TPTP Problem File: SLH0091^1.p

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

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Finite_Fields/0003_Finite_Fields_Preliminary_Results/prob_00923_033365__18095154_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1883 ( 348 unt; 606 typ;   0 def)
%            Number of atoms       : 4034 (1522 equ;   0 cnn)
%            Maximal formula atoms :   11 (   3 avg)
%            Number of connectives : 23608 ( 204   ~;  20   |; 109   &;20918   @)
%                                         (   0 <=>;2357  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   9 avg)
%            Number of types       :   89 (  88 usr)
%            Number of type conns  : 1910 (1910   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  519 ( 518 usr;  13 con; 0-4 aty)
%            Number of variables   : 3457 ( 192   ^;3142   !; 123   ?;3457   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 13:20:53.995
%------------------------------------------------------------------------------
% Could-be-implicit typings (88)
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thf(ty_n_t__List__Olist_I_062_Itf__a_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
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thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
    set_set_list_a_a: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    list_list_list_a: $tType ).

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

thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    list_set_list_a: $tType ).

thf(ty_n_t__List__Olist_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
    list_a_list_a: $tType ).

thf(ty_n_t__List__Olist_I_062_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
    list_list_a_a: $tType ).

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

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
    set_a_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
    set_list_a_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_Itf__a_J_Mtf__a_J_J,type,
    set_set_a_a: $tType ).

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

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    set_set_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
    set_a_a: $tType ).

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

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

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

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

% Explicit typings (518)
thf(sy_c_AbelCoset_OA__RCOSETS_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    a_RCOS6220190738183020281t_unit: partia2670972154091845814t_unit > set_list_a > set_set_list_a ).

thf(sy_c_AbelCoset_OA__RCOSETS_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    a_RCOS7985542791853034265t_unit: partia7496981018696276118t_unit > set_set_list_a > set_set_set_list_a ).

thf(sy_c_AbelCoset_Oa__kernel_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
    a_kern2892906198464180898t_unit: partia2956882679547061052t_unit > partia4960592913263135132t_unit > ( list_list_a > set_list_list_a ) > set_list_list_a ).

thf(sy_c_AbelCoset_Oa__kernel_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    a_kern4140738655600802976it_a_b: partia4556295656693239580t_unit > partia2175431115845679010xt_a_b > ( list_set_list_a > a ) > set_list_set_list_a ).

thf(sy_c_AbelCoset_Oa__kernel_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    a_kern4386753233124370696t_unit: partia2670972154091845814t_unit > partia2956882679547061052t_unit > ( list_a > list_list_a ) > set_list_a ).

thf(sy_c_AbelCoset_Oa__kernel_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    a_kern6921148033081882766t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > ( list_a > list_a ) > set_list_a ).

thf(sy_c_AbelCoset_Oa__kernel_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    a_kern570848875181920430t_unit: partia2670972154091845814t_unit > partia7496981018696276118t_unit > ( list_a > set_list_a ) > set_list_a ).

thf(sy_c_AbelCoset_Oa__kernel_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    a_kern7116238624728830086it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > ( list_a > a ) > set_list_a ).

thf(sy_c_AbelCoset_Oa__kernel_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    a_kern9182406240521667104it_a_b: partia4960592913263135132t_unit > partia2175431115845679010xt_a_b > ( set_list_list_a > a ) > set_set_list_list_a ).

thf(sy_c_AbelCoset_Oa__kernel_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    a_kern2420883521965667502t_unit: partia7496981018696276118t_unit > partia2670972154091845814t_unit > ( set_list_a > list_a ) > set_set_list_a ).

thf(sy_c_AbelCoset_Oa__kernel_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    a_kern5779274734665759910it_a_b: partia7496981018696276118t_unit > partia2175431115845679010xt_a_b > ( set_list_a > a ) > set_set_list_a ).

thf(sy_c_AbelCoset_Oa__kernel_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    a_kern4625283579065039622t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > ( a > list_a ) > set_a ).

thf(sy_c_AbelCoset_Oa__kernel_001tf__a_001tf__b_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    a_kern2981893169186190118t_unit: partia2175431115845679010xt_a_b > partia7496981018696276118t_unit > ( a > set_list_a ) > set_a ).

thf(sy_c_AbelCoset_Oa__kernel_001tf__a_001tf__b_001tf__a_001tf__b,type,
    a_kernel_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > ( a > a ) > set_a ).

thf(sy_c_AbelCoset_Oa__l__coset_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    a_l_co7008843373686234386t_unit: partia2670972154091845814t_unit > list_a > set_list_a > set_list_a ).

thf(sy_c_AbelCoset_Oa__l__coset_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    a_l_co4135707798524667186t_unit: partia7496981018696276118t_unit > set_list_a > set_set_list_a > set_set_list_a ).

thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
    a_l_coset_a_b: partia2175431115845679010xt_a_b > a > set_a > set_a ).

thf(sy_c_AbelCoset_Oa__r__coset_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
    a_r_co4300121960189213440t_unit: partia5333488208502193986t_unit > set_list_list_list_a > list_list_list_a > set_list_list_list_a ).

thf(sy_c_AbelCoset_Oa__r__coset_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    a_r_co7333016316668015878t_unit: partia2956882679547061052t_unit > set_list_list_a > list_list_a > set_list_list_a ).

thf(sy_c_AbelCoset_Oa__r__coset_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
    a_r_co9127925791106265382t_unit: partia4556295656693239580t_unit > set_list_set_list_a > list_set_list_a > set_list_set_list_a ).

thf(sy_c_AbelCoset_Oa__r__coset_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    a_r_co1577572119920843660t_unit: partia2670972154091845814t_unit > set_list_a > list_a > set_list_a ).

thf(sy_c_AbelCoset_Oa__r__coset_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    a_r_co866426763593333676t_unit: partia7496981018696276118t_unit > set_set_list_a > set_list_a > set_set_list_a ).

thf(sy_c_AbelCoset_Oa__r__coset_001tf__a_001tf__b,type,
    a_r_coset_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a ).

thf(sy_c_AbelCoset_Oabelian__group__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    abelia2079122467792593029t_unit: partia2670972154091845814t_unit > partia7496981018696276118t_unit > ( list_a > set_list_a ) > $o ).

thf(sy_c_AbelCoset_Oabelian__subgroup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    abelia6695205329122750356t_unit: set_list_a > partia2670972154091845814t_unit > $o ).

thf(sy_c_AbelCoset_Oadditive__subgroup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    additi4714453376129182166t_unit: set_list_a > partia2670972154091845814t_unit > $o ).

thf(sy_c_AbelCoset_Oadditive__subgroup_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    additi5174913095647240694t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).

thf(sy_c_AbelCoset_Oset__add_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    set_ad92425877771022410t_unit: partia2670972154091845814t_unit > set_list_a > set_list_a > set_list_a ).

thf(sy_c_AbelCoset_Oset__add_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    set_ad2573391977335003242t_unit: partia7496981018696276118t_unit > set_set_list_a > set_set_list_a > set_set_list_a ).

thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
    set_add_a_b: partia2175431115845679010xt_a_b > set_a > set_a > set_a ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_Mt__Product____Type__Ounit_J_J,type,
    partia2412307164297199803t_unit: partia6416617044002294984t_unit > set_li3407770045201608054list_a ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_J_Mt__Product____Type__Ounit_J_J,type,
    partia4325684815206509723t_unit: partia6402430829747179688t_unit > set_li4092901231358442838list_a ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J,type,
    partia5038748322285217333t_unit: partia5333488208502193986t_unit > set_list_list_list_a ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J,type,
    partia7265347635606999311t_unit: partia4556295656693239580t_unit > set_list_set_list_a ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
    partia5361259788508890537t_unit: partia2670972154091845814t_unit > set_list_a ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J,type,
    partia3317168157747563407t_unit: partia4960592913263135132t_unit > set_set_list_list_a ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
    partia5725451877363000582t_unit: ( set_list_list_a > set_list_list_a ) > partia2956882679547061052t_unit > partia2956882679547061052t_unit ).

thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001t__List__Olist_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
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thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
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    member6842060177613954879list_a: list_l7815035709764258326list_a > set_li3407770045201608054list_a > $o ).

thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J_J,type,
    member8950995646279208735list_a: list_l1046759797270158838list_a > set_li4092901231358442838list_a > $o ).

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

thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J,type,
    member6908278366116243871list_a: list_l8227414553961969782list_a > set_li3232697616086161366list_a > $o ).

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

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

thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    member6124916891863447321list_a: list_set_list_list_a > set_li7845362039408639568list_a > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    member334759470184282131list_a: set_list_list_a > set_set_list_list_a > $o ).

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

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

thf(sy_c_member_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_p,type,
    p: list_a ).

thf(sy_v_q,type,
    q: list_a ).

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

% domain_axioms
thf(fact_1_assms_I2_J,axiom,
    member_list_a @ p @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% assms(2)
thf(fact_2_assms_I1_J,axiom,
    subring_a_b @ k @ r ).

% assms(1)
thf(fact_3_assms_I3_J,axiom,
    member_list_a @ q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% assms(3)
thf(fact_4_h_Ois__abelian__group__hom,axiom,
    abelia2079122467792593029t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ).

% h.is_abelian_group_hom
thf(fact_5_h_Ohomh,axiom,
    member4263473470251683292list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( ring_h6188449271506562988t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ).

% h.homh
thf(fact_6_h_Osemiring__axioms,axiom,
    semiri4000464634269493571t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ).

% h.semiring_axioms
thf(fact_7_UP_Osemiring__axioms,axiom,
    semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ k ) ).

% UP.semiring_axioms
thf(fact_8_h_Oabelian__monoid__axioms,axiom,
    abelia3322010900105369177t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ).

% h.abelian_monoid_axioms
thf(fact_9_List_Omap_Ocomp,axiom,
    ! [F: a > a,G: a > a] :
      ( ( comp_l467081738950601185list_a @ ( map_a_a @ F ) @ ( map_a_a @ G ) )
      = ( map_a_a @ ( comp_a_a_a @ F @ G ) ) ) ).

% List.map.comp
thf(fact_10_List_Omap_Ocomp,axiom,
    ! [F: list_a > a,G: a > list_a] :
      ( ( comp_l3988554743985804903list_a @ ( map_list_a_a @ F ) @ ( map_a_list_a @ G ) )
      = ( map_a_a @ ( comp_list_a_a_a @ F @ G ) ) ) ).

% List.map.comp
thf(fact_11_List_Omap_Ocomp,axiom,
    ! [F: a > list_a,G: a > a] :
      ( ( comp_l3924320978405955931list_a @ ( map_a_list_a @ F ) @ ( map_a_a @ G ) )
      = ( map_a_list_a @ ( comp_a_list_a_a @ F @ G ) ) ) ).

% List.map.comp
thf(fact_12_List_Omap_Ocomp,axiom,
    ! [F: list_a > list_a,G: a > list_a] :
      ( ( comp_l5129000637246032609list_a @ ( map_list_a_list_a @ F ) @ ( map_a_list_a @ G ) )
      = ( map_a_list_a @ ( comp_list_a_list_a_a @ F @ G ) ) ) ).

% List.map.comp
thf(fact_13_List_Omap_Ocomp,axiom,
    ! [F: set_list_a > a,G: a > set_list_a] :
      ( ( comp_l2716785710105386567list_a @ ( map_set_list_a_a @ F ) @ ( map_a_set_list_a @ G ) )
      = ( map_a_a @ ( comp_set_list_a_a_a @ F @ G ) ) ) ).

% List.map.comp
thf(fact_14_List_Omap_Ocomp,axiom,
    ! [F: a > set_list_a,G: a > a] :
      ( ( comp_l579487490822090107list_a @ ( map_a_set_list_a @ F ) @ ( map_a_a @ G ) )
      = ( map_a_set_list_a @ ( comp_a_set_list_a_a @ F @ G ) ) ) ).

% List.map.comp
thf(fact_15_List_Omap_Ocomp,axiom,
    ! [F: set_list_a > list_a,G: a > set_list_a] :
      ( ( comp_l3887477203848340161list_a @ ( map_se5127173849416245988list_a @ F ) @ ( map_a_set_list_a @ G ) )
      = ( map_a_list_a @ ( comp_s7923821571832083387st_a_a @ F @ G ) ) ) ).

% List.map.comp
thf(fact_16_List_Omap_Ocomp,axiom,
    ! [F: list_a > set_list_a,G: a > list_a] :
      ( ( comp_l4668593498542294785list_a @ ( map_li2657846702790696612list_a @ F ) @ ( map_a_list_a @ G ) )
      = ( map_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ F @ G ) ) ) ).

% List.map.comp
thf(fact_17_List_Omap_Ocomp,axiom,
    ! [F: a > set_list_a,G: list_a > a] :
      ( ( comp_l4522646598638097921list_a @ ( map_a_set_list_a @ F ) @ ( map_list_a_a @ G ) )
      = ( map_li2657846702790696612list_a @ ( comp_a8195215238713344251list_a @ F @ G ) ) ) ).

% List.map.comp
thf(fact_18_List_Omap_Ocomp,axiom,
    ! [F: set_list_a > set_list_a,G: a > set_list_a] :
      ( ( comp_l1921704731345034977list_a @ ( map_se2668659675339852484list_a @ F ) @ ( map_a_set_list_a @ G ) )
      = ( map_a_set_list_a @ ( comp_s7504242949508807643st_a_a @ F @ G ) ) ) ).

% List.map.comp
thf(fact_19_list_Omap__comp,axiom,
    ! [G: a > a,F: a > a,V: list_a] :
      ( ( map_a_a @ G @ ( map_a_a @ F @ V ) )
      = ( map_a_a @ ( comp_a_a_a @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_20_list_Omap__comp,axiom,
    ! [G: list_a > a,F: a > list_a,V: list_a] :
      ( ( map_list_a_a @ G @ ( map_a_list_a @ F @ V ) )
      = ( map_a_a @ ( comp_list_a_a_a @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_21_list_Omap__comp,axiom,
    ! [G: a > list_a,F: a > a,V: list_a] :
      ( ( map_a_list_a @ G @ ( map_a_a @ F @ V ) )
      = ( map_a_list_a @ ( comp_a_list_a_a @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_22_list_Omap__comp,axiom,
    ! [G: list_a > list_a,F: a > list_a,V: list_a] :
      ( ( map_list_a_list_a @ G @ ( map_a_list_a @ F @ V ) )
      = ( map_a_list_a @ ( comp_list_a_list_a_a @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_23_list_Omap__comp,axiom,
    ! [G: a > set_list_a,F: a > a,V: list_a] :
      ( ( map_a_set_list_a @ G @ ( map_a_a @ F @ V ) )
      = ( map_a_set_list_a @ ( comp_a_set_list_a_a @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_24_list_Omap__comp,axiom,
    ! [G: a > a,F: set_list_a > a,V: list_set_list_a] :
      ( ( map_a_a @ G @ ( map_set_list_a_a @ F @ V ) )
      = ( map_set_list_a_a @ ( comp_a_a_set_list_a @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_25_list_Omap__comp,axiom,
    ! [G: set_list_a > a,F: a > set_list_a,V: list_a] :
      ( ( map_set_list_a_a @ G @ ( map_a_set_list_a @ F @ V ) )
      = ( map_a_a @ ( comp_set_list_a_a_a @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_26_list_Omap__comp,axiom,
    ! [G: list_a > a,F: set_list_a > list_a,V: list_set_list_a] :
      ( ( map_list_a_a @ G @ ( map_se5127173849416245988list_a @ F @ V ) )
      = ( map_set_list_a_a @ ( comp_l2163758027525145287list_a @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_27_list_Omap__comp,axiom,
    ! [G: set_list_list_a > a,F: a > set_list_list_a,V: list_a] :
      ( ( map_se4114524712695348952st_a_a @ G @ ( map_a_132662181483840688list_a @ F @ V ) )
      = ( map_a_a @ ( comp_s4342503745753038023_a_a_a @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_28_list_Omap__comp,axiom,
    ! [G: list_a > set_list_a,F: a > list_a,V: list_a] :
      ( ( map_li2657846702790696612list_a @ G @ ( map_a_list_a @ F @ V ) )
      = ( map_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ G @ F ) @ V ) ) ).

% list.map_comp
thf(fact_29_List_Omap_Ocompositionality,axiom,
    ! [F: a > a,G: a > a,List: list_a] :
      ( ( map_a_a @ F @ ( map_a_a @ G @ List ) )
      = ( map_a_a @ ( comp_a_a_a @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_30_List_Omap_Ocompositionality,axiom,
    ! [F: list_a > a,G: a > list_a,List: list_a] :
      ( ( map_list_a_a @ F @ ( map_a_list_a @ G @ List ) )
      = ( map_a_a @ ( comp_list_a_a_a @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_31_List_Omap_Ocompositionality,axiom,
    ! [F: a > list_a,G: a > a,List: list_a] :
      ( ( map_a_list_a @ F @ ( map_a_a @ G @ List ) )
      = ( map_a_list_a @ ( comp_a_list_a_a @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_32_List_Omap_Ocompositionality,axiom,
    ! [F: list_a > list_a,G: a > list_a,List: list_a] :
      ( ( map_list_a_list_a @ F @ ( map_a_list_a @ G @ List ) )
      = ( map_a_list_a @ ( comp_list_a_list_a_a @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_33_List_Omap_Ocompositionality,axiom,
    ! [F: a > set_list_a,G: a > a,List: list_a] :
      ( ( map_a_set_list_a @ F @ ( map_a_a @ G @ List ) )
      = ( map_a_set_list_a @ ( comp_a_set_list_a_a @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_34_List_Omap_Ocompositionality,axiom,
    ! [F: a > a,G: set_list_a > a,List: list_set_list_a] :
      ( ( map_a_a @ F @ ( map_set_list_a_a @ G @ List ) )
      = ( map_set_list_a_a @ ( comp_a_a_set_list_a @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_35_List_Omap_Ocompositionality,axiom,
    ! [F: set_list_a > a,G: a > set_list_a,List: list_a] :
      ( ( map_set_list_a_a @ F @ ( map_a_set_list_a @ G @ List ) )
      = ( map_a_a @ ( comp_set_list_a_a_a @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_36_List_Omap_Ocompositionality,axiom,
    ! [F: list_a > a,G: set_list_a > list_a,List: list_set_list_a] :
      ( ( map_list_a_a @ F @ ( map_se5127173849416245988list_a @ G @ List ) )
      = ( map_set_list_a_a @ ( comp_l2163758027525145287list_a @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_37_List_Omap_Ocompositionality,axiom,
    ! [F: set_list_list_a > a,G: a > set_list_list_a,List: list_a] :
      ( ( map_se4114524712695348952st_a_a @ F @ ( map_a_132662181483840688list_a @ G @ List ) )
      = ( map_a_a @ ( comp_s4342503745753038023_a_a_a @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_38_List_Omap_Ocompositionality,axiom,
    ! [F: list_a > set_list_a,G: a > list_a,List: list_a] :
      ( ( map_li2657846702790696612list_a @ F @ ( map_a_list_a @ G @ List ) )
      = ( map_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ F @ G ) @ List ) ) ).

% List.map.compositionality
thf(fact_39_map__map,axiom,
    ! [F: a > a,G: a > a,Xs: list_a] :
      ( ( map_a_a @ F @ ( map_a_a @ G @ Xs ) )
      = ( map_a_a @ ( comp_a_a_a @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_40_map__map,axiom,
    ! [F: list_a > a,G: a > list_a,Xs: list_a] :
      ( ( map_list_a_a @ F @ ( map_a_list_a @ G @ Xs ) )
      = ( map_a_a @ ( comp_list_a_a_a @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_41_map__map,axiom,
    ! [F: a > list_a,G: a > a,Xs: list_a] :
      ( ( map_a_list_a @ F @ ( map_a_a @ G @ Xs ) )
      = ( map_a_list_a @ ( comp_a_list_a_a @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_42_map__map,axiom,
    ! [F: list_a > list_a,G: a > list_a,Xs: list_a] :
      ( ( map_list_a_list_a @ F @ ( map_a_list_a @ G @ Xs ) )
      = ( map_a_list_a @ ( comp_list_a_list_a_a @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_43_map__map,axiom,
    ! [F: a > set_list_a,G: a > a,Xs: list_a] :
      ( ( map_a_set_list_a @ F @ ( map_a_a @ G @ Xs ) )
      = ( map_a_set_list_a @ ( comp_a_set_list_a_a @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_44_map__map,axiom,
    ! [F: a > a,G: set_list_a > a,Xs: list_set_list_a] :
      ( ( map_a_a @ F @ ( map_set_list_a_a @ G @ Xs ) )
      = ( map_set_list_a_a @ ( comp_a_a_set_list_a @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_45_map__map,axiom,
    ! [F: set_list_a > a,G: a > set_list_a,Xs: list_a] :
      ( ( map_set_list_a_a @ F @ ( map_a_set_list_a @ G @ Xs ) )
      = ( map_a_a @ ( comp_set_list_a_a_a @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_46_map__map,axiom,
    ! [F: list_a > a,G: set_list_a > list_a,Xs: list_set_list_a] :
      ( ( map_list_a_a @ F @ ( map_se5127173849416245988list_a @ G @ Xs ) )
      = ( map_set_list_a_a @ ( comp_l2163758027525145287list_a @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_47_map__map,axiom,
    ! [F: set_list_list_a > a,G: a > set_list_list_a,Xs: list_a] :
      ( ( map_se4114524712695348952st_a_a @ F @ ( map_a_132662181483840688list_a @ G @ Xs ) )
      = ( map_a_a @ ( comp_s4342503745753038023_a_a_a @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_48_map__map,axiom,
    ! [F: list_a > set_list_a,G: a > list_a,Xs: list_a] :
      ( ( map_li2657846702790696612list_a @ F @ ( map_a_list_a @ G @ Xs ) )
      = ( map_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ F @ G ) @ Xs ) ) ).

% map_map
thf(fact_49_map__comp__map,axiom,
    ! [F: a > a,G: a > a] :
      ( ( comp_l467081738950601185list_a @ ( map_a_a @ F ) @ ( map_a_a @ G ) )
      = ( map_a_a @ ( comp_a_a_a @ F @ G ) ) ) ).

% map_comp_map
thf(fact_50_map__comp__map,axiom,
    ! [F: list_a > a,G: a > list_a] :
      ( ( comp_l3988554743985804903list_a @ ( map_list_a_a @ F ) @ ( map_a_list_a @ G ) )
      = ( map_a_a @ ( comp_list_a_a_a @ F @ G ) ) ) ).

% map_comp_map
thf(fact_51_map__comp__map,axiom,
    ! [F: a > list_a,G: a > a] :
      ( ( comp_l3924320978405955931list_a @ ( map_a_list_a @ F ) @ ( map_a_a @ G ) )
      = ( map_a_list_a @ ( comp_a_list_a_a @ F @ G ) ) ) ).

% map_comp_map
thf(fact_52_map__comp__map,axiom,
    ! [F: list_a > list_a,G: a > list_a] :
      ( ( comp_l5129000637246032609list_a @ ( map_list_a_list_a @ F ) @ ( map_a_list_a @ G ) )
      = ( map_a_list_a @ ( comp_list_a_list_a_a @ F @ G ) ) ) ).

% map_comp_map
thf(fact_53_map__comp__map,axiom,
    ! [F: a > set_list_a,G: a > a] :
      ( ( comp_l579487490822090107list_a @ ( map_a_set_list_a @ F ) @ ( map_a_a @ G ) )
      = ( map_a_set_list_a @ ( comp_a_set_list_a_a @ F @ G ) ) ) ).

% map_comp_map
thf(fact_54_map__comp__map,axiom,
    ! [F: a > a,G: set_list_a > a] :
      ( ( comp_l998799806410342215list_a @ ( map_a_a @ F ) @ ( map_set_list_a_a @ G ) )
      = ( map_set_list_a_a @ ( comp_a_a_set_list_a @ F @ G ) ) ) ).

% map_comp_map
thf(fact_55_map__comp__map,axiom,
    ! [F: set_list_a > a,G: a > set_list_a] :
      ( ( comp_l2716785710105386567list_a @ ( map_set_list_a_a @ F ) @ ( map_a_set_list_a @ G ) )
      = ( map_a_a @ ( comp_set_list_a_a_a @ F @ G ) ) ) ).

% map_comp_map
thf(fact_56_map__comp__map,axiom,
    ! [F: list_a > a,G: set_list_a > list_a] :
      ( ( comp_l5087905814130546893list_a @ ( map_list_a_a @ F ) @ ( map_se5127173849416245988list_a @ G ) )
      = ( map_set_list_a_a @ ( comp_l2163758027525145287list_a @ F @ G ) ) ) ).

% map_comp_map
thf(fact_57_map__comp__map,axiom,
    ! [F: set_list_list_a > a,G: a > set_list_list_a] :
      ( ( comp_l449629555712507085list_a @ ( map_se4114524712695348952st_a_a @ F ) @ ( map_a_132662181483840688list_a @ G ) )
      = ( map_a_a @ ( comp_s4342503745753038023_a_a_a @ F @ G ) ) ) ).

% map_comp_map
thf(fact_58_map__comp__map,axiom,
    ! [F: list_a > set_list_a,G: a > list_a] :
      ( ( comp_l4668593498542294785list_a @ ( map_li2657846702790696612list_a @ F ) @ ( map_a_list_a @ G ) )
      = ( map_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ F @ G ) ) ) ).

% map_comp_map
thf(fact_59_h_Oring__hom__ring__axioms,axiom,
    ring_h6719293541701509614t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ).

% h.ring_hom_ring_axioms
thf(fact_60_UP_Ocgenideal__self,axiom,
    ! [I: list_a] :
      ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ I ) ) ) ).

% UP.cgenideal_self
thf(fact_61_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_62_UP_Oring__hom__restrict,axiom,
    ! [F: list_a > set_list_a,S: partia7496981018696276118t_unit,G: list_a > set_list_a] :
      ( ( member4263473470251683292list_a @ F @ ( ring_h6188449271506562988t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) )
     => ( ! [R: list_a] :
            ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( F @ R )
              = ( G @ R ) ) )
       => ( member4263473470251683292list_a @ G @ ( ring_h6188449271506562988t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) ) ) ) ).

% UP.ring_hom_restrict
thf(fact_63_UP_Oring__hom__restrict,axiom,
    ! [F: list_a > a,S: partia2175431115845679010xt_a_b,G: list_a > a] :
      ( ( member_list_a_a @ F @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ k ) @ S ) )
     => ( ! [R: list_a] :
            ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( F @ R )
              = ( G @ R ) ) )
       => ( member_list_a_a @ G @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ k ) @ S ) ) ) ) ).

% UP.ring_hom_restrict
thf(fact_64_map__ident,axiom,
    ( ( map_a_a
      @ ^ [X: a] : X )
    = ( ^ [Xs2: list_a] : Xs2 ) ) ).

% map_ident
thf(fact_65_map__ident,axiom,
    ( ( map_se2668659675339852484list_a
      @ ^ [X: set_list_a] : X )
    = ( ^ [Xs2: list_set_list_a] : Xs2 ) ) ).

% map_ident
thf(fact_66_rupture__surj__hom_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_h6719293541701509614t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ).

% rupture_surj_hom(2)
thf(fact_67_rupture__surj__hom_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 ) ) )
       => ( member4263473470251683292list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) @ ( ring_h6188449271506562988t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) ) ) ) ) ).

% rupture_surj_hom(1)
thf(fact_68_list_Omap__ident,axiom,
    ! [T: list_a] :
      ( ( map_a_a
        @ ^ [X: a] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_69_list_Omap__ident,axiom,
    ! [T: list_set_list_a] :
      ( ( map_se2668659675339852484list_a
        @ ^ [X: set_list_a] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_70_UP_Ocgenideal__is__principalideal,axiom,
    ! [I: list_a] :
      ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ I ) @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.cgenideal_is_principalideal
thf(fact_71_UP_Oonepideal,axiom,
    princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).

% UP.onepideal
thf(fact_72_polynomial__rupture,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) @ ( map_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) @ ( poly_of_const_a_b @ r ) ) @ P ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ ( var_a_b @ r ) ) )
          = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) ) ) ) ) ).

% polynomial_rupture
thf(fact_73__092_060open_062h_Oeval_A_Imap_A_Irupture__surj_AK_Ap_A_092_060circ_062_Apoly__of__const_J_Aq_J_A_Irupture__surj_AK_Ap_AX_J_A_061_Arupture__surj_AK_Ap_A_IUP_Oeval_A_Imap_Apoly__of__const_Aq_J_AX_J_092_060close_062,axiom,
    ( ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( map_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( poly_of_const_a_b @ r ) ) @ q ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( var_a_b @ r ) ) )
    = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ q ) @ ( var_a_b @ r ) ) ) ) ).

% \<open>h.eval (map (rupture_surj K p \<circ> poly_of_const) q) (rupture_surj K p X) = rupture_surj K p (UP.eval (map poly_of_const q) X)\<close>
thf(fact_74_domain_Orupture__surj__hom_I2_J,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( ring_h5964920862996313302t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ ( polyno8931960069169623149t_unit @ R2 @ K @ P ) @ ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ P ) ) ) ) ) ) ).

% domain.rupture_surj_hom(2)
thf(fact_75_domain_Orupture__surj__hom_I2_J,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( ring_h2829259075339158498t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ ( polyno859807163042199155t_unit @ R2 @ K @ P ) @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ P ) ) ) ) ) ) ).

% domain.rupture_surj_hom(2)
thf(fact_76_domain_Orupture__surj__hom_I2_J,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( ring_h4803640365354032162t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ ( polyno7054224608333822611t_unit @ R2 @ K @ P ) @ ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ P ) ) ) ) ) ) ).

% domain.rupture_surj_hom(2)
thf(fact_77_domain_Orupture__surj__hom_I2_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( ring_h6719293541701509614t_unit @ ( univ_poly_a_b @ R2 @ K ) @ ( polyno5459750281392823787re_a_b @ R2 @ K @ P ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R2 @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R2 @ K ) @ P ) ) ) ) ) ) ).

% domain.rupture_surj_hom(2)
thf(fact_78_domain_Orupture__surj__hom_I1_J,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( member6369349398066642652list_a @ ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ P ) ) @ ( ring_h5921505907431827092t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ ( polyno8931960069169623149t_unit @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.rupture_surj_hom(1)
thf(fact_79_domain_Orupture__surj__hom_I1_J,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( member2776101604912167772list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ P ) ) @ ( ring_h8245347068061950880t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ ( polyno859807163042199155t_unit @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.rupture_surj_hom(1)
thf(fact_80_domain_Orupture__surj__hom_I1_J,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( member2273002913129892700list_a @ ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ P ) ) @ ( ring_h45727557117266912t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ ( polyno7054224608333822611t_unit @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.rupture_surj_hom(1)
thf(fact_81_domain_Orupture__surj__hom_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( member4263473470251683292list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R2 @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R2 @ K ) @ P ) ) @ ( ring_h6188449271506562988t_unit @ ( univ_poly_a_b @ R2 @ K ) @ ( polyno5459750281392823787re_a_b @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.rupture_surj_hom(1)
thf(fact_82_h_Ohom__closed,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( member_set_list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ X2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.hom_closed
thf(fact_83__092_060open_062rupture__surj_AK_Ap_A_IUP_Oeval_A_Imap_Apoly__of__const_Aq_J_AX_J_A_061_Arupture__surj_AK_Ap_Aq_092_060close_062,axiom,
    ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ q ) @ ( var_a_b @ r ) ) )
    = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ q ) ) ).

