853965630390486993t_unit @ R @ Y @ ( mult_l4853965630390486993t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(24)
thf(fact_391_cring_Ocring__simprules_I14_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( mult_l7073676228092353617t_unit @ R @ X @ Y )
            = ( mult_l7073676228092353617t_unit @ R @ Y @ X ) ) ) ) ) ).

% cring.cring_simprules(14)
thf(fact_392_cring_Ocring__simprules_I14_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( mult_a_ring_ext_a_b @ R @ X @ Y )
            = ( mult_a_ring_ext_a_b @ R @ Y @ X ) ) ) ) ) ).

% cring.cring_simprules(14)
thf(fact_393_cring_Ocring__simprules_I14_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( mult_l4853965630390486993t_unit @ R @ X @ Y )
            = ( mult_l4853965630390486993t_unit @ R @ Y @ X ) ) ) ) ) ).

% cring.cring_simprules(14)
thf(fact_394_cring_Ocring__simprules_I11_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z )
              = ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(11)
thf(fact_395_cring_Ocring__simprules_I11_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
              = ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(11)
thf(fact_396_cring_Ocring__simprules_I11_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ Z )
              = ( mult_l4853965630390486993t_unit @ R @ X @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(11)
thf(fact_397_cring_Ocring__simprules_I5_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% cring.cring_simprules(5)
thf(fact_398_cring_Ocring__simprules_I5_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% cring.cring_simprules(5)
thf(fact_399_cring_Ocring__simprules_I5_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% cring.cring_simprules(5)
thf(fact_400_semiring_Osemiring__simprules_I8_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z )
              = ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(8)
thf(fact_401_semiring_Osemiring__simprules_I8_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
              = ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(8)
thf(fact_402_semiring_Osemiring__simprules_I8_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ Z )
              = ( mult_l4853965630390486993t_unit @ R @ X @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(8)
thf(fact_403_semiring_Osemiring__simprules_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% semiring.semiring_simprules(3)
thf(fact_404_semiring_Osemiring__simprules_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% semiring.semiring_simprules(3)
thf(fact_405_semiring_Osemiring__simprules_I3_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% semiring.semiring_simprules(3)
thf(fact_406_univ__poly__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) )
      = nil_a ) ).

% univ_poly_zero
thf(fact_407_univ__poly__zero,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
      = nil_list_a ) ).

% univ_poly_zero
thf(fact_408_ring_Oring__simprules_I19_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( a_inv_8944721093294617173t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
            = ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ) ) ).

% ring.ring_simprules(19)
thf(fact_409_ring_Oring__simprules_I19_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( a_inv_a_b @ R @ ( add_a_b @ R @ X @ Y ) )
            = ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).

% ring.ring_simprules(19)
thf(fact_410_ring_Oring__simprules_I19_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( a_inv_7033018035630854991t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
            = ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ ( a_inv_7033018035630854991t_unit @ R @ Y ) ) ) ) ) ) ).

% ring.ring_simprules(19)
thf(fact_411_ring_Oring__simprules_I18_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
            = Y ) ) ) ) ).

% ring.ring_simprules(18)
thf(fact_412_ring_Oring__simprules_I18_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( add_a_b @ R @ X @ Y ) )
            = Y ) ) ) ) ).

% ring.ring_simprules(18)
thf(fact_413_ring_Oring__simprules_I18_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
            = Y ) ) ) ) ).

% ring.ring_simprules(18)
thf(fact_414_ring_Oring__simprules_I17_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y ) )
            = Y ) ) ) ) ).

% ring.ring_simprules(17)
thf(fact_415_ring_Oring__simprules_I17_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ X @ ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y ) )
            = Y ) ) ) ) ).

% ring.ring_simprules(17)
thf(fact_416_ring_Oring__simprules_I17_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ Y ) )
            = Y ) ) ) ) ).

% ring.ring_simprules(17)
thf(fact_417_cring_Ocring__simprules_I20_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( a_inv_8944721093294617173t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
            = ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ) ) ).

% cring.cring_simprules(20)
thf(fact_418_cring_Ocring__simprules_I20_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( a_inv_a_b @ R @ ( add_a_b @ R @ X @ Y ) )
            = ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).

% cring.cring_simprules(20)
thf(fact_419_cring_Ocring__simprules_I20_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( a_inv_7033018035630854991t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
            = ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ ( a_inv_7033018035630854991t_unit @ R @ Y ) ) ) ) ) ) ).

% cring.cring_simprules(20)
thf(fact_420_cring_Ocring__simprules_I19_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
            = Y ) ) ) ) ).

% cring.cring_simprules(19)
thf(fact_421_cring_Ocring__simprules_I19_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( add_a_b @ R @ X @ Y ) )
            = Y ) ) ) ) ).

% cring.cring_simprules(19)
thf(fact_422_cring_Ocring__simprules_I19_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
            = Y ) ) ) ) ).

% cring.cring_simprules(19)
thf(fact_423_cring_Ocring__simprules_I18_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y ) )
            = Y ) ) ) ) ).

% cring.cring_simprules(18)
thf(fact_424_cring_Ocring__simprules_I18_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ X @ ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y ) )
            = Y ) ) ) ) ).

% cring.cring_simprules(18)
thf(fact_425_cring_Ocring__simprules_I18_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ Y ) )
            = Y ) ) ) ) ).

% cring.cring_simprules(18)
thf(fact_426_abelian__group_Or__neg1,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) )
            = Y ) ) ) ) ).

% abelian_group.r_neg1
thf(fact_427_abelian__group_Or__neg1,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_group_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( add_a_b @ G @ X @ Y ) )
            = Y ) ) ) ) ).

% abelian_group.r_neg1
thf(fact_428_abelian__group_Or__neg1,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( add_li174743652000525320t_unit @ G @ ( a_inv_7033018035630854991t_unit @ G @ X ) @ ( add_li174743652000525320t_unit @ G @ X @ Y ) )
            = Y ) ) ) ) ).

% abelian_group.r_neg1
thf(fact_429_abelian__group_Or__neg2,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( add_li7652885771158616974t_unit @ G @ X @ ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ Y ) )
            = Y ) ) ) ) ).

% abelian_group.r_neg2
thf(fact_430_abelian__group_Or__neg2,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_group_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( add_a_b @ G @ X @ ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ Y ) )
            = Y ) ) ) ) ).

% abelian_group.r_neg2
thf(fact_431_abelian__group_Or__neg2,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( add_li174743652000525320t_unit @ G @ X @ ( add_li174743652000525320t_unit @ G @ ( a_inv_7033018035630854991t_unit @ G @ X ) @ Y ) )
            = Y ) ) ) ) ).

% abelian_group.r_neg2
thf(fact_432_abelian__group_Ominus__add,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( a_inv_8944721093294617173t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ X @ Y ) )
            = ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ ( a_inv_8944721093294617173t_unit @ G @ Y ) ) ) ) ) ) ).

% abelian_group.minus_add
thf(fact_433_abelian__group_Ominus__add,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_group_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( a_inv_a_b @ G @ ( add_a_b @ G @ X @ Y ) )
            = ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ ( a_inv_a_b @ G @ Y ) ) ) ) ) ) ).

% abelian_group.minus_add
thf(fact_434_abelian__group_Ominus__add,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( a_inv_7033018035630854991t_unit @ G @ ( add_li174743652000525320t_unit @ G @ X @ Y ) )
            = ( add_li174743652000525320t_unit @ G @ ( a_inv_7033018035630854991t_unit @ G @ X ) @ ( a_inv_7033018035630854991t_unit @ G @ Y ) ) ) ) ) ) ).

% abelian_group.minus_add
thf(fact_435_domain_Ouniv__poly__is__cring,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.univ_poly_is_cring
thf(fact_436_domain_Ouniv__poly__is__cring,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( cring_5991999922451032090t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_cring
thf(fact_437_ring_Oring__simprules_I25_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% ring.ring_simprules(25)
thf(fact_438_ring_Oring__simprules_I25_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
          = ( zero_a_b @ R ) ) ) ) ).

% ring.ring_simprules(25)
thf(fact_439_ring_Oring__simprules_I25_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% ring.ring_simprules(25)
thf(fact_440_ring_Oring__simprules_I24_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% ring.ring_simprules(24)
thf(fact_441_ring_Oring__simprules_I24_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
          = ( zero_a_b @ R ) ) ) ) ).

% ring.ring_simprules(24)
thf(fact_442_ring_Oring__simprules_I24_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% ring.ring_simprules(24)
thf(fact_443_ring_Oring__simprules_I23_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
              = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).

% ring.ring_simprules(23)
thf(fact_444_ring_Oring__simprules_I23_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
              = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).

% ring.ring_simprules(23)
thf(fact_445_ring_Oring__simprules_I23_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ Z @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
              = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ Z @ X ) @ ( mult_l4853965630390486993t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).

% ring.ring_simprules(23)
thf(fact_446_ring_Oring__simprules_I13_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(13)
thf(fact_447_ring_Oring__simprules_I13_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(13)
thf(fact_448_ring_Oring__simprules_I13_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Z ) @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% ring.ring_simprules(13)
thf(fact_449_Group_Omonoid_Ointro,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ! [X2: list_a,Y3: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
           => ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X2 @ Y3 ) @ ( partia5361259788508890537t_unit @ G ) ) ) )
     => ( ! [X2: list_a,Y3: list_a,Z2: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
             => ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
               => ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X2 @ Y3 ) @ Z2 )
                  = ( mult_l7073676228092353617t_unit @ G @ X2 @ ( mult_l7073676228092353617t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
       => ( ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
               => ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X2 )
                  = X2 ) )
           => ( ! [X2: list_a] :
                  ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
                 => ( ( mult_l7073676228092353617t_unit @ G @ X2 @ ( one_li8328186300101108157t_unit @ G ) )
                    = X2 ) )
             => ( monoid5589397312508706001t_unit @ G ) ) ) ) ) ) ).

% Group.monoid.intro
thf(fact_450_Group_Omonoid_Ointro,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ! [X2: a,Y3: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
           => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
     => ( ! [X2: a,Y3: a,Z2: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
             => ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
               => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
                  = ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
       => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
               => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
                  = X2 ) )
           => ( ! [X2: a] :
                  ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
                 => ( ( mult_a_ring_ext_a_b @ G @ X2 @ ( one_a_ring_ext_a_b @ G ) )
                    = X2 ) )
             => ( monoid8385113658579753027xt_a_b @ G ) ) ) ) ) ) ).

% Group.monoid.intro
thf(fact_451_Group_Omonoid_Ointro,axiom,
    ! [G: partia2956882679547061052t_unit] :
      ( ! [X2: list_list_a,Y3: list_list_a] :
          ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G ) )
           => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ G @ X2 @ Y3 ) @ ( partia2464479390973590831t_unit @ G ) ) ) )
     => ( ! [X2: list_list_a,Y3: list_list_a,Z2: list_list_a] :
            ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G ) )
             => ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ G ) )
               => ( ( mult_l4853965630390486993t_unit @ G @ ( mult_l4853965630390486993t_unit @ G @ X2 @ Y3 ) @ Z2 )
                  = ( mult_l4853965630390486993t_unit @ G @ X2 @ ( mult_l4853965630390486993t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
       => ( ( member_list_list_a @ ( one_li8234411390022467901t_unit @ G ) @ ( partia2464479390973590831t_unit @ G ) )
         => ( ! [X2: list_list_a] :
                ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
               => ( ( mult_l4853965630390486993t_unit @ G @ ( one_li8234411390022467901t_unit @ G ) @ X2 )
                  = X2 ) )
           => ( ! [X2: list_list_a] :
                  ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
                 => ( ( mult_l4853965630390486993t_unit @ G @ X2 @ ( one_li8234411390022467901t_unit @ G ) )
                    = X2 ) )
             => ( monoid5729698748631984209t_unit @ G ) ) ) ) ) ) ).

% Group.monoid.intro
thf(fact_452_monoid_Oinv__unique,axiom,
    ! [G: partia2670972154091845814t_unit,Y: list_a,X: list_a,Y2: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( ( mult_l7073676228092353617t_unit @ G @ Y @ X )
          = ( one_li8328186300101108157t_unit @ G ) )
       => ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y2 )
            = ( one_li8328186300101108157t_unit @ G ) )
         => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
             => ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
               => ( Y = Y2 ) ) ) ) ) ) ) ).

% monoid.inv_unique
thf(fact_453_monoid_Oinv__unique,axiom,
    ! [G: partia2175431115845679010xt_a_b,Y: a,X: a,Y2: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
          = ( one_a_ring_ext_a_b @ G ) )
       => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y2 )
            = ( one_a_ring_ext_a_b @ G ) )
         => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
             => ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
               => ( Y = Y2 ) ) ) ) ) ) ) ).

% monoid.inv_unique
thf(fact_454_monoid_Oinv__unique,axiom,
    ! [G: partia2956882679547061052t_unit,Y: list_list_a,X: list_list_a,Y2: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( ( mult_l4853965630390486993t_unit @ G @ Y @ X )
          = ( one_li8234411390022467901t_unit @ G ) )
       => ( ( ( mult_l4853965630390486993t_unit @ G @ X @ Y2 )
            = ( one_li8234411390022467901t_unit @ G ) )
         => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
             => ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
               => ( Y = Y2 ) ) ) ) ) ) ) ).

% monoid.inv_unique
thf(fact_455_monoid_Oone__unique,axiom,
    ! [G: partia2670972154091845814t_unit,U: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ G ) )
       => ( ! [X2: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
             => ( ( mult_l7073676228092353617t_unit @ G @ U @ X2 )
                = X2 ) )
         => ( U
            = ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).

% monoid.one_unique
thf(fact_456_monoid_Oone__unique,axiom,
    ! [G: partia2175431115845679010xt_a_b,U: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
             => ( ( mult_a_ring_ext_a_b @ G @ U @ X2 )
                = X2 ) )
         => ( U
            = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).

% monoid.one_unique
thf(fact_457_monoid_Oone__unique,axiom,
    ! [G: partia2956882679547061052t_unit,U: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ U @ ( partia2464479390973590831t_unit @ G ) )
       => ( ! [X2: list_list_a] :
              ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
             => ( ( mult_l4853965630390486993t_unit @ G @ U @ X2 )
                = X2 ) )
         => ( U
            = ( one_li8234411390022467901t_unit @ G ) ) ) ) ) ).

% monoid.one_unique
thf(fact_458_monoidI,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ! [X2: list_a,Y3: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
           => ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X2 @ Y3 ) @ ( partia5361259788508890537t_unit @ G ) ) ) )
     => ( ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) )
       => ( ! [X2: list_a,Y3: list_a,Z2: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
             => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
               => ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
                 => ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X2 @ Y3 ) @ Z2 )
                    = ( mult_l7073676228092353617t_unit @ G @ X2 @ ( mult_l7073676228092353617t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
               => ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X2 )
                  = X2 ) )
           => ( ! [X2: list_a] :
                  ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
                 => ( ( mult_l7073676228092353617t_unit @ G @ X2 @ ( one_li8328186300101108157t_unit @ G ) )
                    = X2 ) )
             => ( monoid5589397312508706001t_unit @ G ) ) ) ) ) ) ).

% monoidI
thf(fact_459_monoidI,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ! [X2: a,Y3: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
           => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
     => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ! [X2: a,Y3: a,Z2: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
             => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
               => ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
                 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
                    = ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
               => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
                  = X2 ) )
           => ( ! [X2: a] :
                  ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
                 => ( ( mult_a_ring_ext_a_b @ G @ X2 @ ( one_a_ring_ext_a_b @ G ) )
                    = X2 ) )
             => ( monoid8385113658579753027xt_a_b @ G ) ) ) ) ) ) ).

% monoidI
thf(fact_460_monoidI,axiom,
    ! [G: partia2956882679547061052t_unit] :
      ( ! [X2: list_list_a,Y3: list_list_a] :
          ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G ) )
           => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ G @ X2 @ Y3 ) @ ( partia2464479390973590831t_unit @ G ) ) ) )
     => ( ( member_list_list_a @ ( one_li8234411390022467901t_unit @ G ) @ ( partia2464479390973590831t_unit @ G ) )
       => ( ! [X2: list_list_a,Y3: list_list_a,Z2: list_list_a] :
              ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
             => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G ) )
               => ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ G ) )
                 => ( ( mult_l4853965630390486993t_unit @ G @ ( mult_l4853965630390486993t_unit @ G @ X2 @ Y3 ) @ Z2 )
                    = ( mult_l4853965630390486993t_unit @ G @ X2 @ ( mult_l4853965630390486993t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: list_list_a] :
                ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
               => ( ( mult_l4853965630390486993t_unit @ G @ ( one_li8234411390022467901t_unit @ G ) @ X2 )
                  = X2 ) )
           => ( ! [X2: list_list_a] :
                  ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
                 => ( ( mult_l4853965630390486993t_unit @ G @ X2 @ ( one_li8234411390022467901t_unit @ G ) )
                    = X2 ) )
             => ( monoid5729698748631984209t_unit @ G ) ) ) ) ) ) ).

% monoidI
thf(fact_461_Group_Omonoid__def,axiom,
    ( monoid5589397312508706001t_unit
    = ( ^ [G2: partia2670972154091845814t_unit] :
          ( ! [X3: list_a,Y4: list_a] :
              ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G2 ) )
             => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ G2 ) )
               => ( member_list_a @ ( mult_l7073676228092353617t_unit @ G2 @ X3 @ Y4 ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) )
          & ! [X3: list_a,Y4: list_a,Z3: list_a] :
              ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G2 ) )
             => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ G2 ) )
               => ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ G2 ) )
                 => ( ( mult_l7073676228092353617t_unit @ G2 @ ( mult_l7073676228092353617t_unit @ G2 @ X3 @ Y4 ) @ Z3 )
                    = ( mult_l7073676228092353617t_unit @ G2 @ X3 @ ( mult_l7073676228092353617t_unit @ G2 @ Y4 @ Z3 ) ) ) ) ) )
          & ( member_list_a @ ( one_li8328186300101108157t_unit @ G2 ) @ ( partia5361259788508890537t_unit @ G2 ) )
          & ! [X3: list_a] :
              ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G2 ) )
             => ( ( mult_l7073676228092353617t_unit @ G2 @ ( one_li8328186300101108157t_unit @ G2 ) @ X3 )
                = X3 ) )
          & ! [X3: list_a] :
              ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G2 ) )
             => ( ( mult_l7073676228092353617t_unit @ G2 @ X3 @ ( one_li8328186300101108157t_unit @ G2 ) )
                = X3 ) ) ) ) ) ).

% Group.monoid_def
thf(fact_462_Group_Omonoid__def,axiom,
    ( monoid8385113658579753027xt_a_b
    = ( ^ [G2: partia2175431115845679010xt_a_b] :
          ( ! [X3: a,Y4: a] :
              ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
             => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
               => ( member_a @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) )
          & ! [X3: a,Y4: a,Z3: a] :
              ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
             => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
               => ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ G2 ) )
                 => ( ( mult_a_ring_ext_a_b @ G2 @ ( mult_a_ring_ext_a_b @ G2 @ X3 @ Y4 ) @ Z3 )
                    = ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( mult_a_ring_ext_a_b @ G2 @ Y4 @ Z3 ) ) ) ) ) )
          & ( member_a @ ( one_a_ring_ext_a_b @ G2 ) @ ( partia707051561876973205xt_a_b @ G2 ) )
          & ! [X3: a] :
              ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
             => ( ( mult_a_ring_ext_a_b @ G2 @ ( one_a_ring_ext_a_b @ G2 ) @ X3 )
                = X3 ) )
          & ! [X3: a] :
              ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G2 ) )
             => ( ( mult_a_ring_ext_a_b @ G2 @ X3 @ ( one_a_ring_ext_a_b @ G2 ) )
                = X3 ) ) ) ) ) ).

% Group.monoid_def
thf(fact_463_Group_Omonoid__def,axiom,
    ( monoid5729698748631984209t_unit
    = ( ^ [G2: partia2956882679547061052t_unit] :
          ( ! [X3: list_list_a,Y4: list_list_a] :
              ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G2 ) )
             => ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ G2 ) )
               => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ G2 @ X3 @ Y4 ) @ ( partia2464479390973590831t_unit @ G2 ) ) ) )
          & ! [X3: list_list_a,Y4: list_list_a,Z3: list_list_a] :
              ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G2 ) )
             => ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ G2 ) )
               => ( ( member_list_list_a @ Z3 @ ( partia2464479390973590831t_unit @ G2 ) )
                 => ( ( mult_l4853965630390486993t_unit @ G2 @ ( mult_l4853965630390486993t_unit @ G2 @ X3 @ Y4 ) @ Z3 )
                    = ( mult_l4853965630390486993t_unit @ G2 @ X3 @ ( mult_l4853965630390486993t_unit @ G2 @ Y4 @ Z3 ) ) ) ) ) )
          & ( member_list_list_a @ ( one_li8234411390022467901t_unit @ G2 ) @ ( partia2464479390973590831t_unit @ G2 ) )
          & ! [X3: list_list_a] :
              ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G2 ) )
             => ( ( mult_l4853965630390486993t_unit @ G2 @ ( one_li8234411390022467901t_unit @ G2 ) @ X3 )
                = X3 ) )
          & ! [X3: list_list_a] :
              ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G2 ) )
             => ( ( mult_l4853965630390486993t_unit @ G2 @ X3 @ ( one_li8234411390022467901t_unit @ G2 ) )
                = X3 ) ) ) ) ) ).

% Group.monoid_def
thf(fact_464_domain_Ointegral,axiom,
    ! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
          = ( zero_l4142658623432671053t_unit @ R ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( A
                = ( zero_l4142658623432671053t_unit @ R ) )
              | ( B
                = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).

% domain.integral
thf(fact_465_domain_Ointegral,axiom,
    ! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
      ( ( domain_a_b @ R )
     => ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
          = ( zero_a_b @ R ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( A
                = ( zero_a_b @ R ) )
              | ( B
                = ( zero_a_b @ R ) ) ) ) ) ) ) ).

% domain.integral
thf(fact_466_domain_Ointegral,axiom,
    ! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
          = ( zero_l347298301471573063t_unit @ R ) )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( A
                = ( zero_l347298301471573063t_unit @ R ) )
              | ( B
                = ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).

% domain.integral
thf(fact_467_domain_Om__lcancel,axiom,
    ! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( A
         != ( zero_l4142658623432671053t_unit @ R ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
                  = ( mult_l7073676228092353617t_unit @ R @ A @ C ) )
                = ( B = C ) ) ) ) ) ) ) ).

% domain.m_lcancel
thf(fact_468_domain_Om__lcancel,axiom,
    ! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
      ( ( domain_a_b @ R )
     => ( ( A
         != ( zero_a_b @ R ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
                  = ( mult_a_ring_ext_a_b @ R @ A @ C ) )
                = ( B = C ) ) ) ) ) ) ) ).

% domain.m_lcancel
thf(fact_469_domain_Om__lcancel,axiom,
    ! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( A
         != ( zero_l347298301471573063t_unit @ R ) )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
                  = ( mult_l4853965630390486993t_unit @ R @ A @ C ) )
                = ( B = C ) ) ) ) ) ) ) ).

% domain.m_lcancel
thf(fact_470_domain_Om__rcancel,axiom,
    ! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( A
         != ( zero_l4142658623432671053t_unit @ R ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( ( mult_l7073676228092353617t_unit @ R @ B @ A )
                  = ( mult_l7073676228092353617t_unit @ R @ C @ A ) )
                = ( B = C ) ) ) ) ) ) ) ).

% domain.m_rcancel
thf(fact_471_domain_Om__rcancel,axiom,
    ! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
      ( ( domain_a_b @ R )
     => ( ( A
         != ( zero_a_b @ R ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( ( mult_a_ring_ext_a_b @ R @ B @ A )
                  = ( mult_a_ring_ext_a_b @ R @ C @ A ) )
                = ( B = C ) ) ) ) ) ) ) ).

% domain.m_rcancel
thf(fact_472_domain_Om__rcancel,axiom,
    ! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( A
         != ( zero_l347298301471573063t_unit @ R ) )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( ( mult_l4853965630390486993t_unit @ R @ B @ A )
                  = ( mult_l4853965630390486993t_unit @ R @ C @ A ) )
                = ( B = C ) ) ) ) ) ) ) ).

% domain.m_rcancel
thf(fact_473_domain_Ointegral__iff,axiom,
    ! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
              = ( zero_l4142658623432671053t_unit @ R ) )
            = ( ( A
                = ( zero_l4142658623432671053t_unit @ R ) )
              | ( B
                = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).

% domain.integral_iff
thf(fact_474_domain_Ointegral__iff,axiom,
    ! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
      ( ( domain_a_b @ R )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
              = ( zero_a_b @ R ) )
            = ( ( A
                = ( zero_a_b @ R ) )
              | ( B
                = ( zero_a_b @ R ) ) ) ) ) ) ) ).

% domain.integral_iff
thf(fact_475_domain_Ointegral__iff,axiom,
    ! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
              = ( zero_l347298301471573063t_unit @ R ) )
            = ( ( A
                = ( zero_l347298301471573063t_unit @ R ) )
              | ( B
                = ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).

% domain.integral_iff
thf(fact_476_ring_Oring__simprules_I4_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( a_minu3984020753470702548t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% ring.ring_simprules(4)
thf(fact_477_ring_Oring__simprules_I4_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_a_b @ R )
     => ( ( 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 ) ) ) ) ) ).

% ring.ring_simprules(4)
thf(fact_478_ring_Oring__simprules_I4_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( a_minu2241224857956505934t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% ring.ring_simprules(4)
thf(fact_479_comm__monoidI,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ! [X2: list_a,Y3: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
           => ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X2 @ Y3 ) @ ( partia5361259788508890537t_unit @ G ) ) ) )
     => ( ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) )
       => ( ! [X2: list_a,Y3: list_a,Z2: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
             => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
               => ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
                 => ( ( mult_l7073676228092353617t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ X2 @ Y3 ) @ Z2 )
                    = ( mult_l7073676228092353617t_unit @ G @ X2 @ ( mult_l7073676228092353617t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
               => ( ( mult_l7073676228092353617t_unit @ G @ ( one_li8328186300101108157t_unit @ G ) @ X2 )
                  = X2 ) )
           => ( ! [X2: list_a,Y3: list_a] :
                  ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
                 => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
                   => ( ( mult_l7073676228092353617t_unit @ G @ X2 @ Y3 )
                      = ( mult_l7073676228092353617t_unit @ G @ Y3 @ X2 ) ) ) )
             => ( comm_m1219397618491936389t_unit @ G ) ) ) ) ) ) ).

% comm_monoidI
thf(fact_480_comm__monoidI,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ! [X2: a,Y3: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
           => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) )
     => ( ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ! [X2: a,Y3: a,Z2: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
             => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
               => ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
                 => ( ( mult_a_ring_ext_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 ) @ Z2 )
                    = ( mult_a_ring_ext_a_b @ G @ X2 @ ( mult_a_ring_ext_a_b @ G @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
               => ( ( mult_a_ring_ext_a_b @ G @ ( one_a_ring_ext_a_b @ G ) @ X2 )
                  = X2 ) )
           => ( ! [X2: a,Y3: a] :
                  ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
                 => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
                   => ( ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 )
                      = ( mult_a_ring_ext_a_b @ G @ Y3 @ X2 ) ) ) )
             => ( comm_m952295370001973751xt_a_b @ G ) ) ) ) ) ) ).