% \<open>rupture_surj K p (UP.eval (map poly_of_const q) X) = rupture_surj K p q\<close>
thf(fact_84_ring__hom__ring_Oeval__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H: a > a,K: set_a,A: a,P: list_a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
           => ( ( H @ ( eval_a_b @ R2 @ P @ A ) )
              = ( eval_a_b @ S @ ( map_a_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom
thf(fact_85_ring__hom__ring_Oeval__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H: list_a > a,K: set_list_a,A: list_a,P: list_list_a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
           => ( ( H @ ( eval_l34571156754992824t_unit @ R2 @ P @ A ) )
              = ( eval_a_b @ S @ ( map_list_a_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom
thf(fact_86_ring__hom__ring_Oeval__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H: a > list_a,K: set_a,A: a,P: list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
           => ( ( H @ ( eval_a_b @ R2 @ P @ A ) )
              = ( eval_l34571156754992824t_unit @ S @ ( map_a_list_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom
thf(fact_87_ring__hom__ring_Oeval__hom,axiom,
    ! [R2: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H: list_list_a > a,K: set_list_list_a,A: list_list_a,P: list_list_list_a] :
      ( ( ring_h3841606220870141376it_a_b @ R2 @ S @ H )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R2 ) )
         => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
           => ( ( H @ ( eval_l1088911609197519410t_unit @ R2 @ P @ A ) )
              = ( eval_a_b @ S @ ( map_list_list_a_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom
thf(fact_88_ring__hom__ring_Oeval__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H: list_a > list_a,K: set_list_a,A: list_a,P: list_list_a] :
      ( ( ring_h1334922693953046990t_unit @ R2 @ S @ H )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
           => ( ( H @ ( eval_l34571156754992824t_unit @ R2 @ P @ A ) )
              = ( eval_l34571156754992824t_unit @ S @ ( map_list_a_list_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom
thf(fact_89_ring__hom__ring_Oeval__hom,axiom,
    ! [R2: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b,H: set_list_a > a,K: set_set_list_a,A: set_list_a,P: list_set_list_a] :
      ( ( ring_h1355656666774236646it_a_b @ R2 @ S @ H )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
         => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
           => ( ( H @ ( eval_s5133945360527818456t_unit @ R2 @ P @ A ) )
              = ( eval_a_b @ S @ ( map_set_list_a_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom
thf(fact_90_ring__hom__ring_Oeval__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit,H: a > set_list_a,K: set_a,A: a,P: list_a] :
      ( ( ring_h7781647138149442662t_unit @ R2 @ S @ H )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
           => ( ( H @ ( eval_a_b @ R2 @ P @ A ) )
              = ( eval_s5133945360527818456t_unit @ S @ ( map_a_set_list_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom
thf(fact_91_ring__hom__ring_Oeval__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,H: a > list_list_a,K: set_a,A: a,P: list_a] :
      ( ( ring_h2621993495375454400t_unit @ R2 @ S @ H )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
           => ( ( H @ ( eval_a_b @ R2 @ P @ A ) )
              = ( eval_l1088911609197519410t_unit @ S @ ( map_a_list_list_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom
thf(fact_92_ring__hom__ring_Oeval__hom,axiom,
    ! [R2: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H: list_list_a > list_a,K: set_list_list_a,A: list_list_a,P: list_list_list_a] :
      ( ( ring_h4589914651911841480t_unit @ R2 @ S @ H )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R2 ) )
         => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
           => ( ( H @ ( eval_l1088911609197519410t_unit @ R2 @ P @ A ) )
              = ( eval_l34571156754992824t_unit @ S @ ( map_li1646474281249396926list_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom
thf(fact_93_ring__hom__ring_Oeval__hom,axiom,
    ! [R2: partia4556295656693239580t_unit,S: partia2175431115845679010xt_a_b,H: list_set_list_a > a,K: set_list_set_list_a,A: list_set_list_a,P: list_list_set_list_a] :
      ( ( ring_h4081162582912290272it_a_b @ R2 @ S @ H )
     => ( ( subrin7712475646129147716t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ A @ ( partia7265347635606999311t_unit @ R2 ) )
         => ( ( member352051402189872281list_a @ P @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ K ) ) )
           => ( ( H @ ( eval_l7642959053407759954t_unit @ R2 @ P @ A ) )
              = ( eval_a_b @ S @ ( map_li6310725680953440600st_a_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom
thf(fact_94_domain_Oeval__ring__hom,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,A: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R2 ) )
         => ( ring_h7694777735462631100t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ R2
            @ ^ [P2: list_list_list_a] : ( eval_l1088911609197519410t_unit @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_ring_hom
thf(fact_95_domain_Oeval__ring__hom,axiom,
    ! [R2: partia4556295656693239580t_unit,K: set_list_set_list_a,A: list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R2 )
     => ( ( subrin7712475646129147716t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ A @ ( partia7265347635606999311t_unit @ R2 ) )
         => ( ring_h811738076660924156t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ K ) @ R2
            @ ^ [P2: list_list_set_list_a] : ( eval_l7642959053407759954t_unit @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_ring_hom
thf(fact_96_domain_Oeval__ring__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,A: list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ring_h4589914651911841480t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ R2
            @ ^ [P2: list_list_a] : ( eval_l34571156754992824t_unit @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_ring_hom
thf(fact_97_domain_Oeval__ring__hom,axiom,
    ! [R2: partia4960592913263135132t_unit,K: set_set_list_list_a,A: set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R2 )
     => ( ( subrin3011427358523593924t_unit @ K @ R2 )
       => ( ( member334759470184282131list_a @ A @ ( partia3317168157747563407t_unit @ R2 ) )
         => ( ring_h4765045011162728956t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ K ) @ R2
            @ ^ [P2: list_set_list_list_a] : ( eval_s2941910765802206162t_unit @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_ring_hom
thf(fact_98_domain_Oeval__ring__hom,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,A: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
         => ( ring_h8484543649848848136t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ R2
            @ ^ [P2: list_set_list_a] : ( eval_s5133945360527818456t_unit @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_ring_hom
thf(fact_99_domain_Oeval__ring__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,A: a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R2 @ K ) @ R2
            @ ^ [P2: list_a] : ( eval_a_b @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_ring_hom
thf(fact_100_h_Orcos__imp__homeq,axiom,
    ! [A: list_a,X2: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X2 @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) @ A ) )
       => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ X2 )
          = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ A ) ) ) ) ).

% h.rcos_imp_homeq
thf(fact_101_h_Orcos__eq__homeq,axiom,
    ! [A: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) @ A )
        = ( collect_list_a
          @ ^ [X: list_a] :
              ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
              & ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ X )
                = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ A ) ) ) ) ) ) ).

% h.rcos_eq_homeq
thf(fact_102_carrier__is__subring,axiom,
    subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% carrier_is_subring
thf(fact_103_eval__var,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X2 )
        = X2 ) ) ).

% eval_var
thf(fact_104_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_105_UP_Ouniv__poly__is__domain,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% UP.univ_poly_is_domain
thf(fact_106_UP_Odomain__axioms,axiom,
    domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) ).

% UP.domain_axioms
thf(fact_107_eval__ring__hom,axiom,
    ! [K: set_a,A: a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ r
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ A ) ) ) ) ).

% eval_ring_hom
thf(fact_108_eval__is__hom,axiom,
    ! [K: set_a,A: a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ A )
          @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ K ) @ r ) ) ) ) ).

% eval_is_hom
thf(fact_109_UP_Orupture__surj__hom_I2_J,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ring_h2829259075339158498t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P ) @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) ) ) ) ) ).

% UP.rupture_surj_hom(2)
thf(fact_110_UP_Orupture__surj__hom_I1_J,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( member2776101604912167772list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) ) @ ( ring_h8245347068061950880t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P ) ) ) ) ) ).

% UP.rupture_surj_hom(1)
thf(fact_111_UP_Ocarrier__is__subring,axiom,
    subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).

% UP.carrier_is_subring
thf(fact_112_h_Ocarrier__is__subring,axiom,
    subrin5643252653130547402t_unit @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ).

% h.carrier_is_subring
thf(fact_113_mem__Collect__eq,axiom,
    ! [A: list_a > a,P3: ( list_a > a ) > $o] :
      ( ( member_list_a_a @ A @ ( collect_list_a_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_114_mem__Collect__eq,axiom,
    ! [A: set_list_a > set_list_a,P3: ( set_list_a > set_list_a ) > $o] :
      ( ( member5068272912271824380list_a @ A @ ( collec3925428268116793146list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_115_mem__Collect__eq,axiom,
    ! [A: a > list_a,P3: ( a > list_a ) > $o] :
      ( ( member_a_list_a @ A @ ( collect_a_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_116_mem__Collect__eq,axiom,
    ! [A: a > set_list_a,P3: ( a > set_list_a ) > $o] :
      ( ( member_a_set_list_a @ A @ ( collect_a_set_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_117_mem__Collect__eq,axiom,
    ! [A: a,P3: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_118_mem__Collect__eq,axiom,
    ! [A: list_a,P3: list_a > $o] :
      ( ( member_list_a @ A @ ( collect_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_119_Collect__mem__eq,axiom,
    ! [A2: set_list_a_a] :
      ( ( collect_list_a_a
        @ ^ [X: list_a > a] : ( member_list_a_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_120_Collect__mem__eq,axiom,
    ! [A2: set_se1917860372504128155list_a] :
      ( ( collec3925428268116793146list_a
        @ ^ [X: set_list_a > set_list_a] : ( member5068272912271824380list_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_121_Collect__mem__eq,axiom,
    ! [A2: set_a_list_a] :
      ( ( collect_a_list_a
        @ ^ [X: a > list_a] : ( member_a_list_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_122_Collect__mem__eq,axiom,
    ! [A2: set_a_set_list_a] :
      ( ( collect_a_set_list_a
        @ ^ [X: a > set_list_a] : ( member_a_set_list_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_123_Collect__mem__eq,axiom,
    ! [A2: set_a] :
      ( ( collect_a
        @ ^ [X: a] : ( member_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_124_Collect__mem__eq,axiom,
    ! [A2: set_list_a] :
      ( ( collect_list_a
        @ ^ [X: list_a] : ( member_list_a @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_125_Collect__cong,axiom,
    ! [P3: list_a > $o,Q: list_a > $o] :
      ( ! [X3: list_a] :
          ( ( P3 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_list_a @ P3 )
        = ( collect_list_a @ Q ) ) ) ).

% Collect_cong
thf(fact_126_UP_Oeval__ring__hom,axiom,
    ! [K: set_list_a,A: list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ring_h4589914651911841480t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( univ_poly_a_b @ r @ k )
          @ ^ [P2: list_list_a] : ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ A ) ) ) ) ).

% UP.eval_ring_hom
thf(fact_127_map__norm__in__poly__ring__carrier,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_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ).

% map_norm_in_poly_ring_carrier
thf(fact_128_eval__rewrite,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
          = ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ K ) @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( var_a_b @ r ) ) ) ) ) ).

% eval_rewrite
thf(fact_129_h_Ohomeq__imp__rcos,axiom,
    ! [A: list_a,X2: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ X2 )
            = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ A ) )
         => ( member_list_a @ X2 @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) @ A ) ) ) ) ) ).

% h.homeq_imp_rcos
thf(fact_130_h_Oeval__hom,axiom,
    ! [K: set_list_a,A: list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ A ) )
            = ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( map_li2657846702790696612list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ P ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ A ) ) ) ) ) ) ).

% h.eval_hom
thf(fact_131_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_132_UP_Ocarrier__polynomial__shell,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).

% UP.carrier_polynomial_shell
thf(fact_133_h_Ocarrier__polynomial__shell,axiom,
    ! [K: set_set_list_a,P: list_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K ) ) )
       => ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ) ) ).

% h.carrier_polynomial_shell
thf(fact_134_h_OS_Ozero__closed,axiom,
    member_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ).

% h.S.zero_closed
thf(fact_135_h_Okernel__zero,axiom,
    ! [I: list_a,R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit] :
      ( ( member_list_a @ I @ ( a_kern570848875181920430t_unit @ R2 @ S @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) )
     => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ I )
        = ( zero_s2910681146719230829t_unit @ S ) ) ) ).

% h.kernel_zero
thf(fact_136_ring_Oeval_Ocong,axiom,
    eval_s5133945360527818456t_unit = eval_s5133945360527818456t_unit ).

% ring.eval.cong
thf(fact_137_ring_Oeval_Ocong,axiom,
    eval_l34571156754992824t_unit = eval_l34571156754992824t_unit ).

% ring.eval.cong
thf(fact_138_ring_Oeval_Ocong,axiom,
    eval_a_b = eval_a_b ).

% ring.eval.cong
thf(fact_139_ring_Oeval_Ocong,axiom,
    eval_l1088911609197519410t_unit = eval_l1088911609197519410t_unit ).

% ring.eval.cong
thf(fact_140_ring_Oeval_Ocong,axiom,
    eval_s2941910765802206162t_unit = eval_s2941910765802206162t_unit ).

% ring.eval.cong
thf(fact_141_ring_Opoly__of__const_Ocong,axiom,
    poly_of_const_a_b = poly_of_const_a_b ).

% ring.poly_of_const.cong
thf(fact_142_ring_Opoly__of__const_Ocong,axiom,
    poly_o8716471131768098070t_unit = poly_o8716471131768098070t_unit ).

% ring.poly_of_const.cong
thf(fact_143_ring_Opoly__of__const_Ocong,axiom,
    poly_o3535013565302768950t_unit = poly_o3535013565302768950t_unit ).

% ring.poly_of_const.cong
thf(fact_144_domain_Oeval__rewrite,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( P
            = ( eval_l332302722638357036t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ ( map_li5227692475714150986list_a @ ( poly_o1617770581224298896t_unit @ R2 ) @ P ) @ ( var_li3532061862469730199t_unit @ R2 ) ) ) ) ) ) ).

% domain.eval_rewrite
thf(fact_145_domain_Oeval__rewrite,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( P
            = ( eval_l1088911609197519410t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ ( map_li5729356230488778442list_a @ ( poly_o8716471131768098070t_unit @ R2 ) @ P ) @ ( var_li8453953174693405341t_unit @ R2 ) ) ) ) ) ) ).

% domain.eval_rewrite
thf(fact_146_domain_Oeval__rewrite,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( P
            = ( eval_l7642959053407759954t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ ( map_se1776605471917444810list_a @ ( poly_o3535013565302768950t_unit @ R2 ) @ P ) @ ( var_se6008125447796440765t_unit @ R2 ) ) ) ) ) ) ).

% domain.eval_rewrite
thf(fact_147_domain_Oeval__rewrite,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( P
            = ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ R2 @ K ) @ ( map_a_list_a @ ( poly_of_const_a_b @ R2 ) @ P ) @ ( var_a_b @ R2 ) ) ) ) ) ) ).

% domain.eval_rewrite
thf(fact_148_domain_Opolynomial__rupture,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( ( eval_s1985824173399790924t_unit @ ( polyno8931960069169623149t_unit @ R2 @ K @ P ) @ ( map_li3946859955893575466list_a @ ( comp_l4894990565352033927list_a @ ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ P ) ) @ ( poly_o1617770581224298896t_unit @ R2 ) ) @ P ) @ ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ P ) @ ( var_li3532061862469730199t_unit @ R2 ) ) )
            = ( zero_s6646109095048795361t_unit @ ( polyno8931960069169623149t_unit @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.polynomial_rupture
thf(fact_149_domain_Opolynomial__rupture,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( ( eval_s2941910765802206162t_unit @ ( polyno859807163042199155t_unit @ R2 @ K @ P ) @ ( map_li5777811645572713770list_a @ ( comp_l92744468816104577list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ P ) ) @ ( poly_o8716471131768098070t_unit @ R2 ) ) @ P ) @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ P ) @ ( var_li8453953174693405341t_unit @ R2 ) ) )
            = ( zero_s2920163772466840039t_unit @ ( polyno859807163042199155t_unit @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.polynomial_rupture
thf(fact_150_domain_Opolynomial__rupture,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( ( eval_s6864058807516611570t_unit @ ( polyno7054224608333822611t_unit @ R2 @ K @ P ) @ ( map_se2910130997222928682list_a @ ( comp_l2462527268875606113list_a @ ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ P ) ) @ ( poly_o3535013565302768950t_unit @ R2 ) ) @ P ) @ ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ P ) @ ( var_se6008125447796440765t_unit @ R2 ) ) )
            = ( zero_s8658464781190069767t_unit @ ( polyno7054224608333822611t_unit @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.polynomial_rupture
thf(fact_151_domain_Opolynomial__rupture,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ R2 @ K @ P ) @ ( map_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R2 @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R2 @ K ) @ P ) ) @ ( poly_of_const_a_b @ R2 ) ) @ P ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R2 @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R2 @ K ) @ P ) @ ( var_a_b @ R2 ) ) )
            = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.polynomial_rupture
thf(fact_152_domain_Ouniv__poly__is__domain,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ R2 @ K ) ) ) ) ).

% domain.univ_poly_is_domain
thf(fact_153_domain_Ouniv__poly__is__domain,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) ) ) ).

% domain.univ_poly_is_domain
thf(fact_154_domain_Ouniv__poly__is__domain,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( domain2898972329295444579t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) ) ) ).

% domain.univ_poly_is_domain
thf(fact_155_domain_Ouniv__poly__is__domain,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( domain9211287710782191037t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) ) ) ).

% domain.univ_poly_is_domain
thf(fact_156_domain_Ovar__closed_I1_J,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( member_list_list_a @ ( var_li8453953174693405341t_unit @ R2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) ) ) ) ).

% domain.var_closed(1)
thf(fact_157_domain_Ovar__closed_I1_J,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( member5342144027231129785list_a @ ( var_li3532061862469730199t_unit @ R2 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) ) ) ) ).

% domain.var_closed(1)
thf(fact_158_domain_Ovar__closed_I1_J,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( member5524387281408368019list_a @ ( var_se6008125447796440765t_unit @ R2 ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) ) ) ) ).

% domain.var_closed(1)
thf(fact_159_domain_Ovar__closed_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( member_list_a @ ( var_a_b @ R2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) ) ) ) ).

% domain.var_closed(1)
thf(fact_160_domain_Omap__norm__in__poly__ring__carrier,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( member6842060177613954879list_a @ ( map_li5227692475714150986list_a @ ( poly_o1617770581224298896t_unit @ R2 ) @ P ) @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) ) ) ) ) ) ) ).

% domain.map_norm_in_poly_ring_carrier
thf(fact_161_domain_Omap__norm__in__poly__ring__carrier,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ ( poly_o8716471131768098070t_unit @ R2 ) @ P ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) ) ) ) ) ) ) ).

% domain.map_norm_in_poly_ring_carrier
thf(fact_162_domain_Omap__norm__in__poly__ring__carrier,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( member352051402189872281list_a @ ( map_se1776605471917444810list_a @ ( poly_o3535013565302768950t_unit @ R2 ) @ P ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) ) ) ) ) ) ) ).

% domain.map_norm_in_poly_ring_carrier
thf(fact_163_domain_Omap__norm__in__poly__ring__carrier,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ R2 ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R2 @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) ) ) ) ) ) ) ).

% domain.map_norm_in_poly_ring_carrier
thf(fact_164_h_Oonepideal,axiom,
    princi5566405997746891961t_unit @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ).

% h.onepideal
thf(fact_165_h_Oadditive__subgroup__a__kernel,axiom,
    additi4714453376129182166t_unit @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) @ ( univ_poly_a_b @ r @ k ) ).

% h.additive_subgroup_a_kernel
thf(fact_166_h_Oeval__var,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( var_se6008125447796440765t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ X2 )
        = X2 ) ) ).

% h.eval_var
thf(fact_167_UP_Oeval__cring__hom,axiom,
    ! [K: set_list_a,A: list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ring_h453377649743177125t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( univ_poly_a_b @ r @ k )
          @ ^ [P2: list_list_a] : ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ A ) ) ) ) ).

% UP.eval_cring_hom
thf(fact_168_h_Oquot__mem,axiom,
    ! [X4: set_list_a] :
      ( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) ) )
     => ? [X3: list_a] :
          ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          & ( X4
            = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) @ X3 ) ) ) ) ).

% h.quot_mem
thf(fact_169_h_Oabelian__subgroup__a__kernel,axiom,
    abelia6695205329122750356t_unit @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) @ ( univ_poly_a_b @ r @ k ) ).

% h.abelian_subgroup_a_kernel
thf(fact_170_UP_Ouniv__poly__a__minus__consistent,axiom,
    ! [K: set_list_a,Q2: list_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P @ Q2 )
          = ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P @ Q2 ) ) ) ) ).

% UP.univ_poly_a_minus_consistent
thf(fact_171_h_Oconst__term__def,axiom,
    ! [P: list_set_list_a] :
      ( ( const_3308765751713425893t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P )
      = ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.const_term_def
thf(fact_172_onepideal,axiom,
    principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% onepideal
thf(fact_173_h_Ocgenideal__self,axiom,
    ! [I: set_list_a] :
      ( ( member_set_list_a @ I @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( member_set_list_a @ I @ ( cgenid9032708300698165283t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ I ) ) ) ).

% h.cgenideal_self
thf(fact_174_h_Oring__iso__restrict,axiom,
    ! [F: set_list_a > set_list_a,S: partia7496981018696276118t_unit,G: set_list_a > set_list_a] :
      ( ( member5068272912271824380list_a @ F @ ( ring_i6425008796027319266t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ S ) )
     => ( ! [R: set_list_a] :
            ( ( member_set_list_a @ R @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( F @ R )
              = ( G @ R ) ) )
       => ( member5068272912271824380list_a @ G @ ( ring_i6425008796027319266t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ S ) ) ) ) ).

% h.ring_iso_restrict
thf(fact_175_UP_Oeval__var,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X2 )
        = X2 ) ) ).

% UP.eval_var
thf(fact_176_UP_Oeval__rewrite,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( P
          = ( eval_l1088911609197519410t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( map_li5729356230488778442list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ P ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).

% UP.eval_rewrite
thf(fact_177_UP_Ovar__closed_I1_J,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( member_list_list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ).

% UP.var_closed(1)
thf(fact_178_UP_Omap__norm__in__poly__ring__carrier,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ P ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ) ) ) ).

% UP.map_norm_in_poly_ring_carrier
thf(fact_179_UP_Opolynomial__rupture,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( eval_s2941910765802206162t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P ) @ ( map_li5777811645572713770list_a @ ( comp_l92744468816104577list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P ) @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) ) )
          = ( zero_s2920163772466840039t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P ) ) ) ) ) ).

% UP.polynomial_rupture
thf(fact_180_ring_Oconst__term_Ocong,axiom,
    const_3308765751713425893t_unit = const_3308765751713425893t_unit ).

% ring.const_term.cong
thf(fact_181_ring_Oconst__term_Ocong,axiom,
    const_term_a_b = const_term_a_b ).

% ring.const_term.cong
thf(fact_182_ring_Oconst__term_Ocong,axiom,
    const_6738166269504826821t_unit = const_6738166269504826821t_unit ).

% ring.const_term.cong
thf(fact_183_lift__iso__to__poly__ring,axiom,
    ! [H: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
      ( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R2 @ S ) )
     => ( ( domain_a_b @ R2 )
       => ( ( domain_a_b @ S )
         => ( member_list_a_list_a @ ( map_a_a @ H ) @ ( ring_i7414513579304222626t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ).

% lift_iso_to_poly_ring
thf(fact_184_lift__iso__to__poly__ring,axiom,
    ! [H: list_a > a,R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R2 @ S ) )
     => ( ( domain6553523120543210313t_unit @ R2 )
       => ( ( domain_a_b @ S )
         => ( member7168557129179038582list_a @ ( map_list_a_a @ H ) @ ( ring_i4611353245267337884t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ).

% lift_iso_to_poly_ring
thf(fact_185_lift__iso__to__poly__ring,axiom,
    ! [H: a > list_a,R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
      ( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R2 @ S ) )
     => ( ( domain_a_b @ R2 )
       => ( ( domain6553523120543210313t_unit @ S )
         => ( member6714375691612171394list_a @ ( map_a_list_a @ H ) @ ( ring_i7582117978422105628t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ).

% lift_iso_to_poly_ring
thf(fact_186_lift__iso__to__poly__ring,axiom,
    ! [H: list_list_a > a,R2: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( member_list_list_a_a @ H @ ( ring_i5684343068699926420it_a_b @ R2 @ S ) )
     => ( ( domain7810152921033798211t_unit @ R2 )
       => ( ( domain_a_b @ S )
         => ( member3330872541585647984list_a @ ( map_list_list_a_a @ H ) @ ( ring_i3639271393462710806t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ).

% lift_iso_to_poly_ring
thf(fact_187_lift__iso__to__poly__ring,axiom,
    ! [H: list_a > list_a,R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit] :
      ( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R2 @ S ) )
     => ( ( domain6553523120543210313t_unit @ R2 )
       => ( ( domain6553523120543210313t_unit @ S )
         => ( member8231385768148312316list_a @ ( map_list_a_list_a @ H ) @ ( ring_i6186174840089424918t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ).

% lift_iso_to_poly_ring
thf(fact_188_lift__iso__to__poly__ring,axiom,
    ! [H: set_list_a > a,R2: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( member_set_list_a_a @ H @ ( ring_i8122894263081988538it_a_b @ R2 @ S ) )
     => ( ( domain1617769409708967785t_unit @ R2 )
       => ( ( domain_a_b @ S )
         => ( member1384923570607417238list_a @ ( map_set_list_a_a @ H ) @ ( ring_i5647238599803775676t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ).

% lift_iso_to_poly_ring
thf(fact_189_lift__iso__to__poly__ring,axiom,
    ! [H: a > list_list_a,R2: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit] :
      ( ( member_a_list_list_a @ H @ ( ring_i4464730343205239444t_unit @ R2 @ S ) )
     => ( ( domain_a_b @ R2 )
       => ( ( domain7810152921033798211t_unit @ S )
         => ( member7649053982783128200list_a @ ( map_a_list_list_a @ H ) @ ( ring_i8780440220419004694t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) ) ) ) ) ) ).

% lift_iso_to_poly_ring
thf(fact_190_lift__iso__to__poly__ring,axiom,
    ! [H: a > set_list_a,R2: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit] :
      ( ( member_a_set_list_a @ H @ ( ring_i5325512697602418746t_unit @ R2 @ S ) )
     => ( ( domain_a_b @ R2 )
       => ( ( domain1617769409708967785t_unit @ S )
         => ( member8672638751254680674list_a @ ( map_a_set_list_a @ H ) @ ( ring_i6060622876368992828t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ ( univ_p863672496597069550t_unit @ S @ ( partia141011252114345353t_unit @ S ) ) ) ) ) ) ) ).

% lift_iso_to_poly_ring
thf(fact_191_lift__iso__to__poly__ring,axiom,
    ! [H: list_list_a > list_a,R2: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit] :
      ( ( member7168557129179038582list_a @ H @ ( ring_i4611353245267337884t_unit @ R2 @ S ) )
     => ( ( domain7810152921033798211t_unit @ R2 )
       => ( ( domain6553523120543210313t_unit @ S )
         => ( member2200549709344965110list_a @ ( map_li1646474281249396926list_a @ H ) @ ( ring_i2841386107341831824t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ).

% lift_iso_to_poly_ring
thf(fact_192_lift__iso__to__poly__ring,axiom,
    ! [H: list_set_list_a > a,R2: partia4556295656693239580t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( member2876526313003830672st_a_a @ H @ ( ring_i3830580289522096052it_a_b @ R2 @ S ) )
     => ( ( domain2898972329295444579t_unit @ R2 )
       => ( ( domain_a_b @ S )
         => ( member9188088514701420944list_a @ ( map_li6310725680953440600st_a_a @ H ) @ ( ring_i8873061966688704054t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ).

% lift_iso_to_poly_ring
thf(fact_193_rupture__def,axiom,
    ( polyno7054224608333822611t_unit
    = ( ^ [R3: partia7496981018696276118t_unit,K2: set_set_list_a,P2: list_set_list_a] : ( factRi6954973740111656068t_unit @ ( univ_p863672496597069550t_unit @ R3 @ K2 ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R3 @ K2 ) @ P2 ) ) ) ) ).

% rupture_def
thf(fact_194_rupture__def,axiom,
    ( polyno8931960069169623149t_unit
    = ( ^ [R3: partia2956882679547061052t_unit,K2: set_list_list_a,P2: list_list_list_a] : ( factRi1900087049263378270t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K2 ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K2 ) @ P2 ) ) ) ) ).

% rupture_def
thf(fact_195_rupture__def,axiom,
    ( polyno5459750281392823787re_a_b
    = ( ^ [R3: partia2175431115845679010xt_a_b,K2: set_a,P2: list_a] : ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ R3 @ K2 ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R3 @ K2 ) @ P2 ) ) ) ) ).

% rupture_def
thf(fact_196_rupture__def,axiom,
    ( polyno859807163042199155t_unit
    = ( ^ [R3: partia2670972154091845814t_unit,K2: set_list_a,P2: list_list_a] : ( factRi7259693425559269476t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K2 ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K2 ) @ P2 ) ) ) ) ).

% rupture_def
thf(fact_197_domain_Oconst__term__simprules__shell_I1_J,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( member_list_list_a @ ( const_6243872422735025855t_unit @ R2 @ P ) @ K ) ) ) ) ).

% domain.const_term_simprules_shell(1)
thf(fact_198_domain_Oconst__term__simprules__shell_I1_J,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( member_list_a @ ( const_6738166269504826821t_unit @ R2 @ P ) @ K ) ) ) ) ).

% domain.const_term_simprules_shell(1)
thf(fact_199_domain_Oconst__term__simprules__shell_I1_J,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( member_set_list_a @ ( const_3308765751713425893t_unit @ R2 @ P ) @ K ) ) ) ) ).

% domain.const_term_simprules_shell(1)
thf(fact_200_domain_Oconst__term__simprules__shell_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( member_a @ ( const_term_a_b @ R2 @ P ) @ K ) ) ) ) ).

% domain.const_term_simprules_shell(1)
thf(fact_201_carrier__hom,axiom,
    ! [F: list_a,R2: partia2175431115845679010xt_a_b,H: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
     => ( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R2 @ S ) )
       => ( ( domain_a_b @ R2 )
         => ( ( domain_a_b @ S )
           => ( member_list_a @ ( map_a_a @ H @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ).

% carrier_hom
thf(fact_202_carrier__hom,axiom,
    ! [F: list_list_a,R2: partia2670972154091845814t_unit,H: list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
     => ( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R2 @ S ) )
       => ( ( domain6553523120543210313t_unit @ R2 )
         => ( ( domain_a_b @ S )
           => ( member_list_a @ ( map_list_a_a @ H @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ).

% carrier_hom
thf(fact_203_carrier__hom,axiom,
    ! [F: list_a,R2: partia2175431115845679010xt_a_b,H: a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
     => ( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R2 @ S ) )
       => ( ( domain_a_b @ R2 )
         => ( ( domain6553523120543210313t_unit @ S )
           => ( member_list_list_a @ ( map_a_list_a @ H @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ) ).

% carrier_hom
thf(fact_204_carrier__hom,axiom,
    ! [F: list_list_list_a,R2: partia2956882679547061052t_unit,H: list_list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
     => ( ( member_list_list_a_a @ H @ ( ring_i5684343068699926420it_a_b @ R2 @ S ) )
       => ( ( domain7810152921033798211t_unit @ R2 )
         => ( ( domain_a_b @ S )
           => ( member_list_a @ ( map_list_list_a_a @ H @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ).

% carrier_hom
thf(fact_205_carrier__hom,axiom,
    ! [F: list_list_a,R2: partia2670972154091845814t_unit,H: list_a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
     => ( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R2 @ S ) )
       => ( ( domain6553523120543210313t_unit @ R2 )
         => ( ( domain6553523120543210313t_unit @ S )
           => ( member_list_list_a @ ( map_list_a_list_a @ H @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ) ).

% carrier_hom
thf(fact_206_carrier__hom,axiom,
    ! [F: list_set_list_a,R2: partia7496981018696276118t_unit,H: set_list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member5524387281408368019list_a @ F @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
     => ( ( member_set_list_a_a @ H @ ( ring_i8122894263081988538it_a_b @ R2 @ S ) )
       => ( ( domain1617769409708967785t_unit @ R2 )
         => ( ( domain_a_b @ S )
           => ( member_list_a @ ( map_set_list_a_a @ H @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ).

% carrier_hom
thf(fact_207_carrier__hom,axiom,
    ! [F: list_a,R2: partia2175431115845679010xt_a_b,H: a > list_list_a,S: partia2956882679547061052t_unit] :
      ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
     => ( ( member_a_list_list_a @ H @ ( ring_i4464730343205239444t_unit @ R2 @ S ) )
       => ( ( domain_a_b @ R2 )
         => ( ( domain7810152921033798211t_unit @ S )
           => ( member5342144027231129785list_a @ ( map_a_list_list_a @ H @ F ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) ) ) ) ) ) ) ).

% carrier_hom
thf(fact_208_carrier__hom,axiom,
    ! [F: list_a,R2: partia2175431115845679010xt_a_b,H: a > set_list_a,S: partia7496981018696276118t_unit] :
      ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
     => ( ( member_a_set_list_a @ H @ ( ring_i5325512697602418746t_unit @ R2 @ S ) )
       => ( ( domain_a_b @ R2 )
         => ( ( domain1617769409708967785t_unit @ S )
           => ( member5524387281408368019list_a @ ( map_a_set_list_a @ H @ F ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ S @ ( partia141011252114345353t_unit @ S ) ) ) ) ) ) ) ) ).

% carrier_hom
thf(fact_209_carrier__hom,axiom,
    ! [F: list_list_list_a,R2: partia2956882679547061052t_unit,H: list_list_a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
     => ( ( member7168557129179038582list_a @ H @ ( ring_i4611353245267337884t_unit @ R2 @ S ) )
       => ( ( domain7810152921033798211t_unit @ R2 )
         => ( ( domain6553523120543210313t_unit @ S )
           => ( member_list_list_a @ ( map_li1646474281249396926list_a @ H @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ) ).

% carrier_hom
thf(fact_210_carrier__hom,axiom,
    ! [F: list_list_set_list_a,R2: partia4556295656693239580t_unit,H: list_set_list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member352051402189872281list_a @ F @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) ) )
     => ( ( member2876526313003830672st_a_a @ H @ ( ring_i3830580289522096052it_a_b @ R2 @ S ) )
       => ( ( domain2898972329295444579t_unit @ R2 )
         => ( ( domain_a_b @ S )
           => ( member_list_a @ ( map_li6310725680953440600st_a_a @ H @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ).

% carrier_hom
thf(fact_211_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,Q2: list_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ Q2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ P @ Q2 )
            = ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ P @ Q2 ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_212_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R2: partia4556295656693239580t_unit,K: set_list_set_list_a,Q2: list_list_set_list_a,P: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R2 )
     => ( ( subrin7712475646129147716t_unit @ K @ R2 )
       => ( ( member352051402189872281list_a @ Q2 @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ K ) ) )
         => ( ( a_minu9044011302611522408t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ K ) @ P @ Q2 )
            = ( a_minu9044011302611522408t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) @ P @ Q2 ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_213_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,Q2: list_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ P @ Q2 )
            = ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P @ Q2 ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_214_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R2: partia4960592913263135132t_unit,K: set_set_list_list_a,Q2: list_set_list_list_a,P: list_set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R2 )
     => ( ( subrin3011427358523593924t_unit @ K @ R2 )
       => ( ( member6124916891863447321list_a @ Q2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ K ) ) )
         => ( ( a_minu1178922365601208552t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ K ) @ P @ Q2 )
            = ( a_minu1178922365601208552t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) @ P @ Q2 ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_215_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,Q2: list_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ Q2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( ( a_minu6874796375791416686t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ P @ Q2 )
            = ( a_minu6874796375791416686t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ P @ Q2 ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_216_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,Q2: list_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R2 @ K ) @ P @ Q2 )
            = ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P @ Q2 ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_217_domain_Oeval__cring__hom,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,A: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R2 ) )
         => ( ring_h5732965726171697305t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ R2
            @ ^ [P2: list_list_list_a] : ( eval_l1088911609197519410t_unit @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_cring_hom
thf(fact_218_domain_Oeval__cring__hom,axiom,
    ! [R2: partia4556295656693239580t_unit,K: set_list_set_list_a,A: list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R2 )
     => ( ( subrin7712475646129147716t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ A @ ( partia7265347635606999311t_unit @ R2 ) )
         => ( ring_h6371442481623134425t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ K ) @ R2
            @ ^ [P2: list_list_set_list_a] : ( eval_l7642959053407759954t_unit @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_cring_hom
thf(fact_219_domain_Oeval__cring__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,A: list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ring_h453377649743177125t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ R2
            @ ^ [P2: list_list_a] : ( eval_l34571156754992824t_unit @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_cring_hom
thf(fact_220_domain_Oeval__cring__hom,axiom,
    ! [R2: partia4960592913263135132t_unit,K: set_set_list_list_a,A: set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R2 )
     => ( ( subrin3011427358523593924t_unit @ K @ R2 )
       => ( ( member334759470184282131list_a @ A @ ( partia3317168157747563407t_unit @ R2 ) )
         => ( ring_h1101377379270163417t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ K ) @ R2
            @ ^ [P2: list_set_list_list_a] : ( eval_s2941910765802206162t_unit @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_cring_hom
thf(fact_221_domain_Oeval__cring__hom,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,A: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
         => ( ring_h4012356781940393445t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ R2
            @ ^ [P2: list_set_list_a] : ( eval_s5133945360527818456t_unit @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_cring_hom
thf(fact_222_domain_Oeval__cring__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,A: a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ R2 @ K ) @ R2
            @ ^ [P2: list_a] : ( eval_a_b @ R2 @ P2 @ A ) ) ) ) ) ).

% domain.eval_cring_hom
thf(fact_223_eval__cring__hom,axiom,
    ! [K: set_a,A: a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ r @ K ) @ r
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ A ) ) ) ) ).

% eval_cring_hom
thf(fact_224_UP_OsubdomainI_H,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.subdomainI'
thf(fact_225_UP_Oeval__is__hom,axiom,
    ! [K: set_list_a,A: list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( member7168557129179038582list_a
          @ ^ [P2: list_list_a] : ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P2 @ A )
          @ ( ring_h5031276006722532742t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.eval_is_hom
thf(fact_226_univ__poly__a__minus__consistent,axiom,
    ! [K: set_a,Q2: list_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 )
          = ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) ) ) ) ).

% univ_poly_a_minus_consistent
thf(fact_227_h_Oimg__is__subring,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( subrin5643252653130547402t_unit @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ K ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ).

% h.img_is_subring
thf(fact_228_h_Oabelian__group_Ozero__closed,axiom,
    member_set_list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ).

% h.abelian_group.zero_closed
thf(fact_229_h_Oinj__on__domain,axiom,
    ( ( inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
   => ( ( domain1617769409708967785t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% h.inj_on_domain
thf(fact_230_h_Oconst__term__not__zero,axiom,
    ! [P: list_set_list_a] :
      ( ( ( const_3308765751713425893t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P )
       != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( P != nil_set_list_a ) ) ).

% h.const_term_not_zero
thf(fact_231_h_Oabelian__group_Oa__inv__closed,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( member_set_list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.abelian_group.a_inv_closed
thf(fact_232_h_Oinduced__ring__hom,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ring_h6719293541701509614t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : K
          @ ( univ_poly_a_b @ r @ k ) )
        @ ( polyno5459750281392823787re_a_b @ r @ k @ p )
        @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) ).

% h.induced_ring_hom
thf(fact_233_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_234_UP_Oconst__term__def,axiom,
    ! [P: list_list_a] :
      ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P )
      = ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.const_term_def
thf(fact_235_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_236_UP_Oconst__term__simprules__shell_I1_J,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) @ K ) ) ) ).

% UP.const_term_simprules_shell(1)
thf(fact_237_UP_Ouniv__poly__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( univ_p7953238456130426574t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) ) )
        = ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.univ_poly_consistent
thf(fact_238_UP_Opoly__of__const__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( poly_o8716471131768098070t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) ) )
        = ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.poly_of_const_consistent
thf(fact_239_UP_Osubring__is__domain,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( domain6553523120543210313t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : H2
          @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.subring_is_domain
thf(fact_240_list_Omap__disc__iff,axiom,
    ! [F: a > a,A: list_a] :
      ( ( ( map_a_a @ F @ A )
        = nil_a )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_241_list_Omap__disc__iff,axiom,
    ! [F: list_a > a,A: list_list_a] :
      ( ( ( map_list_a_a @ F @ A )
        = nil_a )
      = ( A = nil_list_a ) ) ).

% list.map_disc_iff
thf(fact_242_list_Omap__disc__iff,axiom,
    ! [F: a > list_a,A: list_a] :
      ( ( ( map_a_list_a @ F @ A )
        = nil_list_a )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_243_list_Omap__disc__iff,axiom,
    ! [F: list_a > list_a,A: list_list_a] :
      ( ( ( map_list_a_list_a @ F @ A )
        = nil_list_a )
      = ( A = nil_list_a ) ) ).

% list.map_disc_iff
thf(fact_244_list_Omap__disc__iff,axiom,
    ! [F: a > set_list_a,A: list_a] :
      ( ( ( map_a_set_list_a @ F @ A )
        = nil_set_list_a )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_245_list_Omap__disc__iff,axiom,
    ! [F: set_list_a > a,A: list_set_list_a] :
      ( ( ( map_set_list_a_a @ F @ A )
        = nil_a )
      = ( A = nil_set_list_a ) ) ).

% list.map_disc_iff
thf(fact_246_list_Omap__disc__iff,axiom,
    ! [F: list_a > set_list_a,A: list_list_a] :
      ( ( ( map_li2657846702790696612list_a @ F @ A )
        = nil_set_list_a )
      = ( A = nil_list_a ) ) ).

% list.map_disc_iff
thf(fact_247_list_Omap__disc__iff,axiom,
    ! [F: set_list_a > list_a,A: list_set_list_a] :
      ( ( ( map_se5127173849416245988list_a @ F @ A )
        = nil_list_a )
      = ( A = nil_set_list_a ) ) ).

% list.map_disc_iff
thf(fact_248_list_Omap__disc__iff,axiom,
    ! [F: a > set_list_list_a,A: list_a] :
      ( ( ( map_a_132662181483840688list_a @ F @ A )
        = nil_set_list_list_a )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_249_list_Omap__disc__iff,axiom,
    ! [F: list_a > set_list_list_a,A: list_list_a] :
      ( ( ( map_li5777811645572713770list_a @ F @ A )
        = nil_set_list_list_a )
      = ( A = nil_list_a ) ) ).

% list.map_disc_iff
thf(fact_250_Nil__is__map__conv,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( nil_a
        = ( map_a_a @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_251_Nil__is__map__conv,axiom,
    ! [F: list_a > a,Xs: list_list_a] :
      ( ( nil_a
        = ( map_list_a_a @ F @ Xs ) )
      = ( Xs = nil_list_a ) ) ).

% Nil_is_map_conv
thf(fact_252_Nil__is__map__conv,axiom,
    ! [F: a > list_a,Xs: list_a] :
      ( ( nil_list_a
        = ( map_a_list_a @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_253_Nil__is__map__conv,axiom,
    ! [F: list_a > list_a,Xs: list_list_a] :
      ( ( nil_list_a
        = ( map_list_a_list_a @ F @ Xs ) )
      = ( Xs = nil_list_a ) ) ).

% Nil_is_map_conv
thf(fact_254_Nil__is__map__conv,axiom,
    ! [F: a > set_list_a,Xs: list_a] :
      ( ( nil_set_list_a
        = ( map_a_set_list_a @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_255_Nil__is__map__conv,axiom,
    ! [F: set_list_a > a,Xs: list_set_list_a] :
      ( ( nil_a
        = ( map_set_list_a_a @ F @ Xs ) )
      = ( Xs = nil_set_list_a ) ) ).

% Nil_is_map_conv
thf(fact_256_Nil__is__map__conv,axiom,
    ! [F: list_a > set_list_a,Xs: list_list_a] :
      ( ( nil_set_list_a
        = ( map_li2657846702790696612list_a @ F @ Xs ) )
      = ( Xs = nil_list_a ) ) ).

% Nil_is_map_conv
thf(fact_257_Nil__is__map__conv,axiom,
    ! [F: set_list_a > list_a,Xs: list_set_list_a] :
      ( ( nil_list_a
        = ( map_se5127173849416245988list_a @ F @ Xs ) )
      = ( Xs = nil_set_list_a ) ) ).

% Nil_is_map_conv
thf(fact_258_Nil__is__map__conv,axiom,
    ! [F: a > set_list_list_a,Xs: list_a] :
      ( ( nil_set_list_list_a
        = ( map_a_132662181483840688list_a @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_259_Nil__is__map__conv,axiom,
    ! [F: list_a > set_list_list_a,Xs: list_list_a] :
      ( ( nil_set_list_list_a
        = ( map_li5777811645572713770list_a @ F @ Xs ) )
      = ( Xs = nil_list_a ) ) ).

% Nil_is_map_conv
thf(fact_260_map__is__Nil__conv,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_261_map__is__Nil__conv,axiom,
    ! [F: list_a > a,Xs: list_list_a] :
      ( ( ( map_list_a_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_list_a ) ) ).

% map_is_Nil_conv
thf(fact_262_map__is__Nil__conv,axiom,
    ! [F: a > list_a,Xs: list_a] :
      ( ( ( map_a_list_a @ F @ Xs )
        = nil_list_a )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_263_map__is__Nil__conv,axiom,
    ! [F: list_a > list_a,Xs: list_list_a] :
      ( ( ( map_list_a_list_a @ F @ Xs )
        = nil_list_a )
      = ( Xs = nil_list_a ) ) ).

% map_is_Nil_conv
thf(fact_264_map__is__Nil__conv,axiom,
    ! [F: a > set_list_a,Xs: list_a] :
      ( ( ( map_a_set_list_a @ F @ Xs )
        = nil_set_list_a )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_265_map__is__Nil__conv,axiom,
    ! [F: set_list_a > a,Xs: list_set_list_a] :
      ( ( ( map_set_list_a_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_set_list_a ) ) ).

% map_is_Nil_conv
thf(fact_266_map__is__Nil__conv,axiom,
    ! [F: list_a > set_list_a,Xs: list_list_a] :
      ( ( ( map_li2657846702790696612list_a @ F @ Xs )
        = nil_set_list_a )
      = ( Xs = nil_list_a ) ) ).

% map_is_Nil_conv
thf(fact_267_map__is__Nil__conv,axiom,
    ! [F: set_list_a > list_a,Xs: list_set_list_a] :
      ( ( ( map_se5127173849416245988list_a @ F @ Xs )
        = nil_list_a )
      = ( Xs = nil_set_list_a ) ) ).

% map_is_Nil_conv
thf(fact_268_map__is__Nil__conv,axiom,
    ! [F: a > set_list_list_a,Xs: list_a] :
      ( ( ( map_a_132662181483840688list_a @ F @ Xs )
        = nil_set_list_list_a )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_269_map__is__Nil__conv,axiom,
    ! [F: list_a > set_list_list_a,Xs: list_list_a] :
      ( ( ( map_li5777811645572713770list_a @ F @ Xs )
        = nil_set_list_list_a )
      = ( Xs = nil_list_a ) ) ).

% map_is_Nil_conv
thf(fact_270_h_Oeval_Osimps_I1_J,axiom,
    ( ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ nil_set_list_a )
    = ( ^ [Uu: set_list_a] : ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.eval.simps(1)
thf(fact_271_UP_Osubdomain__is__domain,axiom,
    ! [H2: set_list_a] :
      ( ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( domain6553523120543210313t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : H2
          @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.subdomain_is_domain
thf(fact_272_UP_Ocanonical__embedding__ring__hom,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ring_h7560679385066609224t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : K
          @ ( univ_poly_a_b @ r @ k ) )
        @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K )
        @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.canonical_embedding_ring_hom
thf(fact_273_UP_Ocanonical__embedding__is__hom,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( member6714375691612171394list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) )
        @ ( ring_h8002040739877300486t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) )
          @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ).

% UP.canonical_embedding_is_hom
thf(fact_274_univ__poly__zero__closed,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) ) ).

% univ_poly_zero_closed
thf(fact_275_univ__poly__zero__closed,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a] : ( member5342144027231129785list_a @ nil_list_list_a @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) ) ).

% univ_poly_zero_closed
thf(fact_276_univ__poly__zero__closed,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a] : ( member5524387281408368019list_a @ nil_set_list_a @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) ) ).

% univ_poly_zero_closed
thf(fact_277_univ__poly__zero__closed,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) ) ).

% univ_poly_zero_closed
thf(fact_278_UP_Orupture__surj__norm__is__hom,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( member5222132204073479906list_a @ ( comp_l92744468816104577list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          @ ( ring_h5771512479963622054t_unit
            @ ( partia9041243232023819264t_unit
              @ ^ [Uu: set_list_a] : K
              @ ( univ_poly_a_b @ r @ k ) )
            @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P ) ) ) ) ) ).

% UP.rupture_surj_norm_is_hom
thf(fact_279_UP_Ozero__closed,axiom,
    member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% UP.zero_closed
thf(fact_280_UP_Oadd_Oinv__closed,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.add.inv_closed
thf(fact_281_UP_Ominus__minus,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) )
        = X2 ) ) ).

% UP.minus_minus
thf(fact_282_UP_Ominus__zero,axiom,
    ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
    = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.minus_zero
thf(fact_283_UP_Ominus__closed,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.minus_closed
thf(fact_284_UP_Oadd_Oinv__eq__1__iff,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 )
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        = ( X2
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.add.inv_eq_1_iff
thf(fact_285_UP_Or__right__minus__eq,axiom,
    ! [A: list_a,B: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          = ( A = B ) ) ) ) ).

% UP.r_right_minus_eq
thf(fact_286_UP_Oeval__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( eval_l34571156754992824t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) ) )
        = ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.eval_consistent
thf(fact_287_h_Ohom__zero,axiom,
    ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
    = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ).

% h.hom_zero
thf(fact_288_univ__poly__zero,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R2 @ K ) )
      = nil_a ) ).

% univ_poly_zero
thf(fact_289_univ__poly__zero,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) )
      = nil_list_a ) ).

% univ_poly_zero
thf(fact_290_univ__poly__zero,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( zero_l7621212060072393831t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) )
      = nil_set_list_a ) ).

% univ_poly_zero
thf(fact_291_univ__poly__zero,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( zero_l317200538825487809t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) )
      = nil_list_list_a ) ).

% univ_poly_zero
thf(fact_292_list_Osimps_I8_J,axiom,
    ! [F: a > a] :
      ( ( map_a_a @ F @ nil_a )
      = nil_a ) ).

% list.simps(8)
thf(fact_293_list_Osimps_I8_J,axiom,
    ! [F: list_a > a] :
      ( ( map_list_a_a @ F @ nil_list_a )
      = nil_a ) ).

% list.simps(8)
thf(fact_294_list_Osimps_I8_J,axiom,
    ! [F: a > list_a] :
      ( ( map_a_list_a @ F @ nil_a )
      = nil_list_a ) ).

% list.simps(8)
thf(fact_295_list_Osimps_I8_J,axiom,
    ! [F: list_a > list_a] :
      ( ( map_list_a_list_a @ F @ nil_list_a )
      = nil_list_a ) ).

% list.simps(8)
thf(fact_296_list_Osimps_I8_J,axiom,
    ! [F: a > set_list_a] :
      ( ( map_a_set_list_a @ F @ nil_a )
      = nil_set_list_a ) ).

% list.simps(8)
thf(fact_297_list_Osimps_I8_J,axiom,
    ! [F: set_list_a > a] :
      ( ( map_set_list_a_a @ F @ nil_set_list_a )
      = nil_a ) ).

% list.simps(8)
thf(fact_298_list_Osimps_I8_J,axiom,
    ! [F: list_a > set_list_a] :
      ( ( map_li2657846702790696612list_a @ F @ nil_list_a )
      = nil_set_list_a ) ).

% list.simps(8)
thf(fact_299_list_Osimps_I8_J,axiom,
    ! [F: set_list_a > list_a] :
      ( ( map_se5127173849416245988list_a @ F @ nil_set_list_a )
      = nil_list_a ) ).

% list.simps(8)
thf(fact_300_list_Osimps_I8_J,axiom,
    ! [F: a > set_list_list_a] :
      ( ( map_a_132662181483840688list_a @ F @ nil_a )
      = nil_set_list_list_a ) ).

% list.simps(8)
thf(fact_301_list_Osimps_I8_J,axiom,
    ! [F: list_a > set_list_list_a] :
      ( ( map_li5777811645572713770list_a @ F @ nil_list_a )
      = nil_set_list_list_a ) ).

% list.simps(8)
thf(fact_302_domain_Ocanonical__embedding__is__hom,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( member8941091444255470082list_a @ ( poly_o1617770581224298896t_unit @ R2 )
          @ ( ring_h6733830324725517050t_unit
            @ ( partia5725451877363000582t_unit
              @ ^ [Uu: set_list_list_a] : K
              @ R2 )
            @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) ) ) ) ).

% domain.canonical_embedding_is_hom
thf(fact_303_domain_Ocanonical__embedding__is__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( member6714375691612171394list_a @ ( poly_o8716471131768098070t_unit @ R2 )
          @ ( ring_h8002040739877300486t_unit
            @ ( partia9041243232023819264t_unit
              @ ^ [Uu: set_list_a] : K
              @ R2 )
            @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) ) ) ) ).

% domain.canonical_embedding_is_hom
thf(fact_304_domain_Ocanonical__embedding__is__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( member_a_list_a @ ( poly_of_const_a_b @ R2 )
          @ ( ring_h405018892823518980t_unit
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : K
              @ R2 )
            @ ( univ_poly_a_b @ R2 @ K ) ) ) ) ) ).

% domain.canonical_embedding_is_hom
thf(fact_305_domain_Ocanonical__embedding__is__hom,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( member6189391442678058114list_a @ ( poly_o3535013565302768950t_unit @ R2 )
          @ ( ring_h7589977632246733126t_unit
            @ ( partia2314006361284536288t_unit
              @ ^ [Uu: set_set_list_a] : K
              @ R2 )
            @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) ) ) ) ).

% domain.canonical_embedding_is_hom
thf(fact_306_domain_Oconst__term__simprules__shell_I4_J,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( ( const_6243872422735025855t_unit @ R2 @ ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ P ) )
            = ( a_inv_7033018035630854991t_unit @ R2 @ ( const_6243872422735025855t_unit @ R2 @ P ) ) ) ) ) ) ).

% domain.const_term_simprules_shell(4)
thf(fact_307_domain_Oconst__term__simprules__shell_I4_J,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( ( const_6738166269504826821t_unit @ R2 @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ P ) )
            = ( a_inv_8944721093294617173t_unit @ R2 @ ( const_6738166269504826821t_unit @ R2 @ P ) ) ) ) ) ) ).

% domain.const_term_simprules_shell(4)
thf(fact_308_domain_Oconst__term__simprules__shell_I4_J,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( ( const_3308765751713425893t_unit @ R2 @ ( a_inv_1838491359387560815t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ P ) )
            = ( a_inv_5715216516650856053t_unit @ R2 @ ( const_3308765751713425893t_unit @ R2 @ P ) ) ) ) ) ) ).

% domain.const_term_simprules_shell(4)
thf(fact_309_domain_Oconst__term__simprules__shell_I4_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( ( const_term_a_b @ R2 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R2 @ K ) @ P ) )
            = ( a_inv_a_b @ R2 @ ( const_term_a_b @ R2 @ P ) ) ) ) ) ) ).

% domain.const_term_simprules_shell(4)
thf(fact_310_domain_Ouniv__poly__a__inv__def_H,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ P )
            = ( map_list_a_list_a @ ( a_inv_8944721093294617173t_unit @ R2 ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_def'
thf(fact_311_domain_Ouniv__poly__a__inv__def_H,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ P )
            = ( map_li8713736314956022724list_a @ ( a_inv_7033018035630854991t_unit @ R2 ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_def'
thf(fact_312_domain_Ouniv__poly__a__inv__def_H,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( ( a_inv_1838491359387560815t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ P )
            = ( map_se2668659675339852484list_a @ ( a_inv_5715216516650856053t_unit @ R2 ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_def'
thf(fact_313_domain_Ouniv__poly__a__inv__def_H,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R2 @ K ) @ P )
            = ( map_a_a @ ( a_inv_a_b @ R2 ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_def'
thf(fact_314_domain_Ouniv__poly__a__inv__consistent,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ P )
            = ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_consistent
thf(fact_315_domain_Ouniv__poly__a__inv__consistent,axiom,
    ! [R2: partia4556295656693239580t_unit,K: set_list_set_list_a,P: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R2 )
     => ( ( subrin7712475646129147716t_unit @ K @ R2 )
       => ( ( member352051402189872281list_a @ P @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ K ) ) )
         => ( ( a_inv_2107337025857688809t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ K ) @ P )
            = ( a_inv_2107337025857688809t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_consistent
thf(fact_316_domain_Ouniv__poly__a__inv__consistent,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ P )
            = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_consistent
thf(fact_317_domain_Ouniv__poly__a__inv__consistent,axiom,
    ! [R2: partia4960592913263135132t_unit,K: set_set_list_list_a,P: list_set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R2 )
     => ( ( subrin3011427358523593924t_unit @ K @ R2 )
       => ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ K ) ) )
         => ( ( a_inv_3465620125702150761t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ K ) @ P )
            = ( a_inv_3465620125702150761t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_consistent
thf(fact_318_domain_Ouniv__poly__a__inv__consistent,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( ( a_inv_1838491359387560815t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ P )
            = ( a_inv_1838491359387560815t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_consistent
thf(fact_319_domain_Ouniv__poly__a__inv__consistent,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R2 @ K ) @ P )
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ P ) ) ) ) ) ).

% domain.univ_poly_a_inv_consistent
thf(fact_320_domain_Ocanonical__embedding__ring__hom,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ring_h1130228327422126652t_unit
          @ ( partia5725451877363000582t_unit
            @ ^ [Uu: set_list_list_a] : K
            @ R2 )
          @ ( univ_p2250591967980070728t_unit @ R2 @ K )
          @ ( poly_o1617770581224298896t_unit @ R2 ) ) ) ) ).

% domain.canonical_embedding_ring_hom
thf(fact_321_domain_Ocanonical__embedding__ring__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ring_h7560679385066609224t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ R2 )
          @ ( univ_p7953238456130426574t_unit @ R2 @ K )
          @ ( poly_o8716471131768098070t_unit @ R2 ) ) ) ) ).

% domain.canonical_embedding_ring_hom
thf(fact_322_domain_Ocanonical__embedding__ring__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ring_h5357930050666032198t_unit
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ R2 )
          @ ( univ_poly_a_b @ R2 @ K )
          @ ( poly_of_const_a_b @ R2 ) ) ) ) ).

% domain.canonical_embedding_ring_hom
thf(fact_323_domain_Ocanonical__embedding__ring__hom,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ring_h2825600532425112200t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : K
            @ R2 )
          @ ( univ_p863672496597069550t_unit @ R2 @ K )
          @ ( poly_o3535013565302768950t_unit @ R2 ) ) ) ) ).

% domain.canonical_embedding_ring_hom
thf(fact_324_carrier__hom_H,axiom,
    ! [F: list_a,R2: partia2175431115845679010xt_a_b,H: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
     => ( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R2 @ S ) )
       => ( ( domain_a_b @ R2 )
         => ( ( domain_a_b @ S )
           => ( ( inj_on_a_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) )
             => ( member_list_a @ ( map_a_a @ H @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ) ).

% carrier_hom'
thf(fact_325_carrier__hom_H,axiom,
    ! [F: list_list_a,R2: partia2670972154091845814t_unit,H: list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
     => ( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R2 @ S ) )
       => ( ( domain6553523120543210313t_unit @ R2 )
         => ( ( domain_a_b @ S )
           => ( ( inj_on_list_a_a @ H @ ( partia5361259788508890537t_unit @ R2 ) )
             => ( member_list_a @ ( map_list_a_a @ H @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ) ).

% carrier_hom'
thf(fact_326_carrier__hom_H,axiom,
    ! [F: list_a,R2: partia2175431115845679010xt_a_b,H: a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
     => ( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R2 @ S ) )
       => ( ( domain_a_b @ R2 )
         => ( ( domain6553523120543210313t_unit @ S )
           => ( ( inj_on_a_list_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) )
             => ( member_list_list_a @ ( map_a_list_a @ H @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ) ) ).

% carrier_hom'
thf(fact_327_carrier__hom_H,axiom,
    ! [F: list_list_list_a,R2: partia2956882679547061052t_unit,H: list_list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
     => ( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R2 @ S ) )
       => ( ( domain7810152921033798211t_unit @ R2 )
         => ( ( domain_a_b @ S )
           => ( ( inj_on_list_list_a_a @ H @ ( partia2464479390973590831t_unit @ R2 ) )
             => ( member_list_a @ ( map_list_list_a_a @ H @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ) ).

% carrier_hom'
thf(fact_328_carrier__hom_H,axiom,
    ! [F: list_list_a,R2: partia2670972154091845814t_unit,H: list_a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
     => ( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R2 @ S ) )
       => ( ( domain6553523120543210313t_unit @ R2 )
         => ( ( domain6553523120543210313t_unit @ S )
           => ( ( inj_on_list_a_list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) )
             => ( member_list_list_a @ ( map_list_a_list_a @ H @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ) ) ).

% carrier_hom'
thf(fact_329_carrier__hom_H,axiom,
    ! [F: list_set_list_a,R2: partia7496981018696276118t_unit,H: set_list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member5524387281408368019list_a @ F @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
     => ( ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R2 @ S ) )
       => ( ( domain1617769409708967785t_unit @ R2 )
         => ( ( domain_a_b @ S )
           => ( ( inj_on_set_list_a_a @ H @ ( partia141011252114345353t_unit @ R2 ) )
             => ( member_list_a @ ( map_set_list_a_a @ H @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ) ).

% carrier_hom'
thf(fact_330_carrier__hom_H,axiom,
    ! [F: list_a,R2: partia2175431115845679010xt_a_b,H: a > list_list_a,S: partia2956882679547061052t_unit] :
      ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
     => ( ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ R2 @ S ) )
       => ( ( domain_a_b @ R2 )
         => ( ( domain7810152921033798211t_unit @ S )
           => ( ( inj_on_a_list_list_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) )
             => ( member5342144027231129785list_a @ ( map_a_list_list_a @ H @ F ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ S @ ( partia2464479390973590831t_unit @ S ) ) ) ) ) ) ) ) ) ).

% carrier_hom'
thf(fact_331_carrier__hom_H,axiom,
    ! [F: list_a,R2: partia2175431115845679010xt_a_b,H: a > set_list_a,S: partia7496981018696276118t_unit] :
      ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
     => ( ( member_a_set_list_a @ H @ ( ring_h6109298854714515236t_unit @ R2 @ S ) )
       => ( ( domain_a_b @ R2 )
         => ( ( domain1617769409708967785t_unit @ S )
           => ( ( inj_on_a_set_list_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) )
             => ( member5524387281408368019list_a @ ( map_a_set_list_a @ H @ F ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ S @ ( partia141011252114345353t_unit @ S ) ) ) ) ) ) ) ) ) ).

% carrier_hom'
thf(fact_332_carrier__hom_H,axiom,
    ! [F: list_list_list_a,R2: partia2956882679547061052t_unit,H: list_list_a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
     => ( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R2 @ S ) )
       => ( ( domain7810152921033798211t_unit @ R2 )
         => ( ( domain6553523120543210313t_unit @ S )
           => ( ( inj_on2455837796590618731list_a @ H @ ( partia2464479390973590831t_unit @ R2 ) )
             => ( member_list_list_a @ ( map_li1646474281249396926list_a @ H @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ S @ ( partia5361259788508890537t_unit @ S ) ) ) ) ) ) ) ) ) ).

% carrier_hom'
thf(fact_333_carrier__hom_H,axiom,
    ! [F: list_list_set_list_a,R2: partia4556295656693239580t_unit,H: list_set_list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member352051402189872281list_a @ F @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) ) )
     => ( ( member2876526313003830672st_a_a @ H @ ( ring_h763817415916081566it_a_b @ R2 @ S ) )
       => ( ( domain2898972329295444579t_unit @ R2 )
         => ( ( domain_a_b @ S )
           => ( ( inj_on4917424479047495557st_a_a @ H @ ( partia7265347635606999311t_unit @ R2 ) )
             => ( member_list_a @ ( map_li6310725680953440600st_a_a @ H @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ S @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ) ) ) ) ) ).