% comm_monoidI
thf(fact_481_comm__monoidI,axiom,
    ! [G: partia2956882679547061052t_unit] :
      ( ! [X2: list_list_a,Y3: list_list_a] :
          ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G ) )
           => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ G @ X2 @ Y3 ) @ ( partia2464479390973590831t_unit @ G ) ) ) )
     => ( ( member_list_list_a @ ( one_li8234411390022467901t_unit @ G ) @ ( partia2464479390973590831t_unit @ G ) )
       => ( ! [X2: list_list_a,Y3: list_list_a,Z2: list_list_a] :
              ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
             => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G ) )
               => ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ G ) )
                 => ( ( mult_l4853965630390486993t_unit @ G @ ( mult_l4853965630390486993t_unit @ G @ X2 @ Y3 ) @ Z2 )
                    = ( mult_l4853965630390486993t_unit @ G @ X2 @ ( mult_l4853965630390486993t_unit @ G @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: list_list_a] :
                ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
               => ( ( mult_l4853965630390486993t_unit @ G @ ( one_li8234411390022467901t_unit @ G ) @ X2 )
                  = X2 ) )
           => ( ! [X2: list_list_a,Y3: list_list_a] :
                  ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
                 => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G ) )
                   => ( ( mult_l4853965630390486993t_unit @ G @ X2 @ Y3 )
                      = ( mult_l4853965630390486993t_unit @ G @ Y3 @ X2 ) ) ) )
             => ( comm_m2156883222399968773t_unit @ G ) ) ) ) ) ) ).

% comm_monoidI
thf(fact_482_ring_Oring__simprules_I12_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X )
          = X ) ) ) ).

% ring.ring_simprules(12)
thf(fact_483_ring_Oring__simprules_I12_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
          = X ) ) ) ).

% ring.ring_simprules(12)
thf(fact_484_ring_Oring__simprules_I12_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) @ X )
          = X ) ) ) ).

% ring.ring_simprules(12)
thf(fact_485_cring_Ocring__simprules_I27_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% cring.cring_simprules(27)
thf(fact_486_cring_Ocring__simprules_I27_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
          = ( zero_a_b @ R ) ) ) ) ).

% cring.cring_simprules(27)
thf(fact_487_cring_Ocring__simprules_I27_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% cring.cring_simprules(27)
thf(fact_488_cring_Ocring__simprules_I26_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% cring.cring_simprules(26)
thf(fact_489_cring_Ocring__simprules_I26_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
          = ( zero_a_b @ R ) ) ) ) ).

% cring.cring_simprules(26)
thf(fact_490_cring_Ocring__simprules_I26_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% cring.cring_simprules(26)
thf(fact_491_cring_Ocring__simprules_I25_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
              = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).

% cring.cring_simprules(25)
thf(fact_492_cring_Ocring__simprules_I25_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
              = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).

% cring.cring_simprules(25)
thf(fact_493_cring_Ocring__simprules_I25_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ Z @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
              = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ Z @ X ) @ ( mult_l4853965630390486993t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).

% cring.cring_simprules(25)
thf(fact_494_cring_Ocring__simprules_I13_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(13)
thf(fact_495_cring_Ocring__simprules_I13_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(13)
thf(fact_496_cring_Ocring__simprules_I13_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Z ) @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% cring.cring_simprules(13)
thf(fact_497_domain_Ouniv__poly__is__abelian__group,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( abelia3891852623213500406t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.univ_poly_is_abelian_group
thf(fact_498_domain_Ouniv__poly__is__abelian__group,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( abelia2778853791629620336t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_abelian_group
thf(fact_499_monoid_Omonoid__comm__monoidI,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ! [X2: list_a,Y3: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
             => ( ( mult_l7073676228092353617t_unit @ G @ X2 @ Y3 )
                = ( mult_l7073676228092353617t_unit @ G @ Y3 @ X2 ) ) ) )
       => ( comm_m1219397618491936389t_unit @ G ) ) ) ).

% monoid.monoid_comm_monoidI
thf(fact_500_monoid_Omonoid__comm__monoidI,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ! [X2: a,Y3: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
             => ( ( mult_a_ring_ext_a_b @ G @ X2 @ Y3 )
                = ( mult_a_ring_ext_a_b @ G @ Y3 @ X2 ) ) ) )
       => ( comm_m952295370001973751xt_a_b @ G ) ) ) ).

% monoid.monoid_comm_monoidI
thf(fact_501_monoid_Omonoid__comm__monoidI,axiom,
    ! [G: partia2956882679547061052t_unit] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ! [X2: list_list_a,Y3: list_list_a] :
            ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G ) )
             => ( ( mult_l4853965630390486993t_unit @ G @ X2 @ Y3 )
                = ( mult_l4853965630390486993t_unit @ G @ Y3 @ X2 ) ) ) )
       => ( comm_m2156883222399968773t_unit @ G ) ) ) ).

% monoid.monoid_comm_monoidI
thf(fact_502_cring_Ocring__simprules_I12_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X )
          = X ) ) ) ).

% cring.cring_simprules(12)
thf(fact_503_cring_Ocring__simprules_I12_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
          = X ) ) ) ).

% cring.cring_simprules(12)
thf(fact_504_cring_Ocring__simprules_I12_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) @ X )
          = X ) ) ) ).

% cring.cring_simprules(12)
thf(fact_505_cring_Ocring__simprules_I4_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( a_minu3984020753470702548t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% cring.cring_simprules(4)
thf(fact_506_cring_Ocring__simprules_I4_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( 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 ) ) ) ) ) ).

% cring.cring_simprules(4)
thf(fact_507_cring_Ocring__simprules_I4_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( a_minu2241224857956505934t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% cring.cring_simprules(4)
thf(fact_508_abelian__group_Ominus__closed,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( member_list_a @ ( a_minu3984020753470702548t_unit @ G @ X @ Y ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).

% abelian_group.minus_closed
thf(fact_509_abelian__group_Ominus__closed,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_group_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( member_a @ ( a_minus_a_b @ G @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).

% abelian_group.minus_closed
thf(fact_510_abelian__group_Ominus__closed,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( member_list_list_a @ ( a_minu2241224857956505934t_unit @ G @ X @ Y ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ).

% abelian_group.minus_closed
thf(fact_511_semiring_Ol__null,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% semiring.l_null
thf(fact_512_semiring_Ol__null,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
          = ( zero_a_b @ R ) ) ) ) ).

% semiring.l_null
thf(fact_513_semiring_Ol__null,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% semiring.l_null
thf(fact_514_semiring_Or__null,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% semiring.r_null
thf(fact_515_semiring_Or__null,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
          = ( zero_a_b @ R ) ) ) ) ).

% semiring.r_null
thf(fact_516_semiring_Or__null,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% semiring.r_null
thf(fact_517_semiring_Ol__distr,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.l_distr
thf(fact_518_semiring_Ol__distr,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.l_distr
thf(fact_519_semiring_Ol__distr,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Z ) @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.l_distr
thf(fact_520_semiring_Or__distr,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
              = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).

% semiring.r_distr
thf(fact_521_semiring_Or__distr,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
              = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).

% semiring.r_distr
thf(fact_522_semiring_Or__distr,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ Z @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
              = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ Z @ X ) @ ( mult_l4853965630390486993t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).

% semiring.r_distr
thf(fact_523_semiring_Osemiring__simprules_I9_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(9)
thf(fact_524_semiring_Osemiring__simprules_I9_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(9)
thf(fact_525_semiring_Osemiring__simprules_I9_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(9)
thf(fact_526_ring_Oring__simprules_I16_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% ring.ring_simprules(16)
thf(fact_527_ring_Oring__simprules_I16_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ X ) )
          = ( zero_a_b @ R ) ) ) ) ).

% ring.ring_simprules(16)
thf(fact_528_ring_Oring__simprules_I16_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ X @ ( a_inv_7033018035630854991t_unit @ R @ X ) )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% ring.ring_simprules(16)
thf(fact_529_ring_Oring__simprules_I9_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ X )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% ring.ring_simprules(9)
thf(fact_530_ring_Oring__simprules_I9_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ X )
          = ( zero_a_b @ R ) ) ) ) ).

% ring.ring_simprules(9)
thf(fact_531_ring_Oring__simprules_I9_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ X )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% ring.ring_simprules(9)
thf(fact_532_cring_Ocring__simprules_I17_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% cring.cring_simprules(17)
thf(fact_533_cring_Ocring__simprules_I17_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ X ) )
          = ( zero_a_b @ R ) ) ) ) ).

% cring.cring_simprules(17)
thf(fact_534_cring_Ocring__simprules_I17_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ X @ ( a_inv_7033018035630854991t_unit @ R @ X ) )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% cring.cring_simprules(17)
thf(fact_535_cring_Ocring__simprules_I9_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ X )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% cring.cring_simprules(9)
thf(fact_536_cring_Ocring__simprules_I9_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ X )
          = ( zero_a_b @ R ) ) ) ) ).

% cring.cring_simprules(9)
thf(fact_537_cring_Ocring__simprules_I9_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ X )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% cring.cring_simprules(9)
thf(fact_538_cring_Osum__zero__eq__neg,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
              = ( zero_l4142658623432671053t_unit @ R ) )
           => ( X
              = ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ) ) ).

% cring.sum_zero_eq_neg
thf(fact_539_cring_Osum__zero__eq__neg,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( cring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( ( add_a_b @ R @ X @ Y )
              = ( zero_a_b @ R ) )
           => ( X
              = ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).

% cring.sum_zero_eq_neg
thf(fact_540_cring_Osum__zero__eq__neg,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( ( add_li174743652000525320t_unit @ R @ X @ Y )
              = ( zero_l347298301471573063t_unit @ R ) )
           => ( X
              = ( a_inv_7033018035630854991t_unit @ R @ Y ) ) ) ) ) ) ).

% cring.sum_zero_eq_neg
thf(fact_541_abelian__group_Ol__neg,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X ) @ X )
          = ( zero_l4142658623432671053t_unit @ G ) ) ) ) ).

% abelian_group.l_neg
thf(fact_542_abelian__group_Ol__neg,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_group_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X ) @ X )
          = ( zero_a_b @ G ) ) ) ) ).

% abelian_group.l_neg
thf(fact_543_abelian__group_Ol__neg,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( add_li174743652000525320t_unit @ G @ ( a_inv_7033018035630854991t_unit @ G @ X ) @ X )
          = ( zero_l347298301471573063t_unit @ G ) ) ) ) ).

% abelian_group.l_neg
thf(fact_544_abelian__group_Or__neg,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
       => ( ( add_li7652885771158616974t_unit @ G @ X @ ( a_inv_8944721093294617173t_unit @ G @ X ) )
          = ( zero_l4142658623432671053t_unit @ G ) ) ) ) ).

% abelian_group.r_neg
thf(fact_545_abelian__group_Or__neg,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_group_a_b @ G )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
       => ( ( add_a_b @ G @ X @ ( a_inv_a_b @ G @ X ) )
          = ( zero_a_b @ G ) ) ) ) ).

% abelian_group.r_neg
thf(fact_546_abelian__group_Or__neg,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ G )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
       => ( ( add_li174743652000525320t_unit @ G @ X @ ( a_inv_7033018035630854991t_unit @ G @ X ) )
          = ( zero_l347298301471573063t_unit @ G ) ) ) ) ).

% abelian_group.r_neg
thf(fact_547_abelian__group_Ominus__equality,axiom,
    ! [G: partia2670972154091845814t_unit,Y: list_a,X: list_a] :
      ( ( abelia3891852623213500406t_unit @ G )
     => ( ( ( add_li7652885771158616974t_unit @ G @ Y @ X )
          = ( zero_l4142658623432671053t_unit @ G ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( a_inv_8944721093294617173t_unit @ G @ X )
              = Y ) ) ) ) ) ).

% abelian_group.minus_equality
thf(fact_548_abelian__group_Ominus__equality,axiom,
    ! [G: partia2175431115845679010xt_a_b,Y: a,X: a] :
      ( ( abelian_group_a_b @ G )
     => ( ( ( add_a_b @ G @ Y @ X )
          = ( zero_a_b @ G ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( a_inv_a_b @ G @ X )
              = Y ) ) ) ) ) ).

% abelian_group.minus_equality
thf(fact_549_abelian__group_Ominus__equality,axiom,
    ! [G: partia2956882679547061052t_unit,Y: list_list_a,X: list_list_a] :
      ( ( abelia2778853791629620336t_unit @ G )
     => ( ( ( add_li174743652000525320t_unit @ G @ Y @ X )
          = ( zero_l347298301471573063t_unit @ G ) )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( a_inv_7033018035630854991t_unit @ G @ X )
              = Y ) ) ) ) ) ).

% abelian_group.minus_equality
thf(fact_550_domain_Ouniv__poly__a__inv__consistent,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P )
            = ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_consistent
thf(fact_551_domain_Ouniv__poly__a__inv__consistent,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_consistent
thf(fact_552_domain_Ouniv__poly__a__inv__consistent,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
            = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_consistent
thf(fact_553_domain_Olong__division__a__inv_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( polyno1727750685288865234t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) @ Q )
              = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).

% domain.long_division_a_inv(2)
thf(fact_554_domain_Olong__division__a__inv_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( 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 ) @ P ) @ Q )
              = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).

% domain.long_division_a_inv(2)
thf(fact_555_domain_Olong__division__a__inv_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( polyno5893782122288709345t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) @ Q )
              = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).

% domain.long_division_a_inv(1)
thf(fact_556_domain_Olong__division__a__inv_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( 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 ) @ P ) @ Q )
              = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pdiv_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).

% domain.long_division_a_inv(1)
thf(fact_557_domain_Ovar__pow__closed,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,N: nat] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ ( var_a_b @ R ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ).

% domain.var_pow_closed
thf(fact_558_domain_Ovar__pow__closed,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,N: nat] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( var_li8453953174693405341t_unit @ R ) @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ).

% domain.var_pow_closed
thf(fact_559_domain_Oconst__term__simprules__shell_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( member_a @ ( const_term_a_b @ R @ P ) @ K ) ) ) ) ).

% domain.const_term_simprules_shell(1)
thf(fact_560_domain_Oconst__term__simprules__shell_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( member_list_a @ ( const_6738166269504826821t_unit @ R @ P ) @ K ) ) ) ) ).

% domain.const_term_simprules_shell(1)
thf(fact_561_ringI,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( abelia3891852623213500406t_unit @ R )
     => ( ( monoid5589397312508706001t_unit @ R )
       => ( ! [X2: list_a,Y3: list_a,Z2: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
                 => ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) @ Z2 )
                    = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: list_a,Y3: list_a,Z2: list_a] :
                ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
                 => ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
                   => ( ( mult_l7073676228092353617t_unit @ R @ Z2 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) )
                      = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ X2 ) @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ Y3 ) ) ) ) ) )
           => ( ring_l6212528067271185461t_unit @ R ) ) ) ) ) ).

% ringI
thf(fact_562_ringI,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( abelian_group_a_b @ R )
     => ( ( monoid8385113658579753027xt_a_b @ R )
       => ( ! [X2: a,Y3: a,Z2: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
                 => ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X2 @ Y3 ) @ Z2 )
                    = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X2 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: a,Y3: a,Z2: a] :
                ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
                 => ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
                   => ( ( mult_a_ring_ext_a_b @ R @ Z2 @ ( add_a_b @ R @ X2 @ Y3 ) )
                      = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z2 @ X2 ) @ ( mult_a_ring_ext_a_b @ R @ Z2 @ Y3 ) ) ) ) ) )
           => ( ring_a_b @ R ) ) ) ) ) ).

% ringI
thf(fact_563_ringI,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( abelia2778853791629620336t_unit @ R )
     => ( ( monoid5729698748631984209t_unit @ R )
       => ( ! [X2: list_list_a,Y3: list_list_a,Z2: list_list_a] :
              ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
               => ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
                 => ( ( mult_l4853965630390486993t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X2 @ Y3 ) @ Z2 )
                    = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X2 @ Z2 ) @ ( mult_l4853965630390486993t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( ! [X2: list_list_a,Y3: list_list_a,Z2: list_list_a] :
                ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
               => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
                 => ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
                   => ( ( mult_l4853965630390486993t_unit @ R @ Z2 @ ( add_li174743652000525320t_unit @ R @ X2 @ Y3 ) )
                      = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ Z2 @ X2 ) @ ( mult_l4853965630390486993t_unit @ R @ Z2 @ Y3 ) ) ) ) ) )
           => ( ring_l1939023646219158831t_unit @ R ) ) ) ) ) ).

% ringI
thf(fact_564_cringI,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( abelia3891852623213500406t_unit @ R )
     => ( ( comm_m1219397618491936389t_unit @ R )
       => ( ! [X2: list_a,Y3: list_a,Z2: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
                 => ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) @ Z2 )
                    = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( cring_3148771470849435808t_unit @ R ) ) ) ) ).

% cringI
thf(fact_565_cringI,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( abelian_group_a_b @ R )
     => ( ( comm_m952295370001973751xt_a_b @ R )
       => ( ! [X2: a,Y3: a,Z2: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
                 => ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X2 @ Y3 ) @ Z2 )
                    = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X2 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( cring_a_b @ R ) ) ) ) ).

% cringI
thf(fact_566_cringI,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( abelia2778853791629620336t_unit @ R )
     => ( ( comm_m2156883222399968773t_unit @ R )
       => ( ! [X2: list_list_a,Y3: list_list_a,Z2: list_list_a] :
              ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R ) )
               => ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
                 => ( ( mult_l4853965630390486993t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X2 @ Y3 ) @ Z2 )
                    = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X2 @ Z2 ) @ ( mult_l4853965630390486993t_unit @ R @ Y3 @ Z2 ) ) ) ) ) )
         => ( cring_5991999922451032090t_unit @ R ) ) ) ) ).

% cringI
thf(fact_567_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_568_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_569_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_570_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_571_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a,P: 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 ) @ P @ Q )
            = ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_572_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a,P: 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 ) @ P @ Q )
            = ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_573_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a,P: 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 ) @ P @ Q )
            = ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_574_domain_Opdiv__pmod,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( P
              = ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).

% domain.pdiv_pmod
thf(fact_575_domain_Opdiv__pmod,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( P
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ ( polynomial_pdiv_a_b @ R @ P @ Q ) ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).

% domain.pdiv_pmod
thf(fact_576_domain_Oconst__term__simprules__shell_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( const_term_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q ) )
              = ( add_a_b @ R @ ( const_term_a_b @ R @ P ) @ ( const_term_a_b @ R @ Q ) ) ) ) ) ) ) ).

% domain.const_term_simprules_shell(3)
thf(fact_577_domain_Oconst__term__simprules__shell_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( const_6738166269504826821t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q ) )
              = ( add_li7652885771158616974t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) @ ( const_6738166269504826821t_unit @ R @ Q ) ) ) ) ) ) ) ).

% domain.const_term_simprules_shell(3)
thf(fact_578_exists__unique__long__division,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( 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 @ P @ Q @ X2 )
                & ! [Y5: produc9164743771328383783list_a] :
                    ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y5 )
                   => ( Y5 = X2 ) ) ) ) ) ) ) ).

% exists_unique_long_division
thf(fact_579_pprimeE_I1_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
         => ( P != nil_a ) ) ) ) ).

% pprimeE(1)
thf(fact_580_ring__primeI,axiom,
    ! [P: a] :
      ( ( P
       != ( zero_a_b @ r ) )
     => ( ( prime_a_ring_ext_a_b @ r @ P )
       => ( ring_ring_prime_a_b @ r @ P ) ) ) ).

% ring_primeI
thf(fact_581_ring__primeE_I3_J,axiom,
    ! [P: a] :
      ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_ring_prime_a_b @ r @ P )
       => ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).

% ring_primeE(3)
thf(fact_582_is__root__poly__mult__imp__is__root,axiom,
    ! [P: list_a,Q: list_a,X: a] :
      ( ( member_list_a @ P @ ( 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 ) ) ) )
       => ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X )
         => ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
            | ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).

% is_root_poly_mult_imp_is_root
thf(fact_583_unitary__monom__eq__var__pow,axiom,
    ! [K: set_a,N: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( monom_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
        = ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ).

% unitary_monom_eq_var_pow
thf(fact_584_domainI,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( ( one_li8328186300101108157t_unit @ R )
         != ( zero_l4142658623432671053t_unit @ R ) )
       => ( ! [A3: list_a,B2: list_a] :
              ( ( ( mult_l7073676228092353617t_unit @ R @ A3 @ B2 )
                = ( zero_l4142658623432671053t_unit @ R ) )
             => ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R ) )
                 => ( ( A3
                      = ( zero_l4142658623432671053t_unit @ R ) )
                    | ( B2
                      = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) )
         => ( domain6553523120543210313t_unit @ R ) ) ) ) ).

% domainI
thf(fact_585_domainI,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( cring_a_b @ R )
     => ( ( ( one_a_ring_ext_a_b @ R )
         != ( zero_a_b @ R ) )
       => ( ! [A3: a,B2: a] :
              ( ( ( mult_a_ring_ext_a_b @ R @ A3 @ B2 )
                = ( zero_a_b @ R ) )
             => ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
                 => ( ( A3
                      = ( zero_a_b @ R ) )
                    | ( B2
                      = ( zero_a_b @ R ) ) ) ) ) )
         => ( domain_a_b @ R ) ) ) ) ).

% domainI
thf(fact_586_domainI,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( ( one_li8234411390022467901t_unit @ R )
         != ( zero_l347298301471573063t_unit @ R ) )
       => ( ! [A3: list_list_a,B2: list_list_a] :
              ( ( ( mult_l4853965630390486993t_unit @ R @ A3 @ B2 )
                = ( zero_l347298301471573063t_unit @ R ) )
             => ( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ R ) )
               => ( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ R ) )
                 => ( ( A3
                      = ( zero_l347298301471573063t_unit @ R ) )
                    | ( B2
                      = ( zero_l347298301471573063t_unit @ R ) ) ) ) ) )
         => ( domain7810152921033798211t_unit @ R ) ) ) ) ).

% domainI
thf(fact_587_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_588_const__term__simprules__shell_I2_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
            = ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).

% const_term_simprules_shell(2)
thf(fact_589_minus__eq,axiom,
    ! [X: a,Y: a] :
      ( ( a_minus_a_b @ r @ X @ Y )
      = ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).

% minus_eq
thf(fact_590_subring__props_I5_J,axiom,
    ! [K: set_a,H3: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ H3 @ K )
       => ( member_a @ ( a_inv_a_b @ r @ H3 ) @ K ) ) ) ).

% subring_props(5)
thf(fact_591_pow__non__zero,axiom,
    ! [X: a,N: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( X
         != ( zero_a_b @ r ) )
       => ( ( pow_a_1026414303147256608_b_nat @ r @ X @ N )
         != ( zero_a_b @ r ) ) ) ) ).

% pow_non_zero
thf(fact_592_add_Oinv__mult__group,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
          = ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).

% add.inv_mult_group
thf(fact_593_add_Oinv__solve__left,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( A
              = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
            = ( C
              = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).

% add.inv_solve_left
thf(fact_594_add_Oinv__solve__left_H,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
              = A )
            = ( C
              = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).

% add.inv_solve_left'
thf(fact_595_add_Oinv__solve__right,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( A
              = ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
            = ( B
              = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).

% add.inv_solve_right
thf(fact_596_add_Oinv__solve__right_H,axiom,
    ! [A: a,B: a,C: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
              = A )
            = ( B
              = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).

% add.inv_solve_right'
thf(fact_597_a__transpose__inv,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( ( add_a_b @ r @ X @ Y )
        = Z )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
              = Y ) ) ) ) ) ).

% a_transpose_inv
thf(fact_598_local_Ominus__add,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
          = ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).

% local.minus_add
thf(fact_599_r__neg1,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
          = Y ) ) ) ).

% r_neg1
thf(fact_600_r__neg2,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
          = Y ) ) ) ).

% r_neg2
thf(fact_601_group__commutes__pow,axiom,
    ! [X: a,Y: a,N: nat] :
      ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
        = ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ Y )
            = ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ) ) ).

% group_commutes_pow
thf(fact_602_nat__pow__comm,axiom,
    ! [X: a,N: nat,M: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) )
        = ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).

% nat_pow_comm
thf(fact_603_nat__pow__distrib,axiom,
    ! [X: a,Y: a,N: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
          = ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ).

% nat_pow_distrib
thf(fact_604_pow__mult__distrib,axiom,
    ! [X: a,Y: a,N: nat] :
      ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
        = ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
            = ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).

% pow_mult_distrib
thf(fact_605_l__minus,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
          = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).

% l_minus
thf(fact_606_r__minus,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
          = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).

% r_minus
thf(fact_607_m__assoc,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
            = ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).

% m_assoc
thf(fact_608_m__comm,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
          = ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).

% m_comm
thf(fact_609_m__lcomm,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
            = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).

% m_lcomm
thf(fact_610_subring__props_I6_J,axiom,
    ! [K: set_a,H1: a,H2: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ H1 @ K )
       => ( ( member_a @ H2 @ K )
         => ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H2 ) @ K ) ) ) ) ).

% subring_props(6)
thf(fact_611_zero__pdivides,axiom,
    ! [P: list_a] :
      ( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
      = ( P = nil_a ) ) ).

% zero_pdivides
thf(fact_612_zero__pdivides__zero,axiom,
    polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).

% zero_pdivides_zero
thf(fact_613_zero__is__prime_I1_J,axiom,
    prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).

% zero_is_prime(1)
thf(fact_614_l__neg,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
        = ( zero_a_b @ r ) ) ) ).

% l_neg
thf(fact_615_minus__equality,axiom,
    ! [Y: a,X: a] :
      ( ( ( add_a_b @ r @ Y @ X )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( a_inv_a_b @ r @ X )
            = Y ) ) ) ) ).

% minus_equality
thf(fact_616_r__neg,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
        = ( zero_a_b @ r ) ) ) ).

% r_neg
thf(fact_617_sum__zero__eq__neg,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ X @ Y )
            = ( zero_a_b @ r ) )
         => ( X
            = ( a_inv_a_b @ r @ Y ) ) ) ) ) ).

% sum_zero_eq_neg
thf(fact_618_local_Ointegral,axiom,
    ! [A: a,B: a] :
      ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( A
              = ( zero_a_b @ r ) )
            | ( B
              = ( zero_a_b @ r ) ) ) ) ) ) ).

% local.integral
thf(fact_619_integral__iff,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
            = ( zero_a_b @ r ) )
          = ( ( A
              = ( zero_a_b @ r ) )
            | ( B
              = ( zero_a_b @ r ) ) ) ) ) ) ).

% integral_iff
thf(fact_620_m__lcancel,axiom,
    ! [A: a,B: a,C: a] :
      ( ( A
       != ( zero_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
                = ( mult_a_ring_ext_a_b @ r @ A @ C ) )
              = ( B = C ) ) ) ) ) ) ).

% m_lcancel
thf(fact_621_m__rcancel,axiom,
    ! [A: a,B: a,C: a] :
      ( ( A
       != ( zero_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( ( mult_a_ring_ext_a_b @ r @ B @ A )
                = ( mult_a_ring_ext_a_b @ r @ C @ A ) )
              = ( B = C ) ) ) ) ) ) ).

% m_rcancel
thf(fact_622_l__distr,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).

% l_distr
thf(fact_623_r__distr,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).

% r_distr
thf(fact_624_square__eq__one,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ( mult_a_ring_ext_a_b @ r @ X @ X )
          = ( one_a_ring_ext_a_b @ r ) )
       => ( ( X
            = ( one_a_ring_ext_a_b @ r ) )
          | ( X
            = ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).

% square_eq_one
thf(fact_625_inv__unique,axiom,
    ! [Y: a,X: a,Y2: a] :
      ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
        = ( one_a_ring_ext_a_b @ r ) )
     => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
          = ( one_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
             => ( Y = Y2 ) ) ) ) ) ) ).

% inv_unique
thf(fact_626_one__unique,axiom,
    ! [U: a] :
      ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ! [X2: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
              = X2 ) )
       => ( U
          = ( one_a_ring_ext_a_b @ r ) ) ) ) ).

% one_unique
thf(fact_627_mult__cong__l,axiom,
    ! [A: a,A4: a,B: a] :
      ( ( associ5860276527279195403xt_a_b @ r @ A @ A4 )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
           => ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A4 @ B ) ) ) ) ) ) ).

% mult_cong_l
thf(fact_628_mult__cong__r,axiom,
    ! [B: a,B3: a,A: a] :
      ( ( associ5860276527279195403xt_a_b @ r @ B @ B3 )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B3 ) ) ) ) ) ) ).

% mult_cong_r
thf(fact_629_monoid__comm__monoidI,axiom,
    ( ! [X2: a,Y3: a] :
        ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y3 )
            = ( mult_a_ring_ext_a_b @ r @ Y3 @ X2 ) ) ) )
   => ( comm_m952295370001973751xt_a_b @ r ) ) ).

% monoid_comm_monoidI
thf(fact_630_pdivides__zero,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ) ).

% pdivides_zero
thf(fact_631_const__term__simprules__shell_I4_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
          = ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P ) ) ) ) ) ).

% const_term_simprules_shell(4)
thf(fact_632_pmod__zero__iff__pdivides,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( 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 @ P @ Q )
              = nil_a )
            = ( polyno5814909790663948098es_a_b @ r @ Q @ P ) ) ) ) ) ).

% pmod_zero_iff_pdivides
thf(fact_633_pprimeE_I3_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a,R3: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
             => ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
               => ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
                  | ( polyno5814909790663948098es_a_b @ r @ P @ R3 ) ) ) ) ) ) ) ) ).

% pprimeE(3)
thf(fact_634_nat__pow__closed,axiom,
    ! [X: a,N: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% nat_pow_closed
thf(fact_635_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_636_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_637_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_638_local_Ominus__zero,axiom,
    ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
    = ( zero_a_b @ r ) ) ).

% local.minus_zero
thf(fact_639_nat__pow__one,axiom,
    ! [N: nat] :
      ( ( pow_a_1026414303147256608_b_nat @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% nat_pow_one
thf(fact_640_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_641_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_642_m__closed,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% m_closed
thf(fact_643_l__null,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
        = ( zero_a_b @ r ) ) ) ).

% l_null
thf(fact_644_r__null,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
        = ( zero_a_b @ r ) ) ) ).

% r_null
thf(fact_645_l__one,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
        = X ) ) ).

% l_one
thf(fact_646_r__one,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
        = X ) ) ).

% r_one
thf(fact_647_ring_Olong__divides_Ocong,axiom,
    polyno2806191415236617128es_a_b = polyno2806191415236617128es_a_b ).

% ring.long_divides.cong
thf(fact_648_ring_Omonom_Ocong,axiom,
    monom_a_b = monom_a_b ).

% ring.monom.cong
thf(fact_649_ring_Ois__root_Ocong,axiom,
    polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).

% ring.is_root.cong
thf(fact_650_domain_Ozero__pdivides,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( polyno8016796738000020810t_unit @ R @ nil_list_a @ P )
        = ( P = nil_list_a ) ) ) ).

% domain.zero_pdivides
thf(fact_651_domain_Ozero__pdivides,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( polyno5814909790663948098es_a_b @ R @ nil_a @ P )
        = ( P = nil_a ) ) ) ).

% domain.zero_pdivides
thf(fact_652_domain_Ozero__pdivides__zero,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( polyno8016796738000020810t_unit @ R @ nil_list_a @ nil_list_a ) ) ).

% domain.zero_pdivides_zero
thf(fact_653_domain_Ozero__pdivides__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( domain_a_b @ R )
     => ( polyno5814909790663948098es_a_b @ R @ nil_a @ nil_a ) ) ).

% domain.zero_pdivides_zero
thf(fact_654_domain_OpprimeE_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R3: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
           => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
             => ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
               => ( ( polyno5814909790663948098es_a_b @ R @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ R3 ) )
                 => ( ( polyno5814909790663948098es_a_b @ R @ P @ Q )
                    | ( polyno5814909790663948098es_a_b @ R @ P @ R3 ) ) ) ) ) ) ) ) ) ).

% domain.pprimeE(3)
thf(fact_655_domain_OpprimeE_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
           => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
             => ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
               => ( ( polyno8016796738000020810t_unit @ R @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ R3 ) )
                 => ( ( polyno8016796738000020810t_unit @ R @ P @ Q )
                    | ( polyno8016796738000020810t_unit @ R @ P @ R3 ) ) ) ) ) ) ) ) ) ).

% domain.pprimeE(3)
thf(fact_656_domain_Opdivides__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( polyno5814909790663948098es_a_b @ R @ P @ nil_a ) ) ) ) ).

% domain.pdivides_zero
thf(fact_657_domain_Opdivides__zero,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( polyno8016796738000020810t_unit @ R @ P @ nil_list_a ) ) ) ) ).

% domain.pdivides_zero
thf(fact_658_domain_Opmod__zero__iff__pdivides,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( ( polyno1727750685288865234t_unit @ R @ P @ Q )
                = nil_list_a )
              = ( polyno8016796738000020810t_unit @ R @ Q @ P ) ) ) ) ) ) ).

% domain.pmod_zero_iff_pdivides
thf(fact_659_domain_Opmod__zero__iff__pdivides,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( 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 @ P @ Q )
                = nil_a )
              = ( polyno5814909790663948098es_a_b @ R @ Q @ P ) ) ) ) ) ) ).

% domain.pmod_zero_iff_pdivides
thf(fact_660_domain_Ois__root__poly__mult__imp__is__root,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,X: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
         => ( ( polyno5142720416380192742t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q ) @ X )
           => ( ( polyno5142720416380192742t_unit @ R @ P @ X )
              | ( polyno5142720416380192742t_unit @ R @ Q @ X ) ) ) ) ) ) ).

% domain.is_root_poly_mult_imp_is_root
thf(fact_661_domain_Ois__root__poly__mult__imp__is__root,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
         => ( ( polyno6951661231331188332t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q ) @ X )
           => ( ( polyno6951661231331188332t_unit @ R @ P @ X )
              | ( polyno6951661231331188332t_unit @ R @ Q @ X ) ) ) ) ) ) ).

% domain.is_root_poly_mult_imp_is_root
thf(fact_662_domain_Ois__root__poly__mult__imp__is__root,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X: a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P @ ( 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 ) ) ) )
         => ( ( polyno4133073214067823460ot_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q ) @ X )
           => ( ( polyno4133073214067823460ot_a_b @ R @ P @ X )
              | ( polyno4133073214067823460ot_a_b @ R @ Q @ X ) ) ) ) ) ) ).

% domain.is_root_poly_mult_imp_is_root
thf(fact_663_domain_OpprimeE_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
           => ( P != nil_a ) ) ) ) ) ).

% domain.pprimeE(1)
thf(fact_664_domain_OpprimeE_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
           => ( P != nil_list_a ) ) ) ) ) ).

% domain.pprimeE(1)
thf(fact_665_domain_Osame__pmod__iff__pdivides,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
             => ( ( ( polyno1727750685288865234t_unit @ R @ A @ Q )
                  = ( polyno1727750685288865234t_unit @ R @ B @ Q ) )
                = ( polyno8016796738000020810t_unit @ R @ Q @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) ) ) ) ) ) ) ) ).

% domain.same_pmod_iff_pdivides
thf(fact_666_domain_Osame__pmod__iff__pdivides,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( 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 ) ) ) ) ) ) ) ) ).

% domain.same_pmod_iff_pdivides
thf(fact_667_domain_Ounitary__monom__eq__var__pow,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,N: nat] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( monom_7446464087056152608t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ N )
          = ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( var_li8453953174693405341t_unit @ R ) @ N ) ) ) ) ).

% domain.unitary_monom_eq_var_pow
thf(fact_668_domain_Ounitary__monom__eq__var__pow,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,N: nat] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( monom_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ N )
          = ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ ( var_a_b @ R ) @ N ) ) ) ) ).

% domain.unitary_monom_eq_var_pow
thf(fact_669_domain_Oexists__unique__long__division,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( 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 @ P @ Q @ X2 )
                  & ! [Y5: produc7709606177366032167list_a] :
                      ( ( polyno6947042923167803568t_unit @ R @ P @ Q @ Y5 )
                     => ( Y5 = X2 ) ) ) ) ) ) ) ) ).

% domain.exists_unique_long_division
thf(fact_670_domain_Oexists__unique__long__division,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( 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 @ P @ Q @ X2 )
                  & ! [Y5: produc9164743771328383783list_a] :
                      ( ( polyno2806191415236617128es_a_b @ R @ P @ Q @ Y5 )
                     => ( Y5 = X2 ) ) ) ) ) ) ) ) ).

% domain.exists_unique_long_division
thf(fact_671_pirreducible__pow__pdivides__iff,axiom,
    ! [K: set_a,P: list_a,Q: list_a,R3: list_a,N: nat] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
             => ( ~ ( polyno5814909790663948098es_a_b @ r @ P @ Q )
               => ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
                  = ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ).

% pirreducible_pow_pdivides_iff
thf(fact_672_long__divisionI,axiom,
    ! [K: set_a,P: list_a,Q: list_a,B: list_a,R3: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( 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 @ P @ Q @ ( produc6837034575241423639list_a @ B @ R3 ) )
             => ( ( produc6837034575241423639list_a @ B @ R3 )
                = ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ) ).

% long_divisionI
thf(fact_673_long__divisionE,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( 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 @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ).

% long_divisionE
thf(fact_674_is__root__imp__pdivides,axiom,
    ! [P: list_a,X: a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
       => ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P ) ) ) ).

% is_root_imp_pdivides
thf(fact_675_subdomainI,axiom,
    ! [H: set_a] :
      ( ( subcring_a_b @ H @ r )
     => ( ( ( one_a_ring_ext_a_b @ r )
         != ( zero_a_b @ r ) )
       => ( ! [H12: a,H22: a] :
              ( ( member_a @ H12 @ H )
             => ( ( member_a @ H22 @ H )
               => ( ( ( mult_a_ring_ext_a_b @ r @ H12 @ H22 )
                    = ( zero_a_b @ r ) )
                 => ( ( H12
                      = ( zero_a_b @ r ) )
                    | ( H22
                      = ( zero_a_b @ r ) ) ) ) ) )
         => ( subdomain_a_b @ H @ r ) ) ) ) ).

% subdomainI
thf(fact_676_pprimeI,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( P != nil_a )
         => ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ! [Q2: list_a,R4: list_a] :
                  ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                 => ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                   => ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R4 ) )
                     => ( ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
                        | ( polyno5814909790663948098es_a_b @ r @ P @ R4 ) ) ) ) )
             => ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).

% pprimeI
thf(fact_677_pprime__iff__pirreducible,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
          = ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).

% pprime_iff_pirreducible
thf(fact_678_poly__add_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ~ ! [P1: list_a,P22: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).

% poly_add.cases
thf(fact_679_normalize_Ocases,axiom,
    ! [X: list_a] :
      ( ( X != nil_a )
     => ~ ! [V: a,Va: list_a] :
            ( X
           != ( cons_a @ V @ Va ) ) ) ).

% normalize.cases
thf(fact_680_poly__mult_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [P22: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ nil_a @ P22 ) )
     => ~ ! [V: a,Va: list_a,P22: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P22 ) ) ) ).

% poly_mult.cases
thf(fact_681_combine_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [K2: a,Ks: list_a,U2: a,Us: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ ( cons_a @ K2 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
     => ( ! [Us: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ nil_a @ Us ) )
       => ~ ! [Ks: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).

% combine.cases
thf(fact_682_carrier__is__subcring,axiom,
    subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% carrier_is_subcring
thf(fact_683_subcringI_H,axiom,
    ! [H: set_a] :
      ( ( subring_a_b @ H @ r )
     => ( subcring_a_b @ H @ r ) ) ).

% subcringI'
thf(fact_684_subcringI,axiom,
    ! [H: set_a] :
      ( ( subring_a_b @ H @ r )
     => ( ! [H12: a,H22: a] :
            ( ( member_a @ H12 @ H )
           => ( ( member_a @ H22 @ H )
             => ( ( mult_a_ring_ext_a_b @ r @ H12 @ H22 )
                = ( mult_a_ring_ext_a_b @ r @ H22 @ H12 ) ) ) )
       => ( subcring_a_b @ H @ r ) ) ) ).

% subcringI
thf(fact_685_pirreducibleE_I1_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
         => ( P != nil_a ) ) ) ) ).

% pirreducibleE(1)
thf(fact_686_pirreducibleE_I2_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
         => ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% pirreducibleE(2)
thf(fact_687_pprimeE_I2_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
         => ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% pprimeE(2)
thf(fact_688_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_689_exists__long__division,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ~ ! [B2: list_a] :
                  ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                 => ! [R4: list_a] :
                      ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                     => ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ).

% exists_long_division
thf(fact_690_pirreducibleE_I3_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a,R3: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
             => ( ( P
                  = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R3 ) )
               => ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                  | ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ) ) ) ).

% pirreducibleE(3)
thf(fact_691_pdivides__imp__is__root,axiom,
    ! [P: list_a,X: a] :
      ( ( P != 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 ) ) @ P )
         => ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ) ).

% pdivides_imp_is_root
thf(fact_692_pirreducibleI,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( P != nil_a )
         => ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ! [Q2: list_a,R4: list_a] :
                  ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                 => ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                   => ( ( P
                        = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R4 ) )
                     => ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                        | ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) )
             => ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).

% pirreducibleI
thf(fact_693_ring_Opoly__mult_Ocases,axiom,
    ! [R: partia2670972154091845814t_unit,X: produc7709606177366032167list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ! [P22: list_list_a] :
            ( X
           != ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
       => ~ ! [V: list_a,Va: list_list_a,P22: list_list_a] :
              ( X
             != ( produc8696003437204565271list_a @ ( cons_list_a @ V @ Va ) @ P22 ) ) ) ) ).

% ring.poly_mult.cases
thf(fact_694_ring_Opoly__mult_Ocases,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: produc9164743771328383783list_a] :
      ( ( ring_a_b @ R )
     => ( ! [P22: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ nil_a @ P22 ) )
       => ~ ! [V: a,Va: list_a,P22: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P22 ) ) ) ) ).

% ring.poly_mult.cases
thf(fact_695_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_696_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_697_monoid_OUnits__closed,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
       => ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).

% monoid.Units_closed
thf(fact_698_monoid_OUnits__closed,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
       => ( member_a @ X @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).

% monoid.Units_closed
thf(fact_699_monoid_OUnits__closed,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ X @ ( units_4903515905731149798t_unit @ G ) )
       => ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G ) ) ) ) ).

% monoid.Units_closed
thf(fact_700_monoid_OUnits__m__closed,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ G ) )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ X @ Y ) @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ).

% monoid.Units_m_closed
thf(fact_701_monoid_OUnits__m__closed,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
       => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
         => ( member_a @ ( mult_a_ring_ext_a_b @ G @ X @ Y ) @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ).

% monoid.Units_m_closed
thf(fact_702_monoid_OUnits__one__closed,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ G ) @ ( units_2932844235741507942t_unit @ G ) ) ) ).

% monoid.Units_one_closed
thf(fact_703_monoid_OUnits__one__closed,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( member_a @ ( one_a_ring_ext_a_b @ G ) @ ( units_a_ring_ext_a_b @ G ) ) ) ).

% monoid.Units_one_closed
thf(fact_704_domain_OpirreducibleE_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
           => ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.pirreducibleE(2)
thf(fact_705_domain_OpirreducibleE_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
           => ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).

% domain.pirreducibleE(2)
thf(fact_706_ring_Odense__repr_Ocases,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( X != nil_list_a )
       => ~ ! [V: list_a,Va: list_list_a] :
              ( X
             != ( cons_list_a @ V @ Va ) ) ) ) ).

% ring.dense_repr.cases
thf(fact_707_ring_Odense__repr_Ocases,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: list_a] :
      ( ( ring_a_b @ R )
     => ( ( X != nil_a )
       => ~ ! [V: a,Va: list_a] :
              ( X
             != ( cons_a @ V @ Va ) ) ) ) ).

% ring.dense_repr.cases
thf(fact_708_domain_OpirreducibleE_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
           => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
             => ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
               => ( ( P
                    = ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ R3 ) )
                 => ( ( member_list_list_a @ Q @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                    | ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ) ) ) ) ).

% domain.pirreducibleE(3)
thf(fact_709_domain_OpirreducibleE_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R3: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
           => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
             => ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
               => ( ( P
                    = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ R3 ) )
                 => ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                    | ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ) ) ) ) ).

% domain.pirreducibleE(3)
thf(fact_710_monoid_OUnits__l__cancel,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y )
                = ( mult_l7073676228092353617t_unit @ G @ X @ Z ) )
              = ( Y = Z ) ) ) ) ) ) ).

% monoid.Units_l_cancel
thf(fact_711_monoid_OUnits__l__cancel,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
                = ( mult_a_ring_ext_a_b @ G @ X @ Z ) )
              = ( Y = Z ) ) ) ) ) ) ).

% monoid.Units_l_cancel
thf(fact_712_monoid_OUnits__l__cancel,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ X @ ( units_4903515905731149798t_unit @ G ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( ( mult_l4853965630390486993t_unit @ G @ X @ Y )
                = ( mult_l4853965630390486993t_unit @ G @ X @ Z ) )
              = ( Y = Z ) ) ) ) ) ) ).

% monoid.Units_l_cancel
thf(fact_713_monoid_OUnits__inv__comm,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( ( mult_l7073676228092353617t_unit @ G @ X @ Y )
          = ( one_li8328186300101108157t_unit @ G ) )
       => ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
         => ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ G ) )
           => ( ( mult_l7073676228092353617t_unit @ G @ Y @ X )
              = ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ) ).

% monoid.Units_inv_comm
thf(fact_714_monoid_OUnits__inv__comm,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( ( mult_a_ring_ext_a_b @ G @ X @ Y )
          = ( one_a_ring_ext_a_b @ G ) )
       => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
         => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ G ) )
           => ( ( mult_a_ring_ext_a_b @ G @ Y @ X )
              = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ) ).

% monoid.Units_inv_comm
thf(fact_715_ring_OUnits__minus__one__closed,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) ) @ ( units_2932844235741507942t_unit @ R ) ) ) ).

% ring.Units_minus_one_closed
thf(fact_716_ring_OUnits__minus__one__closed,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R )
     => ( member_a @ ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) @ ( units_a_ring_ext_a_b @ R ) ) ) ).

% ring.Units_minus_one_closed
thf(fact_717_domain_OpirreducibleI,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( P != nil_list_a )
           => ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
             => ( ! [Q2: list_list_a,R4: list_list_a] :
                    ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                   => ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                     => ( ( P
                          = ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q2 @ R4 ) )
                       => ( ( member_list_list_a @ Q2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                          | ( member_list_list_a @ R4 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) )
               => ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ) ) ).

% domain.pirreducibleI
thf(fact_718_domain_OpirreducibleI,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( P != nil_a )
           => ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
             => ( ! [Q2: list_a,R4: list_a] :
                    ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                   => ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                     => ( ( P
                          = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q2 @ R4 ) )
                       => ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                          | ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) )
               => ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) ) ) ) ) ) ).

% domain.pirreducibleI
thf(fact_719_monoid_OUnits__l__inv__ex,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
       => ? [X2: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
            & ( ( mult_l7073676228092353617t_unit @ G @ X2 @ X )
              = ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).

% monoid.Units_l_inv_ex
thf(fact_720_monoid_OUnits__l__inv__ex,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
       => ? [X2: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
            & ( ( mult_a_ring_ext_a_b @ G @ X2 @ X )
              = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).

% monoid.Units_l_inv_ex
thf(fact_721_monoid_OUnits__l__inv__ex,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ X @ ( units_4903515905731149798t_unit @ G ) )
       => ? [X2: list_list_a] :
            ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
            & ( ( mult_l4853965630390486993t_unit @ G @ X2 @ X )
              = ( one_li8234411390022467901t_unit @ G ) ) ) ) ) ).

% monoid.Units_l_inv_ex
thf(fact_722_monoid_OUnits__r__inv__ex,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
       => ? [X2: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ G ) )
            & ( ( mult_l7073676228092353617t_unit @ G @ X @ X2 )
              = ( one_li8328186300101108157t_unit @ G ) ) ) ) ) ).

% monoid.Units_r_inv_ex
thf(fact_723_monoid_OUnits__r__inv__ex,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
       => ? [X2: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ G ) )
            & ( ( mult_a_ring_ext_a_b @ G @ X @ X2 )
              = ( one_a_ring_ext_a_b @ G ) ) ) ) ) ).

% monoid.Units_r_inv_ex
thf(fact_724_monoid_OUnits__r__inv__ex,axiom,
    ! [G: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ X @ ( units_4903515905731149798t_unit @ G ) )
       => ? [X2: list_list_a] :
            ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ G ) )
            & ( ( mult_l4853965630390486993t_unit @ G @ X @ X2 )
              = ( one_li8234411390022467901t_unit @ G ) ) ) ) ) ).

% monoid.Units_r_inv_ex
thf(fact_725_var__def,axiom,
    ( var_li8453953174693405341t_unit
    = ( ^ [R2: partia2670972154091845814t_unit] : ( cons_list_a @ ( one_li8328186300101108157t_unit @ R2 ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ nil_list_a ) ) ) ) ).

% var_def
thf(fact_726_var__def,axiom,
    ( var_a_b
    = ( ^ [R2: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R2 ) @ ( cons_a @ ( zero_a_b @ R2 ) @ nil_a ) ) ) ) ).

% var_def
thf(fact_727_univ__poly__one,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ K ) )
      = ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ).

% univ_poly_one
thf(fact_728_univ__poly__one,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
      = ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) ) ).

% univ_poly_one
thf(fact_729_domain_OpprimeE_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
           => ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).

% domain.pprimeE(2)
thf(fact_730_domain_OpprimeE_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
           => ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).

% domain.pprimeE(2)
thf(fact_731_domain_OpirreducibleE_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
           => ( P != nil_list_a ) ) ) ) ) ).

% domain.pirreducibleE(1)
thf(fact_732_domain_OpirreducibleE_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
           => ( P != nil_a ) ) ) ) ) ).

% domain.pirreducibleE(1)
thf(fact_733_ring_Omonic__degree__one__root__condition,axiom,
    ! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( polyno6951661231331188332t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ A ) @ nil_list_a ) ) @ B )
          = ( A = B ) ) ) ) ).

% ring.monic_degree_one_root_condition
thf(fact_734_ring_Omonic__degree__one__root__condition,axiom,
    ! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( polyno5142720416380192742t_unit @ R @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ A ) @ nil_list_list_a ) ) @ B )
          = ( A = B ) ) ) ) ).

% ring.monic_degree_one_root_condition
thf(fact_735_ring_Omonic__degree__one__root__condition,axiom,
    ! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
      ( ( ring_a_b @ R )
     => ( ( 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 ) ) ) ) ).

% ring.monic_degree_one_root_condition
thf(fact_736_domain_Oexists__long__division,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( Q != nil_list_a )
             => ~ ! [B2: list_list_a] :
                    ( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                   => ! [R4: list_list_a] :
                        ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                       => ~ ( polyno6947042923167803568t_unit @ R @ P @ Q @ ( produc8696003437204565271list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ) ).

% domain.exists_long_division
thf(fact_737_domain_Oexists__long__division,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( Q != nil_a )
             => ~ ! [B2: list_a] :
                    ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                   => ! [R4: list_a] :
                        ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                       => ~ ( polyno2806191415236617128es_a_b @ R @ P @ Q @ ( produc6837034575241423639list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ) ).

% domain.exists_long_division
thf(fact_738_domain_Opprime__iff__pirreducible,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
            = ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ).

% domain.pprime_iff_pirreducible
thf(fact_739_domain_Opprime__iff__pirreducible,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
            = ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) ) ) ) ).

% domain.pprime_iff_pirreducible
thf(fact_740_domain_Opdivides__imp__is__root,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( P != nil_list_a )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ P )
           => ( polyno6951661231331188332t_unit @ R @ P @ X ) ) ) ) ) ).

% domain.pdivides_imp_is_root
thf(fact_741_domain_Opdivides__imp__is__root,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( P != nil_list_list_a )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( polyno4453881341673752516t_unit @ R @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ nil_list_list_a ) ) @ P )
           => ( polyno5142720416380192742t_unit @ R @ P @ X ) ) ) ) ) ).

% domain.pdivides_imp_is_root
thf(fact_742_domain_Opdivides__imp__is__root,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
      ( ( domain_a_b @ R )
     => ( ( P != 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 ) ) @ P )
           => ( polyno4133073214067823460ot_a_b @ R @ P @ X ) ) ) ) ) ).

% domain.pdivides_imp_is_root
thf(fact_743_domain_OpprimeI,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( P != nil_a )
           => ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
             => ( ! [Q2: list_a,R4: list_a] :
                    ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                   => ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                     => ( ( polyno5814909790663948098es_a_b @ R @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q2 @ R4 ) )
                       => ( ( polyno5814909790663948098es_a_b @ R @ P @ Q2 )
                          | ( polyno5814909790663948098es_a_b @ R @ P @ R4 ) ) ) ) )
               => ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) ) ) ) ) ) ).

% domain.pprimeI
thf(fact_744_domain_OpprimeI,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( P != nil_list_a )
           => ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
             => ( ! [Q2: list_list_a,R4: list_list_a] :
                    ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                   => ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                     => ( ( polyno8016796738000020810t_unit @ R @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q2 @ R4 ) )
                       => ( ( polyno8016796738000020810t_unit @ R @ P @ Q2 )
                          | ( polyno8016796738000020810t_unit @ R @ P @ R4 ) ) ) ) )
               => ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ) ) ).

% domain.pprimeI
thf(fact_745_domain_Olong__divisionE,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( 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 @ P @ Q @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ) ).

% domain.long_divisionE
thf(fact_746_domain_Olong__divisionE,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( 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 @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P @ Q ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ) ).

% domain.long_divisionE
thf(fact_747_domain_Olong__divisionI,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,B: list_list_a,R3: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( 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 @ P @ Q @ ( produc8696003437204565271list_a @ B @ R3 ) )
               => ( ( produc8696003437204565271list_a @ B @ R3 )
                  = ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ) ) ).

% domain.long_divisionI
thf(fact_748_domain_Olong__divisionI,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,B: list_a,R3: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( 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 @ P @ Q @ ( produc6837034575241423639list_a @ B @ R3 ) )
               => ( ( produc6837034575241423639list_a @ B @ R3 )
                  = ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P @ Q ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ) ) ).

% domain.long_divisionI
thf(fact_749_domain_Opirreducible__pow__pdivides__iff,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R3: list_list_a,N: nat] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
           => ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
             => ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
               => ( ~ ( polyno8016796738000020810t_unit @ R @ P @ Q )
                 => ( ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ R3 ) )
                    = ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ) ).