% carrier_hom'
thf(fact_334_domain_Omap__in__poly__ring__carrier,axiom,
    ! [R2: partia2956882679547061052t_unit,P: list_list_list_a,F: list_list_a > list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
       => ( ! [A3: list_list_a] :
              ( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ R2 ) )
             => ( member5342144027231129785list_a @ ( F @ A3 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) ) )
         => ( ! [A3: list_list_a] :
                ( ( A3
                 != ( zero_l347298301471573063t_unit @ R2 ) )
               => ( ( F @ A3 )
                 != nil_list_list_a ) )
           => ( member6842060177613954879list_a @ ( map_li5227692475714150986list_a @ F @ P ) @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) ) ) ) ) ) ) ) ).

% domain.map_in_poly_ring_carrier
thf(fact_335_domain_Omap__in__poly__ring__carrier,axiom,
    ! [R2: partia4556295656693239580t_unit,P: list_list_set_list_a,F: list_set_list_a > list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R2 )
     => ( ( member352051402189872281list_a @ P @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) ) )
       => ( ! [A3: list_set_list_a] :
              ( ( member5524387281408368019list_a @ A3 @ ( partia7265347635606999311t_unit @ R2 ) )
             => ( member352051402189872281list_a @ ( F @ A3 ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) ) ) )
         => ( ! [A3: list_set_list_a] :
                ( ( A3
                 != ( zero_l7621212060072393831t_unit @ R2 ) )
               => ( ( F @ A3 )
                 != nil_list_set_list_a ) )
           => ( member8950995646279208735list_a @ ( map_li7946501270768669258list_a @ F @ P ) @ ( partia4325684815206509723t_unit @ ( univ_p6110687715593860194t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) ) ) ) ) ) ) ) ) ).

% domain.map_in_poly_ring_carrier
thf(fact_336_domain_Omap__in__poly__ring__carrier,axiom,
    ! [R2: partia2670972154091845814t_unit,P: list_list_a,F: list_a > list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
       => ( ! [A3: list_a] :
              ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R2 ) )
             => ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) )
         => ( ! [A3: list_a] :
                ( ( A3
                 != ( zero_l4142658623432671053t_unit @ R2 ) )
               => ( ( F @ A3 )
                 != nil_list_a ) )
           => ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ F @ P ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ) ) ) ) ) ).

% domain.map_in_poly_ring_carrier
thf(fact_337_domain_Omap__in__poly__ring__carrier,axiom,
    ! [R2: partia4960592913263135132t_unit,P: list_set_list_list_a,F: set_list_list_a > list_set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R2 )
     => ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) ) )
       => ( ! [A3: set_list_list_a] :
              ( ( member334759470184282131list_a @ A3 @ ( partia3317168157747563407t_unit @ R2 ) )
             => ( member6124916891863447321list_a @ ( F @ A3 ) @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) ) ) )
         => ( ! [A3: set_list_list_a] :
                ( ( A3
                 != ( zero_s2920163772466840039t_unit @ R2 ) )
               => ( ( F @ A3 )
                 != nil_set_list_list_a ) )
           => ( member6908278366116243871list_a @ ( map_se4724155948646101578list_a @ F @ P ) @ ( partia6954261711924553499t_unit @ ( univ_p7468970815438322146t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) ) ) ) ) ) ) ) ) ).

% domain.map_in_poly_ring_carrier
thf(fact_338_domain_Omap__in__poly__ring__carrier,axiom,
    ! [R2: partia7496981018696276118t_unit,P: list_set_list_a,F: set_list_a > list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
       => ( ! [A3: set_list_a] :
              ( ( member_set_list_a @ A3 @ ( partia141011252114345353t_unit @ R2 ) )
             => ( member5524387281408368019list_a @ ( F @ A3 ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) ) )
         => ( ! [A3: set_list_a] :
                ( ( A3
                 != ( zero_s2910681146719230829t_unit @ R2 ) )
               => ( ( F @ A3 )
                 != nil_set_list_a ) )
           => ( member352051402189872281list_a @ ( map_se1776605471917444810list_a @ F @ P ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) ) ) ) ) ) ) ) ).

% domain.map_in_poly_ring_carrier
thf(fact_339_domain_Omap__in__poly__ring__carrier,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P: list_a,F: a > list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
       => ( ! [A3: a] :
              ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R2 ) )
             => ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) )
         => ( ! [A3: a] :
                ( ( A3
                 != ( zero_a_b @ R2 ) )
               => ( ( F @ A3 )
                 != nil_a ) )
           => ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ) ) ) ) ) ).

% domain.map_in_poly_ring_carrier
thf(fact_340_domain_Orupture__surj__norm__is__hom,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
         => ( member3350673534532848354list_a @ ( comp_l4894990565352033927list_a @ ( a_r_co4300121960189213440t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ ( cgenid4058437189055694115t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ P ) ) @ ( poly_o1617770581224298896t_unit @ R2 ) )
            @ ( ring_h168092891735524890t_unit
              @ ( partia5725451877363000582t_unit
                @ ^ [Uu: set_list_list_a] : K
                @ R2 )
              @ ( polyno8931960069169623149t_unit @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.rupture_surj_norm_is_hom
thf(fact_341_domain_Orupture__surj__norm__is__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
         => ( member5222132204073479906list_a @ ( comp_l92744468816104577list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ P ) ) @ ( poly_o8716471131768098070t_unit @ R2 ) )
            @ ( ring_h5771512479963622054t_unit
              @ ( partia9041243232023819264t_unit
                @ ^ [Uu: set_list_a] : K
                @ R2 )
              @ ( polyno859807163042199155t_unit @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.rupture_surj_norm_is_hom
thf(fact_342_domain_Orupture__surj__norm__is__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
         => ( member_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ R2 @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R2 @ K ) @ P ) ) @ ( poly_of_const_a_b @ R2 ) )
            @ ( ring_h6109298854714515236t_unit
              @ ( partia8674076737563717228xt_a_b
                @ ^ [Uu: set_a] : K
                @ R2 )
              @ ( polyno5459750281392823787re_a_b @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.rupture_surj_norm_is_hom
thf(fact_343_domain_Orupture__surj__norm__is__hom,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
         => ( member3288041530688867042list_a @ ( comp_l2462527268875606113list_a @ ( a_r_co9127925791106265382t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ ( cgenid538262697157091747t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ P ) ) @ ( poly_o3535013565302768950t_unit @ R2 ) )
            @ ( ring_h8230865552078659302t_unit
              @ ( partia2314006361284536288t_unit
                @ ^ [Uu: set_set_list_a] : K
                @ R2 )
              @ ( polyno7054224608333822611t_unit @ R2 @ K @ P ) ) ) ) ) ) ).

% domain.rupture_surj_norm_is_hom
thf(fact_344_h_OFactRing__iso,axiom,
    ( ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
      = ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
   => ( is_rin1671138935573318232t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ).

% h.FactRing_iso
thf(fact_345_h_Ois__root__def,axiom,
    ! [P: list_set_list_a,X2: set_list_a] :
      ( ( polyno4320237611291262604t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P @ X2 )
      = ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        & ( ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P @ X2 )
          = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        & ( P != nil_set_list_a ) ) ) ).

% h.is_root_def
thf(fact_346_h_OFactRing__iso__set,axiom,
    ( ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
      = ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
   => ( member5068272912271824380list_a
      @ ^ [X5: set_list_a] : ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ X5 ) )
      @ ( ring_i6425008796027319266t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.FactRing_iso_set
thf(fact_347_h_Othe__elem__inj,axiom,
    ! [X4: set_list_a,Y2: set_list_a] :
      ( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) ) )
     => ( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) ) )
       => ( ( ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ X4 ) )
            = ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ Y2 ) ) )
         => ( X4 = Y2 ) ) ) ) ).

% h.the_elem_inj
thf(fact_348_UP_Omap__in__poly__ring__carrier,axiom,
    ! [P: list_list_a,F: list_a > list_list_a] :
      ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ! [A3: list_a] :
            ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) )
       => ( ! [A3: list_a] :
              ( ( A3
               != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
             => ( ( F @ A3 )
               != nil_list_a ) )
         => ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ F @ P ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ) ) ) ).

% UP.map_in_poly_ring_carrier
thf(fact_349_UP_Ozero__is__prime_I1_J,axiom,
    prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% UP.zero_is_prime(1)
thf(fact_350_UP_Onorm__map__in__poly__ring__carrier,axiom,
    ! [P: list_list_a,F: list_a > list_list_a] :
      ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ! [A3: list_a] :
            ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) )
       => ( member5342144027231129785list_a @ ( normal1297324897130370429t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ ( map_li5729356230488778442list_a @ F @ P ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ) ) ).

% UP.norm_map_in_poly_ring_carrier
thf(fact_351_UP_Oring__primeE_I1_J,axiom,
    ! [P: list_a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ k ) @ P )
       => ( P
         != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.ring_primeE(1)
thf(fact_352_UP_Oring__iso__imp__img__domain,axiom,
    ! [H: list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ ( univ_poly_a_b @ r @ k ) @ S ) )
     => ( domain_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          @ S ) ) ) ).

% UP.ring_iso_imp_img_domain
thf(fact_353_UP_Oring__iso__imp__img__domain,axiom,
    ! [H: list_a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) )
     => ( domain6553523120543210313t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          @ S ) ) ) ).

% UP.ring_iso_imp_img_domain
thf(fact_354_UP_Oring__iso__imp__img__domain,axiom,
    ! [H: list_a > list_list_a,S: partia2956882679547061052t_unit] :
      ( ( member6714375691612171394list_a @ H @ ( ring_i7582117978422105628t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) )
     => ( domain7810152921033798211t_unit
        @ ( zero_u569140119141447472t_unit
          @ ^ [Uu: list_list_a] : ( H @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          @ S ) ) ) ).

% UP.ring_iso_imp_img_domain
thf(fact_355_UP_Oring__iso__imp__img__domain,axiom,
    ! [H: list_a > set_list_a,S: partia7496981018696276118t_unit] :
      ( ( member4263473470251683292list_a @ H @ ( ring_i6716952747341498306t_unit @ ( univ_poly_a_b @ r @ k ) @ S ) )
     => ( domain1617769409708967785t_unit
        @ ( zero_u3422801854947504854t_unit
          @ ^ [Uu: set_list_a] : ( H @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          @ S ) ) ) ).

% UP.ring_iso_imp_img_domain
thf(fact_356_UP_Osubdomain__iff,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
        = ( domain6553523120543210313t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : H2
            @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.subdomain_iff
thf(fact_357_a__inv__inj,axiom,
    inj_on_a_a @ ( a_inv_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% a_inv_inj
thf(fact_358_ring__hom__restrict,axiom,
    ! [F: a > list_a,S: partia2670972154091845814t_unit,G: a > list_a] :
      ( ( member_a_list_a @ F @ ( ring_h405018892823518980t_unit @ r @ S ) )
     => ( ! [R: a] :
            ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( F @ R )
              = ( G @ R ) ) )
       => ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ r @ S ) ) ) ) ).

% ring_hom_restrict
thf(fact_359_ring__hom__restrict,axiom,
    ! [F: a > set_list_a,S: partia7496981018696276118t_unit,G: a > set_list_a] :
      ( ( member_a_set_list_a @ F @ ( ring_h6109298854714515236t_unit @ r @ S ) )
     => ( ! [R: a] :
            ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( F @ R )
              = ( G @ R ) ) )
       => ( member_a_set_list_a @ G @ ( ring_h6109298854714515236t_unit @ r @ S ) ) ) ) ).

% ring_hom_restrict
thf(fact_360_eval_Osimps_I1_J,axiom,
    ( ( eval_a_b @ r @ nil_a )
    = ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).

% eval.simps(1)
thf(fact_361_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_362_univ__poly__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( univ_poly_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r ) )
        = ( univ_poly_a_b @ r ) ) ) ).

% univ_poly_consistent
thf(fact_363_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_364_poly__of__const__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( poly_of_const_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r ) )
        = ( poly_of_const_a_b @ r ) ) ) ).

% poly_of_const_consistent
thf(fact_365_subring__is__domain,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( domain_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ r ) ) ) ).

% subring_is_domain
thf(fact_366_canonical__embedding__is__hom,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( member_a_list_a @ ( poly_of_const_a_b @ r )
        @ ( ring_h405018892823518980t_unit
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r )
          @ ( univ_poly_a_b @ r @ K ) ) ) ) ).

% canonical_embedding_is_hom
thf(fact_367_canonical__embedding__ring__hom,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ring_h5357930050666032198t_unit
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : K
          @ r )
        @ ( univ_poly_a_b @ r @ K )
        @ ( poly_of_const_a_b @ r ) ) ) ).

% canonical_embedding_ring_hom
thf(fact_368_ring__iso__imp__img__domain,axiom,
    ! [H: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ r @ S ) )
     => ( domain_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_domain
thf(fact_369_ring__iso__imp__img__domain,axiom,
    ! [H: a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ r @ S ) )
     => ( domain6553523120543210313t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_domain
thf(fact_370_ring__iso__imp__img__domain,axiom,
    ! [H: a > list_list_a,S: partia2956882679547061052t_unit] :
      ( ( member_a_list_list_a @ H @ ( ring_i4464730343205239444t_unit @ r @ S ) )
     => ( domain7810152921033798211t_unit
        @ ( zero_u569140119141447472t_unit
          @ ^ [Uu: list_list_a] : ( H @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_domain
thf(fact_371_ring__iso__imp__img__domain,axiom,
    ! [H: a > set_list_a,S: partia7496981018696276118t_unit] :
      ( ( member_a_set_list_a @ H @ ( ring_i5325512697602418746t_unit @ r @ S ) )
     => ( domain1617769409708967785t_unit
        @ ( zero_u3422801854947504854t_unit
          @ ^ [Uu: set_list_a] : ( H @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_domain
thf(fact_372_UP_Oadd_Oinv__inj,axiom,
    inj_on_list_a_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% UP.add.inv_inj
thf(fact_373_univ__poly__a__inv__def_H,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 )
          = ( map_a_a @ ( a_inv_a_b @ r ) @ P ) ) ) ) ).

% univ_poly_a_inv_def'
thf(fact_374_local_Ominus__zero,axiom,
    ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
    = ( zero_a_b @ r ) ) ).

% local.minus_zero
thf(fact_375_h_Oadd_Oinv__inj,axiom,
    inj_on727569427812895985list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ).

% h.add.inv_inj
thf(fact_376_h_Oring__hom__restrict,axiom,
    ! [F: set_list_a > set_list_a,S: partia7496981018696276118t_unit,G: set_list_a > set_list_a] :
      ( ( member5068272912271824380list_a @ F @ ( ring_h6076331213207892940t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ S ) )
     => ( ! [R: set_list_a] :
            ( ( member_set_list_a @ R @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( F @ R )
              = ( G @ R ) ) )
       => ( member5068272912271824380list_a @ G @ ( ring_h6076331213207892940t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ S ) ) ) ) ).

% h.ring_hom_restrict
thf(fact_377_h_Opoly__of__const__consistent,axiom,
    ! [K: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( poly_o3535013565302768950t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : K
            @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = ( poly_o3535013565302768950t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.poly_of_const_consistent
thf(fact_378_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_379_UP_Oa__r__coset__subset__G,axiom,
    ! [H2: set_list_a,X2: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ord_le8861187494160871172list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ X2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.a_r_coset_subset_G
thf(fact_380_UP_Oeval_Osimps_I1_J,axiom,
    ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a )
    = ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.eval.simps(1)
thf(fact_381_UP_Oconst__term__not__zero,axiom,
    ! [P: list_list_a] :
      ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P )
       != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( P != nil_list_a ) ) ).

% UP.const_term_not_zero
thf(fact_382_UP_Oring__primeE_I3_J,axiom,
    ! [P: list_a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ k ) @ P )
       => ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) ) ) ).

% UP.ring_primeE(3)
thf(fact_383_UP_Oring__primeI,axiom,
    ! [P: list_a] :
      ( ( P
       != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ k ) @ P )
       => ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) ) ) ).

% UP.ring_primeI
thf(fact_384_UP_Ouniv__poly__a__inv__consistent,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P )
          = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P ) ) ) ) ).

% UP.univ_poly_a_inv_consistent
thf(fact_385_h_Ouniv__poly__consistent,axiom,
    ! [K: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( univ_p863672496597069550t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : K
            @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.univ_poly_consistent
thf(fact_386_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_387_UP_Ouniv__poly__a__inv__def_H,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P )
          = ( map_list_a_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) ) @ P ) ) ) ) ).

% UP.univ_poly_a_inv_def'
thf(fact_388_UP_Oconst__term__simprules__shell_I4_J,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) ) ) ) ) ).

% UP.const_term_simprules_shell(4)
thf(fact_389_add_Oinv__eq__1__iff,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ( a_inv_a_b @ r @ X2 )
          = ( zero_a_b @ r ) )
        = ( X2
          = ( zero_a_b @ r ) ) ) ) ).

% add.inv_eq_1_iff
thf(fact_390_local_Ominus__minus,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X2 ) )
        = X2 ) ) ).

% local.minus_minus
thf(fact_391_a__inv__closed,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ ( a_inv_a_b @ r @ X2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% a_inv_closed
thf(fact_392_zero__closed,axiom,
    member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% zero_closed
thf(fact_393_rupture__surj__norm__is__hom,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_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) @ ( poly_of_const_a_b @ r ) )
          @ ( ring_h6109298854714515236t_unit
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : K
              @ r )
            @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) ) ) ) ) ).

% rupture_surj_norm_is_hom
thf(fact_394_h_Oimg__is__domain,axiom,
    ( ( domain1617769409708967785t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
   => ( domain1617769409708967785t_unit
      @ ( partia2314006361284536288t_unit
        @ ^ [Uu: set_set_list_a] : ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.img_is_domain
thf(fact_395_h_Othe__elem__hom,axiom,
    ( member5068272912271824380list_a
    @ ^ [X5: set_list_a] : ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ X5 ) )
    @ ( ring_h6076331213207892940t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ).

% h.the_elem_hom
thf(fact_396_eval__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( eval_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r ) )
        = ( eval_a_b @ r ) ) ) ).

% eval_consistent
thf(fact_397_h_OFactRing__iso__set__aux,axiom,
    ( member5068272912271824380list_a
    @ ^ [X5: set_list_a] : ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ X5 ) )
    @ ( ring_i6425008796027319266t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) )
      @ ( partia2314006361284536288t_unit
        @ ^ [Uu: set_set_list_a] : ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.FactRing_iso_set_aux
thf(fact_398_h_Ominus__minus,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 ) )
        = X2 ) ) ).

% h.minus_minus
thf(fact_399_h_Oadd_Oinv__closed,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( member_set_list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.add.inv_closed
thf(fact_400_h_Ominus__zero,axiom,
    ( ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
    = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ).

% h.minus_zero
thf(fact_401_h_Oadd_Oinv__eq__1__iff,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 )
          = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = ( X2
          = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.add.inv_eq_1_iff
thf(fact_402_UP_Oa__coset__add__zero,axiom,
    ! [M: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        = M ) ) ).

% UP.a_coset_add_zero
thf(fact_403_h_Oeval__consistent,axiom,
    ! [K: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( eval_s5133945360527818456t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : K
            @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.eval_consistent
thf(fact_404_h_Ohom__a__inv,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) )
        = ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ X2 ) ) ) ) ).

% h.hom_a_inv
thf(fact_405_h_Othe__elem__simp,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) @ X2 ) ) )
        = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ X2 ) ) ) ).

% h.the_elem_simp
thf(fact_406_ring_Ois__root_Ocong,axiom,
    polyno4320237611291262604t_unit = polyno4320237611291262604t_unit ).

% ring.is_root.cong
thf(fact_407_ring_Ois__root_Ocong,axiom,
    polyno6951661231331188332t_unit = polyno6951661231331188332t_unit ).

% ring.is_root.cong
thf(fact_408_ring_Ois__root_Ocong,axiom,
    polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).

% ring.is_root.cong
thf(fact_409_ring_Onormalize_Ocong,axiom,
    normal1297324897130370429t_unit = normal1297324897130370429t_unit ).

% ring.normalize.cong
thf(fact_410_ring_Onormalize_Ocong,axiom,
    normalize_a_b = normalize_a_b ).

% ring.normalize.cong
thf(fact_411_ring_Onormalize_Ocong,axiom,
    normal637505603836502915t_unit = normal637505603836502915t_unit ).

% ring.normalize.cong
thf(fact_412_ring_Onormalize_Ocong,axiom,
    normal4052021864830707619t_unit = normal4052021864830707619t_unit ).

% ring.normalize.cong
thf(fact_413_domain_Onorm__map__in__poly__ring__carrier,axiom,
    ! [R2: partia2956882679547061052t_unit,P: list_list_list_a,F: list_list_a > list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) )
       => ( ! [A3: list_list_a] :
              ( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ R2 ) )
             => ( member5342144027231129785list_a @ ( F @ A3 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) ) )
         => ( member6842060177613954879list_a @ ( normal5368706450718127095t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ ( map_li5227692475714150986list_a @ F @ P ) ) @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) ) ) ) ) ) ) ) ).

% domain.norm_map_in_poly_ring_carrier
thf(fact_414_domain_Onorm__map__in__poly__ring__carrier,axiom,
    ! [R2: partia4556295656693239580t_unit,P: list_list_set_list_a,F: list_set_list_a > list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R2 )
     => ( ( member352051402189872281list_a @ P @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) ) )
       => ( ! [A3: list_set_list_a] :
              ( ( member5524387281408368019list_a @ A3 @ ( partia7265347635606999311t_unit @ R2 ) )
             => ( member352051402189872281list_a @ ( F @ A3 ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) ) ) )
         => ( member8950995646279208735list_a @ ( normal6046089599045693463t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) @ ( map_li7946501270768669258list_a @ F @ P ) ) @ ( partia4325684815206509723t_unit @ ( univ_p6110687715593860194t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ ( partia7265347635606999311t_unit @ R2 ) ) ) ) ) ) ) ) ) ).

% domain.norm_map_in_poly_ring_carrier
thf(fact_415_domain_Onorm__map__in__poly__ring__carrier,axiom,
    ! [R2: partia2670972154091845814t_unit,P: list_list_a,F: list_a > list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) )
       => ( ! [A3: list_a] :
              ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R2 ) )
             => ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) )
         => ( member5342144027231129785list_a @ ( normal1297324897130370429t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ ( map_li5729356230488778442list_a @ F @ P ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ) ) ) ) ).

% domain.norm_map_in_poly_ring_carrier
thf(fact_416_domain_Onorm__map__in__poly__ring__carrier,axiom,
    ! [R2: partia4960592913263135132t_unit,P: list_set_list_list_a,F: set_list_list_a > list_set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R2 )
     => ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) ) )
       => ( ! [A3: set_list_list_a] :
              ( ( member334759470184282131list_a @ A3 @ ( partia3317168157747563407t_unit @ R2 ) )
             => ( member6124916891863447321list_a @ ( F @ A3 ) @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) ) ) )
         => ( member6908278366116243871list_a @ ( normal7404372698890155415t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) @ ( map_se4724155948646101578list_a @ F @ P ) ) @ ( partia6954261711924553499t_unit @ ( univ_p7468970815438322146t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ ( partia3317168157747563407t_unit @ R2 ) ) ) ) ) ) ) ) ) ).

% domain.norm_map_in_poly_ring_carrier
thf(fact_417_domain_Onorm__map__in__poly__ring__carrier,axiom,
    ! [R2: partia7496981018696276118t_unit,P: list_set_list_a,F: set_list_a > list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) )
       => ( ! [A3: set_list_a] :
              ( ( member_set_list_a @ A3 @ ( partia141011252114345353t_unit @ R2 ) )
             => ( member5524387281408368019list_a @ ( F @ A3 ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) ) )
         => ( member352051402189872281list_a @ ( normal7202597576022193565t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ ( map_se1776605471917444810list_a @ F @ P ) ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ ( partia141011252114345353t_unit @ R2 ) ) ) ) ) ) ) ) ) ).

% domain.norm_map_in_poly_ring_carrier
thf(fact_418_domain_Onorm__map__in__poly__ring__carrier,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P: list_a,F: a > list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) )
       => ( ! [A3: a] :
              ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R2 ) )
             => ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) )
         => ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ ( map_a_list_a @ F @ P ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ) ) ) ) ).

% domain.norm_map_in_poly_ring_carrier
thf(fact_419_h_Osubdomain__is__domain,axiom,
    ! [H2: set_set_list_a] :
      ( ( subdom3220114454046903646t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( domain1617769409708967785t_unit
        @ ( partia2314006361284536288t_unit
          @ ^ [Uu: set_set_list_a] : H2
          @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.subdomain_is_domain
thf(fact_420_UP_Ois__root__def,axiom,
    ! [P: list_list_a,X2: list_a] :
      ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ X2 )
      = ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        & ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ X2 )
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        & ( P != nil_list_a ) ) ) ).

% UP.is_root_def
thf(fact_421_h_Oeval__hom_H,axiom,
    ! [A: list_a,P: list_list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ A ) )
          = ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( map_li2657846702790696612list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ P ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ A ) ) ) ) ) ).

% h.eval_hom'
thf(fact_422_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_423_h_Othe__elem__surj,axiom,
    ( ( image_5749939591322298757list_a
      @ ^ [X5: set_list_a] : ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ X5 ) )
      @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) ) )
    = ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% h.the_elem_surj
thf(fact_424_UP_Oa__rcosetsI,axiom,
    ! [H2: set_list_a,X2: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( member_set_list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ X2 ) @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 ) ) ) ) ).

% UP.a_rcosetsI
thf(fact_425_UP_Oeval__as__unique__hom,axiom,
    ! [K: set_list_a,X2: list_a,H: list_list_a > list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ring_h4589914651911841480t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( univ_poly_a_b @ r @ k ) @ H )
         => ( ! [K3: list_a] :
                ( ( member_list_a @ K3 @ K )
               => ( ( H @ ( cons_list_a @ K3 @ nil_list_a ) )
                  = K3 ) )
           => ( ( ( H @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) )
                = X2 )
             => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
               => ( ( H @ P )
                  = ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ X2 ) ) ) ) ) ) ) ) ).

% UP.eval_as_unique_hom
thf(fact_426_h_Oset__add__ker__hom_I1_J,axiom,
    ! [I2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ I2 @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) )
        = ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ I2 ) ) ) ).

% h.set_add_ker_hom(1)
thf(fact_427_h_Oset__add__ker__hom_I2_J,axiom,
    ! [I2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) @ I2 ) )
        = ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ I2 ) ) ) ).

% h.set_add_ker_hom(2)
thf(fact_428_UP_Onormalize_Ocases,axiom,
    ! [X2: list_list_a] :
      ( ( X2 != nil_list_a )
     => ~ ! [V2: list_a,Va: list_list_a] :
            ( X2
           != ( cons_list_a @ V2 @ Va ) ) ) ).

% UP.normalize.cases
thf(fact_429_normalize_Osimps_I1_J,axiom,
    ( ( normalize_a_b @ r @ nil_a )
    = nil_a ) ).

% normalize.simps(1)
thf(fact_430_list_Oinject,axiom,
    ! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
      ( ( ( cons_list_a @ X21 @ X22 )
        = ( cons_list_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_431_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_432_list_Oinject,axiom,
    ! [X21: set_list_a,X22: list_set_list_a,Y21: set_list_a,Y22: list_set_list_a] :
      ( ( ( cons_set_list_a @ X21 @ X22 )
        = ( cons_set_list_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_433_UP_Onormalize_Osimps_I1_J,axiom,
    ( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a )
    = nil_list_a ) ).

% UP.normalize.simps(1)
thf(fact_434_is__root__def,axiom,
    ! [P: list_a,X2: a] :
      ( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
      = ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
        & ( ( eval_a_b @ r @ P @ X2 )
          = ( zero_a_b @ r ) )
        & ( P != nil_a ) ) ) ).

% is_root_def
thf(fact_435_h_Onormalize_Osimps_I1_J,axiom,
    ( ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ nil_set_list_a )
    = nil_set_list_a ) ).

% h.normalize.simps(1)
thf(fact_436_UP_Oadd__additive__subgroups,axiom,
    ! [H2: set_list_a,K: set_list_a] :
      ( ( additi4714453376129182166t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( ( additi4714453376129182166t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
       => ( additi4714453376129182166t_unit @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ K ) @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.add_additive_subgroups
thf(fact_437_UP_Onormalize__in__carrier,axiom,
    ! [P: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.normalize_in_carrier
thf(fact_438_UP_Oset__add__closed,axiom,
    ! [A2: set_list_a,B2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ A2 @ B2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.set_add_closed
thf(fact_439_UP_Oset__add__comm,axiom,
    ! [I2: set_list_a,J: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ J @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ I2 @ J )
          = ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ J @ I2 ) ) ) ) ).

% UP.set_add_comm
thf(fact_440_UP_Osetadd__subset__G,axiom,
    ! [H2: set_list_a,K: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.setadd_subset_G
thf(fact_441_UP_Opoly__of__const__def,axiom,
    ( ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) )
    = ( ^ [K4: list_a] : ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cons_list_a @ K4 @ nil_list_a ) ) ) ) ).

% UP.poly_of_const_def
thf(fact_442_UP_Oeval__normalize,axiom,
    ! [P: list_list_a,A: list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) @ A )
          = ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ A ) ) ) ) ).

% UP.eval_normalize
thf(fact_443_UP_Oeval__in__carrier,axiom,
    ! [P: list_list_a,X2: list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ X2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.eval_in_carrier
thf(fact_444_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_445_UP_Oconst__term__simprules_I1_J,axiom,
    ! [P: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.const_term_simprules(1)
thf(fact_446_UP_Oa__setmult__rcos__assoc,axiom,
    ! [H2: set_list_a,K: set_list_a,X2: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ X2 ) )
            = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ K ) @ X2 ) ) ) ) ) ).

% UP.a_setmult_rcos_assoc
thf(fact_447_map__eq__conv,axiom,
    ! [F: list_a > set_list_list_a,Xs: list_list_a,G: list_a > set_list_list_a] :
      ( ( ( map_li5777811645572713770list_a @ F @ Xs )
        = ( map_li5777811645572713770list_a @ G @ Xs ) )
      = ( ! [X: list_a] :
            ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_448_map__eq__conv,axiom,
    ! [F: list_a > set_list_a,Xs: list_list_a,G: list_a > set_list_a] :
      ( ( ( map_li2657846702790696612list_a @ F @ Xs )
        = ( map_li2657846702790696612list_a @ G @ Xs ) )
      = ( ! [X: list_a] :
            ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_449_map__eq__conv,axiom,
    ! [F: a > list_a,Xs: list_a,G: a > list_a] :
      ( ( ( map_a_list_a @ F @ Xs )
        = ( map_a_list_a @ G @ Xs ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_450_map__eq__conv,axiom,
    ! [F: a > set_list_a,Xs: list_a,G: a > set_list_a] :
      ( ( ( map_a_set_list_a @ F @ Xs )
        = ( map_a_set_list_a @ G @ Xs ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_451_map__eq__conv,axiom,
    ! [F: a > a,Xs: list_a,G: a > a] :
      ( ( ( map_a_a @ F @ Xs )
        = ( map_a_a @ G @ Xs ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_452_map__eq__conv,axiom,
    ! [F: a > set_list_list_a,Xs: list_a,G: a > set_list_list_a] :
      ( ( ( map_a_132662181483840688list_a @ F @ Xs )
        = ( map_a_132662181483840688list_a @ G @ Xs ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_453_map__eq__conv,axiom,
    ! [F: set_list_a > list_a,Xs: list_set_list_a,G: set_list_a > list_a] :
      ( ( ( map_se5127173849416245988list_a @ F @ Xs )
        = ( map_se5127173849416245988list_a @ G @ Xs ) )
      = ( ! [X: set_list_a] :
            ( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_454_map__eq__conv,axiom,
    ! [F: set_list_a > set_list_list_a,Xs: list_set_list_a,G: set_list_a > set_list_list_a] :
      ( ( ( map_se5810349697548134730list_a @ F @ Xs )
        = ( map_se5810349697548134730list_a @ G @ Xs ) )
      = ( ! [X: set_list_a] :
            ( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_455_map__eq__conv,axiom,
    ! [F: set_list_a > set_list_a,Xs: list_set_list_a,G: set_list_a > set_list_a] :
      ( ( ( map_se2668659675339852484list_a @ F @ Xs )
        = ( map_se2668659675339852484list_a @ G @ Xs ) )
      = ( ! [X: set_list_a] :
            ( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_456_map__eq__conv,axiom,
    ! [F: set_list_a > a,Xs: list_set_list_a,G: set_list_a > a] :
      ( ( ( map_set_list_a_a @ F @ Xs )
        = ( map_set_list_a_a @ G @ Xs ) )
      = ( ! [X: set_list_a] :
            ( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_457_UP_Odetermination__of__hom,axiom,
    ! [K: set_list_a,A2: partia4960592913263135132t_unit,H: list_list_a > set_list_list_a,G: list_list_a > set_list_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ring_h2829259075339158498t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ A2 @ H )
       => ( ( ring_h2829259075339158498t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ A2 @ G )
         => ( ! [K3: list_a] :
                ( ( member_list_a @ K3 @ K )
               => ( ( H @ ( cons_list_a @ K3 @ nil_list_a ) )
                  = ( G @ ( cons_list_a @ K3 @ nil_list_a ) ) ) )
           => ( ( ( H @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) )
                = ( G @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) ) )
             => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
               => ( ( H @ P )
                  = ( G @ P ) ) ) ) ) ) ) ) ).