% domain.pirreducible_pow_pdivides_iff
thf(fact_750_domain_Opirreducible__pow__pdivides__iff,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R3: list_a,N: nat] :
      ( ( domain_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
           => ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
             => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
               => ( ~ ( polyno5814909790663948098es_a_b @ R @ P @ Q )
                 => ( ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ R3 ) )
                    = ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ) ).

% domain.pirreducible_pow_pdivides_iff
thf(fact_751_domain_Ois__root__imp__pdivides,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( polyno5142720416380192742t_unit @ R @ P @ X )
         => ( polyno4453881341673752516t_unit @ R @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ nil_list_list_a ) ) @ P ) ) ) ) ).

% domain.is_root_imp_pdivides
thf(fact_752_domain_Ois__root__imp__pdivides,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( polyno6951661231331188332t_unit @ R @ P @ X )
         => ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ P ) ) ) ) ).

% domain.is_root_imp_pdivides
thf(fact_753_domain_Ois__root__imp__pdivides,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( polyno4133073214067823460ot_a_b @ R @ P @ X )
         => ( polyno5814909790663948098es_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X ) @ nil_a ) ) @ P ) ) ) ) ).

% domain.is_root_imp_pdivides
thf(fact_754_alg__multE_I1_J,axiom,
    ! [X: a,P: list_a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( P != 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 @ P @ X ) ) @ P ) ) ) ) ).

% alg_multE(1)
thf(fact_755_poly__mult__var,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ( P = nil_a )
           => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
              = nil_a ) )
          & ( ( P != nil_a )
           => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
              = ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).

% poly_mult_var
thf(fact_756_subringI,axiom,
    ! [H: set_a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H )
       => ( ! [H4: a] :
              ( ( member_a @ H4 @ H )
             => ( member_a @ ( a_inv_a_b @ r @ H4 ) @ H ) )
         => ( ! [H12: a,H22: a] :
                ( ( member_a @ H12 @ H )
               => ( ( member_a @ H22 @ H )
                 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H22 ) @ H ) ) )
           => ( ! [H12: a,H22: a] :
                  ( ( member_a @ H12 @ H )
                 => ( ( member_a @ H22 @ H )
                   => ( member_a @ ( add_a_b @ r @ H12 @ H22 ) @ H ) ) )
             => ( subring_a_b @ H @ r ) ) ) ) ) ) ).

% subringI
thf(fact_757_ring_OsubdomainI,axiom,
    ! [R: partia2670972154091845814t_unit,H: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subcri7763218559781929323t_unit @ H @ R )
       => ( ( ( one_li8328186300101108157t_unit @ R )
           != ( zero_l4142658623432671053t_unit @ R ) )
         => ( ! [H12: list_a,H22: list_a] :
                ( ( member_list_a @ H12 @ H )
               => ( ( member_list_a @ H22 @ H )
                 => ( ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H22 )
                      = ( zero_l4142658623432671053t_unit @ R ) )
                   => ( ( H12
                        = ( zero_l4142658623432671053t_unit @ R ) )
                      | ( H22
                        = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) )
           => ( subdom7821232466298058046t_unit @ H @ R ) ) ) ) ) ).

% ring.subdomainI
thf(fact_758_ring_OsubdomainI,axiom,
    ! [R: partia2175431115845679010xt_a_b,H: set_a] :
      ( ( ring_a_b @ R )
     => ( ( subcring_a_b @ H @ R )
       => ( ( ( one_a_ring_ext_a_b @ R )
           != ( zero_a_b @ R ) )
         => ( ! [H12: a,H22: a] :
                ( ( member_a @ H12 @ H )
               => ( ( member_a @ H22 @ H )
                 => ( ( ( mult_a_ring_ext_a_b @ R @ H12 @ H22 )
                      = ( zero_a_b @ R ) )
                   => ( ( H12
                        = ( zero_a_b @ R ) )
                      | ( H22
                        = ( zero_a_b @ R ) ) ) ) ) )
           => ( subdomain_a_b @ H @ R ) ) ) ) ) ).

% ring.subdomainI
thf(fact_759_monoid__cancelI,axiom,
    ( ! [A3: a,B2: a,C2: a] :
        ( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A3 )
          = ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
       => ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
             => ( A3 = B2 ) ) ) ) )
   => ( ! [A3: a,B2: a,C2: a] :
          ( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C2 )
            = ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
         => ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
             => ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
               => ( A3 = B2 ) ) ) ) )
     => ( monoid5798828371819920185xt_a_b @ r ) ) ) ).

% monoid_cancelI
thf(fact_760_polynomial__pow__division,axiom,
    ! [P: list_a,N: nat,M: nat] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ M ) ) ) ) ).

% polynomial_pow_division
thf(fact_761_Units__closed,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
     => ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% Units_closed
thf(fact_762_Units__assoc,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
       => ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ).

% Units_assoc
thf(fact_763_Units__pow__closed,axiom,
    ! [X: a,D: nat] :
      ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
     => ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ D ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).

% Units_pow_closed
thf(fact_764_prod__unit__l,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
           => ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).

% prod_unit_l
thf(fact_765_prod__unit__r,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
           => ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).

% prod_unit_r
thf(fact_766_unit__factor,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).

% unit_factor
thf(fact_767_Units__inv__comm,axiom,
    ! [X: a,Y: a] :
      ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
        = ( one_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
            = ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).

% Units_inv_comm
thf(fact_768_Units__cong,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).

% Units_cong
thf(fact_769_ring__irreducibleE_I4_J,axiom,
    ! [R3: a] :
      ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_r999134135267193926le_a_b @ r @ R3 )
       => ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).

% ring_irreducibleE(4)
thf(fact_770_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_771_Units__l__inv__ex,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
     => ? [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
            = ( one_a_ring_ext_a_b @ r ) ) ) ) ).

% Units_l_inv_ex
thf(fact_772_Units__r__inv__ex,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
     => ? [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) ) ) ).

% Units_r_inv_ex
thf(fact_773_ring__associated__iff,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
          = ( ? [X3: a] :
                ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
                & ( A
                  = ( mult_a_ring_ext_a_b @ r @ X3 @ B ) ) ) ) ) ) ) ).

% ring_associated_iff
thf(fact_774_associatedI2_H,axiom,
    ! [A: a,B: a,U: a] :
      ( ( A
        = ( mult_a_ring_ext_a_b @ r @ B @ U ) )
     => ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).

% associatedI2'
thf(fact_775_associatedI2,axiom,
    ! [U: a,A: a,B: a] :
      ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( A
          = ( mult_a_ring_ext_a_b @ r @ B @ U ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
         => ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).

% associatedI2
thf(fact_776_ring__irreducibleE_I5_J,axiom,
    ! [R3: a,A: a,B: a] :
      ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_r999134135267193926le_a_b @ r @ R3 )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( R3
                = ( mult_a_ring_ext_a_b @ r @ A @ B ) )
             => ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
                | ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).

% ring_irreducibleE(5)
thf(fact_777_Units__m__closed,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
       => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).

% Units_m_closed
thf(fact_778_Units__one__closed,axiom,
    member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).

% Units_one_closed
thf(fact_779_Units__l__cancel,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
              = ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
            = ( Y = Z ) ) ) ) ) ).

% Units_l_cancel
thf(fact_780_alg__multE_I2_J,axiom,
    ! [X: a,P: list_a,N: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( P != 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 ) ) @ N ) @ P )
           => ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) ) ) ) ) ) ).

% alg_multE(2)
thf(fact_781_le__alg__mult__imp__pdivides,axiom,
    ! [X: a,P: list_a,N: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ 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 ) ) @ N ) @ P ) ) ) ) ).

% le_alg_mult_imp_pdivides
thf(fact_782_Units__minus__one__closed,axiom,
    member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).

% Units_minus_one_closed
thf(fact_783_ring_Oalg__mult_Ocong,axiom,
    polyno4422430861927485590lt_a_b = polyno4422430861927485590lt_a_b ).

% ring.alg_mult.cong
thf(fact_784_subfieldE_I3_J,axiom,
    ! [K: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subfie1779122896746047282t_unit @ K @ R )
     => ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% subfieldE(3)
thf(fact_785_subfieldE_I3_J,axiom,
    ! [K: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subfield_a_b @ K @ R )
     => ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% subfieldE(3)
thf(fact_786_subfieldE_I3_J,axiom,
    ! [K: set_list_list_a,R: partia2956882679547061052t_unit] :
      ( ( subfie4546268998243038636t_unit @ K @ R )
     => ( ord_le8488217952732425610list_a @ K @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% subfieldE(3)
thf(fact_787_subringE_I1_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subrin6918843898125473962t_unit @ H @ R )
     => ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% subringE(1)
thf(fact_788_subringE_I1_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subring_a_b @ H @ R )
     => ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% subringE(1)
thf(fact_789_subringE_I1_J,axiom,
    ! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
      ( ( subrin3541368690557094692t_unit @ H @ R )
     => ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% subringE(1)
thf(fact_790_subcringE_I1_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subcri7763218559781929323t_unit @ H @ R )
     => ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% subcringE(1)
thf(fact_791_subcringE_I1_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subcring_a_b @ H @ R )
     => ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% subcringE(1)
thf(fact_792_subcringE_I1_J,axiom,
    ! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
      ( ( subcri8676831449680469861t_unit @ H @ R )
     => ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% subcringE(1)
thf(fact_793_subdomainE_I1_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subdom7821232466298058046t_unit @ H @ R )
     => ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% subdomainE(1)
thf(fact_794_subdomainE_I1_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subdomain_a_b @ H @ R )
     => ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% subdomainE(1)
thf(fact_795_subdomainE_I1_J,axiom,
    ! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
      ( ( subdom561091866123308472t_unit @ H @ R )
     => ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% subdomainE(1)
thf(fact_796_ring_OsubringI,axiom,
    ! [R: partia2670972154091845814t_unit,H: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H )
         => ( ! [H4: list_a] :
                ( ( member_list_a @ H4 @ H )
               => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H4 ) @ H ) )
           => ( ! [H12: list_a,H22: list_a] :
                  ( ( member_list_a @ H12 @ H )
                 => ( ( member_list_a @ H22 @ H )
                   => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H22 ) @ H ) ) )
             => ( ! [H12: list_a,H22: list_a] :
                    ( ( member_list_a @ H12 @ H )
                   => ( ( member_list_a @ H22 @ H )
                     => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H22 ) @ H ) ) )
               => ( subrin6918843898125473962t_unit @ H @ R ) ) ) ) ) ) ) ).

% ring.subringI
thf(fact_797_ring_OsubringI,axiom,
    ! [R: partia2175431115845679010xt_a_b,H: set_a] :
      ( ( ring_a_b @ R )
     => ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H )
         => ( ! [H4: a] :
                ( ( member_a @ H4 @ H )
               => ( member_a @ ( a_inv_a_b @ R @ H4 ) @ H ) )
           => ( ! [H12: a,H22: a] :
                  ( ( member_a @ H12 @ H )
                 => ( ( member_a @ H22 @ H )
                   => ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H22 ) @ H ) ) )
             => ( ! [H12: a,H22: a] :
                    ( ( member_a @ H12 @ H )
                   => ( ( member_a @ H22 @ H )
                     => ( member_a @ ( add_a_b @ R @ H12 @ H22 ) @ H ) ) )
               => ( subring_a_b @ H @ R ) ) ) ) ) ) ) ).

% ring.subringI
thf(fact_798_ring_OsubringI,axiom,
    ! [R: partia2956882679547061052t_unit,H: set_list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ H )
         => ( ! [H4: list_list_a] :
                ( ( member_list_list_a @ H4 @ H )
               => ( member_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ H4 ) @ H ) )
           => ( ! [H12: list_list_a,H22: list_list_a] :
                  ( ( member_list_list_a @ H12 @ H )
                 => ( ( member_list_list_a @ H22 @ H )
                   => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ H12 @ H22 ) @ H ) ) )
             => ( ! [H12: list_list_a,H22: list_list_a] :
                    ( ( member_list_list_a @ H12 @ H )
                   => ( ( member_list_list_a @ H22 @ H )
                     => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ H12 @ H22 ) @ H ) ) )
               => ( subrin3541368690557094692t_unit @ H @ R ) ) ) ) ) ) ) ).

% ring.subringI
thf(fact_799_domain_Ole__alg__mult__imp__pdivides,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,P: list_list_a,N: nat] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
         => ( ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ R @ P @ X ) )
           => ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ N ) @ P ) ) ) ) ) ).

% domain.le_alg_mult_imp_pdivides
thf(fact_800_domain_Ole__alg__mult__imp__pdivides,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,P: list_list_list_a,N: nat] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
         => ( ( ord_less_eq_nat @ N @ ( polyno1672195411705137432t_unit @ R @ P @ X ) )
           => ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ nil_list_list_a ) ) @ N ) @ P ) ) ) ) ) ).

% domain.le_alg_mult_imp_pdivides
thf(fact_801_domain_Ole__alg__mult__imp__pdivides,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,P: list_a,N: nat] :
      ( ( domain_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
         => ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ R @ P @ 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 ) ) @ N ) @ P ) ) ) ) ) ).

% domain.le_alg_mult_imp_pdivides
thf(fact_802_domain_Oalg__multE_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,P: list_list_a,N: nat] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
         => ( ( P != nil_list_a )
           => ( ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ N ) @ P )
             => ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ R @ P @ X ) ) ) ) ) ) ) ).

% domain.alg_multE(2)
thf(fact_803_domain_Oalg__multE_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,P: list_list_list_a,N: nat] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
         => ( ( P != nil_list_list_a )
           => ( ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ nil_list_list_a ) ) @ N ) @ P )
             => ( ord_less_eq_nat @ N @ ( polyno1672195411705137432t_unit @ R @ P @ X ) ) ) ) ) ) ) ).

% domain.alg_multE(2)
thf(fact_804_domain_Oalg__multE_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,P: list_a,N: nat] :
      ( ( domain_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
         => ( ( P != 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 ) ) @ N ) @ P )
             => ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ R @ P @ X ) ) ) ) ) ) ) ).

% domain.alg_multE(2)
thf(fact_805_domain_Opolynomial__pow__division,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,N: nat,M: nat] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ N ) @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ M ) ) ) ) ) ).

% domain.polynomial_pow_division
thf(fact_806_domain_Opolynomial__pow__division,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat,M: nat] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ M ) ) ) ) ) ).

% domain.polynomial_pow_division
thf(fact_807_domain_Opolynomial__pow__division,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat,M: nat] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( ord_less_eq_nat @ N @ M )
         => ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ M ) ) ) ) ) ).

% domain.polynomial_pow_division
thf(fact_808_subringE_I2_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subrin6918843898125473962t_unit @ H @ R )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).

% subringE(2)
thf(fact_809_subringE_I2_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subring_a_b @ H @ R )
     => ( member_a @ ( zero_a_b @ R ) @ H ) ) ).

% subringE(2)
thf(fact_810_subringE_I7_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
      ( ( subrin6918843898125473962t_unit @ H @ R )
     => ( ( member_list_a @ H1 @ H )
       => ( ( member_list_a @ H2 @ H )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subringE(7)
thf(fact_811_subringE_I7_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
      ( ( subring_a_b @ H @ R )
     => ( ( member_a @ H1 @ H )
       => ( ( member_a @ H2 @ H )
         => ( member_a @ ( add_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subringE(7)
thf(fact_812_subfieldE_I4_J,axiom,
    ! [K: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ R )
     => ( ( member_list_a @ K1 @ K )
       => ( ( member_list_a @ K22 @ K )
         => ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
            = ( mult_l7073676228092353617t_unit @ R @ K22 @ K1 ) ) ) ) ) ).

% subfieldE(4)
thf(fact_813_subfieldE_I4_J,axiom,
    ! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
      ( ( subfield_a_b @ K @ R )
     => ( ( member_a @ K1 @ K )
       => ( ( member_a @ K22 @ K )
         => ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
            = ( mult_a_ring_ext_a_b @ R @ K22 @ K1 ) ) ) ) ) ).

% subfieldE(4)
thf(fact_814_subringE_I6_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
      ( ( subrin6918843898125473962t_unit @ H @ R )
     => ( ( member_list_a @ H1 @ H )
       => ( ( member_list_a @ H2 @ H )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subringE(6)
thf(fact_815_subringE_I6_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
      ( ( subring_a_b @ H @ R )
     => ( ( member_a @ H1 @ H )
       => ( ( member_a @ H2 @ H )
         => ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subringE(6)
thf(fact_816_subfieldE_I1_J,axiom,
    ! [K: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subfield_a_b @ K @ R )
     => ( subring_a_b @ K @ R ) ) ).

% subfieldE(1)
thf(fact_817_subringE_I3_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subrin6918843898125473962t_unit @ H @ R )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).

% subringE(3)
thf(fact_818_subringE_I3_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subring_a_b @ H @ R )
     => ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).

% subringE(3)
thf(fact_819_subringE_I5_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H3: list_a] :
      ( ( subrin6918843898125473962t_unit @ H @ R )
     => ( ( member_list_a @ H3 @ H )
       => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ H ) ) ) ).

% subringE(5)
thf(fact_820_subringE_I5_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H3: a] :
      ( ( subring_a_b @ H @ R )
     => ( ( member_a @ H3 @ H )
       => ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H ) ) ) ).

% subringE(5)
thf(fact_821_subcringE_I2_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subcri7763218559781929323t_unit @ H @ R )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).

% subcringE(2)
thf(fact_822_subcringE_I2_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subcring_a_b @ H @ R )
     => ( member_a @ ( zero_a_b @ R ) @ H ) ) ).

% subcringE(2)
thf(fact_823_subcringE_I7_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
      ( ( subcri7763218559781929323t_unit @ H @ R )
     => ( ( member_list_a @ H1 @ H )
       => ( ( member_list_a @ H2 @ H )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subcringE(7)
thf(fact_824_subcringE_I7_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
      ( ( subcring_a_b @ H @ R )
     => ( ( member_a @ H1 @ H )
       => ( ( member_a @ H2 @ H )
         => ( member_a @ ( add_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subcringE(7)
thf(fact_825_subcringE_I6_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
      ( ( subcri7763218559781929323t_unit @ H @ R )
     => ( ( member_list_a @ H1 @ H )
       => ( ( member_list_a @ H2 @ H )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subcringE(6)
thf(fact_826_subcringE_I6_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
      ( ( subcring_a_b @ H @ R )
     => ( ( member_a @ H1 @ H )
       => ( ( member_a @ H2 @ H )
         => ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subcringE(6)
thf(fact_827_subcring_Osub__m__comm,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
      ( ( subcri7763218559781929323t_unit @ H @ R )
     => ( ( member_list_a @ H1 @ H )
       => ( ( member_list_a @ H2 @ H )
         => ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 )
            = ( mult_l7073676228092353617t_unit @ R @ H2 @ H1 ) ) ) ) ) ).

% subcring.sub_m_comm
thf(fact_828_subcring_Osub__m__comm,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
      ( ( subcring_a_b @ H @ R )
     => ( ( member_a @ H1 @ H )
       => ( ( member_a @ H2 @ H )
         => ( ( mult_a_ring_ext_a_b @ R @ H1 @ H2 )
            = ( mult_a_ring_ext_a_b @ R @ H2 @ H1 ) ) ) ) ) ).

% subcring.sub_m_comm
thf(fact_829_subfieldE_I2_J,axiom,
    ! [K: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subfield_a_b @ K @ R )
     => ( subcring_a_b @ K @ R ) ) ).

% subfieldE(2)
thf(fact_830_subcringE_I3_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subcri7763218559781929323t_unit @ H @ R )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).

% subcringE(3)
thf(fact_831_subcringE_I3_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subcring_a_b @ H @ R )
     => ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).

% subcringE(3)
thf(fact_832_subcring_Oaxioms_I1_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subcring_a_b @ H @ R )
     => ( subring_a_b @ H @ R ) ) ).

% subcring.axioms(1)
thf(fact_833_subdomainE_I2_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subdom7821232466298058046t_unit @ H @ R )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).

% subdomainE(2)
thf(fact_834_subdomainE_I2_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subdomain_a_b @ H @ R )
     => ( member_a @ ( zero_a_b @ R ) @ H ) ) ).

% subdomainE(2)
thf(fact_835_subcringE_I5_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H3: list_a] :
      ( ( subcri7763218559781929323t_unit @ H @ R )
     => ( ( member_list_a @ H3 @ H )
       => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ H ) ) ) ).

% subcringE(5)
thf(fact_836_subcringE_I5_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H3: a] :
      ( ( subcring_a_b @ H @ R )
     => ( ( member_a @ H3 @ H )
       => ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H ) ) ) ).

% subcringE(5)
thf(fact_837_subdomainE_I7_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
      ( ( subdom7821232466298058046t_unit @ H @ R )
     => ( ( member_list_a @ H1 @ H )
       => ( ( member_list_a @ H2 @ H )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subdomainE(7)
thf(fact_838_subdomainE_I7_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
      ( ( subdomain_a_b @ H @ R )
     => ( ( member_a @ H1 @ H )
       => ( ( member_a @ H2 @ H )
         => ( member_a @ ( add_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subdomainE(7)
thf(fact_839_subdomainE_I6_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
      ( ( subdom7821232466298058046t_unit @ H @ R )
     => ( ( member_list_a @ H1 @ H )
       => ( ( member_list_a @ H2 @ H )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subdomainE(6)
thf(fact_840_subdomainE_I6_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
      ( ( subdomain_a_b @ H @ R )
     => ( ( member_a @ H1 @ H )
       => ( ( member_a @ H2 @ H )
         => ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H2 ) @ H ) ) ) ) ).

% subdomainE(6)
thf(fact_841_subdomainE_I8_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
      ( ( subdom7821232466298058046t_unit @ H @ R )
     => ( ( member_list_a @ H1 @ H )
       => ( ( member_list_a @ H2 @ H )
         => ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 )
            = ( mult_l7073676228092353617t_unit @ R @ H2 @ H1 ) ) ) ) ) ).

% subdomainE(8)
thf(fact_842_subdomainE_I8_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
      ( ( subdomain_a_b @ H @ R )
     => ( ( member_a @ H1 @ H )
       => ( ( member_a @ H2 @ H )
         => ( ( mult_a_ring_ext_a_b @ R @ H1 @ H2 )
            = ( mult_a_ring_ext_a_b @ R @ H2 @ H1 ) ) ) ) ) ).

% subdomainE(8)
thf(fact_843_subfield_Oaxioms_I1_J,axiom,
    ! [K: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subfield_a_b @ K @ R )
     => ( subdomain_a_b @ K @ R ) ) ).

% subfield.axioms(1)
thf(fact_844_subdomainE_I3_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subdom7821232466298058046t_unit @ H @ R )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).

% subdomainE(3)
thf(fact_845_subdomainE_I3_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subdomain_a_b @ H @ R )
     => ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).

% subdomainE(3)
thf(fact_846_subdomainE_I5_J,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H3: list_a] :
      ( ( subdom7821232466298058046t_unit @ H @ R )
     => ( ( member_list_a @ H3 @ H )
       => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ H ) ) ) ).

% subdomainE(5)
thf(fact_847_subdomainE_I5_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H3: a] :
      ( ( subdomain_a_b @ H @ R )
     => ( ( member_a @ H3 @ H )
       => ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H ) ) ) ).

% subdomainE(5)
thf(fact_848_subdomain_Oaxioms_I1_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subdomain_a_b @ H @ R )
     => ( subcring_a_b @ H @ R ) ) ).

% subdomain.axioms(1)
thf(fact_849_domain_Opoly__mult__var,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( ( P = nil_a )
             => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ ( var_a_b @ R ) )
                = nil_a ) )
            & ( ( P != nil_a )
             => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ ( var_a_b @ R ) )
                = ( append_a @ P @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ) ).

% domain.poly_mult_var
thf(fact_850_domain_Opoly__mult__var,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( ( P = nil_list_a )
             => ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ ( var_li8453953174693405341t_unit @ R ) )
                = nil_list_a ) )
            & ( ( P != nil_list_a )
             => ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ ( var_li8453953174693405341t_unit @ R ) )
                = ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ) ) ).

% domain.poly_mult_var
thf(fact_851_subfieldE_I5_J,axiom,
    ! [K: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ R )
     => ( ( member_list_a @ K1 @ K )
       => ( ( member_list_a @ K22 @ K )
         => ( ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
              = ( zero_l4142658623432671053t_unit @ R ) )
           => ( ( K1
                = ( zero_l4142658623432671053t_unit @ R ) )
              | ( K22
                = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).

% subfieldE(5)
thf(fact_852_subfieldE_I5_J,axiom,
    ! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
      ( ( subfield_a_b @ K @ R )
     => ( ( member_a @ K1 @ K )
       => ( ( member_a @ K22 @ K )
         => ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
              = ( zero_a_b @ R ) )
           => ( ( K1
                = ( zero_a_b @ R ) )
              | ( K22
                = ( zero_a_b @ R ) ) ) ) ) ) ) ).

% subfieldE(5)
thf(fact_853_ring_Ocarrier__is__subring,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).

% ring.carrier_is_subring
thf(fact_854_ring_Ocarrier__is__subring,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R )
     => ( subring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).

% ring.carrier_is_subring
thf(fact_855_ring_Ocarrier__is__subring,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( subrin3541368690557094692t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).

% ring.carrier_is_subring
thf(fact_856_subfieldE_I6_J,axiom,
    ! [K: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subfie1779122896746047282t_unit @ K @ R )
     => ( ( one_li8328186300101108157t_unit @ R )
       != ( zero_l4142658623432671053t_unit @ R ) ) ) ).

% subfieldE(6)
thf(fact_857_subfieldE_I6_J,axiom,
    ! [K: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subfield_a_b @ K @ R )
     => ( ( one_a_ring_ext_a_b @ R )
       != ( zero_a_b @ R ) ) ) ).

% subfieldE(6)
thf(fact_858_subdomain_Osubintegral,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H2: list_a] :
      ( ( subdom7821232466298058046t_unit @ H @ R )
     => ( ( member_list_a @ H1 @ H )
       => ( ( member_list_a @ H2 @ H )
         => ( ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 )
              = ( zero_l4142658623432671053t_unit @ R ) )
           => ( ( H1
                = ( zero_l4142658623432671053t_unit @ R ) )
              | ( H2
                = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).

% subdomain.subintegral
thf(fact_859_subdomain_Osubintegral,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H2: a] :
      ( ( subdomain_a_b @ H @ R )
     => ( ( member_a @ H1 @ H )
       => ( ( member_a @ H2 @ H )
         => ( ( ( mult_a_ring_ext_a_b @ R @ H1 @ H2 )
              = ( zero_a_b @ R ) )
           => ( ( H1
                = ( zero_a_b @ R ) )
              | ( H2
                = ( zero_a_b @ R ) ) ) ) ) ) ) ).

% subdomain.subintegral
thf(fact_860_cring_Ocarrier__is__subcring,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).

% cring.carrier_is_subcring
thf(fact_861_cring_Ocarrier__is__subcring,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( cring_a_b @ R )
     => ( subcring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).

% cring.carrier_is_subcring
thf(fact_862_cring_Ocarrier__is__subcring,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( subcri8676831449680469861t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).

% cring.carrier_is_subcring
thf(fact_863_subdomain_Osub__one__not__zero,axiom,
    ! [H: set_list_a,R: partia2670972154091845814t_unit] :
      ( ( subdom7821232466298058046t_unit @ H @ R )
     => ( ( one_li8328186300101108157t_unit @ R )
       != ( zero_l4142658623432671053t_unit @ R ) ) ) ).

% subdomain.sub_one_not_zero
thf(fact_864_subdomain_Osub__one__not__zero,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subdomain_a_b @ H @ R )
     => ( ( one_a_ring_ext_a_b @ R )
       != ( zero_a_b @ R ) ) ) ).

% subdomain.sub_one_not_zero
thf(fact_865_cring_OsubcringI_H,axiom,
    ! [R: partia2670972154091845814t_unit,H: set_list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ H @ R )
       => ( subcri7763218559781929323t_unit @ H @ R ) ) ) ).

% cring.subcringI'
thf(fact_866_cring_OsubcringI_H,axiom,
    ! [R: partia2175431115845679010xt_a_b,H: set_a] :
      ( ( cring_a_b @ R )
     => ( ( subring_a_b @ H @ R )
       => ( subcring_a_b @ H @ R ) ) ) ).

% cring.subcringI'
thf(fact_867_domain_OsubdomainI_H,axiom,
    ! [R: partia2670972154091845814t_unit,H: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ H @ R )
       => ( subdom7821232466298058046t_unit @ H @ R ) ) ) ).

% domain.subdomainI'
thf(fact_868_domain_OsubdomainI_H,axiom,
    ! [R: partia2175431115845679010xt_a_b,H: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ H @ R )
       => ( subdomain_a_b @ H @ R ) ) ) ).

% domain.subdomainI'
thf(fact_869_ring_OsubcringI,axiom,
    ! [R: partia2670972154091845814t_unit,H: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ H @ R )
       => ( ! [H12: list_a,H22: list_a] :
              ( ( member_list_a @ H12 @ H )
             => ( ( member_list_a @ H22 @ H )
               => ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H22 )
                  = ( mult_l7073676228092353617t_unit @ R @ H22 @ H12 ) ) ) )
         => ( subcri7763218559781929323t_unit @ H @ R ) ) ) ) ).

% ring.subcringI
thf(fact_870_ring_OsubcringI,axiom,
    ! [R: partia2175431115845679010xt_a_b,H: set_a] :
      ( ( ring_a_b @ R )
     => ( ( subring_a_b @ H @ R )
       => ( ! [H12: a,H22: a] :
              ( ( member_a @ H12 @ H )
             => ( ( member_a @ H22 @ H )
               => ( ( mult_a_ring_ext_a_b @ R @ H12 @ H22 )
                  = ( mult_a_ring_ext_a_b @ R @ H22 @ H12 ) ) ) )
         => ( subcring_a_b @ H @ R ) ) ) ) ).

% ring.subcringI
thf(fact_871_domain_Oalg__multE_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
         => ( ( P != nil_list_a )
           => ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ R @ P @ X ) ) @ P ) ) ) ) ) ).

% domain.alg_multE(1)
thf(fact_872_domain_Oalg__multE_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
         => ( ( P != nil_list_list_a )
           => ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X ) @ nil_list_list_a ) ) @ ( polyno1672195411705137432t_unit @ R @ P @ X ) ) @ P ) ) ) ) ) ).

% domain.alg_multE(1)
thf(fact_873_domain_Oalg__multE_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
         => ( ( P != 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 @ P @ X ) ) @ P ) ) ) ) ) ).

% domain.alg_multE(1)
thf(fact_874_units__of__pow,axiom,
    ! [X: a,N: nat] :
      ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
     => ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ r ) @ X @ N )
        = ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ).

% units_of_pow
thf(fact_875_append1__eq__conv,axiom,
    ! [Xs: list_a,X: a,Ys: list_a,Y: a] :
      ( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
        = ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_876_const__term__explicit,axiom,
    ! [P: list_a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( P != nil_a )
       => ( ( ( const_term_a_b @ r @ P )
            = A )
         => ~ ! [P3: list_a] :
                ( ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( partia707051561876973205xt_a_b @ r ) )
               => ( P
                 != ( append_a @ P3 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).

% const_term_explicit
thf(fact_877_const__term__eq__last,axiom,
    ! [P: list_a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( const_term_a_b @ r @ ( append_a @ P @ ( cons_a @ A @ nil_a ) ) )
          = A ) ) ) ).

% const_term_eq_last
thf(fact_878_add_Oone__in__subset,axiom,
    ! [H: set_a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( H != bot_bot_set_a )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ H )
             => ( member_a @ ( a_inv_a_b @ r @ X2 ) @ H ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ H )
               => ! [Xa2: a] :
                    ( ( member_a @ Xa2 @ H )
                   => ( member_a @ ( add_a_b @ r @ X2 @ Xa2 ) @ H ) ) )
           => ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).

% add.one_in_subset
thf(fact_879_carrier__not__empty,axiom,
    ( ( partia707051561876973205xt_a_b @ r )
   != bot_bot_set_a ) ).

% carrier_not_empty
thf(fact_880_subring__props_I4_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( K != bot_bot_set_a ) ) ).

% subring_props(4)
thf(fact_881_list_Oinject,axiom,
    ! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
      ( ( ( cons_a @ X21 @ X22 )
        = ( cons_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_882_const__term__simprules_I1_J,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ ( const_term_a_b @ r @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% const_term_simprules(1)
thf(fact_883_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = nil_a )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% append_is_Nil_conv
thf(fact_884_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( nil_a
        = ( append_a @ Xs @ Ys ) )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% Nil_is_append_conv
thf(fact_885_self__append__conv2,axiom,
    ! [Y: list_a,Xs: list_a] :
      ( ( Y
        = ( append_a @ Xs @ Y ) )
      = ( Xs = nil_a ) ) ).

% self_append_conv2
thf(fact_886_append__self__conv2,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_a ) ) ).

% append_self_conv2
thf(fact_887_self__append__conv,axiom,
    ! [Y: list_a,Ys: list_a] :
      ( ( Y
        = ( append_a @ Y @ Ys ) )
      = ( Ys = nil_a ) ) ).

% self_append_conv
thf(fact_888_append__self__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_a ) ) ).

% append_self_conv
thf(fact_889_append__Nil2,axiom,
    ! [Xs: list_a] :
      ( ( append_a @ Xs @ nil_a )
      = Xs ) ).

% append_Nil2
thf(fact_890_append_Oright__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ A @ nil_a )
      = A ) ).

% append.right_neutral
thf(fact_891_units__of__units,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( units_8735880885477018085t_unit @ ( units_6477118173342999439t_unit @ G ) )
      = ( units_2932844235741507942t_unit @ G ) ) ).

% units_of_units
thf(fact_892_units__of__units,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( units_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
      = ( units_a_ring_ext_a_b @ G ) ) ).

% units_of_units
thf(fact_893_set__empty,axiom,
    ! [Xs: list_a] :
      ( ( ( set_a2 @ Xs )
        = bot_bot_set_a )
      = ( Xs = nil_a ) ) ).

% set_empty
thf(fact_894_set__empty,axiom,
    ! [Xs: list_list_a] :
      ( ( ( set_list_a2 @ Xs )
        = bot_bot_set_list_a )
      = ( Xs = nil_list_a ) ) ).

% set_empty
thf(fact_895_set__empty2,axiom,
    ! [Xs: list_a] :
      ( ( bot_bot_set_a
        = ( set_a2 @ Xs ) )
      = ( Xs = nil_a ) ) ).

% set_empty2
thf(fact_896_set__empty2,axiom,
    ! [Xs: list_list_a] :
      ( ( bot_bot_set_list_a
        = ( set_list_a2 @ Xs ) )
      = ( Xs = nil_list_a ) ) ).

% set_empty2
thf(fact_897_monom__in__carrier,axiom,
    ! [A: a,N: nat] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% monom_in_carrier
thf(fact_898_list_Oset__intros_I2_J,axiom,
    ! [Y: list_a,X22: list_list_a,X21: list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ X22 ) )
     => ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_899_list_Oset__intros_I2_J,axiom,
    ! [Y: list_list_a,X22: list_list_list_a,X21: list_list_a] :
      ( ( member_list_list_a @ Y @ ( set_list_list_a2 @ X22 ) )
     => ( member_list_list_a @ Y @ ( set_list_list_a2 @ ( cons_list_list_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_900_list_Oset__intros_I2_J,axiom,
    ! [Y: a,X22: list_a,X21: a] :
      ( ( member_a @ Y @ ( set_a2 @ X22 ) )
     => ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_901_list_Oset__intros_I1_J,axiom,
    ! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_902_list_Oset__intros_I1_J,axiom,
    ! [X21: list_list_a,X22: list_list_list_a] : ( member_list_list_a @ X21 @ ( set_list_list_a2 @ ( cons_list_list_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_903_list_Oset__intros_I1_J,axiom,
    ! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_904_list_Oset__cases,axiom,
    ! [E: list_a,A: list_list_a] :
      ( ( member_list_a @ E @ ( set_list_a2 @ A ) )
     => ( ! [Z22: list_list_a] :
            ( A
           != ( cons_list_a @ E @ Z22 ) )
       => ~ ! [Z1: list_a,Z22: list_list_a] :
              ( ( A
                = ( cons_list_a @ Z1 @ Z22 ) )
             => ~ ( member_list_a @ E @ ( set_list_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_905_list_Oset__cases,axiom,
    ! [E: list_list_a,A: list_list_list_a] :
      ( ( member_list_list_a @ E @ ( set_list_list_a2 @ A ) )
     => ( ! [Z22: list_list_list_a] :
            ( A
           != ( cons_list_list_a @ E @ Z22 ) )
       => ~ ! [Z1: list_list_a,Z22: list_list_list_a] :
              ( ( A
                = ( cons_list_list_a @ Z1 @ Z22 ) )
             => ~ ( member_list_list_a @ E @ ( set_list_list_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_906_list_Oset__cases,axiom,
    ! [E: a,A: list_a] :
      ( ( member_a @ E @ ( set_a2 @ A ) )
     => ( ! [Z22: list_a] :
            ( A
           != ( cons_a @ E @ Z22 ) )
       => ~ ! [Z1: a,Z22: list_a] :
              ( ( A
                = ( cons_a @ Z1 @ Z22 ) )
             => ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_907_set__ConsD,axiom,
    ! [Y: list_a,X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_908_set__ConsD,axiom,
    ! [Y: list_list_a,X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ Y @ ( set_list_list_a2 @ ( cons_list_list_a @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_list_list_a @ Y @ ( set_list_list_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_909_set__ConsD,axiom,
    ! [Y: a,X: a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_910_empty__set,axiom,
    ( bot_bot_set_a
    = ( set_a2 @ nil_a ) ) ).

% empty_set
thf(fact_911_empty__set,axiom,
    ( bot_bot_set_list_a
    = ( set_list_a2 @ nil_list_a ) ) ).

% empty_set
thf(fact_912_subringE_I4_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subring_a_b @ H @ R )
     => ( H != bot_bot_set_a ) ) ).

% subringE(4)
thf(fact_913_set__subset__Cons,axiom,
    ! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_914_split__list__first__prop__iff,axiom,
    ! [Xs: list_a,P2: a > $o] :
      ( ( ? [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
            & ( P2 @ X3 ) ) )
      = ( ? [Ys2: list_a,X3: a] :
            ( ? [Zs: list_a] :
                ( Xs
                = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs ) ) )
            & ( P2 @ X3 )
            & ! [Y4: a] :
                ( ( member_a @ Y4 @ ( set_a2 @ Ys2 ) )
               => ~ ( P2 @ Y4 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_915_split__list__last__prop__iff,axiom,
    ! [Xs: list_a,P2: a > $o] :
      ( ( ? [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
            & ( P2 @ X3 ) ) )
      = ( ? [Ys2: list_a,X3: a,Zs: list_a] :
            ( ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs ) ) )
            & ( P2 @ X3 )
            & ! [Y4: a] :
                ( ( member_a @ Y4 @ ( set_a2 @ Zs ) )
               => ~ ( P2 @ Y4 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_916_in__set__conv__decomp__first,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
      = ( ? [Ys2: list_list_a,Zs: list_list_a] :
            ( ( Xs
              = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs ) ) )
            & ~ ( member_list_a @ X @ ( set_list_a2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_917_in__set__conv__decomp__first,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
      = ( ? [Ys2: list_list_list_a,Zs: list_list_list_a] :
            ( ( Xs
              = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X @ Zs ) ) )
            & ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_918_in__set__conv__decomp__first,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [Ys2: list_a,Zs: list_a] :
            ( ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X @ Zs ) ) )
            & ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_919_in__set__conv__decomp__last,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
      = ( ? [Ys2: list_list_a,Zs: list_list_a] :
            ( ( Xs
              = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs ) ) )
            & ~ ( member_list_a @ X @ ( set_list_a2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_920_in__set__conv__decomp__last,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
      = ( ? [Ys2: list_list_list_a,Zs: list_list_list_a] :
            ( ( Xs
              = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X @ Zs ) ) )
            & ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_921_in__set__conv__decomp__last,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [Ys2: list_a,Zs: list_a] :
            ( ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X @ Zs ) ) )
            & ~ ( member_a @ X @ ( set_a2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_922_split__list__first__propE,axiom,
    ! [Xs: list_a,P2: a > $o] :
      ( ? [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
          & ( P2 @ X4 ) )
     => ~ ! [Ys3: list_a,X2: a] :
            ( ? [Zs2: list_a] :
                ( Xs
                = ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
           => ( ( P2 @ X2 )
             => ~ ! [Xa: a] :
                    ( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_923_split__list__last__propE,axiom,
    ! [Xs: list_a,P2: a > $o] :
      ( ? [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
          & ( P2 @ X4 ) )
     => ~ ! [Ys3: list_a,X2: a,Zs2: list_a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
           => ( ( P2 @ X2 )
             => ~ ! [Xa: a] :
                    ( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_924_split__list__first__prop,axiom,
    ! [Xs: list_a,P2: a > $o] :
      ( ? [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
          & ( P2 @ X4 ) )
     => ? [Ys3: list_a,X2: a] :
          ( ? [Zs2: list_a] :
              ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
          & ( P2 @ X2 )
          & ! [Xa: a] :
              ( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_925_split__list__last__prop,axiom,
    ! [Xs: list_a,P2: a > $o] :
      ( ? [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
          & ( P2 @ X4 ) )
     => ? [Ys3: list_a,X2: a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
          & ( P2 @ X2 )
          & ! [Xa: a] :
              ( ( member_a @ Xa @ ( set_a2 @ Zs2 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_926_in__set__conv__decomp,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
      = ( ? [Ys2: list_list_a,Zs: list_list_a] :
            ( Xs
            = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_927_in__set__conv__decomp,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
      = ( ? [Ys2: list_list_list_a,Zs: list_list_list_a] :
            ( Xs
            = ( append_list_list_a @ Ys2 @ ( cons_list_list_a @ X @ Zs ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_928_in__set__conv__decomp,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [Ys2: list_a,Zs: list_a] :
            ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X @ Zs ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_929_append__Cons__eq__iff,axiom,
    ! [X: list_a,Xs: list_list_a,Ys: list_list_a,Xs2: list_list_a,Ys4: list_list_a] :
      ( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
       => ( ( ( append_list_a @ Xs @ ( cons_list_a @ X @ Ys ) )
            = ( append_list_a @ Xs2 @ ( cons_list_a @ X @ Ys4 ) ) )
          = ( ( Xs = Xs2 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_930_append__Cons__eq__iff,axiom,
    ! [X: list_list_a,Xs: list_list_list_a,Ys: list_list_list_a,Xs2: list_list_list_a,Ys4: list_list_list_a] :
      ( ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
     => ( ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Ys ) )
       => ( ( ( append_list_list_a @ Xs @ ( cons_list_list_a @ X @ Ys ) )
            = ( append_list_list_a @ Xs2 @ ( cons_list_list_a @ X @ Ys4 ) ) )
          = ( ( Xs = Xs2 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_931_append__Cons__eq__iff,axiom,
    ! [X: a,Xs: list_a,Ys: list_a,Xs2: list_a,Ys4: list_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs ) )
     => ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
       => ( ( ( append_a @ Xs @ ( cons_a @ X @ Ys ) )
            = ( append_a @ Xs2 @ ( cons_a @ X @ Ys4 ) ) )
          = ( ( Xs = Xs2 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_932_split__list__propE,axiom,
    ! [Xs: list_a,P2: a > $o] :
      ( ? [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
          & ( P2 @ X4 ) )
     => ~ ! [Ys3: list_a,X2: a] :
            ( ? [Zs2: list_a] :
                ( Xs
                = ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
           => ~ ( P2 @ X2 ) ) ) ).

% split_list_propE
thf(fact_933_split__list__first,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ? [Ys3: list_list_a,Zs2: list_list_a] :
          ( ( Xs
            = ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) )
          & ~ ( member_list_a @ X @ ( set_list_a2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_934_split__list__first,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
     => ? [Ys3: list_list_list_a,Zs2: list_list_list_a] :
          ( ( Xs
            = ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X @ Zs2 ) ) )
          & ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_935_split__list__first,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ? [Ys3: list_a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
          & ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_936_split__list__prop,axiom,
    ! [Xs: list_a,P2: a > $o] :
      ( ? [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
          & ( P2 @ X4 ) )
     => ? [Ys3: list_a,X2: a] :
          ( ? [Zs2: list_a] :
              ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ X2 @ Zs2 ) ) )
          & ( P2 @ X2 ) ) ) ).

% split_list_prop
thf(fact_937_split__list__last,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ? [Ys3: list_list_a,Zs2: list_list_a] :
          ( ( Xs
            = ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) )
          & ~ ( member_list_a @ X @ ( set_list_a2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_938_split__list__last,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
     => ? [Ys3: list_list_list_a,Zs2: list_list_list_a] :
          ( ( Xs
            = ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X @ Zs2 ) ) )
          & ~ ( member_list_list_a @ X @ ( set_list_list_a2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_939_split__list__last,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ? [Ys3: list_a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
          & ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_940_split__list,axiom,
    ! [X: list_a,Xs: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
     => ? [Ys3: list_list_a,Zs2: list_list_a] :
          ( Xs
          = ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_941_split__list,axiom,
    ! [X: list_list_a,Xs: list_list_list_a] :
      ( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
     => ? [Ys3: list_list_list_a,Zs2: list_list_list_a] :
          ( Xs
          = ( append_list_list_a @ Ys3 @ ( cons_list_list_a @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_942_split__list,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ? [Ys3: list_a,Zs2: list_a] :
          ( Xs
          = ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) ) ) ).

% split_list
thf(fact_943_subcringE_I4_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subcring_a_b @ H @ R )
     => ( H != bot_bot_set_a ) ) ).

% subcringE(4)
thf(fact_944_subdomainE_I4_J,axiom,
    ! [H: set_a,R: partia2175431115845679010xt_a_b] :
      ( ( subdomain_a_b @ H @ R )
     => ( H != bot_bot_set_a ) ) ).

% subdomainE(4)
thf(fact_945_units__of__mult,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( mult_l6995149843440949818t_unit @ ( units_6477118173342999439t_unit @ G ) )
      = ( mult_l7073676228092353617t_unit @ G ) ) ).

% units_of_mult
thf(fact_946_units__of__mult,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( mult_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
      = ( mult_a_ring_ext_a_b @ G ) ) ).

% units_of_mult
thf(fact_947_units__of__one,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( one_li6878281577851457998t_unit @ ( units_6477118173342999439t_unit @ G ) )
      = ( one_li8328186300101108157t_unit @ G ) ) ).

% units_of_one
thf(fact_948_units__of__one,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( one_a_Product_unit @ ( units_8174867845824275201xt_a_b @ G ) )
      = ( one_a_ring_ext_a_b @ G ) ) ).

% units_of_one
thf(fact_949_units__of__carrier,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( partia7074150537345710456t_unit @ ( units_6477118173342999439t_unit @ G ) )
      = ( units_2932844235741507942t_unit @ G ) ) ).

% units_of_carrier
thf(fact_950_units__of__carrier,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( partia6735698275553448452t_unit @ ( units_8174867845824275201xt_a_b @ G ) )
      = ( units_a_ring_ext_a_b @ G ) ) ).

% units_of_carrier
thf(fact_951_monoid_Ocarrier__not__empty,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( partia5361259788508890537t_unit @ G )
       != bot_bot_set_list_a ) ) ).

% monoid.carrier_not_empty
thf(fact_952_monoid_Ocarrier__not__empty,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( partia707051561876973205xt_a_b @ G )
       != bot_bot_set_a ) ) ).

% monoid.carrier_not_empty
thf(fact_953_monoid_Ocarrier__not__empty,axiom,
    ! [G: partia2956882679547061052t_unit] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( partia2464479390973590831t_unit @ G )
       != bot_bo1875519244922727510list_a ) ) ).

% monoid.carrier_not_empty
thf(fact_954_not__Cons__self2,axiom,
    ! [X: a,Xs: list_a] :
      ( ( cons_a @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_955_ring_Omonom__in__carrier,axiom,
    ! [R: partia2670972154091845814t_unit,A: list_a,N: nat] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
       => ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( monom_7446464087056152608t_unit @ R @ A @ N ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).

% ring.monom_in_carrier
thf(fact_956_ring_Omonom__in__carrier,axiom,
    ! [R: partia2175431115845679010xt_a_b,A: a,N: nat] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ R @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).

% ring.monom_in_carrier
thf(fact_957_ring_Omonom__in__carrier,axiom,
    ! [R: partia2956882679547061052t_unit,A: list_list_a,N: nat] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
       => ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ ( monom_4043874212805408666t_unit @ R @ A @ N ) ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).

% ring.monom_in_carrier
thf(fact_958_cring_Oconst__term__simprules_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a] :
      ( ( cring_3148771470849435808t_unit @ R )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
       => ( member_list_a @ ( const_6738166269504826821t_unit @ R @ P ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).

% cring.const_term_simprules(1)
thf(fact_959_cring_Oconst__term__simprules_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a] :
      ( ( cring_a_b @ R )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
       => ( member_a @ ( const_term_a_b @ R @ P ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).

% cring.const_term_simprules(1)
thf(fact_960_cring_Oconst__term__simprules_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
      ( ( cring_5991999922451032090t_unit @ R )
     => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
       => ( member_list_list_a @ ( const_6243872422735025855t_unit @ R @ P ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).

% cring.const_term_simprules(1)
thf(fact_961_monoid_Ounits__of__pow,axiom,
    ! [G: partia8223610829204095565t_unit,X: a,N: nat] :
      ( ( monoid2746444814143937472t_unit @ G )
     => ( ( member_a @ X @ ( units_a_Product_unit @ G ) )
       => ( ( pow_a_1875594501834816709it_nat @ ( units_7501539392726747778t_unit @ G ) @ X @ N )
          = ( pow_a_1875594501834816709it_nat @ G @ X @ N ) ) ) ) ).

% monoid.units_of_pow
thf(fact_962_monoid_Ounits__of__pow,axiom,
    ! [G: partia2670972154091845814t_unit,X: list_a,N: nat] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
       => ( ( pow_li8657086744513738943it_nat @ ( units_6477118173342999439t_unit @ G ) @ X @ N )
          = ( pow_li1142815632869257134it_nat @ G @ X @ N ) ) ) ) ).

% monoid.units_of_pow
thf(fact_963_monoid_Ounits__of__pow,axiom,
    ! [G: partia2175431115845679010xt_a_b,X: a,N: nat] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
       => ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ G ) @ X @ N )
          = ( pow_a_1026414303147256608_b_nat @ G @ X @ N ) ) ) ) ).

% monoid.units_of_pow
thf(fact_964_list_Odistinct_I1_J,axiom,
    ! [X21: a,X22: list_a] :
      ( nil_a
     != ( cons_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_965_list_OdiscI,axiom,
    ! [List: list_a,X21: a,X22: list_a] :
      ( ( List
        = ( cons_a @ X21 @ X22 ) )
     => ( List != nil_a ) ) ).

% list.discI
thf(fact_966_list_Oexhaust,axiom,
    ! [Y: list_a] :
      ( ( Y != nil_a )
     => ~ ! [X212: a,X222: list_a] :
            ( Y
           != ( cons_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_967_transpose_Ocases,axiom,
    ! [X: list_list_a] :
      ( ( X != nil_list_a )
     => ( ! [Xss: list_list_a] :
            ( X
           != ( cons_list_a @ nil_a @ Xss ) )
       => ~ ! [X2: a,Xs3: list_a,Xss: list_list_a] :
              ( X
             != ( cons_list_a @ ( cons_a @ X2 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_968_remdups__adj_Ocases,axiom,
    ! [X: list_a] :
      ( ( X != nil_a )
     => ( ! [X2: a] :
            ( X
           != ( cons_a @ X2 @ nil_a ) )
       => ~ ! [X2: a,Y3: a,Xs3: list_a] :
              ( X
             != ( cons_a @ X2 @ ( cons_a @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_969_neq__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
      = ( ? [Y4: a,Ys2: list_a] :
            ( Xs
            = ( cons_a @ Y4 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_970_list__induct2_H,axiom,
    ! [P2: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
      ( ( P2 @ nil_a @ nil_a )
     => ( ! [X2: a,Xs3: list_a] : ( P2 @ ( cons_a @ X2 @ Xs3 ) @ nil_a )
       => ( ! [Y3: a,Ys3: list_a] : ( P2 @ nil_a @ ( cons_a @ Y3 @ Ys3 ) )
         => ( ! [X2: a,Xs3: list_a,Y3: a,Ys3: list_a] :
                ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_971_list__nonempty__induct,axiom,
    ! [Xs: list_a,P2: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X2: a] : ( P2 @ ( cons_a @ X2 @ nil_a ) )
       => ( ! [X2: a,Xs3: list_a] :
              ( ( Xs3 != nil_a )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( cons_a @ X2 @ Xs3 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_972_Cons__eq__appendI,axiom,
    ! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs3: list_a] :
      ( ( ( cons_a @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_a @ Xs1 @ Zs3 ) )
       => ( ( cons_a @ X @ Xs )
          = ( append_a @ Ys @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_973_append__Cons,axiom,
    ! [X: a,Xs: list_a,Ys: list_a] :
      ( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
      = ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_974_append__Nil,axiom,
    ! [Ys: list_a] :
      ( ( append_a @ nil_a @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_975_append_Oleft__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ nil_a @ A )
      = A ) ).

% append.left_neutral
thf(fact_976_eq__Nil__appendI,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_a @ nil_a @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_977_ring_Oconst__term__eq__last,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,A: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( const_6738166269504826821t_unit @ R @ ( append_list_a @ P @ ( cons_list_a @ A @ nil_list_a ) ) )
            = A ) ) ) ) ).

% ring.const_term_eq_last
thf(fact_978_ring_Oconst__term__eq__last,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,A: a] :
      ( ( ring_a_b @ R )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( const_term_a_b @ R @ ( append_a @ P @ ( cons_a @ A @ nil_a ) ) )
            = A ) ) ) ) ).

% ring.const_term_eq_last
thf(fact_979_ring_Oconst__term__eq__last,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,A: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( const_6243872422735025855t_unit @ R @ ( append_list_list_a @ P @ ( cons_list_list_a @ A @ nil_list_list_a ) ) )
            = A ) ) ) ) ).

% ring.const_term_eq_last
thf(fact_980_ring_Oconst__term__explicit,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,A: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( P != nil_list_a )
         => ( ( ( const_6738166269504826821t_unit @ R @ P )
              = A )
           => ~ ! [P3: list_list_a] :
                  ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ R ) )
                 => ( P
                   != ( append_list_a @ P3 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ) ).