% UP.determination_of_hom
thf(fact_458_list_Oset__map,axiom,
    ! [F: a > a,V: list_a] :
      ( ( set_a2 @ ( map_a_a @ F @ V ) )
      = ( image_a_a @ F @ ( set_a2 @ V ) ) ) ).

% list.set_map
thf(fact_459_list_Oset__map,axiom,
    ! [F: a > list_a,V: list_a] :
      ( ( set_list_a2 @ ( map_a_list_a @ F @ V ) )
      = ( image_a_list_a @ F @ ( set_a2 @ V ) ) ) ).

% list.set_map
thf(fact_460_list_Oset__map,axiom,
    ! [F: list_a > a,V: list_list_a] :
      ( ( set_a2 @ ( map_list_a_a @ F @ V ) )
      = ( image_list_a_a @ F @ ( set_list_a2 @ V ) ) ) ).

% list.set_map
thf(fact_461_list_Oset__map,axiom,
    ! [F: list_a > list_a,V: list_list_a] :
      ( ( set_list_a2 @ ( map_list_a_list_a @ F @ V ) )
      = ( image_list_a_list_a @ F @ ( set_list_a2 @ V ) ) ) ).

% list.set_map
thf(fact_462_list_Oset__map,axiom,
    ! [F: set_list_a > a,V: list_set_list_a] :
      ( ( set_a2 @ ( map_set_list_a_a @ F @ V ) )
      = ( image_set_list_a_a @ F @ ( set_set_list_a2 @ V ) ) ) ).

% list.set_map
thf(fact_463_list_Oset__map,axiom,
    ! [F: a > set_list_a,V: list_a] :
      ( ( set_set_list_a2 @ ( map_a_set_list_a @ F @ V ) )
      = ( image_a_set_list_a @ F @ ( set_a2 @ V ) ) ) ).

% list.set_map
thf(fact_464_list_Oset__map,axiom,
    ! [F: a > set_list_list_a,V: list_a] :
      ( ( set_set_list_list_a2 @ ( map_a_132662181483840688list_a @ F @ V ) )
      = ( image_2939653550459479921list_a @ F @ ( set_a2 @ V ) ) ) ).

% list.set_map
thf(fact_465_list_Oset__map,axiom,
    ! [F: set_list_a > list_a,V: list_set_list_a] :
      ( ( set_list_a2 @ ( map_se5127173849416245988list_a @ F @ V ) )
      = ( image_7934165218391885221list_a @ F @ ( set_set_list_a2 @ V ) ) ) ).

% list.set_map
thf(fact_466_list_Oset__map,axiom,
    ! [F: list_a > set_list_a,V: list_list_a] :
      ( ( set_set_list_a2 @ ( map_li2657846702790696612list_a @ F @ V ) )
      = ( image_5464838071766335845list_a @ F @ ( set_list_a2 @ V ) ) ) ).

% list.set_map
thf(fact_467_list_Oset__map,axiom,
    ! [F: list_a > set_list_list_a,V: list_list_a] :
      ( ( set_set_list_list_a2 @ ( map_li5777811645572713770list_a @ F @ V ) )
      = ( image_8589081332878056299list_a @ F @ ( set_list_a2 @ V ) ) ) ).

% list.set_map
thf(fact_468_normalize__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( normalize_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r ) )
        = ( normalize_a_b @ r ) ) ) ).

% normalize_consistent
thf(fact_469_UP_Onormalize__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( normal637505603836502915t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) ) )
        = ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.normalize_consistent
thf(fact_470_h_Onormalize__consistent,axiom,
    ! [K: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( normal4052021864830707619t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : K
            @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.normalize_consistent
thf(fact_471_inj__on__Cons1,axiom,
    ! [X2: list_a,A2: set_list_list_a] : ( inj_on3375637900297828593list_a @ ( cons_list_a @ X2 ) @ A2 ) ).

% inj_on_Cons1
thf(fact_472_inj__on__Cons1,axiom,
    ! [X2: a,A2: set_list_a] : ( inj_on_list_a_list_a @ ( cons_a @ X2 ) @ A2 ) ).

% inj_on_Cons1
thf(fact_473_inj__on__Cons1,axiom,
    ! [X2: set_list_a,A2: set_list_set_list_a] : ( inj_on320113931233832177list_a @ ( cons_set_list_a @ X2 ) @ A2 ) ).

% inj_on_Cons1
thf(fact_474_set__subset__Cons,axiom,
    ! [Xs: list_list_a,X2: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_475_set__subset__Cons,axiom,
    ! [Xs: list_set_list_a,X2: set_list_a] : ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Xs ) @ ( set_set_list_a2 @ ( cons_set_list_a @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_476_set__subset__Cons,axiom,
    ! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_477_list_Oset__intros_I2_J,axiom,
    ! [Y: list_a > a,X22: list_list_a_a,X21: list_a > a] :
      ( ( member_list_a_a @ Y @ ( set_list_a_a2 @ X22 ) )
     => ( member_list_a_a @ Y @ ( set_list_a_a2 @ ( cons_list_a_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_478_list_Oset__intros_I2_J,axiom,
    ! [Y: set_list_a > set_list_a,X22: list_s476705121730489333list_a,X21: set_list_a > set_list_a] :
      ( ( member5068272912271824380list_a @ Y @ ( set_se4893213719394700432list_a @ X22 ) )
     => ( member5068272912271824380list_a @ Y @ ( set_se4893213719394700432list_a @ ( cons_s3960069323163066159list_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_479_list_Oset__intros_I2_J,axiom,
    ! [Y: a > list_a,X22: list_a_list_a,X21: a > list_a] :
      ( ( member_a_list_a @ Y @ ( set_a_list_a2 @ X22 ) )
     => ( member_a_list_a @ Y @ ( set_a_list_a2 @ ( cons_a_list_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_480_list_Oset__intros_I2_J,axiom,
    ! [Y: a > set_list_a,X22: list_a_set_list_a,X21: a > set_list_a] :
      ( ( member_a_set_list_a @ Y @ ( set_a_set_list_a2 @ X22 ) )
     => ( member_a_set_list_a @ Y @ ( set_a_set_list_a2 @ ( cons_a_set_list_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_481_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_482_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_483_list_Oset__intros_I2_J,axiom,
    ! [Y: set_list_a,X22: list_set_list_a,X21: set_list_a] :
      ( ( member_set_list_a @ Y @ ( set_set_list_a2 @ X22 ) )
     => ( member_set_list_a @ Y @ ( set_set_list_a2 @ ( cons_set_list_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_484_list_Oset__intros_I1_J,axiom,
    ! [X21: list_a > a,X22: list_list_a_a] : ( member_list_a_a @ X21 @ ( set_list_a_a2 @ ( cons_list_a_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_485_list_Oset__intros_I1_J,axiom,
    ! [X21: set_list_a > set_list_a,X22: list_s476705121730489333list_a] : ( member5068272912271824380list_a @ X21 @ ( set_se4893213719394700432list_a @ ( cons_s3960069323163066159list_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_486_list_Oset__intros_I1_J,axiom,
    ! [X21: a > list_a,X22: list_a_list_a] : ( member_a_list_a @ X21 @ ( set_a_list_a2 @ ( cons_a_list_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_487_list_Oset__intros_I1_J,axiom,
    ! [X21: a > set_list_a,X22: list_a_set_list_a] : ( member_a_set_list_a @ X21 @ ( set_a_set_list_a2 @ ( cons_a_set_list_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_488_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_489_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_490_list_Oset__intros_I1_J,axiom,
    ! [X21: set_list_a,X22: list_set_list_a] : ( member_set_list_a @ X21 @ ( set_set_list_a2 @ ( cons_set_list_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_491_list_Oset__cases,axiom,
    ! [E: list_a > a,A: list_list_a_a] :
      ( ( member_list_a_a @ E @ ( set_list_a_a2 @ A ) )
     => ( ! [Z2: list_list_a_a] :
            ( A
           != ( cons_list_a_a @ E @ Z2 ) )
       => ~ ! [Z1: list_a > a,Z2: list_list_a_a] :
              ( ( A
                = ( cons_list_a_a @ Z1 @ Z2 ) )
             => ~ ( member_list_a_a @ E @ ( set_list_a_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_492_list_Oset__cases,axiom,
    ! [E: set_list_a > set_list_a,A: list_s476705121730489333list_a] :
      ( ( member5068272912271824380list_a @ E @ ( set_se4893213719394700432list_a @ A ) )
     => ( ! [Z2: list_s476705121730489333list_a] :
            ( A
           != ( cons_s3960069323163066159list_a @ E @ Z2 ) )
       => ~ ! [Z1: set_list_a > set_list_a,Z2: list_s476705121730489333list_a] :
              ( ( A
                = ( cons_s3960069323163066159list_a @ Z1 @ Z2 ) )
             => ~ ( member5068272912271824380list_a @ E @ ( set_se4893213719394700432list_a @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_493_list_Oset__cases,axiom,
    ! [E: a > list_a,A: list_a_list_a] :
      ( ( member_a_list_a @ E @ ( set_a_list_a2 @ A ) )
     => ( ! [Z2: list_a_list_a] :
            ( A
           != ( cons_a_list_a @ E @ Z2 ) )
       => ~ ! [Z1: a > list_a,Z2: list_a_list_a] :
              ( ( A
                = ( cons_a_list_a @ Z1 @ Z2 ) )
             => ~ ( member_a_list_a @ E @ ( set_a_list_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_494_list_Oset__cases,axiom,
    ! [E: a > set_list_a,A: list_a_set_list_a] :
      ( ( member_a_set_list_a @ E @ ( set_a_set_list_a2 @ A ) )
     => ( ! [Z2: list_a_set_list_a] :
            ( A
           != ( cons_a_set_list_a @ E @ Z2 ) )
       => ~ ! [Z1: a > set_list_a,Z2: list_a_set_list_a] :
              ( ( A
                = ( cons_a_set_list_a @ Z1 @ Z2 ) )
             => ~ ( member_a_set_list_a @ E @ ( set_a_set_list_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_495_list_Oset__cases,axiom,
    ! [E: list_a,A: list_list_a] :
      ( ( member_list_a @ E @ ( set_list_a2 @ A ) )
     => ( ! [Z2: list_list_a] :
            ( A
           != ( cons_list_a @ E @ Z2 ) )
       => ~ ! [Z1: list_a,Z2: list_list_a] :
              ( ( A
                = ( cons_list_a @ Z1 @ Z2 ) )
             => ~ ( member_list_a @ E @ ( set_list_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_496_list_Oset__cases,axiom,
    ! [E: a,A: list_a] :
      ( ( member_a @ E @ ( set_a2 @ A ) )
     => ( ! [Z2: list_a] :
            ( A
           != ( cons_a @ E @ Z2 ) )
       => ~ ! [Z1: a,Z2: list_a] :
              ( ( A
                = ( cons_a @ Z1 @ Z2 ) )
             => ~ ( member_a @ E @ ( set_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_497_list_Oset__cases,axiom,
    ! [E: set_list_a,A: list_set_list_a] :
      ( ( member_set_list_a @ E @ ( set_set_list_a2 @ A ) )
     => ( ! [Z2: list_set_list_a] :
            ( A
           != ( cons_set_list_a @ E @ Z2 ) )
       => ~ ! [Z1: set_list_a,Z2: list_set_list_a] :
              ( ( A
                = ( cons_set_list_a @ Z1 @ Z2 ) )
             => ~ ( member_set_list_a @ E @ ( set_set_list_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_498_set__ConsD,axiom,
    ! [Y: list_a > a,X2: list_a > a,Xs: list_list_a_a] :
      ( ( member_list_a_a @ Y @ ( set_list_a_a2 @ ( cons_list_a_a @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member_list_a_a @ Y @ ( set_list_a_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_499_set__ConsD,axiom,
    ! [Y: set_list_a > set_list_a,X2: set_list_a > set_list_a,Xs: list_s476705121730489333list_a] :
      ( ( member5068272912271824380list_a @ Y @ ( set_se4893213719394700432list_a @ ( cons_s3960069323163066159list_a @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member5068272912271824380list_a @ Y @ ( set_se4893213719394700432list_a @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_500_set__ConsD,axiom,
    ! [Y: a > list_a,X2: a > list_a,Xs: list_a_list_a] :
      ( ( member_a_list_a @ Y @ ( set_a_list_a2 @ ( cons_a_list_a @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member_a_list_a @ Y @ ( set_a_list_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_501_set__ConsD,axiom,
    ! [Y: a > set_list_a,X2: a > set_list_a,Xs: list_a_set_list_a] :
      ( ( member_a_set_list_a @ Y @ ( set_a_set_list_a2 @ ( cons_a_set_list_a @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member_a_set_list_a @ Y @ ( set_a_set_list_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_502_set__ConsD,axiom,
    ! [Y: list_a,X2: list_a,Xs: list_list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member_list_a @ Y @ ( set_list_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_503_set__ConsD,axiom,
    ! [Y: a,X2: a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_504_set__ConsD,axiom,
    ! [Y: set_list_a,X2: set_list_a,Xs: list_set_list_a] :
      ( ( member_set_list_a @ Y @ ( set_set_list_a2 @ ( cons_set_list_a @ X2 @ Xs ) ) )
     => ( ( Y = X2 )
        | ( member_set_list_a @ Y @ ( set_set_list_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_505_not__Cons__self2,axiom,
    ! [X2: list_a,Xs: list_list_a] :
      ( ( cons_list_a @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_506_not__Cons__self2,axiom,
    ! [X2: a,Xs: list_a] :
      ( ( cons_a @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_507_not__Cons__self2,axiom,
    ! [X2: set_list_a,Xs: list_set_list_a] :
      ( ( cons_set_list_a @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_508_transpose_Ocases,axiom,
    ! [X2: list_list_list_a] :
      ( ( X2 != nil_list_list_a )
     => ( ! [Xss: list_list_list_a] :
            ( X2
           != ( cons_list_list_a @ nil_list_a @ Xss ) )
       => ~ ! [X3: list_a,Xs3: list_list_a,Xss: list_list_list_a] :
              ( X2
             != ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_509_transpose_Ocases,axiom,
    ! [X2: list_list_set_list_a] :
      ( ( X2 != nil_list_set_list_a )
     => ( ! [Xss: list_list_set_list_a] :
            ( X2
           != ( cons_list_set_list_a @ nil_set_list_a @ Xss ) )
       => ~ ! [X3: set_list_a,Xs3: list_set_list_a,Xss: list_list_set_list_a] :
              ( X2
             != ( cons_list_set_list_a @ ( cons_set_list_a @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_510_transpose_Ocases,axiom,
    ! [X2: list_list_a] :
      ( ( X2 != nil_list_a )
     => ( ! [Xss: list_list_a] :
            ( X2
           != ( cons_list_a @ nil_a @ Xss ) )
       => ~ ! [X3: a,Xs3: list_a,Xss: list_list_a] :
              ( X2
             != ( cons_list_a @ ( cons_a @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_511_the__elem__set,axiom,
    ! [X2: list_a] :
      ( ( the_elem_list_a @ ( set_list_a2 @ ( cons_list_a @ X2 @ nil_list_a ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_512_the__elem__set,axiom,
    ! [X2: a] :
      ( ( the_elem_a @ ( set_a2 @ ( cons_a @ X2 @ nil_a ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_513_the__elem__set,axiom,
    ! [X2: set_list_a] :
      ( ( the_elem_set_list_a @ ( set_set_list_a2 @ ( cons_set_list_a @ X2 @ nil_set_list_a ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_514_subset__code_I1_J,axiom,
    ! [Xs: list_list_a_a,B2: set_list_a_a] :
      ( ( ord_le6942402695062981877st_a_a @ ( set_list_a_a2 @ Xs ) @ B2 )
      = ( ! [X: list_a > a] :
            ( ( member_list_a_a @ X @ ( set_list_a_a2 @ Xs ) )
           => ( member_list_a_a @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_515_subset__code_I1_J,axiom,
    ! [Xs: list_s476705121730489333list_a,B2: set_se1917860372504128155list_a] :
      ( ( ord_le8635267389821127419list_a @ ( set_se4893213719394700432list_a @ Xs ) @ B2 )
      = ( ! [X: set_list_a > set_list_a] :
            ( ( member5068272912271824380list_a @ X @ ( set_se4893213719394700432list_a @ Xs ) )
           => ( member5068272912271824380list_a @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_516_subset__code_I1_J,axiom,
    ! [Xs: list_a_list_a,B2: set_a_list_a] :
      ( ( ord_le50412136050534657list_a @ ( set_a_list_a2 @ Xs ) @ B2 )
      = ( ! [X: a > list_a] :
            ( ( member_a_list_a @ X @ ( set_a_list_a2 @ Xs ) )
           => ( member_a_list_a @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_517_subset__code_I1_J,axiom,
    ! [Xs: list_a_set_list_a,B2: set_a_set_list_a] :
      ( ( ord_le1723303916063593697list_a @ ( set_a_set_list_a2 @ Xs ) @ B2 )
      = ( ! [X: a > set_list_a] :
            ( ( member_a_set_list_a @ X @ ( set_a_set_list_a2 @ Xs ) )
           => ( member_a_set_list_a @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_518_subset__code_I1_J,axiom,
    ! [Xs: list_list_a,B2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B2 )
      = ( ! [X: list_a] :
            ( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
           => ( member_list_a @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_519_subset__code_I1_J,axiom,
    ! [Xs: list_set_list_a,B2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Xs ) @ B2 )
      = ( ! [X: set_list_a] :
            ( ( member_set_list_a @ X @ ( set_set_list_a2 @ Xs ) )
           => ( member_set_list_a @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_520_subset__code_I1_J,axiom,
    ! [Xs: list_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Xs ) )
           => ( member_a @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_521_list_Odistinct_I1_J,axiom,
    ! [X21: list_a,X22: list_list_a] :
      ( nil_list_a
     != ( cons_list_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_522_list_Odistinct_I1_J,axiom,
    ! [X21: a,X22: list_a] :
      ( nil_a
     != ( cons_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_523_list_Odistinct_I1_J,axiom,
    ! [X21: set_list_a,X22: list_set_list_a] :
      ( nil_set_list_a
     != ( cons_set_list_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_524_list_OdiscI,axiom,
    ! [List: list_list_a,X21: list_a,X22: list_list_a] :
      ( ( List
        = ( cons_list_a @ X21 @ X22 ) )
     => ( List != nil_list_a ) ) ).

% list.discI
thf(fact_525_list_OdiscI,axiom,
    ! [List: list_a,X21: a,X22: list_a] :
      ( ( List
        = ( cons_a @ X21 @ X22 ) )
     => ( List != nil_a ) ) ).

% list.discI
thf(fact_526_list_OdiscI,axiom,
    ! [List: list_set_list_a,X21: set_list_a,X22: list_set_list_a] :
      ( ( List
        = ( cons_set_list_a @ X21 @ X22 ) )
     => ( List != nil_set_list_a ) ) ).

% list.discI
thf(fact_527_list_Oexhaust,axiom,
    ! [Y: list_list_a] :
      ( ( Y != nil_list_a )
     => ~ ! [X212: list_a,X222: list_list_a] :
            ( Y
           != ( cons_list_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_528_list_Oexhaust,axiom,
    ! [Y: list_a] :
      ( ( Y != nil_a )
     => ~ ! [X212: a,X222: list_a] :
            ( Y
           != ( cons_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_529_list_Oexhaust,axiom,
    ! [Y: list_set_list_a] :
      ( ( Y != nil_set_list_a )
     => ~ ! [X212: set_list_a,X222: list_set_list_a] :
            ( Y
           != ( cons_set_list_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_530_min__list_Ocases,axiom,
    ! [X2: list_set_list_a] :
      ( ! [X3: set_list_a,Xs3: list_set_list_a] :
          ( X2
         != ( cons_set_list_a @ X3 @ Xs3 ) )
     => ( X2 = nil_set_list_a ) ) ).

% min_list.cases
thf(fact_531_remdups__adj_Ocases,axiom,
    ! [X2: list_list_a] :
      ( ( X2 != nil_list_a )
     => ( ! [X3: list_a] :
            ( X2
           != ( cons_list_a @ X3 @ nil_list_a ) )
       => ~ ! [X3: list_a,Y3: list_a,Xs3: list_list_a] :
              ( X2
             != ( cons_list_a @ X3 @ ( cons_list_a @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_532_remdups__adj_Ocases,axiom,
    ! [X2: list_a] :
      ( ( X2 != nil_a )
     => ( ! [X3: a] :
            ( X2
           != ( cons_a @ X3 @ nil_a ) )
       => ~ ! [X3: a,Y3: a,Xs3: list_a] :
              ( X2
             != ( cons_a @ X3 @ ( cons_a @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_533_remdups__adj_Ocases,axiom,
    ! [X2: list_set_list_a] :
      ( ( X2 != nil_set_list_a )
     => ( ! [X3: set_list_a] :
            ( X2
           != ( cons_set_list_a @ X3 @ nil_set_list_a ) )
       => ~ ! [X3: set_list_a,Y3: set_list_a,Xs3: list_set_list_a] :
              ( X2
             != ( cons_set_list_a @ X3 @ ( cons_set_list_a @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_534_neq__Nil__conv,axiom,
    ! [Xs: list_list_a] :
      ( ( Xs != nil_list_a )
      = ( ? [Y4: list_a,Ys: list_list_a] :
            ( Xs
            = ( cons_list_a @ Y4 @ Ys ) ) ) ) ).

% neq_Nil_conv
thf(fact_535_neq__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
      = ( ? [Y4: a,Ys: list_a] :
            ( Xs
            = ( cons_a @ Y4 @ Ys ) ) ) ) ).

% neq_Nil_conv
thf(fact_536_neq__Nil__conv,axiom,
    ! [Xs: list_set_list_a] :
      ( ( Xs != nil_set_list_a )
      = ( ? [Y4: set_list_a,Ys: list_set_list_a] :
            ( Xs
            = ( cons_set_list_a @ Y4 @ Ys ) ) ) ) ).