% ring.const_term_explicit
thf(fact_981_ring_Oconst__term__explicit,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,A: a] :
      ( ( ring_a_b @ R )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( P != nil_a )
         => ( ( ( const_term_a_b @ R @ P )
              = A )
           => ~ ! [P3: list_a] :
                  ( ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( partia707051561876973205xt_a_b @ R ) )
                 => ( P
                   != ( append_a @ P3 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ) ).

% ring.const_term_explicit
thf(fact_982_ring_Oconst__term__explicit,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,A: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( P != nil_list_list_a )
         => ( ( ( const_6243872422735025855t_unit @ R @ P )
              = A )
           => ~ ! [P3: list_list_list_a] :
                  ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P3 ) @ ( partia2464479390973590831t_unit @ R ) )
                 => ( P
                   != ( append_list_list_a @ P3 @ ( cons_list_list_a @ A @ nil_list_list_a ) ) ) ) ) ) ) ) ).

% ring.const_term_explicit
thf(fact_983_rev__induct,axiom,
    ! [P2: list_a > $o,Xs: list_a] :
      ( ( P2 @ nil_a )
     => ( ! [X2: a,Xs3: list_a] :
            ( ( P2 @ Xs3 )
           => ( P2 @ ( append_a @ Xs3 @ ( cons_a @ X2 @ nil_a ) ) ) )
       => ( P2 @ Xs ) ) ) ).

% rev_induct
thf(fact_984_rev__exhaust,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ~ ! [Ys3: list_a,Y3: a] :
            ( Xs
           != ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).

% rev_exhaust
thf(fact_985_Cons__eq__append__conv,axiom,
    ! [X: a,Xs: list_a,Ys: list_a,Zs3: list_a] :
      ( ( ( cons_a @ X @ Xs )
        = ( append_a @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_a )
          & ( ( cons_a @ X @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_a] :
            ( ( ( cons_a @ X @ Ys5 )
              = Ys )
            & ( Xs
              = ( append_a @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_986_append__eq__Cons__conv,axiom,
    ! [Ys: list_a,Zs3: list_a,X: a,Xs: list_a] :
      ( ( ( append_a @ Ys @ Zs3 )
        = ( cons_a @ X @ Xs ) )
      = ( ( ( Ys = nil_a )
          & ( Zs3
            = ( cons_a @ X @ Xs ) ) )
        | ? [Ys5: list_a] :
            ( ( Ys
              = ( cons_a @ X @ Ys5 ) )
            & ( ( append_a @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_987_rev__nonempty__induct,axiom,
    ! [Xs: list_a,P2: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X2: a] : ( P2 @ ( cons_a @ X2 @ nil_a ) )
       => ( ! [X2: a,Xs3: list_a] :
              ( ( Xs3 != nil_a )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( append_a @ Xs3 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_988_shuffles_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ( ! [Ys3: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
     => ( ! [Xs3: list_a] :
            ( X
           != ( produc6837034575241423639list_a @ Xs3 @ nil_a ) )
       => ~ ! [X2: a,Xs3: list_a,Y3: a,Ys3: list_a] :
              ( X
             != ( produc6837034575241423639list_a @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_989_exp__base__closed,axiom,
    ! [X: a,N: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% exp_base_closed
thf(fact_990_factors__mult,axiom,
    ! [Fa: list_a,A: a,Fb: list_a,B: a] :
      ( ( factor5638265376665762323xt_a_b @ r @ Fa @ A )
     => ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
           => ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).

% factors_mult
thf(fact_991_eval__append__aux,axiom,
    ! [P: list_a,B: a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( eval_a_b @ r @ ( append_a @ P @ ( cons_a @ B @ nil_a ) ) @ A )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ A ) @ B ) ) ) ) ) ).

% eval_append_aux
thf(fact_992_line__extension__smult__closed,axiom,
    ! [K: set_a,E2: set_a,A: a,K3: a,U: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ! [K2: a,V: a] :
            ( ( member_a @ K2 @ K )
           => ( ( member_a @ V @ E2 )
             => ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ V ) @ E2 ) ) )
       => ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ K3 @ K )
             => ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E2 ) )
               => ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E2 ) ) ) ) ) ) ) ) ).

% line_extension_smult_closed
thf(fact_993_eval_Osimps_I1_J,axiom,
    ( ( eval_a_b @ r @ nil_a )
    = ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).

% eval.simps(1)
thf(fact_994_line__extension__in__carrier,axiom,
    ! [K: set_a,A: a,E2: set_a] :
      ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% line_extension_in_carrier
thf(fact_995_line__extension__mem__iff,axiom,
    ! [U: a,K: set_a,A: a,E2: set_a] :
      ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E2 ) )
      = ( ? [X3: a] :
            ( ( member_a @ X3 @ K )
            & ? [Y4: a] :
                ( ( member_a @ Y4 @ E2 )
                & ( U
                  = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y4 ) ) ) ) ) ) ).

% line_extension_mem_iff
thf(fact_996_eval__var,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X )
        = X ) ) ).

% eval_var
thf(fact_997_const__term__def,axiom,
    ! [P: list_a] :
      ( ( const_term_a_b @ r @ P )
      = ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).

% const_term_def
thf(fact_998_eval__in__carrier,axiom,
    ! [P: list_a,X: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% eval_in_carrier
thf(fact_999_factors__closed,axiom,
    ! [Fs: list_a,A: a] :
      ( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% factors_closed
thf(fact_1000_is__root__def,axiom,
    ! [P: list_a,X: a] :
      ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
      = ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
        & ( ( eval_a_b @ r @ P @ X )
          = ( zero_a_b @ r ) )
        & ( P != nil_a ) ) ) ).

% is_root_def
thf(fact_1001_eval__monom,axiom,
    ! [B: a,A: a,N: nat] :
      ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( eval_a_b @ r @ ( monom_a_b @ r @ B @ N ) @ A )
          = ( mult_a_ring_ext_a_b @ r @ B @ ( pow_a_1026414303147256608_b_nat @ r @ A @ N ) ) ) ) ) ).

% eval_monom
thf(fact_1002_pdivides__imp__root__sharing,axiom,
    ! [P: list_a,Q: list_a,A: a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( eval_a_b @ r @ P @ A )
              = ( zero_a_b @ r ) )
           => ( ( eval_a_b @ r @ Q @ A )
              = ( zero_a_b @ r ) ) ) ) ) ) ).

% pdivides_imp_root_sharing
thf(fact_1003_ring_Oexp__base_Ocong,axiom,
    polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).

% ring.exp_base.cong
thf(fact_1004_ring_Oeval_Ocong,axiom,
    eval_a_b = eval_a_b ).

% ring.eval.cong
thf(fact_1005_ring_Oeval_Osimps_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( eval_l34571156754992824t_unit @ R @ nil_list_a )
        = ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% ring.eval.simps(1)
thf(fact_1006_ring_Oeval_Osimps_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R )
     => ( ( eval_a_b @ R @ nil_a )
        = ( ^ [Uu: a] : ( zero_a_b @ R ) ) ) ) ).

% ring.eval.simps(1)
thf(fact_1007_ring_Oconst__term__def,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( const_6738166269504826821t_unit @ R @ P )
        = ( eval_l34571156754992824t_unit @ R @ P @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% ring.const_term_def
thf(fact_1008_ring_Oconst__term__def,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a] :
      ( ( ring_a_b @ R )
     => ( ( const_term_a_b @ R @ P )
        = ( eval_a_b @ R @ P @ ( zero_a_b @ R ) ) ) ) ).

% ring.const_term_def
thf(fact_1009_ring_Oeval__var,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( eval_l34571156754992824t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) @ X )
          = X ) ) ) ).

% ring.eval_var
thf(fact_1010_ring_Oeval__var,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( eval_a_b @ R @ ( var_a_b @ R ) @ X )
          = X ) ) ) ).

% ring.eval_var
thf(fact_1011_ring_Oeval__var,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( eval_l1088911609197519410t_unit @ R @ ( var_li3532061862469730199t_unit @ R ) @ X )
          = X ) ) ) ).

% ring.eval_var
thf(fact_1012_ring_Oeval__in__carrier,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( eval_l34571156754992824t_unit @ R @ P @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% ring.eval_in_carrier
thf(fact_1013_ring_Oeval__in__carrier,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
      ( ( ring_a_b @ R )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( eval_a_b @ R @ P @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% ring.eval_in_carrier
thf(fact_1014_ring_Oeval__in__carrier,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( eval_l1088911609197519410t_unit @ R @ P @ X ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% ring.eval_in_carrier
thf(fact_1015_ring_Ois__root__def,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( polyno6951661231331188332t_unit @ R @ P @ X )
        = ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
          & ( ( eval_l34571156754992824t_unit @ R @ P @ X )
            = ( zero_l4142658623432671053t_unit @ R ) )
          & ( P != nil_list_a ) ) ) ) ).

% ring.is_root_def
thf(fact_1016_ring_Ois__root__def,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( polyno5142720416380192742t_unit @ R @ P @ X )
        = ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
          & ( ( eval_l1088911609197519410t_unit @ R @ P @ X )
            = ( zero_l347298301471573063t_unit @ R ) )
          & ( P != nil_list_list_a ) ) ) ) ).

% ring.is_root_def
thf(fact_1017_ring_Ois__root__def,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
      ( ( ring_a_b @ R )
     => ( ( polyno4133073214067823460ot_a_b @ R @ P @ X )
        = ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
          & ( ( eval_a_b @ R @ P @ X )
            = ( zero_a_b @ R ) )
          & ( P != nil_a ) ) ) ) ).

% ring.is_root_def
thf(fact_1018_ring_Oeval__monom,axiom,
    ! [R: partia2670972154091845814t_unit,B: list_a,A: list_a,N: nat] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( eval_l34571156754992824t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ B @ N ) @ A )
            = ( mult_l7073676228092353617t_unit @ R @ B @ ( pow_li1142815632869257134it_nat @ R @ A @ N ) ) ) ) ) ) ).

% ring.eval_monom
thf(fact_1019_ring_Oeval__monom,axiom,
    ! [R: partia2175431115845679010xt_a_b,B: a,A: a,N: nat] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( eval_a_b @ R @ ( monom_a_b @ R @ B @ N ) @ A )
            = ( mult_a_ring_ext_a_b @ R @ B @ ( pow_a_1026414303147256608_b_nat @ R @ A @ N ) ) ) ) ) ) ).

% ring.eval_monom
thf(fact_1020_ring_Oeval__monom,axiom,
    ! [R: partia2956882679547061052t_unit,B: list_list_a,A: list_list_a,N: nat] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( eval_l1088911609197519410t_unit @ R @ ( monom_4043874212805408666t_unit @ R @ B @ N ) @ A )
            = ( mult_l4853965630390486993t_unit @ R @ B @ ( pow_li488931774710091566it_nat @ R @ A @ N ) ) ) ) ) ) ).

% ring.eval_monom
thf(fact_1021_domain_Opdivides__imp__root__sharing,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,A: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( polyno4453881341673752516t_unit @ R @ P @ Q )
         => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( ( eval_l1088911609197519410t_unit @ R @ P @ A )
                = ( zero_l347298301471573063t_unit @ R ) )
             => ( ( eval_l1088911609197519410t_unit @ R @ Q @ A )
                = ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).

% domain.pdivides_imp_root_sharing
thf(fact_1022_domain_Opdivides__imp__root__sharing,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( polyno5814909790663948098es_a_b @ R @ P @ Q )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( ( eval_a_b @ R @ P @ A )
                = ( zero_a_b @ R ) )
             => ( ( eval_a_b @ R @ Q @ A )
                = ( zero_a_b @ R ) ) ) ) ) ) ) ).

% domain.pdivides_imp_root_sharing
thf(fact_1023_domain_Opdivides__imp__root__sharing,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( polyno8016796738000020810t_unit @ R @ P @ Q )
         => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( ( eval_l34571156754992824t_unit @ R @ P @ A )
                = ( zero_l4142658623432671053t_unit @ R ) )
             => ( ( eval_l34571156754992824t_unit @ R @ Q @ A )
                = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).

% domain.pdivides_imp_root_sharing
thf(fact_1024_ring_Oexp__base__closed,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,N: nat] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ R @ X @ N ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).

% ring.exp_base_closed
thf(fact_1025_ring_Oexp__base__closed,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,N: nat] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ ( polyno6819740552565085946t_unit @ R @ X @ N ) ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).

% ring.exp_base_closed
thf(fact_1026_ring_Oexp__base__closed,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,N: nat] :
      ( ( ring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ R @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).

% ring.exp_base_closed
thf(fact_1027_ring_Oeval__append__aux,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,B: list_a,A: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( eval_l34571156754992824t_unit @ R @ ( append_list_a @ P @ ( cons_list_a @ B @ nil_list_a ) ) @ A )
              = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ ( eval_l34571156754992824t_unit @ R @ P @ A ) @ A ) @ B ) ) ) ) ) ) ).

% ring.eval_append_aux
thf(fact_1028_ring_Oeval__append__aux,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,B: a,A: a] :
      ( ( ring_a_b @ R )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( eval_a_b @ R @ ( append_a @ P @ ( cons_a @ B @ nil_a ) ) @ A )
              = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ ( eval_a_b @ R @ P @ A ) @ A ) @ B ) ) ) ) ) ) ).

% ring.eval_append_aux
thf(fact_1029_ring_Oeval__append__aux,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,B: list_list_a,A: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( eval_l1088911609197519410t_unit @ R @ ( append_list_list_a @ P @ ( cons_list_list_a @ B @ nil_list_list_a ) ) @ A )
              = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ ( eval_l1088911609197519410t_unit @ R @ P @ A ) @ A ) @ B ) ) ) ) ) ) ).

% ring.eval_append_aux
thf(fact_1030_eval__append,axiom,
    ! [P: list_a,Q: list_a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( eval_a_b @ r @ ( append_a @ P @ Q ) @ A )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( pow_a_1026414303147256608_b_nat @ r @ A @ ( size_size_list_a @ Q ) ) ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).

% eval_append
thf(fact_1031_monoid_Ofactors__mult,axiom,
    ! [G: partia2670972154091845814t_unit,Fa: list_list_a,A: list_a,Fb: list_list_a,B: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( factor7181967632740204193t_unit @ G @ Fa @ A )
       => ( ( factor7181967632740204193t_unit @ G @ Fb @ B )
         => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fa ) @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fb ) @ ( partia5361259788508890537t_unit @ G ) )
             => ( factor7181967632740204193t_unit @ G @ ( append_list_a @ Fa @ Fb ) @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) ) ) ) ) ) ) ).

% monoid.factors_mult
thf(fact_1032_monoid_Ofactors__mult,axiom,
    ! [G: partia2175431115845679010xt_a_b,Fa: list_a,A: a,Fb: list_a,B: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( factor5638265376665762323xt_a_b @ G @ Fa @ A )
       => ( ( factor5638265376665762323xt_a_b @ G @ Fb @ B )
         => ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ G ) )
             => ( factor5638265376665762323xt_a_b @ G @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ G @ A @ B ) ) ) ) ) ) ) ).

% monoid.factors_mult
thf(fact_1033_monoid_Ofactors__mult,axiom,
    ! [G: partia2956882679547061052t_unit,Fa: list_list_list_a,A: list_list_a,Fb: list_list_list_a,B: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( factor7945547698214394401t_unit @ G @ Fa @ A )
       => ( ( factor7945547698214394401t_unit @ G @ Fb @ B )
         => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Fa ) @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Fb ) @ ( partia2464479390973590831t_unit @ G ) )
             => ( factor7945547698214394401t_unit @ G @ ( append_list_list_a @ Fa @ Fb ) @ ( mult_l4853965630390486993t_unit @ G @ A @ B ) ) ) ) ) ) ) ).

% monoid.factors_mult
thf(fact_1034_ring_Oline__extension__smult__closed,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,E2: set_list_a,A: list_a,K3: list_a,U: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ! [K2: list_a,V: list_a] :
              ( ( member_list_a @ K2 @ K )
             => ( ( member_list_a @ V @ E2 )
               => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K2 @ V ) @ E2 ) ) )
         => ( ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( member_list_a @ K3 @ K )
               => ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ R @ K @ A @ E2 ) )
                 => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ K3 @ U ) @ ( embedd5150658419831591667t_unit @ R @ K @ A @ E2 ) ) ) ) ) ) ) ) ) ).

% ring.line_extension_smult_closed
thf(fact_1035_ring_Oline__extension__smult__closed,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,E2: set_list_list_a,A: list_list_a,K3: list_list_a,U: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ! [K2: list_list_a,V: list_list_a] :
              ( ( member_list_list_a @ K2 @ K )
             => ( ( member_list_list_a @ V @ E2 )
               => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ K2 @ V ) @ E2 ) ) )
         => ( ( ord_le8488217952732425610list_a @ E2 @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( member_list_list_a @ K3 @ K )
               => ( ( member_list_list_a @ U @ ( embedd3735808041618263277t_unit @ R @ K @ A @ E2 ) )
                 => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ K3 @ U ) @ ( embedd3735808041618263277t_unit @ R @ K @ A @ E2 ) ) ) ) ) ) ) ) ) ).

% ring.line_extension_smult_closed
thf(fact_1036_ring_Oline__extension__smult__closed,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,E2: set_a,A: a,K3: a,U: a] :
      ( ( ring_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ! [K2: a,V: a] :
              ( ( member_a @ K2 @ K )
             => ( ( member_a @ V @ E2 )
               => ( member_a @ ( mult_a_ring_ext_a_b @ R @ K2 @ V ) @ E2 ) ) )
         => ( ( ord_less_eq_set_a @ E2 @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( member_a @ K3 @ K )
               => ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ R @ K @ A @ E2 ) )
                 => ( member_a @ ( mult_a_ring_ext_a_b @ R @ K3 @ U ) @ ( embedd971793762689825387on_a_b @ R @ K @ A @ E2 ) ) ) ) ) ) ) ) ) ).

% ring.line_extension_smult_closed
thf(fact_1037_eval__as__unique__hom,axiom,
    ! [K: set_a,X: a,H3: list_a > a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H3 )
         => ( ! [K2: a] :
                ( ( member_a @ K2 @ K )
               => ( ( H3 @ ( cons_a @ K2 @ nil_a ) )
                  = K2 ) )
           => ( ( ( H3 @ ( var_a_b @ r ) )
                = X )
             => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
               => ( ( H3 @ P )
                  = ( eval_a_b @ r @ P @ X ) ) ) ) ) ) ) ) ).

% eval_as_unique_hom
thf(fact_1038_univ__poly__a__inv__length,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
          = ( size_size_list_a @ P ) ) ) ) ).

% univ_poly_a_inv_length
thf(fact_1039_determination__of__hom,axiom,
    ! [K: set_a,A2: partia2175431115845679010xt_a_b,H3: list_a > a,G3: list_a > a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ A2 @ H3 )
       => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ A2 @ G3 )
         => ( ! [K2: a] :
                ( ( member_a @ K2 @ K )
               => ( ( H3 @ ( cons_a @ K2 @ nil_a ) )
                  = ( G3 @ ( cons_a @ K2 @ nil_a ) ) ) )
           => ( ( ( H3 @ ( var_a_b @ r ) )
                = ( G3 @ ( var_a_b @ r ) ) )
             => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
               => ( ( H3 @ P )
                  = ( G3 @ P ) ) ) ) ) ) ) ) ).

% determination_of_hom
thf(fact_1040_list__induct2,axiom,
    ! [Xs: list_a,Ys: list_a,P2: list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( P2 @ nil_a @ nil_a )
       => ( ! [X2: a,Xs3: list_a,Y3: a,Ys3: list_a] :
              ( ( ( size_size_list_a @ Xs3 )
                = ( size_size_list_a @ Ys3 ) )
             => ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1041_list__induct3,axiom,
    ! [Xs: list_a,Ys: list_a,Zs3: list_a,P2: list_a > list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( P2 @ nil_a @ nil_a @ nil_a )
         => ( ! [X2: a,Xs3: list_a,Y3: a,Ys3: list_a,Z2: a,Zs2: list_a] :
                ( ( ( size_size_list_a @ Xs3 )
                  = ( size_size_list_a @ Ys3 ) )
               => ( ( ( size_size_list_a @ Ys3 )
                    = ( size_size_list_a @ Zs2 ) )
                 => ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_1042_list__induct4,axiom,
    ! [Xs: list_a,Ys: list_a,Zs3: list_a,Ws: list_a,P2: list_a > list_a > list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( size_size_list_a @ Ys )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( ( size_size_list_a @ Zs3 )
            = ( size_size_list_a @ Ws ) )
         => ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
           => ( ! [X2: a,Xs3: list_a,Y3: a,Ys3: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_a @ Xs3 )
                    = ( size_size_list_a @ Ys3 ) )
                 => ( ( ( size_size_list_a @ Ys3 )
                      = ( size_size_list_a @ Zs2 ) )
                   => ( ( ( size_size_list_a @ Zs2 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P2 @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_a @ X2 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1043_impossible__Cons,axiom,
    ! [Xs: list_a,Ys: list_a,X: a] :
      ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
     => ( Xs
       != ( cons_a @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_1044_same__length__different,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
       => ? [Pre: list_a,X2: a,Xs4: list_a,Y3: a,Ys6: list_a] :
            ( ( X2 != Y3 )
            & ( Xs
              = ( append_a @ Pre @ ( append_a @ ( cons_a @ X2 @ nil_a ) @ Xs4 ) ) )
            & ( Ys
              = ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys6 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1045_domain_Ouniv__poly__a__inv__length,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) )
            = ( size_size_list_a @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_length
thf(fact_1046_domain_Ouniv__poly__a__inv__length,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) )
            = ( size_s349497388124573686list_a @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_length
thf(fact_1047_domain_Odetermination__of__hom,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,A2: partia2175431115845679010xt_a_b,H3: list_a > a,G3: list_a > a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ A2 @ H3 )
         => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ A2 @ G3 )
           => ( ! [K2: a] :
                  ( ( member_a @ K2 @ K )
                 => ( ( H3 @ ( cons_a @ K2 @ nil_a ) )
                    = ( G3 @ ( cons_a @ K2 @ nil_a ) ) ) )
             => ( ( ( H3 @ ( var_a_b @ R ) )
                  = ( G3 @ ( var_a_b @ R ) ) )
               => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                 => ( ( H3 @ P )
                    = ( G3 @ P ) ) ) ) ) ) ) ) ) ).

% domain.determination_of_hom
thf(fact_1048_monoid_Oassociated__sym,axiom,
    ! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
       => ( associ5860276527279195403xt_a_b @ G @ B @ A ) ) ) ).

% monoid.associated_sym
thf(fact_1049_monoid_Oassociated__sym,axiom,
    ! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( associ8407585678920448409t_unit @ G @ A @ B )
       => ( associ8407585678920448409t_unit @ G @ B @ A ) ) ) ).

% monoid.associated_sym
thf(fact_1050_monoid__cancel_Oaxioms_I1_J,axiom,
    ! [G: partia2670972154091845814t_unit] :
      ( ( monoid4303264861975686087t_unit @ G )
     => ( monoid5589397312508706001t_unit @ G ) ) ).

% monoid_cancel.axioms(1)
thf(fact_1051_monoid__cancel_Oaxioms_I1_J,axiom,
    ! [G: partia2175431115845679010xt_a_b] :
      ( ( monoid5798828371819920185xt_a_b @ G )
     => ( monoid8385113658579753027xt_a_b @ G ) ) ).

% monoid_cancel.axioms(1)
thf(fact_1052_domain_Oeval__as__unique__hom,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,X: list_a,H3: list_list_a > list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( ring_h4589914651911841480t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ R @ H3 )
           => ( ! [K2: list_a] :
                  ( ( member_list_a @ K2 @ K )
                 => ( ( H3 @ ( cons_list_a @ K2 @ nil_list_a ) )
                    = K2 ) )
             => ( ( ( H3 @ ( var_li8453953174693405341t_unit @ R ) )
                  = X )
               => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
                 => ( ( H3 @ P )
                    = ( eval_l34571156754992824t_unit @ R @ P @ X ) ) ) ) ) ) ) ) ) ).

% domain.eval_as_unique_hom
thf(fact_1053_domain_Oeval__as__unique__hom,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,X: a,H3: list_a > a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ R @ H3 )
           => ( ! [K2: a] :
                  ( ( member_a @ K2 @ K )
                 => ( ( H3 @ ( cons_a @ K2 @ nil_a ) )
                    = K2 ) )
             => ( ( ( H3 @ ( var_a_b @ R ) )
                  = X )
               => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
                 => ( ( H3 @ P )
                    = ( eval_a_b @ R @ P @ X ) ) ) ) ) ) ) ) ) ).

% domain.eval_as_unique_hom
thf(fact_1054_domain_Oeval__as__unique__hom,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,X: list_list_a,H3: list_list_list_a > list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( ring_h7694777735462631100t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ R @ H3 )
           => ( ! [K2: list_list_a] :
                  ( ( member_list_list_a @ K2 @ K )
                 => ( ( H3 @ ( cons_list_list_a @ K2 @ nil_list_list_a ) )
                    = K2 ) )
             => ( ( ( H3 @ ( var_li3532061862469730199t_unit @ R ) )
                  = X )
               => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
                 => ( ( H3 @ P )
                    = ( eval_l1088911609197519410t_unit @ R @ P @ X ) ) ) ) ) ) ) ) ) ).

% domain.eval_as_unique_hom
thf(fact_1055_ring_Oeval__append,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( eval_l34571156754992824t_unit @ R @ ( append_list_a @ P @ Q ) @ A )
              = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ ( eval_l34571156754992824t_unit @ R @ P @ A ) @ ( pow_li1142815632869257134it_nat @ R @ A @ ( size_s349497388124573686list_a @ Q ) ) ) @ ( eval_l34571156754992824t_unit @ R @ Q @ A ) ) ) ) ) ) ) ).

% ring.eval_append
thf(fact_1056_ring_Oeval__append,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a] :
      ( ( ring_a_b @ R )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( eval_a_b @ R @ ( append_a @ P @ Q ) @ A )
              = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ ( eval_a_b @ R @ P @ A ) @ ( pow_a_1026414303147256608_b_nat @ R @ A @ ( size_size_list_a @ Q ) ) ) @ ( eval_a_b @ R @ Q @ A ) ) ) ) ) ) ) ).

% ring.eval_append
thf(fact_1057_ring_Oeval__append,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,A: list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Q ) @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( eval_l1088911609197519410t_unit @ R @ ( append_list_list_a @ P @ Q ) @ A )
              = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ ( eval_l1088911609197519410t_unit @ R @ P @ A ) @ ( pow_li488931774710091566it_nat @ R @ A @ ( size_s2403821588304063868list_a @ Q ) ) ) @ ( eval_l1088911609197519410t_unit @ R @ Q @ A ) ) ) ) ) ) ) ).

% ring.eval_append
thf(fact_1058_ring_Osubring__props_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ K ) ) ) ).

% ring.subring_props(2)
thf(fact_1059_ring_Osubring__props_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( ring_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( member_a @ ( zero_a_b @ R ) @ K ) ) ) ).

% ring.subring_props(2)
thf(fact_1060_ring_Osubring__props_I7_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,H1: list_a,H2: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_a @ H1 @ K )
         => ( ( member_list_a @ H2 @ K )
           => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H2 ) @ K ) ) ) ) ) ).

% ring.subring_props(7)
thf(fact_1061_ring_Osubring__props_I7_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,H1: a,H2: a] :
      ( ( ring_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_a @ H1 @ K )
         => ( ( member_a @ H2 @ K )
           => ( member_a @ ( add_a_b @ R @ H1 @ H2 ) @ K ) ) ) ) ) ).

% ring.subring_props(7)
thf(fact_1062_ring_Osubring__props_I4_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( ring_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( K != bot_bot_set_a ) ) ) ).

% ring.subring_props(4)
thf(fact_1063_ring_Osubring__props_I4_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( K != bot_bot_set_list_a ) ) ) ).

% ring.subring_props(4)
thf(fact_1064_ring_Osubring__props_I6_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,H1: list_a,H2: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_a @ H1 @ K )
         => ( ( member_list_a @ H2 @ K )
           => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 ) @ K ) ) ) ) ) ).

% ring.subring_props(6)
thf(fact_1065_ring_Osubring__props_I6_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,H1: a,H2: a] :
      ( ( ring_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_a @ H1 @ K )
         => ( ( member_a @ H2 @ K )
           => ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H2 ) @ K ) ) ) ) ) ).

% ring.subring_props(6)
thf(fact_1066_monoid_Oassoc__subst,axiom,
    ! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,F: list_a > list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( associ8407585678920448409t_unit @ G @ A @ B )
       => ( ! [A3: list_a,B2: list_a] :
              ( ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ G ) )
                & ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ G ) )
                & ( associ8407585678920448409t_unit @ G @ A3 @ B2 ) )
             => ( ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ G ) )
                & ( member_list_a @ ( F @ B2 ) @ ( partia5361259788508890537t_unit @ G ) )
                & ( associ8407585678920448409t_unit @ G @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
         => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
             => ( associ8407585678920448409t_unit @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).

% monoid.assoc_subst
thf(fact_1067_monoid_Oassoc__subst,axiom,
    ! [G: partia2175431115845679010xt_a_b,A: a,B: a,F: a > a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
       => ( ! [A3: a,B2: a] :
              ( ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ G ) )
                & ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ G ) )
                & ( associ5860276527279195403xt_a_b @ G @ A3 @ B2 ) )
             => ( ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ G ) )
                & ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ G ) )
                & ( associ5860276527279195403xt_a_b @ G @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
             => ( associ5860276527279195403xt_a_b @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).

% monoid.assoc_subst
thf(fact_1068_monoid_Oassoc__subst,axiom,
    ! [G: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,F: list_list_a > list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( associ5603075271488036121t_unit @ G @ A @ B )
       => ( ! [A3: list_list_a,B2: list_list_a] :
              ( ( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ G ) )
                & ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ G ) )
                & ( associ5603075271488036121t_unit @ G @ A3 @ B2 ) )
             => ( ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ G ) )
                & ( member_list_list_a @ ( F @ B2 ) @ ( partia2464479390973590831t_unit @ G ) )
                & ( associ5603075271488036121t_unit @ G @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
         => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
             => ( associ5603075271488036121t_unit @ G @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ) ).

% monoid.assoc_subst
thf(fact_1069_monoid_Oassociated__refl,axiom,
    ! [G: partia2670972154091845814t_unit,A: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
       => ( associ8407585678920448409t_unit @ G @ A @ A ) ) ) ).

% monoid.associated_refl
thf(fact_1070_monoid_Oassociated__refl,axiom,
    ! [G: partia2175431115845679010xt_a_b,A: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
       => ( associ5860276527279195403xt_a_b @ G @ A @ A ) ) ) ).

% monoid.associated_refl
thf(fact_1071_monoid_Oassociated__refl,axiom,
    ! [G: partia2956882679547061052t_unit,A: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
       => ( associ5603075271488036121t_unit @ G @ A @ A ) ) ) ).

% monoid.associated_refl
thf(fact_1072_monoid_Oassociated__trans,axiom,
    ! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( associ8407585678920448409t_unit @ G @ A @ B )
       => ( ( associ8407585678920448409t_unit @ G @ B @ C )
         => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
             => ( associ8407585678920448409t_unit @ G @ A @ C ) ) ) ) ) ) ).

% monoid.associated_trans
thf(fact_1073_monoid_Oassociated__trans,axiom,
    ! [G: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
       => ( ( associ5860276527279195403xt_a_b @ G @ B @ C )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
             => ( associ5860276527279195403xt_a_b @ G @ A @ C ) ) ) ) ) ) ).

% monoid.associated_trans
thf(fact_1074_monoid_Oassociated__trans,axiom,
    ! [G: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( associ5603075271488036121t_unit @ G @ A @ B )
       => ( ( associ5603075271488036121t_unit @ G @ B @ C )
         => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G ) )
             => ( associ5603075271488036121t_unit @ G @ A @ C ) ) ) ) ) ) ).

% monoid.associated_trans
thf(fact_1075_ring_Osubring__props_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ K ) ) ) ).

% ring.subring_props(3)
thf(fact_1076_ring_Osubring__props_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( ring_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( member_a @ ( one_a_ring_ext_a_b @ R ) @ K ) ) ) ).

% ring.subring_props(3)
thf(fact_1077_ring_Osubring__props_I5_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,H3: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ( member_list_a @ H3 @ K )
         => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ K ) ) ) ) ).

% ring.subring_props(5)
thf(fact_1078_ring_Osubring__props_I5_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,H3: a] :
      ( ( ring_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ( member_a @ H3 @ K )
         => ( member_a @ ( a_inv_a_b @ R @ H3 ) @ K ) ) ) ) ).

% ring.subring_props(5)
thf(fact_1079_monoid_OUnits__assoc,axiom,
    ! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
       => ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
         => ( associ5860276527279195403xt_a_b @ G @ A @ B ) ) ) ) ).

% monoid.Units_assoc
thf(fact_1080_monoid_OUnits__assoc,axiom,
    ! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
       => ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) )
         => ( associ8407585678920448409t_unit @ G @ A @ B ) ) ) ) ).

% monoid.Units_assoc
thf(fact_1081_monoid__cancel_Ol__cancel,axiom,
    ! [G: partia2670972154091845814t_unit,C: list_a,A: list_a,B: list_a] :
      ( ( monoid4303264861975686087t_unit @ G )
     => ( ( ( mult_l7073676228092353617t_unit @ G @ C @ A )
          = ( mult_l7073676228092353617t_unit @ G @ C @ B ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
             => ( A = B ) ) ) ) ) ) ).

% monoid_cancel.l_cancel
thf(fact_1082_monoid__cancel_Ol__cancel,axiom,
    ! [G: partia2175431115845679010xt_a_b,C: a,A: a,B: a] :
      ( ( monoid5798828371819920185xt_a_b @ G )
     => ( ( ( mult_a_ring_ext_a_b @ G @ C @ A )
          = ( mult_a_ring_ext_a_b @ G @ C @ B ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
             => ( A = B ) ) ) ) ) ) ).

% monoid_cancel.l_cancel
thf(fact_1083_monoid__cancel_Ol__cancel,axiom,
    ! [G: partia2956882679547061052t_unit,C: list_list_a,A: list_list_a,B: list_list_a] :
      ( ( monoid576229335242748231t_unit @ G )
     => ( ( ( mult_l4853965630390486993t_unit @ G @ C @ A )
          = ( mult_l4853965630390486993t_unit @ G @ C @ B ) )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G ) )
             => ( A = B ) ) ) ) ) ) ).

% monoid_cancel.l_cancel
thf(fact_1084_monoid__cancel_Or__cancel,axiom,
    ! [G: partia2670972154091845814t_unit,A: list_a,C: list_a,B: list_a] :
      ( ( monoid4303264861975686087t_unit @ G )
     => ( ( ( mult_l7073676228092353617t_unit @ G @ A @ C )
          = ( mult_l7073676228092353617t_unit @ G @ B @ C ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
         => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
             => ( A = B ) ) ) ) ) ) ).

% monoid_cancel.r_cancel
thf(fact_1085_monoid__cancel_Or__cancel,axiom,
    ! [G: partia2175431115845679010xt_a_b,A: a,C: a,B: a] :
      ( ( monoid5798828371819920185xt_a_b @ G )
     => ( ( ( mult_a_ring_ext_a_b @ G @ A @ C )
          = ( mult_a_ring_ext_a_b @ G @ B @ C ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
         => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
             => ( A = B ) ) ) ) ) ) ).

% monoid_cancel.r_cancel
thf(fact_1086_monoid__cancel_Or__cancel,axiom,
    ! [G: partia2956882679547061052t_unit,A: list_list_a,C: list_list_a,B: list_list_a] :
      ( ( monoid576229335242748231t_unit @ G )
     => ( ( ( mult_l4853965630390486993t_unit @ G @ A @ C )
          = ( mult_l4853965630390486993t_unit @ G @ B @ C ) )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
         => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G ) )
             => ( A = B ) ) ) ) ) ) ).

% monoid_cancel.r_cancel
thf(fact_1087_ring_Osubring__props_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R )
     => ( ( subfie1779122896746047282t_unit @ K @ R )
       => ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).

% ring.subring_props(1)
thf(fact_1088_ring_Osubring__props_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( ring_a_b @ R )
     => ( ( subfield_a_b @ K @ R )
       => ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).

% ring.subring_props(1)
thf(fact_1089_ring_Osubring__props_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( ring_l1939023646219158831t_unit @ R )
     => ( ( subfie4546268998243038636t_unit @ K @ R )
       => ( ord_le8488217952732425610list_a @ K @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).

% ring.subring_props(1)
thf(fact_1090_monoid_Oprod__unit__l,axiom,
    ! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
      ( ( monoid5589397312508706001t_unit @ G )
     => ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( units_2932844235741507942t_unit @ G ) )
       => ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
         => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
           => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
             => ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ) ).

% monoid.prod_unit_l
thf(fact_1091_monoid_Oprod__unit__l,axiom,
    ! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( units_a_ring_ext_a_b @ G ) )
       => ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
             => ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ) ).

% monoid.prod_unit_l
thf(fact_1092_monoid_Oprod__unit__l,axiom,
    ! [G: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ G @ A @ B ) @ ( units_4903515905731149798t_unit @ G ) )
       => ( ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ G ) )
         => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
             => ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ G ) ) ) ) ) ) ) ).

% monoid.prod_unit_l
thf(fact_1093_monoid_Oprod__unit__r,axiom,
    ! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
      ( ( monoid8385113658579753027xt_a_b @ G )
     => ( ( member_a @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( units_a_ring_ext_a_b @ G ) )
       => ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
           => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
             => ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ) ).

% monoid.prod_unit_r
thf(fact_1094_monoid_Oprod__unit__r,axiom,
    ! [G: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
      ( ( monoid5729698748631984209t_unit @ G )
     => ( ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ G @ A @ B ) @ ( units_4903515905731149798t_unit @ G ) )
       => ( ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ G ) )
         => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
           => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
             => ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ G ) ) ) ) ) ) ) ).

% monoid.prod_unit_r
thf(fact_1095_associated__polynomials__imp__same__length,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
           => ( ( size_size_list_a @ P )
              = ( size_size_list_a @ Q ) ) ) ) ) ) ).

% associated_polynomials_imp_same_length
thf(fact_1096_associated__polynomials__imp__same__is__root,axiom,
    ! [P: list_a,Q: list_a,X: a] :
      ( ( member_list_a @ P @ ( 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 ) ) ) )
       => ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
         => ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
            = ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).

% associated_polynomials_imp_same_is_root
thf(fact_1097_subring__degree__one__associatedI,axiom,
    ! [K: set_a,A: a,A4: a,B: a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_a @ A @ K )
       => ( ( member_a @ A4 @ K )
         => ( ( member_a @ B @ K )
           => ( ( ( mult_a_ring_ext_a_b @ r @ A @ A4 )
                = ( one_a_ring_ext_a_b @ r ) )
             => ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ r @ A4 @ B ) @ nil_a ) ) ) ) ) ) ) ) ).

% subring_degree_one_associatedI
thf(fact_1098_ee__sym,axiom,
    ! [As: list_a,Bs: list_a] :
      ( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( essent8953798148185448568xt_a_b @ r @ Bs @ As ) ) ) ) ).

% ee_sym
thf(fact_1099_ee__trans,axiom,
    ! [As: list_a,Bs: list_a,Cs: list_a] :
      ( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
     => ( ( essent8953798148185448568xt_a_b @ r @ Bs @ Cs )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ r ) )
             => ( essent8953798148185448568xt_a_b @ r @ As @ Cs ) ) ) ) ) ) ).

% ee_trans
thf(fact_1100_combine__append__zero,axiom,
    ! [Us2: list_a,Ks2: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( embedded_combine_a_b @ r @ ( append_a @ Ks2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Us2 )
        = ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) ) ) ).

% combine_append_zero
thf(fact_1101_ee__length,axiom,
    ! [As: list_a,Bs: list_a] :
      ( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
     => ( ( size_size_list_a @ As )
        = ( size_size_list_a @ Bs ) ) ) ).

% ee_length
thf(fact_1102_combine_Osimps_I2_J,axiom,
    ! [Us2: list_a] :
      ( ( embedded_combine_a_b @ r @ nil_a @ Us2 )
      = ( zero_a_b @ r ) ) ).

% combine.simps(2)
thf(fact_1103_combine_Osimps_I3_J,axiom,
    ! [Ks2: list_a] :
      ( ( embedded_combine_a_b @ r @ Ks2 @ nil_a )
      = ( zero_a_b @ r ) ) ).

% combine.simps(3)
thf(fact_1104_combine_Osimps_I1_J,axiom,
    ! [K3: a,Ks2: list_a,U: a,Us2: list_a] :
      ( ( embedded_combine_a_b @ r @ ( cons_a @ K3 @ Ks2 ) @ ( cons_a @ U @ Us2 ) )
      = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K3 @ U ) @ ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) ) ) ).

% combine.simps(1)
thf(fact_1105_combine__eq__eval,axiom,
    ! [Ks2: list_a,X: a] :
      ( ( embedded_combine_a_b @ r @ Ks2 @ ( polyno2922411391617481336se_a_b @ r @ X @ ( size_size_list_a @ Ks2 ) ) )
      = ( eval_a_b @ r @ Ks2 @ X ) ) ).

% combine_eq_eval
thf(fact_1106_combine_Oelims,axiom,
    ! [X: list_a,Xa3: list_a,Y: a] :
      ( ( ( embedded_combine_a_b @ r @ X @ Xa3 )
        = Y )
     => ( ! [K2: a,Ks: list_a] :
            ( ( X
              = ( cons_a @ K2 @ Ks ) )
           => ! [U2: a,Us: list_a] :
                ( ( Xa3
                  = ( cons_a @ U2 @ Us ) )
               => ( Y
                 != ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K2 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks @ Us ) ) ) ) )
       => ( ( ( X = nil_a )
           => ( Y
             != ( zero_a_b @ r ) ) )
         => ~ ( ( Xa3 = nil_a )
             => ( Y
               != ( zero_a_b @ r ) ) ) ) ) ) ).

% combine.elims
thf(fact_1107_combine__append,axiom,
    ! [Ks2: list_a,Us2: list_a,Ks3: list_a,Vs: list_a] :
      ( ( ( size_size_list_a @ Ks2 )
        = ( size_size_list_a @ Us2 ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
             => ( ( add_a_b @ r @ ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) @ ( embedded_combine_a_b @ r @ Ks3 @ Vs ) )
                = ( embedded_combine_a_b @ r @ ( append_a @ Ks2 @ Ks3 ) @ ( append_a @ Us2 @ Vs ) ) ) ) ) ) ) ) ).

% combine_append
thf(fact_1108_combine__in__carrier,axiom,
    ! [Ks2: list_a,Us2: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% combine_in_carrier
thf(fact_1109_ee__refl,axiom,
    ! [As: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( essent8953798148185448568xt_a_b @ r @ As @ As ) ) ).

% ee_refl
thf(fact_1110_cgenideal__pirreducible,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
         => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Q )
           => ( ( member_list_a @ Q @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
             => ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) ) ) ) ) ) ).

% cgenideal_pirreducible
thf(fact_1111_associated__polynomials__imp__same__roots,axiom,
    ! [P: list_a,Q: list_a] :
      ( ( member_list_a @ P @ ( 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 ) ) ) )
       => ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
         => ( ( polynomial_roots_a_b @ r @ P )
            = ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ).

% associated_polynomials_imp_same_roots
thf(fact_1112_a__lcos__m__assoc,axiom,
    ! [M2: set_a,G3: a,H3: a] :
      ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ G3 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ H3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( a_l_coset_a_b @ r @ G3 @ ( a_l_coset_a_b @ r @ H3 @ M2 ) )
            = ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G3 @ H3 ) @ M2 ) ) ) ) ) ).

% a_lcos_m_assoc
thf(fact_1113_a__l__coset__subset__G,axiom,
    ! [H: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% a_l_coset_subset_G
thf(fact_1114_a__lcos__mult__one,axiom,
    ! [M2: set_a] :
      ( ( ord_less_eq_set_a @ M2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M2 )
        = M2 ) ) ).

% a_lcos_mult_one
thf(fact_1115_roots__inclI,axiom,
    ! [P: list_a,Q: list_a] :
      ( ( member_list_a @ P @ ( 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 )
         => ( ! [A3: a] :
                ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
               => ( ( P != 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 @ A3 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ A3 ) ) @ Q ) ) )
           => ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).

% roots_inclI
thf(fact_1116_pdivides__imp__roots__incl,axiom,
    ! [P: list_a,Q: list_a] :
      ( ( member_list_a @ P @ ( 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 @ P @ Q )
           => ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).

% pdivides_imp_roots_incl
thf(fact_1117_cgenideal__self,axiom,
    ! [I: a] :
      ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).

% cgenideal_self
thf(fact_1118_ideal__eq__carrier__iff,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ( partia707051561876973205xt_a_b @ r )
          = ( cgenid547466209912283029xt_a_b @ r @ A ) )
        = ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).

% ideal_eq_carrier_iff
thf(fact_1119_associated__iff__same__ideal,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
          = ( ( cgenid547466209912283029xt_a_b @ r @ A )
            = ( cgenid547466209912283029xt_a_b @ r @ B ) ) ) ) ) ).

% associated_iff_same_ideal
thf(fact_1120_cgenideal__is__principalideal,axiom,
    ! [I: a] :
      ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
     => ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).

% cgenideal_is_principalideal
thf(fact_1121_roots__inclI_H,axiom,
    ! [P: list_a,M: multiset_a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ! [A3: a] :
            ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( P != nil_a )
             => ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ A3 ) @ ( count_a @ M @ A3 ) ) ) )
       => ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ M ) ) ) ).

% roots_inclI'
thf(fact_1122_const__term__zero,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( P != nil_a )
         => ( ( ( const_term_a_b @ r @ P )
              = ( zero_a_b @ r ) )
           => ~ ! [P3: list_a] :
                  ( ( polynomial_a_b @ r @ K @ P3 )
                 => ( ( P3 != nil_a )
                   => ( P
                     != ( append_a @ P3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).

% const_term_zero
thf(fact_1123_Span__mem__iff__length__version,axiom,
    ! [K: set_a,Us2: list_a,A: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
          = ( ? [Ks4: list_a] :
                ( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K )
                & ( ( size_size_list_a @ Ks4 )
                  = ( size_size_list_a @ Us2 ) )
                & ( A
                  = ( embedded_combine_a_b @ r @ Ks4 @ Us2 ) ) ) ) ) ) ) ).

% Span_mem_iff_length_version
thf(fact_1124_polynomial__incl,axiom,
    ! [K: set_a,P: list_a] :
      ( ( polynomial_a_b @ r @ K @ P )
     => ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K ) ) ).

% polynomial_incl
thf(fact_1125_var__closed_I2_J,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( polynomial_a_b @ r @ K @ ( var_a_b @ r ) ) ) ).

% var_closed(2)
thf(fact_1126_Span__in__carrier,axiom,
    ! [K: set_a,Us2: list_a] :
      ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% Span_in_carrier
thf(fact_1127_eval__poly__in__carrier,axiom,
    ! [K: set_a,P: list_a,X: a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
         => ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% eval_poly_in_carrier
thf(fact_1128_mono__Span__subset,axiom,
    ! [K: set_a,Us2: list_a,Vs: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ).

% mono_Span_subset
thf(fact_1129_mono__Span__sublist,axiom,
    ! [K: set_a,Us2: list_a,Vs: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( set_a2 @ Vs ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ).

% mono_Span_sublist
thf(fact_1130_Span__same__set,axiom,
    ! [K: set_a,Us2: list_a,Vs: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( set_a2 @ Us2 )
            = ( set_a2 @ Vs ) )
         => ( ( embedded_Span_a_b @ r @ K @ Us2 )
            = ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ).

% Span_same_set
thf(fact_1131_Span__base__incl,axiom,
    ! [K: set_a,Us2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ).

% Span_base_incl
thf(fact_1132_Span__subgroup__props_I1_J,axiom,
    ! [K: set_a,Us2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% Span_subgroup_props(1)
thf(fact_1133_Span__subgroup__props_I2_J,axiom,
    ! [K: set_a,Us2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( zero_a_b @ r ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ).

% Span_subgroup_props(2)
thf(fact_1134_Span__subgroup__props_I3_J,axiom,
    ! [K: set_a,Us2: list_a,V1: a,V2: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ V1 @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
         => ( ( member_a @ V2 @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
           => ( member_a @ ( add_a_b @ r @ V1 @ V2 ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ).

% Span_subgroup_props(3)
thf(fact_1135_Span__smult__closed,axiom,
    ! [K: set_a,Us2: list_a,K3: a,V3: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ K3 @ K )
         => ( ( member_a @ V3 @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
           => ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V3 ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ).

% Span_smult_closed
thf(fact_1136_mono__Span,axiom,
    ! [K: set_a,Us2: list_a,U: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ ( cons_a @ U @ Us2 ) ) ) ) ) ) ).

% mono_Span
thf(fact_1137_Span__subgroup__props_I4_J,axiom,
    ! [K: set_a,Us2: list_a,V3: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ V3 @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
         => ( member_a @ ( a_inv_a_b @ r @ V3 ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ).

% Span_subgroup_props(4)
thf(fact_1138_mono__Span__append_I2_J,axiom,
    ! [K: set_a,Us2: list_a,Vs: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ ( append_a @ Vs @ Us2 ) ) ) ) ) ) ).

% mono_Span_append(2)
thf(fact_1139_mono__Span__append_I1_J,axiom,
    ! [K: set_a,Us2: list_a,Vs: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) ) ) ) ) ) ).

% mono_Span_append(1)
thf(fact_1140_alg__mult__eq__count__roots,axiom,
    ! [P: list_a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( polyno4422430861927485590lt_a_b @ r @ P )
        = ( count_a @ ( polynomial_roots_a_b @ r @ P ) ) ) ) ).

% alg_mult_eq_count_roots
thf(fact_1141_zero__is__polynomial,axiom,
    ! [K: set_a] : ( polynomial_a_b @ r @ K @ nil_a ) ).

% zero_is_polynomial
thf(fact_1142_carrier__polynomial,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P ) ) ) ).

% carrier_polynomial
thf(fact_1143_polynomial__in__carrier,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% polynomial_in_carrier
thf(fact_1144_one__is__polynomial,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( polynomial_a_b @ r @ K @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) ) ) ).

% one_is_polynomial
thf(fact_1145_Span__is__subalgebra,axiom,
    ! [K: set_a,Us2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( embedd9027525575939734154ra_a_b @ K @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ r ) ) ) ).

% Span_is_subalgebra
thf(fact_1146_Span__subalgebraI,axiom,
    ! [K: set_a,E2: set_a,Us2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd9027525575939734154ra_a_b @ K @ E2 @ r )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ E2 )
         => ( ! [V4: set_a] :
                ( ( embedd9027525575939734154ra_a_b @ K @ V4 @ r )
               => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ V4 )
                 => ( ord_less_eq_set_a @ E2 @ V4 ) ) )
           => ( E2
              = ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ).

% Span_subalgebraI
thf(fact_1147_subalgebra__Span__incl,axiom,
    ! [K: set_a,V5: set_a,Us2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd9027525575939734154ra_a_b @ K @ V5 @ r )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ V5 )
         => ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ V5 ) ) ) ) ).

% subalgebra_Span_incl
thf(fact_1148_subalgebra__in__carrier,axiom,
    ! [K: set_a,V5: set_a] :
      ( ( embedd9027525575939734154ra_a_b @ K @ V5 @ r )
     => ( ord_less_eq_set_a @ V5 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% subalgebra_in_carrier
thf(fact_1149_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_1150_Span__finite__dimension,axiom,
    ! [K: set_a,Us2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( embedd8708762675212832759on_a_b @ r @ K @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ).

% Span_finite_dimension
thf(fact_1151_Span__append__eq__set__add,axiom,
    ! [K: set_a,Us2: list_a,Vs: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( embedded_Span_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) )
            = ( set_add_a_b @ r @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ) ).

% Span_append_eq_set_add
thf(fact_1152_Span__strict__incl,axiom,
    ! [K: set_a,Us2: list_a,Vs: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ord_less_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
           => ? [X2: a] :
                ( ( member_a @ X2 @ ( set_a2 @ Vs ) )
                & ~ ( member_a @ X2 @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ) ).

% Span_strict_incl
thf(fact_1153_telescopic__base__dim_I1_J,axiom,
    ! [K: set_a,F2: set_a,E2: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( subfield_a_b @ F2 @ r )
       => ( ( embedd8708762675212832759on_a_b @ r @ K @ F2 )
         => ( ( embedd8708762675212832759on_a_b @ r @ F2 @ E2 )
           => ( embedd8708762675212832759on_a_b @ r @ K @ E2 ) ) ) ) ) ).

% telescopic_base_dim(1)
thf(fact_1154_set__add__closed,axiom,
    ! [A2: set_a,B4: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B4 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% set_add_closed
thf(fact_1155_set__add__comm,axiom,
    ! [I2: set_a,J: set_a] :
      ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( set_add_a_b @ r @ I2 @ J )
          = ( set_add_a_b @ r @ J @ I2 ) ) ) ) ).

% set_add_comm
thf(fact_1156_setadd__subset__G,axiom,
    ! [H: set_a,K: set_a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% setadd_subset_G
thf(fact_1157_sum__space__dim_I1_J,axiom,
    ! [K: set_a,E2: set_a,F2: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd8708762675212832759on_a_b @ r @ K @ E2 )
       => ( ( embedd8708762675212832759on_a_b @ r @ K @ F2 )
         => ( embedd8708762675212832759on_a_b @ r @ K @ ( set_add_a_b @ r @ E2 @ F2 ) ) ) ) ) ).

% sum_space_dim(1)
thf(fact_1158_finite__dimension__imp__subalgebra,axiom,
    ! [K: set_a,E2: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd8708762675212832759on_a_b @ r @ K @ E2 )
       => ( embedd9027525575939734154ra_a_b @ K @ E2 @ r ) ) ) ).

% finite_dimension_imp_subalgebra
thf(fact_1159_subalbegra__incl__imp__finite__dimension,axiom,
    ! [K: set_a,E2: set_a,V5: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd8708762675212832759on_a_b @ r @ K @ E2 )
       => ( ( embedd9027525575939734154ra_a_b @ K @ V5 @ r )
         => ( ( ord_less_eq_set_a @ V5 @ E2 )
           => ( embedd8708762675212832759on_a_b @ r @ K @ V5 ) ) ) ) ) ).