% neq_Nil_conv
thf(fact_537_list__induct2_H,axiom,
    ! [P3: list_list_a > list_list_a > $o,Xs: list_list_a,Ys2: list_list_a] :
      ( ( P3 @ nil_list_a @ nil_list_a )
     => ( ! [X3: list_a,Xs3: list_list_a] : ( P3 @ ( cons_list_a @ X3 @ Xs3 ) @ nil_list_a )
       => ( ! [Y3: list_a,Ys3: list_list_a] : ( P3 @ nil_list_a @ ( cons_list_a @ Y3 @ Ys3 ) )
         => ( ! [X3: list_a,Xs3: list_list_a,Y3: list_a,Ys3: list_list_a] :
                ( ( P3 @ Xs3 @ Ys3 )
               => ( P3 @ ( cons_list_a @ X3 @ Xs3 ) @ ( cons_list_a @ Y3 @ Ys3 ) ) )
           => ( P3 @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_538_list__induct2_H,axiom,
    ! [P3: list_list_a > list_a > $o,Xs: list_list_a,Ys2: list_a] :
      ( ( P3 @ nil_list_a @ nil_a )
     => ( ! [X3: list_a,Xs3: list_list_a] : ( P3 @ ( cons_list_a @ X3 @ Xs3 ) @ nil_a )
       => ( ! [Y3: a,Ys3: list_a] : ( P3 @ nil_list_a @ ( cons_a @ Y3 @ Ys3 ) )
         => ( ! [X3: list_a,Xs3: list_list_a,Y3: a,Ys3: list_a] :
                ( ( P3 @ Xs3 @ Ys3 )
               => ( P3 @ ( cons_list_a @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
           => ( P3 @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_539_list__induct2_H,axiom,
    ! [P3: list_list_a > list_set_list_a > $o,Xs: list_list_a,Ys2: list_set_list_a] :
      ( ( P3 @ nil_list_a @ nil_set_list_a )
     => ( ! [X3: list_a,Xs3: list_list_a] : ( P3 @ ( cons_list_a @ X3 @ Xs3 ) @ nil_set_list_a )
       => ( ! [Y3: set_list_a,Ys3: list_set_list_a] : ( P3 @ nil_list_a @ ( cons_set_list_a @ Y3 @ Ys3 ) )
         => ( ! [X3: list_a,Xs3: list_list_a,Y3: set_list_a,Ys3: list_set_list_a] :
                ( ( P3 @ Xs3 @ Ys3 )
               => ( P3 @ ( cons_list_a @ X3 @ Xs3 ) @ ( cons_set_list_a @ Y3 @ Ys3 ) ) )
           => ( P3 @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_540_list__induct2_H,axiom,
    ! [P3: list_a > list_list_a > $o,Xs: list_a,Ys2: list_list_a] :
      ( ( P3 @ nil_a @ nil_list_a )
     => ( ! [X3: a,Xs3: list_a] : ( P3 @ ( cons_a @ X3 @ Xs3 ) @ nil_list_a )
       => ( ! [Y3: list_a,Ys3: list_list_a] : ( P3 @ nil_a @ ( cons_list_a @ Y3 @ Ys3 ) )
         => ( ! [X3: a,Xs3: list_a,Y3: list_a,Ys3: list_list_a] :
                ( ( P3 @ Xs3 @ Ys3 )
               => ( P3 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_list_a @ Y3 @ Ys3 ) ) )
           => ( P3 @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_541_list__induct2_H,axiom,
    ! [P3: list_a > list_a > $o,Xs: list_a,Ys2: list_a] :
      ( ( P3 @ nil_a @ nil_a )
     => ( ! [X3: a,Xs3: list_a] : ( P3 @ ( cons_a @ X3 @ Xs3 ) @ nil_a )
       => ( ! [Y3: a,Ys3: list_a] : ( P3 @ nil_a @ ( cons_a @ Y3 @ Ys3 ) )
         => ( ! [X3: a,Xs3: list_a,Y3: a,Ys3: list_a] :
                ( ( P3 @ Xs3 @ Ys3 )
               => ( P3 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
           => ( P3 @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_542_list__induct2_H,axiom,
    ! [P3: list_a > list_set_list_a > $o,Xs: list_a,Ys2: list_set_list_a] :
      ( ( P3 @ nil_a @ nil_set_list_a )
     => ( ! [X3: a,Xs3: list_a] : ( P3 @ ( cons_a @ X3 @ Xs3 ) @ nil_set_list_a )
       => ( ! [Y3: set_list_a,Ys3: list_set_list_a] : ( P3 @ nil_a @ ( cons_set_list_a @ Y3 @ Ys3 ) )
         => ( ! [X3: a,Xs3: list_a,Y3: set_list_a,Ys3: list_set_list_a] :
                ( ( P3 @ Xs3 @ Ys3 )
               => ( P3 @ ( cons_a @ X3 @ Xs3 ) @ ( cons_set_list_a @ Y3 @ Ys3 ) ) )
           => ( P3 @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_543_list__induct2_H,axiom,
    ! [P3: list_set_list_a > list_list_a > $o,Xs: list_set_list_a,Ys2: list_list_a] :
      ( ( P3 @ nil_set_list_a @ nil_list_a )
     => ( ! [X3: set_list_a,Xs3: list_set_list_a] : ( P3 @ ( cons_set_list_a @ X3 @ Xs3 ) @ nil_list_a )
       => ( ! [Y3: list_a,Ys3: list_list_a] : ( P3 @ nil_set_list_a @ ( cons_list_a @ Y3 @ Ys3 ) )
         => ( ! [X3: set_list_a,Xs3: list_set_list_a,Y3: list_a,Ys3: list_list_a] :
                ( ( P3 @ Xs3 @ Ys3 )
               => ( P3 @ ( cons_set_list_a @ X3 @ Xs3 ) @ ( cons_list_a @ Y3 @ Ys3 ) ) )
           => ( P3 @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_544_list__induct2_H,axiom,
    ! [P3: list_set_list_a > list_a > $o,Xs: list_set_list_a,Ys2: list_a] :
      ( ( P3 @ nil_set_list_a @ nil_a )
     => ( ! [X3: set_list_a,Xs3: list_set_list_a] : ( P3 @ ( cons_set_list_a @ X3 @ Xs3 ) @ nil_a )
       => ( ! [Y3: a,Ys3: list_a] : ( P3 @ nil_set_list_a @ ( cons_a @ Y3 @ Ys3 ) )
         => ( ! [X3: set_list_a,Xs3: list_set_list_a,Y3: a,Ys3: list_a] :
                ( ( P3 @ Xs3 @ Ys3 )
               => ( P3 @ ( cons_set_list_a @ X3 @ Xs3 ) @ ( cons_a @ Y3 @ Ys3 ) ) )
           => ( P3 @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_545_list__induct2_H,axiom,
    ! [P3: list_set_list_a > list_set_list_a > $o,Xs: list_set_list_a,Ys2: list_set_list_a] :
      ( ( P3 @ nil_set_list_a @ nil_set_list_a )
     => ( ! [X3: set_list_a,Xs3: list_set_list_a] : ( P3 @ ( cons_set_list_a @ X3 @ Xs3 ) @ nil_set_list_a )
       => ( ! [Y3: set_list_a,Ys3: list_set_list_a] : ( P3 @ nil_set_list_a @ ( cons_set_list_a @ Y3 @ Ys3 ) )
         => ( ! [X3: set_list_a,Xs3: list_set_list_a,Y3: set_list_a,Ys3: list_set_list_a] :
                ( ( P3 @ Xs3 @ Ys3 )
               => ( P3 @ ( cons_set_list_a @ X3 @ Xs3 ) @ ( cons_set_list_a @ Y3 @ Ys3 ) ) )
           => ( P3 @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_546_list__nonempty__induct,axiom,
    ! [Xs: list_list_a,P3: list_list_a > $o] :
      ( ( Xs != nil_list_a )
     => ( ! [X3: list_a] : ( P3 @ ( cons_list_a @ X3 @ nil_list_a ) )
       => ( ! [X3: list_a,Xs3: list_list_a] :
              ( ( Xs3 != nil_list_a )
             => ( ( P3 @ Xs3 )
               => ( P3 @ ( cons_list_a @ X3 @ Xs3 ) ) ) )
         => ( P3 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_547_list__nonempty__induct,axiom,
    ! [Xs: list_a,P3: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X3: a] : ( P3 @ ( cons_a @ X3 @ nil_a ) )
       => ( ! [X3: a,Xs3: list_a] :
              ( ( Xs3 != nil_a )
             => ( ( P3 @ Xs3 )
               => ( P3 @ ( cons_a @ X3 @ Xs3 ) ) ) )
         => ( P3 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_548_list__nonempty__induct,axiom,
    ! [Xs: list_set_list_a,P3: list_set_list_a > $o] :
      ( ( Xs != nil_set_list_a )
     => ( ! [X3: set_list_a] : ( P3 @ ( cons_set_list_a @ X3 @ nil_set_list_a ) )
       => ( ! [X3: set_list_a,Xs3: list_set_list_a] :
              ( ( Xs3 != nil_set_list_a )
             => ( ( P3 @ Xs3 )
               => ( P3 @ ( cons_set_list_a @ X3 @ Xs3 ) ) ) )
         => ( P3 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_549_list_Omap__cong,axiom,
    ! [X2: list_list_a,Ya: list_list_a,F: list_a > set_list_list_a,G: list_a > set_list_list_a] :
      ( ( X2 = Ya )
     => ( ! [Z: list_a] :
            ( ( member_list_a @ Z @ ( set_list_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_li5777811645572713770list_a @ F @ X2 )
          = ( map_li5777811645572713770list_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_550_list_Omap__cong,axiom,
    ! [X2: list_list_a,Ya: list_list_a,F: list_a > set_list_a,G: list_a > set_list_a] :
      ( ( X2 = Ya )
     => ( ! [Z: list_a] :
            ( ( member_list_a @ Z @ ( set_list_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_li2657846702790696612list_a @ F @ X2 )
          = ( map_li2657846702790696612list_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_551_list_Omap__cong,axiom,
    ! [X2: list_a,Ya: list_a,F: a > list_a,G: a > list_a] :
      ( ( X2 = Ya )
     => ( ! [Z: a] :
            ( ( member_a @ Z @ ( set_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_a_list_a @ F @ X2 )
          = ( map_a_list_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_552_list_Omap__cong,axiom,
    ! [X2: list_a,Ya: list_a,F: a > set_list_a,G: a > set_list_a] :
      ( ( X2 = Ya )
     => ( ! [Z: a] :
            ( ( member_a @ Z @ ( set_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_a_set_list_a @ F @ X2 )
          = ( map_a_set_list_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_553_list_Omap__cong,axiom,
    ! [X2: list_a,Ya: list_a,F: a > a,G: a > a] :
      ( ( X2 = Ya )
     => ( ! [Z: a] :
            ( ( member_a @ Z @ ( set_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_a_a @ F @ X2 )
          = ( map_a_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_554_list_Omap__cong,axiom,
    ! [X2: list_a,Ya: list_a,F: a > set_list_list_a,G: a > set_list_list_a] :
      ( ( X2 = Ya )
     => ( ! [Z: a] :
            ( ( member_a @ Z @ ( set_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_a_132662181483840688list_a @ F @ X2 )
          = ( map_a_132662181483840688list_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_555_list_Omap__cong,axiom,
    ! [X2: list_set_list_a,Ya: list_set_list_a,F: set_list_a > list_a,G: set_list_a > list_a] :
      ( ( X2 = Ya )
     => ( ! [Z: set_list_a] :
            ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_se5127173849416245988list_a @ F @ X2 )
          = ( map_se5127173849416245988list_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_556_list_Omap__cong,axiom,
    ! [X2: list_set_list_a,Ya: list_set_list_a,F: set_list_a > set_list_list_a,G: set_list_a > set_list_list_a] :
      ( ( X2 = Ya )
     => ( ! [Z: set_list_a] :
            ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_se5810349697548134730list_a @ F @ X2 )
          = ( map_se5810349697548134730list_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_557_list_Omap__cong,axiom,
    ! [X2: list_set_list_a,Ya: list_set_list_a,F: set_list_a > set_list_a,G: set_list_a > set_list_a] :
      ( ( X2 = Ya )
     => ( ! [Z: set_list_a] :
            ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_se2668659675339852484list_a @ F @ X2 )
          = ( map_se2668659675339852484list_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_558_list_Omap__cong,axiom,
    ! [X2: list_set_list_a,Ya: list_set_list_a,F: set_list_a > a,G: set_list_a > a] :
      ( ( X2 = Ya )
     => ( ! [Z: set_list_a] :
            ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_set_list_a_a @ F @ X2 )
          = ( map_set_list_a_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_559_list_Omap__cong0,axiom,
    ! [X2: list_list_a,F: list_a > set_list_list_a,G: list_a > set_list_list_a] :
      ( ! [Z: list_a] :
          ( ( member_list_a @ Z @ ( set_list_a2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_li5777811645572713770list_a @ F @ X2 )
        = ( map_li5777811645572713770list_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_560_list_Omap__cong0,axiom,
    ! [X2: list_list_a,F: list_a > set_list_a,G: list_a > set_list_a] :
      ( ! [Z: list_a] :
          ( ( member_list_a @ Z @ ( set_list_a2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_li2657846702790696612list_a @ F @ X2 )
        = ( map_li2657846702790696612list_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_561_list_Omap__cong0,axiom,
    ! [X2: list_a,F: a > list_a,G: a > list_a] :
      ( ! [Z: a] :
          ( ( member_a @ Z @ ( set_a2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_a_list_a @ F @ X2 )
        = ( map_a_list_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_562_list_Omap__cong0,axiom,
    ! [X2: list_a,F: a > set_list_a,G: a > set_list_a] :
      ( ! [Z: a] :
          ( ( member_a @ Z @ ( set_a2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_a_set_list_a @ F @ X2 )
        = ( map_a_set_list_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_563_list_Omap__cong0,axiom,
    ! [X2: list_a,F: a > a,G: a > a] :
      ( ! [Z: a] :
          ( ( member_a @ Z @ ( set_a2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_a_a @ F @ X2 )
        = ( map_a_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_564_list_Omap__cong0,axiom,
    ! [X2: list_a,F: a > set_list_list_a,G: a > set_list_list_a] :
      ( ! [Z: a] :
          ( ( member_a @ Z @ ( set_a2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_a_132662181483840688list_a @ F @ X2 )
        = ( map_a_132662181483840688list_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_565_list_Omap__cong0,axiom,
    ! [X2: list_set_list_a,F: set_list_a > list_a,G: set_list_a > list_a] :
      ( ! [Z: set_list_a] :
          ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_se5127173849416245988list_a @ F @ X2 )
        = ( map_se5127173849416245988list_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_566_list_Omap__cong0,axiom,
    ! [X2: list_set_list_a,F: set_list_a > set_list_list_a,G: set_list_a > set_list_list_a] :
      ( ! [Z: set_list_a] :
          ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_se5810349697548134730list_a @ F @ X2 )
        = ( map_se5810349697548134730list_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_567_list_Omap__cong0,axiom,
    ! [X2: list_set_list_a,F: set_list_a > set_list_a,G: set_list_a > set_list_a] :
      ( ! [Z: set_list_a] :
          ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_se2668659675339852484list_a @ F @ X2 )
        = ( map_se2668659675339852484list_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_568_list_Omap__cong0,axiom,
    ! [X2: list_set_list_a,F: set_list_a > a,G: set_list_a > a] :
      ( ! [Z: set_list_a] :
          ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ X2 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_set_list_a_a @ F @ X2 )
        = ( map_set_list_a_a @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_569_list_Oinj__map__strong,axiom,
    ! [X2: list_list_a,Xa: list_list_a,F: list_a > set_list_list_a,Fa: list_a > set_list_list_a] :
      ( ! [Z: list_a,Za: list_a] :
          ( ( member_list_a @ Z @ ( set_list_a2 @ X2 ) )
         => ( ( member_list_a @ Za @ ( set_list_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_li5777811645572713770list_a @ F @ X2 )
          = ( map_li5777811645572713770list_a @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_570_list_Oinj__map__strong,axiom,
    ! [X2: list_list_a,Xa: list_list_a,F: list_a > set_list_a,Fa: list_a > set_list_a] :
      ( ! [Z: list_a,Za: list_a] :
          ( ( member_list_a @ Z @ ( set_list_a2 @ X2 ) )
         => ( ( member_list_a @ Za @ ( set_list_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_li2657846702790696612list_a @ F @ X2 )
          = ( map_li2657846702790696612list_a @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_571_list_Oinj__map__strong,axiom,
    ! [X2: list_a,Xa: list_a,F: a > list_a,Fa: a > list_a] :
      ( ! [Z: a,Za: a] :
          ( ( member_a @ Z @ ( set_a2 @ X2 ) )
         => ( ( member_a @ Za @ ( set_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_a_list_a @ F @ X2 )
          = ( map_a_list_a @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_572_list_Oinj__map__strong,axiom,
    ! [X2: list_a,Xa: list_a,F: a > set_list_a,Fa: a > set_list_a] :
      ( ! [Z: a,Za: a] :
          ( ( member_a @ Z @ ( set_a2 @ X2 ) )
         => ( ( member_a @ Za @ ( set_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_a_set_list_a @ F @ X2 )
          = ( map_a_set_list_a @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_573_list_Oinj__map__strong,axiom,
    ! [X2: list_a,Xa: list_a,F: a > a,Fa: a > a] :
      ( ! [Z: a,Za: a] :
          ( ( member_a @ Z @ ( set_a2 @ X2 ) )
         => ( ( member_a @ Za @ ( set_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_a_a @ F @ X2 )
          = ( map_a_a @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_574_list_Oinj__map__strong,axiom,
    ! [X2: list_a,Xa: list_a,F: a > set_list_list_a,Fa: a > set_list_list_a] :
      ( ! [Z: a,Za: a] :
          ( ( member_a @ Z @ ( set_a2 @ X2 ) )
         => ( ( member_a @ Za @ ( set_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_a_132662181483840688list_a @ F @ X2 )
          = ( map_a_132662181483840688list_a @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_575_list_Oinj__map__strong,axiom,
    ! [X2: list_set_list_a,Xa: list_set_list_a,F: set_list_a > list_a,Fa: set_list_a > list_a] :
      ( ! [Z: set_list_a,Za: set_list_a] :
          ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ X2 ) )
         => ( ( member_set_list_a @ Za @ ( set_set_list_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_se5127173849416245988list_a @ F @ X2 )
          = ( map_se5127173849416245988list_a @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_576_list_Oinj__map__strong,axiom,
    ! [X2: list_set_list_a,Xa: list_set_list_a,F: set_list_a > set_list_list_a,Fa: set_list_a > set_list_list_a] :
      ( ! [Z: set_list_a,Za: set_list_a] :
          ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ X2 ) )
         => ( ( member_set_list_a @ Za @ ( set_set_list_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_se5810349697548134730list_a @ F @ X2 )
          = ( map_se5810349697548134730list_a @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_577_list_Oinj__map__strong,axiom,
    ! [X2: list_set_list_a,Xa: list_set_list_a,F: set_list_a > set_list_a,Fa: set_list_a > set_list_a] :
      ( ! [Z: set_list_a,Za: set_list_a] :
          ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ X2 ) )
         => ( ( member_set_list_a @ Za @ ( set_set_list_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_se2668659675339852484list_a @ F @ X2 )
          = ( map_se2668659675339852484list_a @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_578_list_Oinj__map__strong,axiom,
    ! [X2: list_set_list_a,Xa: list_set_list_a,F: set_list_a > a,Fa: set_list_a > a] :
      ( ! [Z: set_list_a,Za: set_list_a] :
          ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ X2 ) )
         => ( ( member_set_list_a @ Za @ ( set_set_list_a2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_set_list_a_a @ F @ X2 )
          = ( map_set_list_a_a @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_579_list_Omap__ident__strong,axiom,
    ! [T: list_list_a_a,F: ( list_a > a ) > list_a > a] :
      ( ! [Z: list_a > a] :
          ( ( member_list_a_a @ Z @ ( set_list_a_a2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_li8656678886064826500st_a_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_580_list_Omap__ident__strong,axiom,
    ! [T: list_s476705121730489333list_a,F: ( set_list_a > set_list_a ) > set_list_a > set_list_a] :
      ( ! [Z: set_list_a > set_list_a] :
          ( ( member5068272912271824380list_a @ Z @ ( set_se4893213719394700432list_a @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_se4554261891618027152list_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_581_list_Omap__ident__strong,axiom,
    ! [T: list_a_list_a,F: ( a > list_a ) > a > list_a] :
      ( ! [Z: a > list_a] :
          ( ( member_a_list_a @ Z @ ( set_a_list_a2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_a_9223042522881502876list_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_582_list_Omap__ident__strong,axiom,
    ! [T: list_a_set_list_a,F: ( a > set_list_a ) > a > set_list_a] :
      ( ! [Z: a > set_list_a] :
          ( ( member_a_set_list_a @ Z @ ( set_a_set_list_a2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_a_8844646024031188572list_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_583_list_Omap__ident__strong,axiom,
    ! [T: list_list_a,F: list_a > list_a] :
      ( ! [Z: list_a] :
          ( ( member_list_a @ Z @ ( set_list_a2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_list_a_list_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_584_list_Omap__ident__strong,axiom,
    ! [T: list_a,F: a > a] :
      ( ! [Z: a] :
          ( ( member_a @ Z @ ( set_a2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_a_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_585_list_Omap__ident__strong,axiom,
    ! [T: list_set_list_a,F: set_list_a > set_list_a] :
      ( ! [Z: set_list_a] :
          ( ( member_set_list_a @ Z @ ( set_set_list_a2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_se2668659675339852484list_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_586_map__ext,axiom,
    ! [Xs: list_list_a,F: list_a > set_list_list_a,G: list_a > set_list_list_a] :
      ( ! [X3: list_a] :
          ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_li5777811645572713770list_a @ F @ Xs )
        = ( map_li5777811645572713770list_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_587_map__ext,axiom,
    ! [Xs: list_list_a,F: list_a > set_list_a,G: list_a > set_list_a] :
      ( ! [X3: list_a] :
          ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_li2657846702790696612list_a @ F @ Xs )
        = ( map_li2657846702790696612list_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_588_map__ext,axiom,
    ! [Xs: list_a,F: a > list_a,G: a > list_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_a_list_a @ F @ Xs )
        = ( map_a_list_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_589_map__ext,axiom,
    ! [Xs: list_a,F: a > set_list_a,G: a > set_list_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_a_set_list_a @ F @ Xs )
        = ( map_a_set_list_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_590_map__ext,axiom,
    ! [Xs: list_a,F: a > a,G: a > a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_a_a @ F @ Xs )
        = ( map_a_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_591_map__ext,axiom,
    ! [Xs: list_a,F: a > set_list_list_a,G: a > set_list_list_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_a_132662181483840688list_a @ F @ Xs )
        = ( map_a_132662181483840688list_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_592_map__ext,axiom,
    ! [Xs: list_set_list_a,F: set_list_a > list_a,G: set_list_a > list_a] :
      ( ! [X3: set_list_a] :
          ( ( member_set_list_a @ X3 @ ( set_set_list_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_se5127173849416245988list_a @ F @ Xs )
        = ( map_se5127173849416245988list_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_593_map__ext,axiom,
    ! [Xs: list_set_list_a,F: set_list_a > set_list_list_a,G: set_list_a > set_list_list_a] :
      ( ! [X3: set_list_a] :
          ( ( member_set_list_a @ X3 @ ( set_set_list_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_se5810349697548134730list_a @ F @ Xs )
        = ( map_se5810349697548134730list_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_594_map__ext,axiom,
    ! [Xs: list_set_list_a,F: set_list_a > set_list_a,G: set_list_a > set_list_a] :
      ( ! [X3: set_list_a] :
          ( ( member_set_list_a @ X3 @ ( set_set_list_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_se2668659675339852484list_a @ F @ Xs )
        = ( map_se2668659675339852484list_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_595_map__ext,axiom,
    ! [Xs: list_set_list_a,F: set_list_a > a,G: set_list_a > a] :
      ( ! [X3: set_list_a] :
          ( ( member_set_list_a @ X3 @ ( set_set_list_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_set_list_a_a @ F @ Xs )
        = ( map_set_list_a_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_596_map__idI,axiom,
    ! [Xs: list_list_a_a,F: ( list_a > a ) > list_a > a] :
      ( ! [X3: list_a > a] :
          ( ( member_list_a_a @ X3 @ ( set_list_a_a2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_li8656678886064826500st_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_597_map__idI,axiom,
    ! [Xs: list_s476705121730489333list_a,F: ( set_list_a > set_list_a ) > set_list_a > set_list_a] :
      ( ! [X3: set_list_a > set_list_a] :
          ( ( member5068272912271824380list_a @ X3 @ ( set_se4893213719394700432list_a @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_se4554261891618027152list_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_598_map__idI,axiom,
    ! [Xs: list_a_list_a,F: ( a > list_a ) > a > list_a] :
      ( ! [X3: a > list_a] :
          ( ( member_a_list_a @ X3 @ ( set_a_list_a2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_a_9223042522881502876list_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_599_map__idI,axiom,
    ! [Xs: list_a_set_list_a,F: ( a > set_list_a ) > a > set_list_a] :
      ( ! [X3: a > set_list_a] :
          ( ( member_a_set_list_a @ X3 @ ( set_a_set_list_a2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_a_8844646024031188572list_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_600_map__idI,axiom,
    ! [Xs: list_list_a,F: list_a > list_a] :
      ( ! [X3: list_a] :
          ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_list_a_list_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_601_map__idI,axiom,
    ! [Xs: list_a,F: a > a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_602_map__idI,axiom,
    ! [Xs: list_set_list_a,F: set_list_a > set_list_a] :
      ( ! [X3: set_list_a] :
          ( ( member_set_list_a @ X3 @ ( set_set_list_a2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_se2668659675339852484list_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_603_map__cong,axiom,
    ! [Xs: list_list_a,Ys2: list_list_a,F: list_a > set_list_list_a,G: list_a > set_list_list_a] :
      ( ( Xs = Ys2 )
     => ( ! [X3: list_a] :
            ( ( member_list_a @ X3 @ ( set_list_a2 @ Ys2 ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_li5777811645572713770list_a @ F @ Xs )
          = ( map_li5777811645572713770list_a @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_604_map__cong,axiom,
    ! [Xs: list_list_a,Ys2: list_list_a,F: list_a > set_list_a,G: list_a > set_list_a] :
      ( ( Xs = Ys2 )
     => ( ! [X3: list_a] :
            ( ( member_list_a @ X3 @ ( set_list_a2 @ Ys2 ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_li2657846702790696612list_a @ F @ Xs )
          = ( map_li2657846702790696612list_a @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_605_map__cong,axiom,
    ! [Xs: list_a,Ys2: list_a,F: a > list_a,G: a > list_a] :
      ( ( Xs = Ys2 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Ys2 ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_a_list_a @ F @ Xs )
          = ( map_a_list_a @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_606_map__cong,axiom,
    ! [Xs: list_a,Ys2: list_a,F: a > set_list_a,G: a > set_list_a] :
      ( ( Xs = Ys2 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Ys2 ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_a_set_list_a @ F @ Xs )
          = ( map_a_set_list_a @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_607_map__cong,axiom,
    ! [Xs: list_a,Ys2: list_a,F: a > a,G: a > a] :
      ( ( Xs = Ys2 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Ys2 ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_a_a @ F @ Xs )
          = ( map_a_a @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_608_map__cong,axiom,
    ! [Xs: list_a,Ys2: list_a,F: a > set_list_list_a,G: a > set_list_list_a] :
      ( ( Xs = Ys2 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Ys2 ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_a_132662181483840688list_a @ F @ Xs )
          = ( map_a_132662181483840688list_a @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_609_map__cong,axiom,
    ! [Xs: list_set_list_a,Ys2: list_set_list_a,F: set_list_a > list_a,G: set_list_a > list_a] :
      ( ( Xs = Ys2 )
     => ( ! [X3: set_list_a] :
            ( ( member_set_list_a @ X3 @ ( set_set_list_a2 @ Ys2 ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_se5127173849416245988list_a @ F @ Xs )
          = ( map_se5127173849416245988list_a @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_610_map__cong,axiom,
    ! [Xs: list_set_list_a,Ys2: list_set_list_a,F: set_list_a > set_list_list_a,G: set_list_a > set_list_list_a] :
      ( ( Xs = Ys2 )
     => ( ! [X3: set_list_a] :
            ( ( member_set_list_a @ X3 @ ( set_set_list_a2 @ Ys2 ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_se5810349697548134730list_a @ F @ Xs )
          = ( map_se5810349697548134730list_a @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_611_map__cong,axiom,
    ! [Xs: list_set_list_a,Ys2: list_set_list_a,F: set_list_a > set_list_a,G: set_list_a > set_list_a] :
      ( ( Xs = Ys2 )
     => ( ! [X3: set_list_a] :
            ( ( member_set_list_a @ X3 @ ( set_set_list_a2 @ Ys2 ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_se2668659675339852484list_a @ F @ Xs )
          = ( map_se2668659675339852484list_a @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_612_map__cong,axiom,
    ! [Xs: list_set_list_a,Ys2: list_set_list_a,F: set_list_a > a,G: set_list_a > a] :
      ( ( Xs = Ys2 )
     => ( ! [X3: set_list_a] :
            ( ( member_set_list_a @ X3 @ ( set_set_list_a2 @ Ys2 ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_set_list_a_a @ F @ Xs )
          = ( map_set_list_a_a @ G @ Ys2 ) ) ) ) ).

% map_cong
thf(fact_613_ex__map__conv,axiom,
    ! [Ys2: list_set_list_list_a,F: list_a > set_list_list_a] :
      ( ( ? [Xs2: list_list_a] :
            ( Ys2
            = ( map_li5777811645572713770list_a @ F @ Xs2 ) ) )
      = ( ! [X: set_list_list_a] :
            ( ( member334759470184282131list_a @ X @ ( set_set_list_list_a2 @ Ys2 ) )
           => ? [Y4: list_a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_614_ex__map__conv,axiom,
    ! [Ys2: list_set_list_list_a,F: set_list_a > set_list_list_a] :
      ( ( ? [Xs2: list_set_list_a] :
            ( Ys2
            = ( map_se5810349697548134730list_a @ F @ Xs2 ) ) )
      = ( ! [X: set_list_list_a] :
            ( ( member334759470184282131list_a @ X @ ( set_set_list_list_a2 @ Ys2 ) )
           => ? [Y4: set_list_a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_615_ex__map__conv,axiom,
    ! [Ys2: list_set_list_list_a,F: a > set_list_list_a] :
      ( ( ? [Xs2: list_a] :
            ( Ys2
            = ( map_a_132662181483840688list_a @ F @ Xs2 ) ) )
      = ( ! [X: set_list_list_a] :
            ( ( member334759470184282131list_a @ X @ ( set_set_list_list_a2 @ Ys2 ) )
           => ? [Y4: a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_616_ex__map__conv,axiom,
    ! [Ys2: list_list_a,F: a > list_a] :
      ( ( ? [Xs2: list_a] :
            ( Ys2
            = ( map_a_list_a @ F @ Xs2 ) ) )
      = ( ! [X: list_a] :
            ( ( member_list_a @ X @ ( set_list_a2 @ Ys2 ) )
           => ? [Y4: a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_617_ex__map__conv,axiom,
    ! [Ys2: list_list_a,F: set_list_a > list_a] :
      ( ( ? [Xs2: list_set_list_a] :
            ( Ys2
            = ( map_se5127173849416245988list_a @ F @ Xs2 ) ) )
      = ( ! [X: list_a] :
            ( ( member_list_a @ X @ ( set_list_a2 @ Ys2 ) )
           => ? [Y4: set_list_a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_618_ex__map__conv,axiom,
    ! [Ys2: list_a,F: a > a] :
      ( ( ? [Xs2: list_a] :
            ( Ys2
            = ( map_a_a @ F @ Xs2 ) ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Ys2 ) )
           => ? [Y4: a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_619_ex__map__conv,axiom,
    ! [Ys2: list_a,F: set_list_a > a] :
      ( ( ? [Xs2: list_set_list_a] :
            ( Ys2
            = ( map_set_list_a_a @ F @ Xs2 ) ) )
      = ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Ys2 ) )
           => ? [Y4: set_list_a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_620_ex__map__conv,axiom,
    ! [Ys2: list_set_list_a,F: list_a > set_list_a] :
      ( ( ? [Xs2: list_list_a] :
            ( Ys2
            = ( map_li2657846702790696612list_a @ F @ Xs2 ) ) )
      = ( ! [X: set_list_a] :
            ( ( member_set_list_a @ X @ ( set_set_list_a2 @ Ys2 ) )
           => ? [Y4: list_a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_621_ex__map__conv,axiom,
    ! [Ys2: list_set_list_a,F: a > set_list_a] :
      ( ( ? [Xs2: list_a] :
            ( Ys2
            = ( map_a_set_list_a @ F @ Xs2 ) ) )
      = ( ! [X: set_list_a] :
            ( ( member_set_list_a @ X @ ( set_set_list_a2 @ Ys2 ) )
           => ? [Y4: a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_622_ex__map__conv,axiom,
    ! [Ys2: list_set_list_a,F: set_list_a > set_list_a] :
      ( ( ? [Xs2: list_set_list_a] :
            ( Ys2
            = ( map_se2668659675339852484list_a @ F @ Xs2 ) ) )
      = ( ! [X: set_list_a] :
            ( ( member_set_list_a @ X @ ( set_set_list_a2 @ Ys2 ) )
           => ? [Y4: set_list_a] :
                ( X
                = ( F @ Y4 ) ) ) ) ) ).

% ex_map_conv
thf(fact_623_list_Osimps_I9_J,axiom,
    ! [F: a > a,X21: a,X22: list_a] :
      ( ( map_a_a @ F @ ( cons_a @ X21 @ X22 ) )
      = ( cons_a @ ( F @ X21 ) @ ( map_a_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_624_list_Osimps_I9_J,axiom,
    ! [F: list_a > a,X21: list_a,X22: list_list_a] :
      ( ( map_list_a_a @ F @ ( cons_list_a @ X21 @ X22 ) )
      = ( cons_a @ ( F @ X21 ) @ ( map_list_a_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_625_list_Osimps_I9_J,axiom,
    ! [F: a > list_a,X21: a,X22: list_a] :
      ( ( map_a_list_a @ F @ ( cons_a @ X21 @ X22 ) )
      = ( cons_list_a @ ( F @ X21 ) @ ( map_a_list_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_626_list_Osimps_I9_J,axiom,
    ! [F: list_a > list_a,X21: list_a,X22: list_list_a] :
      ( ( map_list_a_list_a @ F @ ( cons_list_a @ X21 @ X22 ) )
      = ( cons_list_a @ ( F @ X21 ) @ ( map_list_a_list_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_627_list_Osimps_I9_J,axiom,
    ! [F: a > set_list_a,X21: a,X22: list_a] :
      ( ( map_a_set_list_a @ F @ ( cons_a @ X21 @ X22 ) )
      = ( cons_set_list_a @ ( F @ X21 ) @ ( map_a_set_list_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_628_list_Osimps_I9_J,axiom,
    ! [F: set_list_a > a,X21: set_list_a,X22: list_set_list_a] :
      ( ( map_set_list_a_a @ F @ ( cons_set_list_a @ X21 @ X22 ) )
      = ( cons_a @ ( F @ X21 ) @ ( map_set_list_a_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_629_list_Osimps_I9_J,axiom,
    ! [F: list_a > set_list_a,X21: list_a,X22: list_list_a] :
      ( ( map_li2657846702790696612list_a @ F @ ( cons_list_a @ X21 @ X22 ) )
      = ( cons_set_list_a @ ( F @ X21 ) @ ( map_li2657846702790696612list_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_630_list_Osimps_I9_J,axiom,
    ! [F: a > set_list_list_a,X21: a,X22: list_a] :
      ( ( map_a_132662181483840688list_a @ F @ ( cons_a @ X21 @ X22 ) )
      = ( cons_set_list_list_a @ ( F @ X21 ) @ ( map_a_132662181483840688list_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_631_list_Osimps_I9_J,axiom,
    ! [F: set_list_a > list_a,X21: set_list_a,X22: list_set_list_a] :
      ( ( map_se5127173849416245988list_a @ F @ ( cons_set_list_a @ X21 @ X22 ) )
      = ( cons_list_a @ ( F @ X21 ) @ ( map_se5127173849416245988list_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_632_list_Osimps_I9_J,axiom,
    ! [F: list_a > set_list_list_a,X21: list_a,X22: list_list_a] :
      ( ( map_li5777811645572713770list_a @ F @ ( cons_list_a @ X21 @ X22 ) )
      = ( cons_set_list_list_a @ ( F @ X21 ) @ ( map_li5777811645572713770list_a @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_633_Cons__eq__map__D,axiom,
    ! [X2: a,Xs: list_a,F: a > a,Ys2: list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( map_a_a @ F @ Ys2 ) )
     => ? [Z: a,Zs: list_a] :
          ( ( Ys2
            = ( cons_a @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_a_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_634_Cons__eq__map__D,axiom,
    ! [X2: list_a,Xs: list_list_a,F: a > list_a,Ys2: list_a] :
      ( ( ( cons_list_a @ X2 @ Xs )
        = ( map_a_list_a @ F @ Ys2 ) )
     => ? [Z: a,Zs: list_a] :
          ( ( Ys2
            = ( cons_a @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_a_list_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_635_Cons__eq__map__D,axiom,
    ! [X2: a,Xs: list_a,F: list_a > a,Ys2: list_list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( map_list_a_a @ F @ Ys2 ) )
     => ? [Z: list_a,Zs: list_list_a] :
          ( ( Ys2
            = ( cons_list_a @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_list_a_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_636_Cons__eq__map__D,axiom,
    ! [X2: list_a,Xs: list_list_a,F: list_a > list_a,Ys2: list_list_a] :
      ( ( ( cons_list_a @ X2 @ Xs )
        = ( map_list_a_list_a @ F @ Ys2 ) )
     => ? [Z: list_a,Zs: list_list_a] :
          ( ( Ys2
            = ( cons_list_a @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_list_a_list_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_637_Cons__eq__map__D,axiom,
    ! [X2: a,Xs: list_a,F: set_list_a > a,Ys2: list_set_list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( map_set_list_a_a @ F @ Ys2 ) )
     => ? [Z: set_list_a,Zs: list_set_list_a] :
          ( ( Ys2
            = ( cons_set_list_a @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_set_list_a_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_638_Cons__eq__map__D,axiom,
    ! [X2: set_list_a,Xs: list_set_list_a,F: a > set_list_a,Ys2: list_a] :
      ( ( ( cons_set_list_a @ X2 @ Xs )
        = ( map_a_set_list_a @ F @ Ys2 ) )
     => ? [Z: a,Zs: list_a] :
          ( ( Ys2
            = ( cons_a @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_a_set_list_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_639_Cons__eq__map__D,axiom,
    ! [X2: set_list_list_a,Xs: list_set_list_list_a,F: a > set_list_list_a,Ys2: list_a] :
      ( ( ( cons_set_list_list_a @ X2 @ Xs )
        = ( map_a_132662181483840688list_a @ F @ Ys2 ) )
     => ? [Z: a,Zs: list_a] :
          ( ( Ys2
            = ( cons_a @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_a_132662181483840688list_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_640_Cons__eq__map__D,axiom,
    ! [X2: list_a,Xs: list_list_a,F: set_list_a > list_a,Ys2: list_set_list_a] :
      ( ( ( cons_list_a @ X2 @ Xs )
        = ( map_se5127173849416245988list_a @ F @ Ys2 ) )
     => ? [Z: set_list_a,Zs: list_set_list_a] :
          ( ( Ys2
            = ( cons_set_list_a @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_se5127173849416245988list_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_641_Cons__eq__map__D,axiom,
    ! [X2: set_list_a,Xs: list_set_list_a,F: list_a > set_list_a,Ys2: list_list_a] :
      ( ( ( cons_set_list_a @ X2 @ Xs )
        = ( map_li2657846702790696612list_a @ F @ Ys2 ) )
     => ? [Z: list_a,Zs: list_list_a] :
          ( ( Ys2
            = ( cons_list_a @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_li2657846702790696612list_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_642_Cons__eq__map__D,axiom,
    ! [X2: set_list_list_a,Xs: list_set_list_list_a,F: list_a > set_list_list_a,Ys2: list_list_a] :
      ( ( ( cons_set_list_list_a @ X2 @ Xs )
        = ( map_li5777811645572713770list_a @ F @ Ys2 ) )
     => ? [Z: list_a,Zs: list_list_a] :
          ( ( Ys2
            = ( cons_list_a @ Z @ Zs ) )
          & ( X2
            = ( F @ Z ) )
          & ( Xs
            = ( map_li5777811645572713770list_a @ F @ Zs ) ) ) ) ).

% Cons_eq_map_D
thf(fact_643_map__eq__Cons__D,axiom,
    ! [F: a > a,Xs: list_a,Y: a,Ys2: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = ( cons_a @ Y @ Ys2 ) )
     => ? [Z: a,Zs: list_a] :
          ( ( Xs
            = ( cons_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_a_a @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_644_map__eq__Cons__D,axiom,
    ! [F: a > list_a,Xs: list_a,Y: list_a,Ys2: list_list_a] :
      ( ( ( map_a_list_a @ F @ Xs )
        = ( cons_list_a @ Y @ Ys2 ) )
     => ? [Z: a,Zs: list_a] :
          ( ( Xs
            = ( cons_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_a_list_a @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_645_map__eq__Cons__D,axiom,
    ! [F: list_a > a,Xs: list_list_a,Y: a,Ys2: list_a] :
      ( ( ( map_list_a_a @ F @ Xs )
        = ( cons_a @ Y @ Ys2 ) )
     => ? [Z: list_a,Zs: list_list_a] :
          ( ( Xs
            = ( cons_list_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_list_a_a @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_646_map__eq__Cons__D,axiom,
    ! [F: list_a > list_a,Xs: list_list_a,Y: list_a,Ys2: list_list_a] :
      ( ( ( map_list_a_list_a @ F @ Xs )
        = ( cons_list_a @ Y @ Ys2 ) )
     => ? [Z: list_a,Zs: list_list_a] :
          ( ( Xs
            = ( cons_list_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_list_a_list_a @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_647_map__eq__Cons__D,axiom,
    ! [F: set_list_a > a,Xs: list_set_list_a,Y: a,Ys2: list_a] :
      ( ( ( map_set_list_a_a @ F @ Xs )
        = ( cons_a @ Y @ Ys2 ) )
     => ? [Z: set_list_a,Zs: list_set_list_a] :
          ( ( Xs
            = ( cons_set_list_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_set_list_a_a @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_648_map__eq__Cons__D,axiom,
    ! [F: a > set_list_a,Xs: list_a,Y: set_list_a,Ys2: list_set_list_a] :
      ( ( ( map_a_set_list_a @ F @ Xs )
        = ( cons_set_list_a @ Y @ Ys2 ) )
     => ? [Z: a,Zs: list_a] :
          ( ( Xs
            = ( cons_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_a_set_list_a @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_649_map__eq__Cons__D,axiom,
    ! [F: a > set_list_list_a,Xs: list_a,Y: set_list_list_a,Ys2: list_set_list_list_a] :
      ( ( ( map_a_132662181483840688list_a @ F @ Xs )
        = ( cons_set_list_list_a @ Y @ Ys2 ) )
     => ? [Z: a,Zs: list_a] :
          ( ( Xs
            = ( cons_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_a_132662181483840688list_a @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_650_map__eq__Cons__D,axiom,
    ! [F: set_list_a > list_a,Xs: list_set_list_a,Y: list_a,Ys2: list_list_a] :
      ( ( ( map_se5127173849416245988list_a @ F @ Xs )
        = ( cons_list_a @ Y @ Ys2 ) )
     => ? [Z: set_list_a,Zs: list_set_list_a] :
          ( ( Xs
            = ( cons_set_list_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_se5127173849416245988list_a @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_651_map__eq__Cons__D,axiom,
    ! [F: list_a > set_list_a,Xs: list_list_a,Y: set_list_a,Ys2: list_set_list_a] :
      ( ( ( map_li2657846702790696612list_a @ F @ Xs )
        = ( cons_set_list_a @ Y @ Ys2 ) )
     => ? [Z: list_a,Zs: list_list_a] :
          ( ( Xs
            = ( cons_list_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_li2657846702790696612list_a @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_652_map__eq__Cons__D,axiom,
    ! [F: list_a > set_list_list_a,Xs: list_list_a,Y: set_list_list_a,Ys2: list_set_list_list_a] :
      ( ( ( map_li5777811645572713770list_a @ F @ Xs )
        = ( cons_set_list_list_a @ Y @ Ys2 ) )
     => ? [Z: list_a,Zs: list_list_a] :
          ( ( Xs
            = ( cons_list_a @ Z @ Zs ) )
          & ( ( F @ Z )
            = Y )
          & ( ( map_li5777811645572713770list_a @ F @ Zs )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_653_Cons__eq__map__conv,axiom,
    ! [X2: a,Xs: list_a,F: a > a,Ys2: list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( map_a_a @ F @ Ys2 ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Ys2
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_a_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_654_Cons__eq__map__conv,axiom,
    ! [X2: list_a,Xs: list_list_a,F: a > list_a,Ys2: list_a] :
      ( ( ( cons_list_a @ X2 @ Xs )
        = ( map_a_list_a @ F @ Ys2 ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Ys2
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_a_list_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_655_Cons__eq__map__conv,axiom,
    ! [X2: a,Xs: list_a,F: list_a > a,Ys2: list_list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( map_list_a_a @ F @ Ys2 ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Ys2
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_list_a_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_656_Cons__eq__map__conv,axiom,
    ! [X2: list_a,Xs: list_list_a,F: list_a > list_a,Ys2: list_list_a] :
      ( ( ( cons_list_a @ X2 @ Xs )
        = ( map_list_a_list_a @ F @ Ys2 ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Ys2
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_list_a_list_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_657_Cons__eq__map__conv,axiom,
    ! [X2: a,Xs: list_a,F: set_list_a > a,Ys2: list_set_list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( map_set_list_a_a @ F @ Ys2 ) )
      = ( ? [Z3: set_list_a,Zs2: list_set_list_a] :
            ( ( Ys2
              = ( cons_set_list_a @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_set_list_a_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_658_Cons__eq__map__conv,axiom,
    ! [X2: set_list_a,Xs: list_set_list_a,F: a > set_list_a,Ys2: list_a] :
      ( ( ( cons_set_list_a @ X2 @ Xs )
        = ( map_a_set_list_a @ F @ Ys2 ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Ys2
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_a_set_list_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_659_Cons__eq__map__conv,axiom,
    ! [X2: set_list_list_a,Xs: list_set_list_list_a,F: a > set_list_list_a,Ys2: list_a] :
      ( ( ( cons_set_list_list_a @ X2 @ Xs )
        = ( map_a_132662181483840688list_a @ F @ Ys2 ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Ys2
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_a_132662181483840688list_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_660_Cons__eq__map__conv,axiom,
    ! [X2: list_a,Xs: list_list_a,F: set_list_a > list_a,Ys2: list_set_list_a] :
      ( ( ( cons_list_a @ X2 @ Xs )
        = ( map_se5127173849416245988list_a @ F @ Ys2 ) )
      = ( ? [Z3: set_list_a,Zs2: list_set_list_a] :
            ( ( Ys2
              = ( cons_set_list_a @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_se5127173849416245988list_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_661_Cons__eq__map__conv,axiom,
    ! [X2: set_list_a,Xs: list_set_list_a,F: list_a > set_list_a,Ys2: list_list_a] :
      ( ( ( cons_set_list_a @ X2 @ Xs )
        = ( map_li2657846702790696612list_a @ F @ Ys2 ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Ys2
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_li2657846702790696612list_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_662_Cons__eq__map__conv,axiom,
    ! [X2: set_list_list_a,Xs: list_set_list_list_a,F: list_a > set_list_list_a,Ys2: list_list_a] :
      ( ( ( cons_set_list_list_a @ X2 @ Xs )
        = ( map_li5777811645572713770list_a @ F @ Ys2 ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Ys2
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( X2
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_li5777811645572713770list_a @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_663_map__eq__Cons__conv,axiom,
    ! [F: a > a,Xs: list_a,Y: a,Ys2: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = ( cons_a @ Y @ Ys2 ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Xs
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_a_a @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_664_map__eq__Cons__conv,axiom,
    ! [F: a > list_a,Xs: list_a,Y: list_a,Ys2: list_list_a] :
      ( ( ( map_a_list_a @ F @ Xs )
        = ( cons_list_a @ Y @ Ys2 ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Xs
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_a_list_a @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_665_map__eq__Cons__conv,axiom,
    ! [F: list_a > a,Xs: list_list_a,Y: a,Ys2: list_a] :
      ( ( ( map_list_a_a @ F @ Xs )
        = ( cons_a @ Y @ Ys2 ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Xs
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_list_a_a @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_666_map__eq__Cons__conv,axiom,
    ! [F: list_a > list_a,Xs: list_list_a,Y: list_a,Ys2: list_list_a] :
      ( ( ( map_list_a_list_a @ F @ Xs )
        = ( cons_list_a @ Y @ Ys2 ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Xs
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_list_a_list_a @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_667_map__eq__Cons__conv,axiom,
    ! [F: set_list_a > a,Xs: list_set_list_a,Y: a,Ys2: list_a] :
      ( ( ( map_set_list_a_a @ F @ Xs )
        = ( cons_a @ Y @ Ys2 ) )
      = ( ? [Z3: set_list_a,Zs2: list_set_list_a] :
            ( ( Xs
              = ( cons_set_list_a @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_set_list_a_a @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_668_map__eq__Cons__conv,axiom,
    ! [F: a > set_list_a,Xs: list_a,Y: set_list_a,Ys2: list_set_list_a] :
      ( ( ( map_a_set_list_a @ F @ Xs )
        = ( cons_set_list_a @ Y @ Ys2 ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Xs
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_a_set_list_a @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_669_map__eq__Cons__conv,axiom,
    ! [F: a > set_list_list_a,Xs: list_a,Y: set_list_list_a,Ys2: list_set_list_list_a] :
      ( ( ( map_a_132662181483840688list_a @ F @ Xs )
        = ( cons_set_list_list_a @ Y @ Ys2 ) )
      = ( ? [Z3: a,Zs2: list_a] :
            ( ( Xs
              = ( cons_a @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_a_132662181483840688list_a @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_670_map__eq__Cons__conv,axiom,
    ! [F: set_list_a > list_a,Xs: list_set_list_a,Y: list_a,Ys2: list_list_a] :
      ( ( ( map_se5127173849416245988list_a @ F @ Xs )
        = ( cons_list_a @ Y @ Ys2 ) )
      = ( ? [Z3: set_list_a,Zs2: list_set_list_a] :
            ( ( Xs
              = ( cons_set_list_a @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_se5127173849416245988list_a @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_671_map__eq__Cons__conv,axiom,
    ! [F: list_a > set_list_a,Xs: list_list_a,Y: set_list_a,Ys2: list_set_list_a] :
      ( ( ( map_li2657846702790696612list_a @ F @ Xs )
        = ( cons_set_list_a @ Y @ Ys2 ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Xs
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_li2657846702790696612list_a @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_672_map__eq__Cons__conv,axiom,
    ! [F: list_a > set_list_list_a,Xs: list_list_a,Y: set_list_list_a,Ys2: list_set_list_list_a] :
      ( ( ( map_li5777811645572713770list_a @ F @ Xs )
        = ( cons_set_list_list_a @ Y @ Ys2 ) )
      = ( ? [Z3: list_a,Zs2: list_list_a] :
            ( ( Xs
              = ( cons_list_a @ Z3 @ Zs2 ) )
            & ( ( F @ Z3 )
              = Y )
            & ( ( map_li5777811645572713770list_a @ F @ Zs2 )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_673_image__set,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( image_a_a @ F @ ( set_a2 @ Xs ) )
      = ( set_a2 @ ( map_a_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_674_image__set,axiom,
    ! [F: list_a > a,Xs: list_list_a] :
      ( ( image_list_a_a @ F @ ( set_list_a2 @ Xs ) )
      = ( set_a2 @ ( map_list_a_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_675_image__set,axiom,
    ! [F: a > list_a,Xs: list_a] :
      ( ( image_a_list_a @ F @ ( set_a2 @ Xs ) )
      = ( set_list_a2 @ ( map_a_list_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_676_image__set,axiom,
    ! [F: list_a > list_a,Xs: list_list_a] :
      ( ( image_list_a_list_a @ F @ ( set_list_a2 @ Xs ) )
      = ( set_list_a2 @ ( map_list_a_list_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_677_image__set,axiom,
    ! [F: a > set_list_a,Xs: list_a] :
      ( ( image_a_set_list_a @ F @ ( set_a2 @ Xs ) )
      = ( set_set_list_a2 @ ( map_a_set_list_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_678_image__set,axiom,
    ! [F: set_list_a > a,Xs: list_set_list_a] :
      ( ( image_set_list_a_a @ F @ ( set_set_list_a2 @ Xs ) )
      = ( set_a2 @ ( map_set_list_a_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_679_image__set,axiom,
    ! [F: list_a > set_list_a,Xs: list_list_a] :
      ( ( image_5464838071766335845list_a @ F @ ( set_list_a2 @ Xs ) )
      = ( set_set_list_a2 @ ( map_li2657846702790696612list_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_680_image__set,axiom,
    ! [F: a > set_list_list_a,Xs: list_a] :
      ( ( image_2939653550459479921list_a @ F @ ( set_a2 @ Xs ) )
      = ( set_set_list_list_a2 @ ( map_a_132662181483840688list_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_681_image__set,axiom,
    ! [F: set_list_a > list_a,Xs: list_set_list_a] :
      ( ( image_7934165218391885221list_a @ F @ ( set_set_list_a2 @ Xs ) )
      = ( set_list_a2 @ ( map_se5127173849416245988list_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_682_image__set,axiom,
    ! [F: list_a > set_list_list_a,Xs: list_list_a] :
      ( ( image_8589081332878056299list_a @ F @ ( set_list_a2 @ Xs ) )
      = ( set_set_list_list_a2 @ ( map_li5777811645572713770list_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_683_ring__hom__ring_Oeval__hom_H,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H: a > a,A: a,P: list_a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( H @ ( eval_a_b @ R2 @ P @ A ) )
            = ( eval_a_b @ S @ ( map_a_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom'
thf(fact_684_ring__hom__ring_Oeval__hom_H,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H: list_a > a,A: list_a,P: list_list_a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ( H @ ( eval_l34571156754992824t_unit @ R2 @ P @ A ) )
            = ( eval_a_b @ S @ ( map_list_a_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom'
thf(fact_685_ring__hom__ring_Oeval__hom_H,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H: a > list_a,A: a,P: list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( H @ ( eval_a_b @ R2 @ P @ A ) )
            = ( eval_l34571156754992824t_unit @ S @ ( map_a_list_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom'
thf(fact_686_ring__hom__ring_Oeval__hom_H,axiom,
    ! [R2: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H: list_list_a > a,A: list_list_a,P: list_list_list_a] :
      ( ( ring_h3841606220870141376it_a_b @ R2 @ S @ H )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R2 ) )
       => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R2 ) )
         => ( ( H @ ( eval_l1088911609197519410t_unit @ R2 @ P @ A ) )
            = ( eval_a_b @ S @ ( map_list_list_a_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom'
thf(fact_687_ring__hom__ring_Oeval__hom_H,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H: list_a > list_a,A: list_a,P: list_list_a] :
      ( ( ring_h1334922693953046990t_unit @ R2 @ S @ H )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ( H @ ( eval_l34571156754992824t_unit @ R2 @ P @ A ) )
            = ( eval_l34571156754992824t_unit @ S @ ( map_list_a_list_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom'
thf(fact_688_ring__hom__ring_Oeval__hom_H,axiom,
    ! [R2: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b,H: set_list_a > a,A: set_list_a,P: list_set_list_a] :
      ( ( ring_h1355656666774236646it_a_b @ R2 @ S @ H )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R2 ) )
       => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ R2 ) )
         => ( ( H @ ( eval_s5133945360527818456t_unit @ R2 @ P @ A ) )
            = ( eval_a_b @ S @ ( map_set_list_a_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom'
thf(fact_689_ring__hom__ring_Oeval__hom_H,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit,H: a > set_list_a,A: a,P: list_a] :
      ( ( ring_h7781647138149442662t_unit @ R2 @ S @ H )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( H @ ( eval_a_b @ R2 @ P @ A ) )
            = ( eval_s5133945360527818456t_unit @ S @ ( map_a_set_list_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom'
thf(fact_690_ring__hom__ring_Oeval__hom_H,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,H: a > list_list_a,A: a,P: list_a] :
      ( ( ring_h2621993495375454400t_unit @ R2 @ S @ H )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( H @ ( eval_a_b @ R2 @ P @ A ) )
            = ( eval_l1088911609197519410t_unit @ S @ ( map_a_list_list_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom'
thf(fact_691_ring__hom__ring_Oeval__hom_H,axiom,
    ! [R2: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H: list_list_a > list_a,A: list_list_a,P: list_list_list_a] :
      ( ( ring_h4589914651911841480t_unit @ R2 @ S @ H )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R2 ) )
       => ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R2 ) )
         => ( ( H @ ( eval_l1088911609197519410t_unit @ R2 @ P @ A ) )
            = ( eval_l34571156754992824t_unit @ S @ ( map_li1646474281249396926list_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom'
thf(fact_692_ring__hom__ring_Oeval__hom_H,axiom,
    ! [R2: partia4556295656693239580t_unit,S: partia2175431115845679010xt_a_b,H: list_set_list_a > a,A: list_set_list_a,P: list_list_set_list_a] :
      ( ( ring_h4081162582912290272it_a_b @ R2 @ S @ H )
     => ( ( member5524387281408368019list_a @ A @ ( partia7265347635606999311t_unit @ R2 ) )
       => ( ( ord_le1961058059034850666list_a @ ( set_list_set_list_a2 @ P ) @ ( partia7265347635606999311t_unit @ R2 ) )
         => ( ( H @ ( eval_l7642959053407759954t_unit @ R2 @ P @ A ) )
            = ( eval_a_b @ S @ ( map_li6310725680953440600st_a_a @ H @ P ) @ ( H @ A ) ) ) ) ) ) ).

% ring_hom_ring.eval_hom'
thf(fact_693_domain_Odetermination__of__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,A2: partia4960592913263135132t_unit,H: list_list_a > set_list_list_a,G: list_list_a > set_list_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( ring_h2829259075339158498t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ A2 @ H )
         => ( ( ring_h2829259075339158498t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ A2 @ G )
           => ( ! [K3: list_a] :
                  ( ( member_list_a @ K3 @ K )
                 => ( ( H @ ( cons_list_a @ K3 @ nil_list_a ) )
                    = ( G @ ( cons_list_a @ K3 @ nil_list_a ) ) ) )
             => ( ( ( H @ ( var_li8453953174693405341t_unit @ R2 ) )
                  = ( G @ ( var_li8453953174693405341t_unit @ R2 ) ) )
               => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
                 => ( ( H @ P )
                    = ( G @ P ) ) ) ) ) ) ) ) ) ).

% domain.determination_of_hom
thf(fact_694_domain_Odetermination__of__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,A2: partia2956882679547061052t_unit,H: list_a > list_list_a,G: list_a > list_list_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( ring_h7560679385066609224t_unit @ ( univ_poly_a_b @ R2 @ K ) @ A2 @ H )
         => ( ( ring_h7560679385066609224t_unit @ ( univ_poly_a_b @ R2 @ K ) @ A2 @ G )
           => ( ! [K3: a] :
                  ( ( member_a @ K3 @ K )
                 => ( ( H @ ( cons_a @ K3 @ nil_a ) )
                    = ( G @ ( cons_a @ K3 @ nil_a ) ) ) )
             => ( ( ( H @ ( var_a_b @ R2 ) )
                  = ( G @ ( var_a_b @ R2 ) ) )
               => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
                 => ( ( H @ P )
                    = ( G @ P ) ) ) ) ) ) ) ) ) ).

% domain.determination_of_hom
thf(fact_695_domain_Odetermination__of__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,A2: partia7496981018696276118t_unit,H: list_a > set_list_a,G: list_a > set_list_a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( ring_h6719293541701509614t_unit @ ( univ_poly_a_b @ R2 @ K ) @ A2 @ H )
         => ( ( ring_h6719293541701509614t_unit @ ( univ_poly_a_b @ R2 @ K ) @ A2 @ G )
           => ( ! [K3: a] :
                  ( ( member_a @ K3 @ K )
                 => ( ( H @ ( cons_a @ K3 @ nil_a ) )
                    = ( G @ ( cons_a @ K3 @ nil_a ) ) ) )
             => ( ( ( H @ ( var_a_b @ R2 ) )
                  = ( G @ ( var_a_b @ R2 ) ) )
               => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
                 => ( ( H @ P )
                    = ( G @ P ) ) ) ) ) ) ) ) ) ).

% domain.determination_of_hom
thf(fact_696_domain_Odetermination__of__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,A2: partia2175431115845679010xt_a_b,H: list_a > a,G: list_a > a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R2 @ K ) @ A2 @ H )
         => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R2 @ K ) @ A2 @ G )
           => ( ! [K3: a] :
                  ( ( member_a @ K3 @ K )
                 => ( ( H @ ( cons_a @ K3 @ nil_a ) )
                    = ( G @ ( cons_a @ K3 @ nil_a ) ) ) )
             => ( ( ( H @ ( var_a_b @ R2 ) )
                  = ( G @ ( var_a_b @ R2 ) ) )
               => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
                 => ( ( H @ P )
                    = ( G @ P ) ) ) ) ) ) ) ) ) ).

% domain.determination_of_hom
thf(fact_697_domain_Oeval__as__unique__hom,axiom,
    ! [R2: partia2956882679547061052t_unit,K: set_list_list_a,X2: list_list_a,H: list_list_list_a > list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R2 )
     => ( ( subrin3541368690557094692t_unit @ K @ R2 )
       => ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R2 ) )
         => ( ( ring_h7694777735462631100t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) @ R2 @ H )
           => ( ! [K3: list_list_a] :
                  ( ( member_list_list_a @ K3 @ K )
                 => ( ( H @ ( cons_list_list_a @ K3 @ nil_list_list_a ) )
                    = K3 ) )
             => ( ( ( H @ ( var_li3532061862469730199t_unit @ R2 ) )
                  = X2 )
               => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ K ) ) )
                 => ( ( H @ P )
                    = ( eval_l1088911609197519410t_unit @ R2 @ P @ X2 ) ) ) ) ) ) ) ) ) ).

% domain.eval_as_unique_hom
thf(fact_698_domain_Oeval__as__unique__hom,axiom,
    ! [R2: partia4556295656693239580t_unit,K: set_list_set_list_a,X2: list_set_list_a,H: list_list_set_list_a > list_set_list_a,P: list_list_set_list_a] :
      ( ( domain2898972329295444579t_unit @ R2 )
     => ( ( subrin7712475646129147716t_unit @ K @ R2 )
       => ( ( member5524387281408368019list_a @ X2 @ ( partia7265347635606999311t_unit @ R2 ) )
         => ( ( ring_h811738076660924156t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ K ) @ R2 @ H )
           => ( ! [K3: list_set_list_a] :
                  ( ( member5524387281408368019list_a @ K3 @ K )
                 => ( ( H @ ( cons_list_set_list_a @ K3 @ nil_list_set_list_a ) )
                    = K3 ) )
             => ( ( ( H @ ( var_li7697098674259343287t_unit @ R2 ) )
                  = X2 )
               => ( ( member352051402189872281list_a @ P @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R2 @ K ) ) )
                 => ( ( H @ P )
                    = ( eval_l7642959053407759954t_unit @ R2 @ P @ X2 ) ) ) ) ) ) ) ) ) ).

% domain.eval_as_unique_hom
thf(fact_699_domain_Oeval__as__unique__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,K: set_list_a,X2: list_a,H: list_list_a > list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( subrin6918843898125473962t_unit @ K @ R2 )
       => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ( ring_h4589914651911841480t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) @ R2 @ H )
           => ( ! [K3: list_a] :
                  ( ( member_list_a @ K3 @ K )
                 => ( ( H @ ( cons_list_a @ K3 @ nil_list_a ) )
                    = K3 ) )
             => ( ( ( H @ ( var_li8453953174693405341t_unit @ R2 ) )
                  = X2 )
               => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ K ) ) )
                 => ( ( H @ P )
                    = ( eval_l34571156754992824t_unit @ R2 @ P @ X2 ) ) ) ) ) ) ) ) ) ).

% domain.eval_as_unique_hom
thf(fact_700_domain_Oeval__as__unique__hom,axiom,
    ! [R2: partia4960592913263135132t_unit,K: set_set_list_list_a,X2: set_list_list_a,H: list_set_list_list_a > set_list_list_a,P: list_set_list_list_a] :
      ( ( domain7421296078544666595t_unit @ R2 )
     => ( ( subrin3011427358523593924t_unit @ K @ R2 )
       => ( ( member334759470184282131list_a @ X2 @ ( partia3317168157747563407t_unit @ R2 ) )
         => ( ( ring_h4765045011162728956t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ K ) @ R2 @ H )
           => ( ! [K3: set_list_list_a] :
                  ( ( member334759470184282131list_a @ K3 @ K )
                 => ( ( H @ ( cons_set_list_list_a @ K3 @ nil_set_list_list_a ) )
                    = K3 ) )
             => ( ( ( H @ ( var_se2996050386653789495t_unit @ R2 ) )
                  = X2 )
               => ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R2 @ K ) ) )
                 => ( ( H @ P )
                    = ( eval_s2941910765802206162t_unit @ R2 @ P @ X2 ) ) ) ) ) ) ) ) ) ).

% domain.eval_as_unique_hom
thf(fact_701_domain_Oeval__as__unique__hom,axiom,
    ! [R2: partia7496981018696276118t_unit,K: set_set_list_a,X2: set_list_a,H: list_set_list_a > set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R2 )
     => ( ( subrin5643252653130547402t_unit @ K @ R2 )
       => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R2 ) )
         => ( ( ring_h8484543649848848136t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) @ R2 @ H )
           => ( ! [K3: set_list_a] :
                  ( ( member_set_list_a @ K3 @ K )
                 => ( ( H @ ( cons_set_list_a @ K3 @ nil_set_list_a ) )
                    = K3 ) )
             => ( ( ( H @ ( var_se6008125447796440765t_unit @ R2 ) )
                  = X2 )
               => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R2 @ K ) ) )
                 => ( ( H @ P )
                    = ( eval_s5133945360527818456t_unit @ R2 @ P @ X2 ) ) ) ) ) ) ) ) ) ).

% domain.eval_as_unique_hom
thf(fact_702_domain_Oeval__as__unique__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,K: set_a,X2: a,H: list_a > a,P: list_a] :
      ( ( domain_a_b @ R2 )
     => ( ( subring_a_b @ K @ R2 )
       => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R2 @ K ) @ R2 @ H )
           => ( ! [K3: a] :
                  ( ( member_a @ K3 @ K )
                 => ( ( H @ ( cons_a @ K3 @ nil_a ) )
                    = K3 ) )
             => ( ( ( H @ ( var_a_b @ R2 ) )
                  = X2 )
               => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R2 @ K ) ) )
                 => ( ( H @ P )
                    = ( eval_a_b @ R2 @ P @ X2 ) ) ) ) ) ) ) ) ) ).

% domain.eval_as_unique_hom
thf(fact_703_UP_Oexp__base__closed,axiom,
    ! [X2: list_a,N: nat] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.exp_base_closed
thf(fact_704_ring__irreducibleE_I1_J,axiom,
    ! [R4: a] :
      ( ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_r999134135267193926le_a_b @ r @ R4 )
       => ( R4
         != ( zero_a_b @ r ) ) ) ) ).

% ring_irreducibleE(1)
thf(fact_705_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H: a > a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H )
     => ( member_set_a_a
        @ ^ [X5: set_a] : ( the_elem_a @ ( image_a_a @ H @ X5 ) )
        @ ( ring_i4751279688611836608it_a_b @ ( factRing_a_b @ R2 @ ( a_kernel_a_b_a_b @ R2 @ S @ H ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_a_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_706_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H )
     => ( member_set_a_list_a
        @ ^ [X5: set_a] : ( the_elem_list_a @ ( image_a_list_a @ H @ X5 ) )
        @ ( ring_i2253880877871059912t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H ) )
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : ( image_a_list_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_707_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H )
     => ( member_set_list_a_a
        @ ^ [X5: set_list_a] : ( the_elem_a @ ( image_list_a_a @ H @ X5 ) )
        @ ( ring_i8122894263081988538it_a_b @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_list_a_a @ H @ ( partia5361259788508890537t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_708_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H: list_a > list_a] :
      ( ( ring_h1334922693953046990t_unit @ R2 @ S @ H )
     => ( member5910328476188217884list_a
        @ ^ [X5: set_list_a] : ( the_elem_list_a @ ( image_list_a_list_a @ H @ X5 ) )
        @ ( ring_i8566987394125245378t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern6921148033081882766t_unit @ R2 @ S @ H ) )
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : ( image_list_a_list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_709_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b,H: set_list_a > a] :
      ( ( ring_h1355656666774236646it_a_b @ R2 @ S @ H )
     => ( member9025319375365623606st_a_a
        @ ^ [X5: set_set_list_a] : ( the_elem_a @ ( image_set_list_a_a @ H @ X5 ) )
        @ ( ring_i974745849744036314it_a_b @ ( factRi5994137953073989130t_unit @ R2 @ ( a_kern5779274734665759910it_a_b @ R2 @ S @ H ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_set_list_a_a @ H @ ( partia141011252114345353t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_710_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit,H: a > set_list_a] :
      ( ( ring_h7781647138149442662t_unit @ R2 @ S @ H )
     => ( member1188894495758700418list_a
        @ ^ [X5: set_a] : ( the_elem_set_list_a @ ( image_a_set_list_a @ H @ X5 ) )
        @ ( ring_i3023096418695644136t_unit @ ( factRing_a_b @ R2 @ ( a_kern2981893169186190118t_unit @ R2 @ S @ H ) )
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : ( image_a_set_list_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_711_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,H: list_a > list_list_a] :
      ( ( ring_h7560679385066609224t_unit @ R2 @ S @ H )
     => ( member1709587767443891362list_a
        @ ^ [X5: set_list_a] : ( the_elem_list_list_a @ ( image_8260866953997875467list_a @ H @ X5 ) )
        @ ( ring_i2750902443338887740t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern4386753233124370696t_unit @ R2 @ S @ H ) )
          @ ( partia5725451877363000582t_unit
            @ ^ [Uu: set_list_list_a] : ( image_8260866953997875467list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_712_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia7496981018696276118t_unit,S: partia2670972154091845814t_unit,H: set_list_a > list_a] :
      ( ( ring_h8569328188485256686t_unit @ R2 @ S @ H )
     => ( member7949344002864897596list_a
        @ ^ [X5: set_set_list_a] : ( the_elem_list_a @ ( image_7934165218391885221list_a @ H @ X5 ) )
        @ ( ring_i3146865591392574434t_unit @ ( factRi5994137953073989130t_unit @ R2 @ ( a_kern2420883521965667502t_unit @ R2 @ S @ H ) )
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : ( image_7934165218391885221list_a @ H @ ( partia141011252114345353t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_713_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia4556295656693239580t_unit,S: partia2175431115845679010xt_a_b,H: list_set_list_a > a] :
      ( ( ring_h4081162582912290272it_a_b @ R2 @ S @ H )
     => ( member1625632663891872560st_a_a
        @ ^ [X5: set_list_set_list_a] : ( the_elem_a @ ( image_9117717049929079833st_a_a @ H @ X5 ) )
        @ ( ring_i490517679781779796it_a_b @ ( factRi6954973740111656068t_unit @ R2 @ ( a_kern4140738655600802976it_a_b @ R2 @ S @ H ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_9117717049929079833st_a_a @ H @ ( partia7265347635606999311t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_714_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia4960592913263135132t_unit,S: partia2175431115845679010xt_a_b,H: set_list_list_a > a] :
      ( ( ring_h9122830167833154400it_a_b @ R2 @ S @ H )
     => ( member7286988230774018224st_a_a
        @ ^ [X5: set_set_list_list_a] : ( the_elem_a @ ( image_6921516081670988185st_a_a @ H @ X5 ) )
        @ ( ring_i8354037051245829844it_a_b @ ( factRi2253925452506102276t_unit @ R2 @ ( a_kern9182406240521667104it_a_b @ R2 @ S @ H ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_6921516081670988185st_a_a @ H @ ( partia3317168157747563407t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_715_h_Osubdomain__iff,axiom,
    ! [H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( subdom3220114454046903646t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
        = ( domain1617769409708967785t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : H2
            @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.subdomain_iff
thf(fact_716_UP_Oa__rcos__assoc__lcos,axiom,
    ! [H2: set_list_a,K: set_list_a,X2: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ X2 ) @ K )
            = ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ K ) ) ) ) ) ) ).