% subalbegra_incl_imp_finite_dimension
thf(fact_1160_set__add__zero,axiom,
    ! [A2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( set_add_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ A2 )
        = A2 ) ) ).

% set_add_zero
thf(fact_1161_poly__mult__append__zero__lcancel,axiom,
    ! [K: set_a,P: list_a,Q: list_a,R3: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( polynomial_a_b @ r @ K @ Q )
         => ( ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
              = ( append_a @ R3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
           => ( ( poly_mult_a_b @ r @ P @ Q )
              = R3 ) ) ) ) ) ).

% poly_mult_append_zero_lcancel
thf(fact_1162_poly__mult_Osimps_I1_J,axiom,
    ! [P23: list_a] :
      ( ( poly_mult_a_b @ r @ nil_a @ P23 )
      = nil_a ) ).

% poly_mult.simps(1)
thf(fact_1163_poly__mult__closed,axiom,
    ! [K: set_a,P12: list_a,P23: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P12 )
       => ( ( polynomial_a_b @ r @ K @ P23 )
         => ( polynomial_a_b @ r @ K @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) ) ) ) ).

% poly_mult_closed
thf(fact_1164_poly__mult__comm,axiom,
    ! [P12: list_a,P23: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_mult_a_b @ r @ P12 @ P23 )
          = ( poly_mult_a_b @ r @ P23 @ P12 ) ) ) ) ).

% poly_mult_comm
thf(fact_1165_poly__mult__in__carrier,axiom,
    ! [P12: list_a,P23: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% poly_mult_in_carrier
thf(fact_1166_poly__mult__integral,axiom,
    ! [K: set_a,P12: list_a,P23: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P12 )
       => ( ( polynomial_a_b @ r @ K @ P23 )
         => ( ( ( poly_mult_a_b @ r @ P12 @ P23 )
              = nil_a )
           => ( ( P12 = nil_a )
              | ( P23 = nil_a ) ) ) ) ) ) ).

% poly_mult_integral
thf(fact_1167_zeropideal,axiom,
    principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).

% zeropideal
thf(fact_1168_one__zeroI,axiom,
    ( ( ( partia707051561876973205xt_a_b @ r )
      = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
   => ( ( one_a_ring_ext_a_b @ r )
      = ( zero_a_b @ r ) ) ) ).

% one_zeroI
thf(fact_1169_one__zeroD,axiom,
    ( ( ( one_a_ring_ext_a_b @ r )
      = ( zero_a_b @ r ) )
   => ( ( partia707051561876973205xt_a_b @ r )
      = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).

% one_zeroD
thf(fact_1170_carrier__one__zero,axiom,
    ( ( ( partia707051561876973205xt_a_b @ r )
      = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
    = ( ( one_a_ring_ext_a_b @ r )
      = ( zero_a_b @ r ) ) ) ).

% carrier_one_zero
thf(fact_1171_carrier__one__not__zero,axiom,
    ( ( ( partia707051561876973205xt_a_b @ r )
     != ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
    = ( ( one_a_ring_ext_a_b @ r )
     != ( zero_a_b @ r ) ) ) ).

% carrier_one_not_zero
thf(fact_1172_poly__mult__zero_I2_J,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_mult_a_b @ r @ P @ nil_a )
        = nil_a ) ) ).

% poly_mult_zero(2)
thf(fact_1173_poly__mult__zero_I1_J,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_mult_a_b @ r @ nil_a @ P )
        = nil_a ) ) ).

% poly_mult_zero(1)
thf(fact_1174_poly__mult__is__polynomial,axiom,
    ! [K: set_a,P12: list_a,P23: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K )
         => ( polynomial_a_b @ r @ K @ ( poly_mult_a_b @ r @ P12 @ P23 ) ) ) ) ) ).

% poly_mult_is_polynomial
thf(fact_1175_poly__mult__monom__assoc,axiom,
    ! [P: list_a,Q: list_a,A: a,N: nat] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P ) @ Q )
            = ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).

% poly_mult_monom_assoc
thf(fact_1176_poly__mult__semiassoc,axiom,
    ! [P: list_a,Q: list_a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P ) @ Q )
            = ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).

% poly_mult_semiassoc
thf(fact_1177_eval__poly__mult,axiom,
    ! [P: list_a,Q: list_a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( eval_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) @ A )
            = ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).

% eval_poly_mult
thf(fact_1178_const__term__simprules_I2_J,axiom,
    ! [P: list_a,Q: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( const_term_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) )
          = ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).

% const_term_simprules(2)
thf(fact_1179_poly__mult__one_I2_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( poly_mult_a_b @ r @ P @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) )
          = P ) ) ) ).

% poly_mult_one(2)
thf(fact_1180_poly__mult__one_I1_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( poly_mult_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) @ P )
          = P ) ) ) ).

% poly_mult_one(1)
thf(fact_1181_poly__mult__append__zero__rcancel,axiom,
    ! [K: set_a,P: list_a,Q: list_a,R3: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( polynomial_a_b @ r @ K @ Q )
         => ( ( ( poly_mult_a_b @ r @ P @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
              = ( append_a @ R3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
           => ( ( poly_mult_a_b @ r @ P @ Q )
              = R3 ) ) ) ) ) ).

% poly_mult_append_zero_rcancel
thf(fact_1182_monom__eq__var__pow,axiom,
    ! [K: set_a,A: a,N: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
       => ( ( monom_a_b @ r @ A @ N )
          = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A @ nil_a ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ) ) ).

% monom_eq_var_pow
thf(fact_1183_Span__mem__imp__non__trivial__combine,axiom,
    ! [K: set_a,Us2: list_a,A: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
         => ~ ! [K2: a] :
                ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
               => ! [Ks: list_a] :
                    ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ K )
                   => ( ( ( size_size_list_a @ Ks )
                        = ( size_size_list_a @ Us2 ) )
                     => ( ( embedded_combine_a_b @ r @ ( cons_a @ K2 @ Ks ) @ ( cons_a @ A @ Us2 ) )
                       != ( zero_a_b @ r ) ) ) ) ) ) ) ) ).

% Span_mem_imp_non_trivial_combine
thf(fact_1184_lead__coeff__not__zero,axiom,
    ! [K: set_a,A: a,P: list_a] :
      ( ( polynomial_a_b @ r @ K @ ( cons_a @ A @ P ) )
     => ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).

% lead_coeff_not_zero
thf(fact_1185_subfield__m__inv__simprule,axiom,
    ! [K: set_a,K3: a,A: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ K3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ A ) @ K )
           => ( member_a @ A @ K ) ) ) ) ) ).

% subfield_m_inv_simprule
thf(fact_1186_lead__coeff__in__carrier,axiom,
    ! [K: set_a,A: a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ ( cons_a @ A @ P ) )
       => ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).

% lead_coeff_in_carrier
thf(fact_1187_ring__irreducibleI,axiom,
    ! [R3: a] :
      ( ( member_a @ R3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
     => ( ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) )
       => ( ! [A3: a,B2: a] :
              ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
             => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
               => ( ( R3
                    = ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) )
                 => ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
                    | ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
         => ( ring_r999134135267193926le_a_b @ r @ R3 ) ) ) ) ).

% ring_irreducibleI
thf(fact_1188_Span__m__inv__simprule,axiom,
    ! [K: set_a,Us2: list_a,K3: a,A: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ K3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ A ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
             => ( member_a @ A @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ) ).

% Span_m_inv_simprule
thf(fact_1189_Span__mem__iff,axiom,
    ! [K: set_a,Us2: list_a,A: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
            = ( ? [X3: a] :
                  ( ( member_a @ X3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
                  & ? [Ks4: list_a] :
                      ( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ K )
                      & ( ( embedded_combine_a_b @ r @ ( cons_a @ X3 @ Ks4 ) @ ( cons_a @ A @ Us2 ) )
                        = ( zero_a_b @ r ) ) ) ) ) ) ) ) ) ).

% Span_mem_iff
thf(fact_1190_associated__polynomials__iff,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
            = ( ? [X3: a] :
                  ( ( member_a @ X3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
                  & ( P
                    = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ X3 @ nil_a ) @ Q ) ) ) ) ) ) ) ) ).

% associated_polynomials_iff
thf(fact_1191_const__is__polynomial,axiom,
    ! [A: a,K: set_a] :
      ( ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
     => ( polynomial_a_b @ r @ K @ ( cons_a @ A @ nil_a ) ) ) ).

% const_is_polynomial
thf(fact_1192_monom__is__polynomial,axiom,
    ! [K: set_a,A: a,N: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
       => ( polynomial_a_b @ r @ K @ ( monom_a_b @ r @ A @ N ) ) ) ) ).

% monom_is_polynomial
thf(fact_1193_primeideal__iff__prime,axiom,
    ! [P: a] :
      ( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
     => ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r )
        = ( ring_ring_prime_a_b @ r @ P ) ) ) ).

% primeideal_iff_prime
thf(fact_1194_zeroprimeideal__domainI,axiom,
    ( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
   => ( domain_a_b @ r ) ) ).

% zeroprimeideal_domainI
thf(fact_1195_zeroprimeideal,axiom,
    primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).

% zeroprimeideal
thf(fact_1196_var__carr,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( member_list_a @ ( var_a_b @ r ) @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) @ bot_bot_set_list_a ) ) ) ) ).

% var_carr
thf(fact_1197_domain__eq__zeroprimeideal,axiom,
    ( ( domain_a_b @ r )
    = ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ) ) ).

% domain_eq_zeroprimeideal
thf(fact_1198_var__pow__carr,axiom,
    ! [K: set_a,N: nat] :
      ( ( subring_a_b @ K @ r )
     => ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) @ bot_bot_set_list_a ) ) ) ) ).

% var_pow_carr
thf(fact_1199_maximalideal__prime,axiom,
    ! [I2: set_a] :
      ( ( maximalideal_a_b @ I2 @ r )
     => ( primeideal_a_b @ I2 @ r ) ) ).

% maximalideal_prime
thf(fact_1200_poly__mult__monom,axiom,
    ! [P: list_a,A: a,N: nat] :
      ( ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P )
     => ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
       => ( ( ( P = nil_a )
           => ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
              = nil_a ) )
          & ( ( P != nil_a )
           => ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
              = ( append_a @ ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ) ).

% poly_mult_monom
thf(fact_1201_local_Omonom__def,axiom,
    ! [A: a,N: nat] :
      ( ( monom_a_b @ r @ A @ N )
      = ( cons_a @ A @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ).

% local.monom_def
thf(fact_1202_append__is__polynomial,axiom,
    ! [K: set_a,P: list_a,N: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( P != nil_a )
         => ( polynomial_a_b @ r @ K @ ( append_a @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).

% append_is_polynomial
thf(fact_1203_poly__mult__replicate__zero_I1_J,axiom,
    ! [P: list_a,N: nat] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_mult_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
        = nil_a ) ) ).

% poly_mult_replicate_zero(1)
thf(fact_1204_poly__mult__replicate__zero_I2_J,axiom,
    ! [P: list_a,N: nat] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_mult_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
        = nil_a ) ) ).

% poly_mult_replicate_zero(2)
thf(fact_1205_combine__replicate,axiom,
    ! [Us2: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( embedded_combine_a_b @ r @ ( replicate_a @ ( size_size_list_a @ Us2 ) @ ( zero_a_b @ r ) ) @ Us2 )
        = ( zero_a_b @ r ) ) ) ).

% combine_replicate
thf(fact_1206_poly__mult__prepend__replicate__zero,axiom,
    ! [P12: list_a,P23: list_a,N: nat] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_mult_a_b @ r @ P12 @ P23 )
          = ( poly_mult_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P12 ) @ P23 ) ) ) ) ).

% poly_mult_prepend_replicate_zero
thf(fact_1207_combine__append__replicate,axiom,
    ! [Us2: list_a,Ks2: list_a,N: nat] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( embedded_combine_a_b @ r @ ( append_a @ Ks2 @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) @ Us2 )
        = ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) ) ) ).

% combine_append_replicate
thf(fact_1208_eval__replicate,axiom,
    ! [P: list_a,A: a,N: nat] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( eval_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) @ A )
          = ( eval_a_b @ r @ P @ A ) ) ) ) ).

% eval_replicate
thf(fact_1209_poly__add__monom,axiom,
    ! [P: list_a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
       => ( ( poly_add_a_b @ r @ ( monom_a_b @ r @ A @ ( size_size_list_a @ P ) ) @ P )
          = ( cons_a @ A @ P ) ) ) ) ).

% poly_add_monom
thf(fact_1210_poly__mult__var_H_I2_J,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_mult_a_b @ r @ P @ ( var_a_b @ r ) )
        = ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).

% poly_mult_var'(2)
thf(fact_1211_normalize_Osimps_I1_J,axiom,
    ( ( normalize_a_b @ r @ nil_a )
    = nil_a ) ).

% normalize.simps(1)
thf(fact_1212_normalize__polynomial,axiom,
    ! [K: set_a,P: list_a] :
      ( ( polynomial_a_b @ r @ K @ P )
     => ( ( normalize_a_b @ r @ P )
        = P ) ) ).

% normalize_polynomial
thf(fact_1213_local_Onormalize__idem,axiom,
    ! [P: list_a,Q: list_a] :
      ( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P ) @ Q ) )
      = ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ).

% local.normalize_idem
thf(fact_1214_normalize__length__le,axiom,
    ! [P: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) ) ).

% normalize_length_le
thf(fact_1215_poly__add__closed,axiom,
    ! [K: set_a,P12: list_a,P23: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P12 )
       => ( ( polynomial_a_b @ r @ K @ P23 )
         => ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P12 @ P23 ) ) ) ) ) ).

% poly_add_closed
thf(fact_1216_normalize__in__carrier,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ r @ P ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% normalize_in_carrier
thf(fact_1217_poly__add__in__carrier,axiom,
    ! [P12: list_a,P23: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% poly_add_in_carrier
thf(fact_1218_poly__add__comm,axiom,
    ! [P12: list_a,P23: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ P12 @ P23 )
          = ( poly_add_a_b @ r @ P23 @ P12 ) ) ) ) ).

% poly_add_comm
thf(fact_1219_normalize__gives__polynomial,axiom,
    ! [P: list_a,K: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K )
     => ( polynomial_a_b @ r @ K @ ( normalize_a_b @ r @ P ) ) ) ).

% normalize_gives_polynomial
thf(fact_1220_poly__add__zero_I1_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( poly_add_a_b @ r @ P @ nil_a )
          = P ) ) ) ).

% poly_add_zero(1)
thf(fact_1221_poly__add__zero_I2_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( poly_add_a_b @ r @ nil_a @ P )
          = P ) ) ) ).

% poly_add_zero(2)
thf(fact_1222_poly__mult__l__distr,axiom,
    ! [K: set_a,P12: list_a,P23: list_a,P32: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P12 )
       => ( ( polynomial_a_b @ r @ K @ P23 )
         => ( ( polynomial_a_b @ r @ K @ P32 )
           => ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P12 @ P23 ) @ P32 )
              = ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P12 @ P32 ) @ ( poly_mult_a_b @ r @ P23 @ P32 ) ) ) ) ) ) ) ).

% poly_mult_l_distr
thf(fact_1223_poly__mult__r__distr,axiom,
    ! [K: set_a,P12: list_a,P23: list_a,P32: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P12 )
       => ( ( polynomial_a_b @ r @ K @ P23 )
         => ( ( polynomial_a_b @ r @ K @ P32 )
           => ( ( poly_mult_a_b @ r @ P12 @ ( poly_add_a_b @ r @ P23 @ P32 ) )
              = ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P12 @ P23 ) @ ( poly_mult_a_b @ r @ P12 @ P32 ) ) ) ) ) ) ) ).

% poly_mult_r_distr
thf(fact_1224_normalize__replicate__zero,axiom,
    ! [N: nat,P: list_a] :
      ( ( normalize_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
      = ( normalize_a_b @ r @ P ) ) ).

% normalize_replicate_zero
thf(fact_1225_poly__mult__normalize,axiom,
    ! [P12: list_a,P23: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_mult_a_b @ r @ P12 @ P23 )
          = ( poly_mult_a_b @ r @ ( normalize_a_b @ r @ P12 ) @ P23 ) ) ) ) ).

% poly_mult_normalize
thf(fact_1226_poly__mult__l__distr_H,axiom,
    ! [P12: list_a,P23: list_a,P32: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P12 @ P23 ) @ P32 )
            = ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P12 @ P32 ) @ ( poly_mult_a_b @ r @ P23 @ P32 ) ) ) ) ) ) ).

% poly_mult_l_distr'
thf(fact_1227_poly__mult__r__distr_H,axiom,
    ! [P12: list_a,P23: list_a,P32: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( poly_mult_a_b @ r @ P12 @ ( poly_add_a_b @ r @ P23 @ P32 ) )
            = ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P12 @ P23 ) @ ( poly_mult_a_b @ r @ P12 @ P32 ) ) ) ) ) ) ).

% poly_mult_r_distr'
thf(fact_1228_poly__add__normalize__aux,axiom,
    ! [P12: list_a,P23: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ P12 @ P23 )
          = ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P12 ) @ P23 ) ) ) ) ).

% poly_add_normalize_aux
thf(fact_1229_poly__add__normalize_I2_J,axiom,
    ! [P12: list_a,P23: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ P12 @ P23 )
          = ( poly_add_a_b @ r @ P12 @ ( normalize_a_b @ r @ P23 ) ) ) ) ) ).

% poly_add_normalize(2)
thf(fact_1230_poly__add__normalize_I3_J,axiom,
    ! [P12: list_a,P23: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ P12 @ P23 )
          = ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P12 ) @ ( normalize_a_b @ r @ P23 ) ) ) ) ) ).

% poly_add_normalize(3)
thf(fact_1231_eval__normalize,axiom,
    ! [P: list_a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( eval_a_b @ r @ ( normalize_a_b @ r @ P ) @ A )
          = ( eval_a_b @ r @ P @ A ) ) ) ) ).

% eval_normalize
thf(fact_1232_poly__add__is__polynomial,axiom,
    ! [K: set_a,P12: list_a,P23: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K )
         => ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P12 @ P23 ) ) ) ) ) ).

% poly_add_is_polynomial
thf(fact_1233_poly__add__replicate__zero_I2_J,axiom,
    ! [K: set_a,P: list_a,N: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
          = P ) ) ) ).

% poly_add_replicate_zero(2)
thf(fact_1234_poly__add__replicate__zero_I1_J,axiom,
    ! [K: set_a,P: list_a,N: nat] :
      ( ( subring_a_b @ K @ r )
     => ( ( polynomial_a_b @ r @ K @ P )
       => ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
          = P ) ) ) ).

% poly_add_replicate_zero(1)
thf(fact_1235_eval__poly__add,axiom,
    ! [P: list_a,Q: list_a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
            = ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).

% eval_poly_add
thf(fact_1236_poly__add__zero_H_I2_J,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_add_a_b @ r @ nil_a @ P )
        = ( normalize_a_b @ r @ P ) ) ) ).

% poly_add_zero'(2)
thf(fact_1237_poly__add__zero_H_I1_J,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_add_a_b @ r @ P @ nil_a )
        = ( normalize_a_b @ r @ P ) ) ) ).

% poly_add_zero'(1)
thf(fact_1238_const__term__simprules_I3_J,axiom,
    ! [P: list_a,Q: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( const_term_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) )
          = ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).

% const_term_simprules(3)
thf(fact_1239_eval__poly__add__aux,axiom,
    ! [P: list_a,Q: list_a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( size_size_list_a @ P )
            = ( size_size_list_a @ Q ) )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
              = ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ) ).

% eval_poly_add_aux
thf(fact_1240_poly__add__replicate__zero_H_I2_J,axiom,
    ! [P: list_a,N: nat] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
        = ( normalize_a_b @ r @ P ) ) ) ).

% poly_add_replicate_zero'(2)
thf(fact_1241_poly__add__replicate__zero_H_I1_J,axiom,
    ! [P: list_a,N: nat] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
        = ( normalize_a_b @ r @ P ) ) ) ).

% poly_add_replicate_zero'(1)
thf(fact_1242_poly__mult__one_H_I1_J,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_mult_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) @ P )
        = ( normalize_a_b @ r @ P ) ) ) ).

% poly_mult_one'(1)
thf(fact_1243_poly__mult__one_H_I2_J,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_mult_a_b @ r @ P @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) )
        = ( normalize_a_b @ r @ P ) ) ) ).

% poly_mult_one'(2)
thf(fact_1244_poly__mult__append__zero,axiom,
    ! [P: list_a,Q: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
          = ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).

% poly_mult_append_zero
thf(fact_1245_poly__add__append__zero,axiom,
    ! [P: list_a,Q: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
          = ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).

% poly_add_append_zero
thf(fact_1246_poly__add__append__replicate,axiom,
    ! [P: list_a,Q: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ ( append_a @ P @ ( replicate_a @ ( size_size_list_a @ Q ) @ ( zero_a_b @ r ) ) ) @ Q )
          = ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ) ) ).

% poly_add_append_replicate
thf(fact_1247_poly__mult__var_H_I1_J,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( poly_mult_a_b @ r @ ( var_a_b @ r ) @ P )
        = ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).

% poly_mult_var'(1)
thf(fact_1248_poly__mult__monom_H,axiom,
    ! [P: list_a,A: a,N: nat] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
          = ( normalize_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ A ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).

% poly_mult_monom'
thf(fact_1249_field__intro2,axiom,
    ( ( ( zero_a_b @ r )
     != ( one_a_ring_ext_a_b @ r ) )
   => ( ! [X2: a] :
          ( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
         => ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) ) )
     => ( field_a_b @ r ) ) ) ).

% field_intro2
thf(fact_1250_cring__fieldI2,axiom,
    ( ( ( zero_a_b @ r )
     != ( one_a_ring_ext_a_b @ r ) )
   => ( ! [A3: a] :
          ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( A3
             != ( zero_a_b @ r ) )
           => ? [X4: a] :
                ( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
                & ( ( mult_a_ring_ext_a_b @ r @ A3 @ X4 )
                  = ( one_a_ring_ext_a_b @ r ) ) ) ) )
     => ( field_a_b @ r ) ) ) ).

% cring_fieldI2
thf(fact_1251_zeromaximalideal__eq__field,axiom,
    ( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
    = ( field_a_b @ r ) ) ).

% zeromaximalideal_eq_field
thf(fact_1252_zeromaximalideal__fieldI,axiom,
    ( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
   => ( field_a_b @ r ) ) ).

% zeromaximalideal_fieldI
thf(fact_1253_combine__r__distr,axiom,
    ! [Ks2: list_a,Us2: list_a,K3: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ K3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ K3 @ ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) )
            = ( embedded_combine_a_b @ r @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ K3 ) @ Ks2 ) @ Us2 ) ) ) ) ) ).

% combine_r_distr
thf(fact_1254_cring__fieldI,axiom,
    ( ( ( units_a_ring_ext_a_b @ r )
      = ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
   => ( field_a_b @ r ) ) ).

% cring_fieldI
thf(fact_1255_cgenideal__eq__genideal,axiom,
    ! [I: a] :
      ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( cgenid547466209912283029xt_a_b @ r @ I )
        = ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).

% cgenideal_eq_genideal
thf(fact_1256_univ__poly__not__field,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_not_field
thf(fact_1257_norm__map__in__poly__ring__carrier,axiom,
    ! [P: list_a,F: a > list_a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ! [A3: a] :
            ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_a_list_a @ F @ P ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).

% norm_map_in_poly_ring_carrier
thf(fact_1258_genideal__self,axiom,
    ! [S: set_a] :
      ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).

% genideal_self
thf(fact_1259_subset__Idl__subset,axiom,
    ! [I2: set_a,H: set_a] :
      ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ H @ I2 )
       => ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).

% subset_Idl_subset
thf(fact_1260_genideal__self_H,axiom,
    ! [I: a] :
      ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).

% genideal_self'
thf(fact_1261_genideal__zero,axiom,
    ( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
    = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).

% genideal_zero
thf(fact_1262_map__in__poly__ring__carrier,axiom,
    ! [P: list_a,F: a > list_a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ! [A3: a] :
            ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [A3: a] :
              ( ( A3
               != ( zero_a_b @ r ) )
             => ( ( F @ A3 )
               != nil_a ) )
         => ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).

% map_in_poly_ring_carrier
thf(fact_1263_Idl__subset__ideal_H,axiom,
    ! [A: a,B: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
          = ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).

% Idl_subset_ideal'
thf(fact_1264_genideal__one,axiom,
    ( ( genideal_a_b @ r @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
    = ( partia707051561876973205xt_a_b @ r ) ) ).

% genideal_one
thf(fact_1265_rupture__is__field__iff__pirreducible,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) )
          = ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).

% rupture_is_field_iff_pirreducible
thf(fact_1266_dependent__imp__non__trivial__combine,axiom,
    ! [K: set_a,Us2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ~ ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
         => ~ ! [Ks: list_a] :
                ( ( ( size_size_list_a @ Ks )
                  = ( size_size_list_a @ Us2 ) )
               => ( ( ( embedded_combine_a_b @ r @ Ks @ Us2 )
                    = ( zero_a_b @ r ) )
                 => ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ K )
                   => ( ( set_a2 @ Ks )
                      = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ) ) ) ).

% dependent_imp_non_trivial_combine
thf(fact_1267_independent__backwards_I2_J,axiom,
    ! [K: set_a,U: a,Us2: list_a] :
      ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ Us2 ) )
     => ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 ) ) ).

% independent_backwards(2)
thf(fact_1268_li__Nil,axiom,
    ! [K: set_a] : ( embedd5208550302661555450nt_a_b @ r @ K @ nil_a ) ).

% li_Nil
thf(fact_1269_independent__backwards_I3_J,axiom,
    ! [K: set_a,U: a,Us2: list_a] :
      ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ Us2 ) )
     => ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% independent_backwards(3)
thf(fact_1270_independent__backwards_I1_J,axiom,
    ! [K: set_a,U: a,Us2: list_a] :
      ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ Us2 ) )
     => ~ ( member_a @ U @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ).

% independent_backwards(1)
thf(fact_1271_independent__split_I1_J,axiom,
    ! [K: set_a,Us2: list_a,Vs: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) )
       => ( embedd5208550302661555450nt_a_b @ r @ K @ Vs ) ) ) ).

% independent_split(1)
thf(fact_1272_independent__split_I2_J,axiom,
    ! [K: set_a,Us2: list_a,Vs: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) )
       => ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 ) ) ) ).

% independent_split(2)
thf(fact_1273_independent__in__carrier,axiom,
    ! [K: set_a,Us2: list_a] :
      ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
     => ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% independent_in_carrier
thf(fact_1274_li__Cons,axiom,
    ! [U: a,K: set_a,Us2: list_a] :
      ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ~ ( member_a @ U @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
       => ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
         => ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ Us2 ) ) ) ) ) ).

% li_Cons
thf(fact_1275_independent__same__set,axiom,
    ! [K: set_a,Us2: list_a,Vs: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ( set_a2 @ Us2 )
          = ( set_a2 @ Vs ) )
       => ( ( ( size_size_list_a @ Us2 )
            = ( size_size_list_a @ Vs ) )
         => ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
           => ( embedd5208550302661555450nt_a_b @ r @ K @ Vs ) ) ) ) ) ).

% independent_same_set

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( formal4452980811800949548iv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ f @ g ) )
    = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( formal4452980811800949548iv_a_b @ r @ f ) @ ( formal4452980811800949548iv_a_b @ r @ g ) ) ) ).

%------------------------------------------------------------------------------