% UP.a_rcos_assoc_lcos
thf(fact_717_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H )
     => ( ( image_set_a_list_a
          @ ^ [X5: set_a] : ( the_elem_list_a @ ( image_a_list_a @ H @ X5 ) )
          @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H ) ) ) )
        = ( image_a_list_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_718_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia2956882679547061052t_unit,S: partia4960592913263135132t_unit,H: list_list_a > set_list_list_a] :
      ( ( ring_h2829259075339158498t_unit @ R2 @ S @ H )
     => ( ( image_452714708507473285list_a
          @ ^ [X5: set_list_list_a] : ( the_el9027665213549201119list_a @ ( image_551801017575455717list_a @ H @ X5 ) )
          @ ( partia3317168157747563407t_unit @ ( factRi7259693425559269476t_unit @ R2 @ ( a_kern2892906198464180898t_unit @ R2 @ S @ H ) ) ) )
        = ( image_551801017575455717list_a @ H @ ( partia2464479390973590831t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_719_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,H: list_a > list_list_a] :
      ( ( ring_h7560679385066609224t_unit @ R2 @ S @ H )
     => ( ( image_8997752375301837099list_a
          @ ^ [X5: set_list_a] : ( the_elem_list_list_a @ ( image_8260866953997875467list_a @ H @ X5 ) )
          @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern4386753233124370696t_unit @ R2 @ S @ H ) ) ) )
        = ( image_8260866953997875467list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_720_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H: list_a > set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H )
     => ( ( image_5749939591322298757list_a
          @ ^ [X5: set_list_a] : ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ H @ X5 ) )
          @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H ) ) ) )
        = ( image_5464838071766335845list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_721_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H )
     => ( ( image_set_list_a_a
          @ ^ [X5: set_list_a] : ( the_elem_a @ ( image_list_a_a @ H @ X5 ) )
          @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H ) ) ) )
        = ( image_list_a_a @ H @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_722_h_Oring__hom__imp__img__ring,axiom,
    ! [H: set_list_a > list_list_a,S: partia2956882679547061052t_unit] :
      ( ( member1709587767443891362list_a @ H @ ( ring_h7162840334539070758t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ S ) )
     => ( ring_l1939023646219158831t_unit
        @ ( zero_u569140119141447472t_unit
          @ ^ [Uu: list_list_a] : ( H @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
          @ ( partia5725451877363000582t_unit
            @ ^ [Uu: set_list_list_a] : ( image_8997752375301837099list_a @ H @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
            @ S ) ) ) ) ).

% h.ring_hom_imp_img_ring
thf(fact_723_h_Oring__hom__imp__img__ring,axiom,
    ! [H: set_list_a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member5910328476188217884list_a @ H @ ( ring_h8038483918290310060t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ S ) )
     => ( ring_l6212528067271185461t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : ( image_7934165218391885221list_a @ H @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
            @ S ) ) ) ) ).

% h.ring_hom_imp_img_ring
thf(fact_724_h_Oring__hom__imp__img__ring,axiom,
    ! [H: set_list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ S ) )
     => ( ring_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_set_list_a_a @ H @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
            @ S ) ) ) ) ).

% h.ring_hom_imp_img_ring
thf(fact_725_h_Oring__hom__imp__img__ring,axiom,
    ! [H: set_list_a > set_list_a,S: partia7496981018696276118t_unit] :
      ( ( member5068272912271824380list_a @ H @ ( ring_h6076331213207892940t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ S ) )
     => ( ring_s8247141995668492373t_unit
        @ ( zero_u3422801854947504854t_unit
          @ ^ [Uu: set_list_a] : ( H @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : ( image_5749939591322298757list_a @ H @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
            @ S ) ) ) ) ).

% h.ring_hom_imp_img_ring
thf(fact_726_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H: a > a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H )
     => ( ( ( image_a_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member_set_a_a
          @ ^ [X5: set_a] : ( the_elem_a @ ( image_a_a @ H @ X5 ) )
          @ ( ring_i4751279688611836608it_a_b @ ( factRing_a_b @ R2 @ ( a_kernel_a_b_a_b @ R2 @ S @ H ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_727_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H )
     => ( ( ( image_list_a_a @ H @ ( partia5361259788508890537t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member_set_list_a_a
          @ ^ [X5: set_list_a] : ( the_elem_a @ ( image_list_a_a @ H @ X5 ) )
          @ ( ring_i8122894263081988538it_a_b @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_728_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H )
     => ( ( ( image_a_list_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia5361259788508890537t_unit @ S ) )
       => ( member_set_a_list_a
          @ ^ [X5: set_a] : ( the_elem_list_a @ ( image_a_list_a @ H @ X5 ) )
          @ ( ring_i2253880877871059912t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_729_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H: list_a > list_a] :
      ( ( ring_h1334922693953046990t_unit @ R2 @ S @ H )
     => ( ( ( image_list_a_list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) )
          = ( partia5361259788508890537t_unit @ S ) )
       => ( member5910328476188217884list_a
          @ ^ [X5: set_list_a] : ( the_elem_list_a @ ( image_list_a_list_a @ H @ X5 ) )
          @ ( ring_i8566987394125245378t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern6921148033081882766t_unit @ R2 @ S @ H ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_730_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b,H: set_list_a > a] :
      ( ( ring_h1355656666774236646it_a_b @ R2 @ S @ H )
     => ( ( ( image_set_list_a_a @ H @ ( partia141011252114345353t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member9025319375365623606st_a_a
          @ ^ [X5: set_set_list_a] : ( the_elem_a @ ( image_set_list_a_a @ H @ X5 ) )
          @ ( ring_i974745849744036314it_a_b @ ( factRi5994137953073989130t_unit @ R2 @ ( a_kern5779274734665759910it_a_b @ R2 @ S @ H ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_731_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit,H: a > set_list_a] :
      ( ( ring_h7781647138149442662t_unit @ R2 @ S @ H )
     => ( ( ( image_a_set_list_a @ H @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia141011252114345353t_unit @ S ) )
       => ( member1188894495758700418list_a
          @ ^ [X5: set_a] : ( the_elem_set_list_a @ ( image_a_set_list_a @ H @ X5 ) )
          @ ( ring_i3023096418695644136t_unit @ ( factRing_a_b @ R2 @ ( a_kern2981893169186190118t_unit @ R2 @ S @ H ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_732_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia4556295656693239580t_unit,S: partia2175431115845679010xt_a_b,H: list_set_list_a > a] :
      ( ( ring_h4081162582912290272it_a_b @ R2 @ S @ H )
     => ( ( ( image_9117717049929079833st_a_a @ H @ ( partia7265347635606999311t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member1625632663891872560st_a_a
          @ ^ [X5: set_list_set_list_a] : ( the_elem_a @ ( image_9117717049929079833st_a_a @ H @ X5 ) )
          @ ( ring_i490517679781779796it_a_b @ ( factRi6954973740111656068t_unit @ R2 @ ( a_kern4140738655600802976it_a_b @ R2 @ S @ H ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_733_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,H: list_a > list_list_a] :
      ( ( ring_h7560679385066609224t_unit @ R2 @ S @ H )
     => ( ( ( image_8260866953997875467list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) )
          = ( partia2464479390973590831t_unit @ S ) )
       => ( member1709587767443891362list_a
          @ ^ [X5: set_list_a] : ( the_elem_list_list_a @ ( image_8260866953997875467list_a @ H @ X5 ) )
          @ ( ring_i2750902443338887740t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern4386753233124370696t_unit @ R2 @ S @ H ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_734_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H: list_a > set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H )
     => ( ( ( image_5464838071766335845list_a @ H @ ( partia5361259788508890537t_unit @ R2 ) )
          = ( partia141011252114345353t_unit @ S ) )
       => ( member5068272912271824380list_a
          @ ^ [X5: set_list_a] : ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ H @ X5 ) )
          @ ( ring_i6425008796027319266t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_735_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia4960592913263135132t_unit,S: partia2175431115845679010xt_a_b,H: set_list_list_a > a] :
      ( ( ring_h9122830167833154400it_a_b @ R2 @ S @ H )
     => ( ( ( image_6921516081670988185st_a_a @ H @ ( partia3317168157747563407t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member7286988230774018224st_a_a
          @ ^ [X5: set_set_list_list_a] : ( the_elem_a @ ( image_6921516081670988185st_a_a @ H @ X5 ) )
          @ ( ring_i8354037051245829844it_a_b @ ( factRi2253925452506102276t_unit @ R2 @ ( a_kern9182406240521667104it_a_b @ R2 @ S @ H ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_736_normalize_Ocases,axiom,
    ! [X2: list_a] :
      ( ( X2 != nil_a )
     => ~ ! [V2: a,Va: list_a] :
            ( X2
           != ( cons_a @ V2 @ Va ) ) ) ).

% normalize.cases
thf(fact_737_h_Onormalize_Ocases,axiom,
    ! [X2: list_set_list_a] :
      ( ( X2 != nil_set_list_a )
     => ~ ! [V2: set_list_a,Va: list_set_list_a] :
            ( X2
           != ( cons_set_list_a @ V2 @ Va ) ) ) ).

% h.normalize.cases
thf(fact_738_local_Oring__axioms,axiom,
    ring_a_b @ r ).

% local.ring_axioms
thf(fact_739_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_740_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_741_eval__in__carrier,axiom,
    ! [P: list_a,X2: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( eval_a_b @ r @ P @ X2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% eval_in_carrier
thf(fact_742_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_743_UP_Oring__axioms,axiom,
    ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ k ) ).

% UP.ring_axioms
thf(fact_744_subring__is__ring,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( ring_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ r ) ) ) ).

% subring_is_ring
thf(fact_745_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_746_poly__of__const__def,axiom,
    ( ( poly_of_const_a_b @ r )
    = ( ^ [K4: a] : ( normalize_a_b @ r @ ( cons_a @ K4 @ nil_a ) ) ) ) ).

% poly_of_const_def
thf(fact_747_ring__incl__imp__subring,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ r ) )
       => ( subring_a_b @ H2 @ r ) ) ) ).

% ring_incl_imp_subring
thf(fact_748_subring__iff,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( subring_a_b @ H2 @ r )
        = ( ring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ r ) ) ) ) ).

% subring_iff
thf(fact_749_h_Oring__axioms,axiom,
    ring_s8247141995668492373t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ).

% h.ring_axioms
thf(fact_750_UP_Ouniv__poly__is__ring,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ring_l1939023646219158831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% UP.univ_poly_is_ring
thf(fact_751_UP_Osubring__is__ring,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( ring_l6212528067271185461t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : H2
          @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.subring_is_ring
thf(fact_752_ring__iso__imp__img__ring,axiom,
    ! [H: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ r @ S ) )
     => ( ring_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_ring
thf(fact_753_ring__iso__imp__img__ring,axiom,
    ! [H: a > list_list_a,S: partia2956882679547061052t_unit] :
      ( ( member_a_list_list_a @ H @ ( ring_i4464730343205239444t_unit @ r @ S ) )
     => ( ring_l1939023646219158831t_unit
        @ ( zero_u569140119141447472t_unit
          @ ^ [Uu: list_list_a] : ( H @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_ring
thf(fact_754_ring__iso__imp__img__ring,axiom,
    ! [H: a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ r @ S ) )
     => ( ring_l6212528067271185461t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_ring
thf(fact_755_ring__iso__imp__img__ring,axiom,
    ! [H: a > set_list_a,S: partia7496981018696276118t_unit] :
      ( ( member_a_set_list_a @ H @ ( ring_i5325512697602418746t_unit @ r @ S ) )
     => ( ring_s8247141995668492373t_unit
        @ ( zero_u3422801854947504854t_unit
          @ ^ [Uu: set_list_a] : ( H @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_ring
thf(fact_756_UP_Oa__l__coset__subset__G,axiom,
    ! [H2: set_list_a,X2: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ H2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.a_l_coset_subset_G
thf(fact_757_h_Osubring__is__ring,axiom,
    ! [H2: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ring_s8247141995668492373t_unit
        @ ( partia2314006361284536288t_unit
          @ ^ [Uu: set_set_list_a] : H2
          @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.subring_is_ring
thf(fact_758_UP_Oa__lcos__mult__one,axiom,
    ! [M: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ M )
        = M ) ) ).

% UP.a_lcos_mult_one
thf(fact_759_determination__of__hom,axiom,
    ! [K: set_a,A2: partia2956882679547061052t_unit,H: list_a > list_list_a,G: list_a > list_list_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( ring_h7560679385066609224t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ H )
       => ( ( ring_h7560679385066609224t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ G )
         => ( ! [K3: a] :
                ( ( member_a @ K3 @ K )
               => ( ( H @ ( cons_a @ K3 @ nil_a ) )
                  = ( G @ ( cons_a @ K3 @ nil_a ) ) ) )
           => ( ( ( H @ ( var_a_b @ r ) )
                = ( G @ ( var_a_b @ r ) ) )
             => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
               => ( ( H @ P )
                  = ( G @ P ) ) ) ) ) ) ) ) ).

% determination_of_hom
thf(fact_760_determination__of__hom,axiom,
    ! [K: set_a,A2: partia7496981018696276118t_unit,H: list_a > set_list_a,G: list_a > set_list_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( ring_h6719293541701509614t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ H )
       => ( ( ring_h6719293541701509614t_unit @ ( univ_poly_a_b @ r @ K ) @ A2 @ G )
         => ( ! [K3: a] :
                ( ( member_a @ K3 @ K )
               => ( ( H @ ( cons_a @ K3 @ nil_a ) )
                  = ( G @ ( cons_a @ K3 @ nil_a ) ) ) )
           => ( ( ( H @ ( var_a_b @ r ) )
                = ( G @ ( var_a_b @ r ) ) )
             => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
               => ( ( H @ P )
                  = ( G @ P ) ) ) ) ) ) ) ) ).

% determination_of_hom
thf(fact_761_determination__of__hom,axiom,
    ! [K: set_a,A2: partia2175431115845679010xt_a_b,H: list_a > a,G: list_a > a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ A2 @ H )
       => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ A2 @ G )
         => ( ! [K3: a] :
                ( ( member_a @ K3 @ K )
               => ( ( H @ ( cons_a @ K3 @ nil_a ) )
                  = ( G @ ( cons_a @ K3 @ nil_a ) ) ) )
           => ( ( ( H @ ( var_a_b @ r ) )
                = ( G @ ( var_a_b @ r ) ) )
             => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
               => ( ( H @ P )
                  = ( G @ P ) ) ) ) ) ) ) ) ).

% determination_of_hom
thf(fact_762_h_Onormalize__in__carrier,axiom,
    ! [P: list_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.normalize_in_carrier
thf(fact_763_h_Oeval__in__carrier,axiom,
    ! [P: list_set_list_a,X2: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( member_set_list_a @ ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P @ X2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.eval_in_carrier
thf(fact_764_UP_Osubring__iff,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
        = ( ring_l6212528067271185461t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : H2
            @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.subring_iff
thf(fact_765_UP_Oring__incl__imp__subring,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ring_l6212528067271185461t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : H2
            @ ( univ_poly_a_b @ r @ k ) ) )
       => ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.ring_incl_imp_subring
thf(fact_766_ring__hom__imp__img__ring,axiom,
    ! [H: a > list_list_a,S: partia2956882679547061052t_unit] :
      ( ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ r @ S ) )
     => ( ring_l1939023646219158831t_unit
        @ ( zero_u569140119141447472t_unit
          @ ^ [Uu: list_list_a] : ( H @ ( zero_a_b @ r ) )
          @ ( partia5725451877363000582t_unit
            @ ^ [Uu: set_list_list_a] : ( image_a_list_list_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
            @ S ) ) ) ) ).

% ring_hom_imp_img_ring
thf(fact_767_ring__hom__imp__img__ring,axiom,
    ! [H: a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ r @ S ) )
     => ( ring_l6212528067271185461t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H @ ( zero_a_b @ r ) )
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : ( image_a_list_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
            @ S ) ) ) ) ).

% ring_hom_imp_img_ring
thf(fact_768_ring__hom__imp__img__ring,axiom,
    ! [H: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ r @ S ) )
     => ( ring_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H @ ( zero_a_b @ r ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_a_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
            @ S ) ) ) ) ).

% ring_hom_imp_img_ring
thf(fact_769_ring__hom__imp__img__ring,axiom,
    ! [H: a > set_list_a,S: partia7496981018696276118t_unit] :
      ( ( member_a_set_list_a @ H @ ( ring_h6109298854714515236t_unit @ r @ S ) )
     => ( ring_s8247141995668492373t_unit
        @ ( zero_u3422801854947504854t_unit
          @ ^ [Uu: set_list_a] : ( H @ ( zero_a_b @ r ) )
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : ( image_a_set_list_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
            @ S ) ) ) ) ).

% ring_hom_imp_img_ring
thf(fact_770_UP_Oring__iso__imp__img__ring,axiom,
    ! [H: list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ ( univ_poly_a_b @ r @ k ) @ S ) )
     => ( ring_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          @ S ) ) ) ).

% UP.ring_iso_imp_img_ring
thf(fact_771_h_Opoly__of__const__def,axiom,
    ( ( poly_o3535013565302768950t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
    = ( ^ [K4: set_list_a] : ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( cons_set_list_a @ K4 @ nil_set_list_a ) ) ) ) ).

% h.poly_of_const_def
thf(fact_772_h_Oeval__normalize,axiom,
    ! [P: list_set_list_a,A: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P ) @ A )
          = ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P @ A ) ) ) ) ).

% h.eval_normalize
thf(fact_773_h_Osubring__iff,axiom,
    ! [H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( subrin5643252653130547402t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
        = ( ring_s8247141995668492373t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : H2
            @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.subring_iff
thf(fact_774_h_Oring__incl__imp__subring,axiom,
    ! [H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ring_s8247141995668492373t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : H2
            @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( subrin5643252653130547402t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.ring_incl_imp_subring
thf(fact_775_eval__as__unique__hom,axiom,
    ! [K: set_a,X2: a,H: list_a > a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H )
         => ( ! [K3: a] :
                ( ( member_a @ K3 @ K )
               => ( ( H @ ( cons_a @ K3 @ nil_a ) )
                  = K3 ) )
           => ( ( ( H @ ( var_a_b @ r ) )
                = X2 )
             => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
               => ( ( H @ P )
                  = ( eval_a_b @ r @ P @ X2 ) ) ) ) ) ) ) ) ).

% eval_as_unique_hom
thf(fact_776_h_Oimg__is__ring,axiom,
    ( ring_s8247141995668492373t_unit
    @ ( partia2314006361284536288t_unit
      @ ^ [Uu: set_set_list_a] : ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
      @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ).

% h.img_is_ring
thf(fact_777_h_Oexp__base__closed,axiom,
    ! [X2: set_list_a,N: nat] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ ( polyno2667201621303875488t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ N ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.exp_base_closed
thf(fact_778_exp__base__closed,axiom,
    ! [X2: a,N: nat] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X2 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% exp_base_closed
thf(fact_779_subdomain__iff,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( subdomain_a_b @ H2 @ r )
        = ( domain_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ r ) ) ) ) ).

% subdomain_iff
thf(fact_780_h_Oa__lcos__mult__one,axiom,
    ! [M: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ M @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( a_l_co4135707798524667186t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ M )
        = M ) ) ).

% h.a_lcos_mult_one
thf(fact_781_h_Osubalgebra__in__carrier,axiom,
    ! [K: set_set_list_a,V3: set_set_list_a] :
      ( ( embedd3586486045337765042t_unit @ K @ V3 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ord_le8877086941679407844list_a @ V3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.subalgebra_in_carrier
thf(fact_782_h_Ocarrier__is__subalgebra,axiom,
    ! [K: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( embedd3586486045337765042t_unit @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ).

% h.carrier_is_subalgebra
thf(fact_783_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_784_a__rcos__assoc__lcos,axiom,
    ! [H2: set_a,K: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ H2 @ X2 ) @ K )
            = ( set_add_a_b @ r @ H2 @ ( a_l_coset_a_b @ r @ X2 @ K ) ) ) ) ) ) ).

% a_rcos_assoc_lcos
thf(fact_785_a__setmult__rcos__assoc,axiom,
    ! [H2: set_a,K: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( set_add_a_b @ r @ H2 @ ( a_r_coset_a_b @ r @ K @ X2 ) )
            = ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H2 @ K ) @ X2 ) ) ) ) ) ).

% a_setmult_rcos_assoc
thf(fact_786_a__r__coset__subset__G,axiom,
    ! [H2: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H2 @ X2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% a_r_coset_subset_G
thf(fact_787_set__add__closed,axiom,
    ! [A2: set_a,B2: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% set_add_closed
thf(fact_788_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_789_setadd__subset__G,axiom,
    ! [H2: set_a,K: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H2 @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% setadd_subset_G
thf(fact_790_a__l__coset__subset__G,axiom,
    ! [H2: set_a,X2: a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X2 @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% a_l_coset_subset_G
thf(fact_791_zero__is__prime_I1_J,axiom,
    prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).

% zero_is_prime(1)
thf(fact_792_subdomainI_H,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( subdomain_a_b @ H2 @ r ) ) ).

% subdomainI'
thf(fact_793_a__lcos__mult__one,axiom,
    ! [M: set_a] :
      ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
        = M ) ) ).

% a_lcos_mult_one
thf(fact_794_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_795_subdomain__is__domain,axiom,
    ! [H2: set_a] :
      ( ( subdomain_a_b @ H2 @ r )
     => ( domain_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ r ) ) ) ).

% subdomain_is_domain
thf(fact_796_h_Oa__setmult__rcos__assoc,axiom,
    ! [H2: set_set_list_a,K: set_set_list_a,X2: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 @ ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ X2 ) )
            = ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 @ K ) @ X2 ) ) ) ) ) ).

% h.a_setmult_rcos_assoc
thf(fact_797_h_Oa__r__coset__subset__G,axiom,
    ! [H2: set_set_list_a,X2: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ord_le8877086941679407844list_a @ ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 @ X2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.a_r_coset_subset_G
thf(fact_798_h_Oset__add__closed,axiom,
    ! [A2: set_set_list_a,B2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ B2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ord_le8877086941679407844list_a @ ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A2 @ B2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.set_add_closed
thf(fact_799_h_Oset__add__comm,axiom,
    ! [I2: set_set_list_a,J: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ I2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ J @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ I2 @ J )
          = ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ J @ I2 ) ) ) ) ).

% h.set_add_comm
thf(fact_800_h_Osetadd__subset__G,axiom,
    ! [H2: set_set_list_a,K: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ord_le8877086941679407844list_a @ ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 @ K ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.setadd_subset_G
thf(fact_801_h_Oa__rcos__assoc__lcos,axiom,
    ! [H2: set_set_list_a,K: set_set_list_a,X2: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 @ X2 ) @ K )
            = ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 @ ( a_l_co4135707798524667186t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ K ) ) ) ) ) ) ).

% h.a_rcos_assoc_lcos
thf(fact_802_h_Oa__l__coset__subset__G,axiom,
    ! [H2: set_set_list_a,X2: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ord_le8877086941679407844list_a @ ( a_l_co4135707798524667186t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ H2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.a_l_coset_subset_G
thf(fact_803_a__coset__add__zero,axiom,
    ! [M: set_a] :
      ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( a_r_coset_a_b @ r @ M @ ( zero_a_b @ r ) )
        = M ) ) ).

% a_coset_add_zero
thf(fact_804_h_Oa__coset__add__zero,axiom,
    ! [M: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ M @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ M @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = M ) ) ).

% h.a_coset_add_zero
thf(fact_805_h_Oimg__is__subalgebra,axiom,
    ! [K: set_list_a,V3: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( embedd1768981623711841426t_unit @ K @ V3 @ ( univ_poly_a_b @ r @ k ) )
       => ( embedd3586486045337765042t_unit @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ K ) @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ V3 ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.img_is_subalgebra
thf(fact_806_h_Oa__rcosetsI,axiom,
    ! [H2: set_set_list_a,X2: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( member8857465052274545133list_a @ ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 @ X2 ) @ ( a_RCOS7985542791853034265t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 ) ) ) ) ).

% h.a_rcosetsI
thf(fact_807_h_Oee__sym,axiom,
    ! [As: list_set_list_a,Bs: list_set_list_a] :
      ( ( essent1568894057571901190t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ As @ Bs )
     => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ As ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Bs ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( essent1568894057571901190t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Bs @ As ) ) ) ) ).

% h.ee_sym
thf(fact_808_h_Oee__trans,axiom,
    ! [As: list_set_list_a,Bs: list_set_list_a,Cs: list_set_list_a] :
      ( ( essent1568894057571901190t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ As @ Bs )
     => ( ( essent1568894057571901190t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Bs @ Cs )
       => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ As ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Bs ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Cs ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
             => ( essent1568894057571901190t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ As @ Cs ) ) ) ) ) ) ).

% h.ee_trans
thf(fact_809_h_Oconst__term__eq__last,axiom,
    ! [P: list_set_list_a,A: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( const_3308765751713425893t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( append_set_list_a @ P @ ( cons_set_list_a @ A @ nil_set_list_a ) ) )
          = A ) ) ) ).

% h.const_term_eq_last
thf(fact_810_UP_Osubalgebra__in__carrier,axiom,
    ! [K: set_list_a,V3: set_list_a] :
      ( ( embedd1768981623711841426t_unit @ K @ V3 @ ( univ_poly_a_b @ r @ k ) )
     => ( ord_le8861187494160871172list_a @ V3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.subalgebra_in_carrier
thf(fact_811_UP_Ocarrier__is__subalgebra,axiom,
    ! [K: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( embedd1768981623711841426t_unit @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.carrier_is_subalgebra
thf(fact_812_h_Onormalize__idem,axiom,
    ! [P: list_set_list_a,Q2: list_set_list_a] :
      ( ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( append_set_list_a @ ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P ) @ Q2 ) )
      = ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( append_set_list_a @ P @ Q2 ) ) ) ).

% h.normalize_idem
thf(fact_813_h_Oconst__term__explicit,axiom,
    ! [P: list_set_list_a,A: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( P != nil_set_list_a )
       => ( ( ( const_3308765751713425893t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P )
            = A )
         => ~ ! [P4: list_set_list_a] :
                ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P4 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
               => ( P
                 != ( append_set_list_a @ P4 @ ( cons_set_list_a @ A @ nil_set_list_a ) ) ) ) ) ) ) ).

% h.const_term_explicit
thf(fact_814_h_Oee__refl,axiom,
    ! [As: list_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ As ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( essent1568894057571901190t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ As @ As ) ) ).

% h.ee_refl
thf(fact_815_h_Oadd__additive__subgroups,axiom,
    ! [H2: set_set_list_a,K: set_set_list_a] :
      ( ( additi5174913095647240694t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( additi5174913095647240694t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
       => ( additi5174913095647240694t_unit @ ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 @ K ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.add_additive_subgroups
thf(fact_816_h_Ofactors__closed,axiom,
    ! [Fs: list_set_list_a,A: set_list_a] :
      ( ( factor7234795573716641441t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Fs @ A )
     => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Fs ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.factors_closed
thf(fact_817_h_Oline__extension__consistent,axiom,
    ! [K: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( embedd5951720228509767443t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : K
            @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = ( embedd5951720228509767443t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.line_extension_consistent
thf(fact_818_h_Opoly__add__append__zero,axiom,
    ! [P: list_set_list_a,Q2: list_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Q2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( append_set_list_a @ P @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ nil_set_list_a ) ) @ ( append_set_list_a @ Q2 @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ nil_set_list_a ) ) )
          = ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( append_set_list_a @ ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P @ Q2 ) @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ nil_set_list_a ) ) ) ) ) ) ).

% h.poly_add_append_zero
thf(fact_819_h_Omonic__degree__one__root__condition,axiom,
    ! [A: set_list_a,B: set_list_a] :
      ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( polyno4320237611291262604t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A ) @ nil_set_list_a ) ) @ B )
        = ( A = B ) ) ) ).

% h.monic_degree_one_root_condition
thf(fact_820_local_Onormalize__idem,axiom,
    ! [P: list_a,Q2: list_a] :
      ( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P ) @ Q2 ) )
      = ( normalize_a_b @ r @ ( append_a @ P @ Q2 ) ) ) ).

% local.normalize_idem
thf(fact_821_UP_Onormalize__idem,axiom,
    ! [P: list_list_a,Q2: list_list_a] :
      ( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ ( append_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) @ Q2 ) )
      = ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ ( append_list_a @ P @ Q2 ) ) ) ).

% UP.normalize_idem
thf(fact_822_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 )
         => ~ ! [P4: list_a] :
                ( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
               => ( P
                 != ( append_a @ P4 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).

% const_term_explicit
thf(fact_823_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_824_h_Oline__extension__in__carrier,axiom,
    ! [K: set_set_list_a,A: set_list_a,E2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( ord_le8877086941679407844list_a @ E2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ord_le8877086941679407844list_a @ ( embedd5951720228509767443t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ A @ E2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ) ).

% h.line_extension_in_carrier
thf(fact_825_h_Opoly__add__comm,axiom,
    ! [P1: list_set_list_a,P22: list_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P1 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P22 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P1 @ P22 )
          = ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P22 @ P1 ) ) ) ) ).

% h.poly_add_comm
thf(fact_826_h_Opoly__add__in__carrier,axiom,
    ! [P1: list_set_list_a,P22: list_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P1 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P22 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P1 @ P22 ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.poly_add_in_carrier
thf(fact_827_UP_Oconst__term__eq__last,axiom,
    ! [P: list_list_a,A: list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ ( append_list_a @ P @ ( cons_list_a @ A @ nil_list_a ) ) )
          = A ) ) ) ).

% UP.const_term_eq_last
thf(fact_828_UP_Oconst__term__explicit,axiom,
    ! [P: list_list_a,A: list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( P != nil_list_a )
       => ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P )
            = A )
         => ~ ! [P4: list_list_a] :
                ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
               => ( P
                 != ( append_list_a @ P4 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ).

% UP.const_term_explicit
thf(fact_829_h_Opoly__add__normalize_I3_J,axiom,
    ! [P1: list_set_list_a,P22: list_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P1 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P22 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P1 @ P22 )
          = ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P1 ) @ ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P22 ) ) ) ) ) ).

% h.poly_add_normalize(3)
thf(fact_830_h_Opoly__add__normalize_I2_J,axiom,
    ! [P1: list_set_list_a,P22: list_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P1 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P22 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P1 @ P22 )
          = ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P1 @ ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P22 ) ) ) ) ) ).

% h.poly_add_normalize(2)
thf(fact_831_h_Opoly__add__normalize__aux,axiom,
    ! [P1: list_set_list_a,P22: list_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P1 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P22 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P1 @ P22 )
          = ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P1 ) @ P22 ) ) ) ) ).

% h.poly_add_normalize_aux
thf(fact_832_h_Opoly__add__zero_H_I2_J,axiom,
    ! [P: list_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ nil_set_list_a @ P )
        = ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P ) ) ) ).

% h.poly_add_zero'(2)
thf(fact_833_h_Opoly__add__zero_H_I1_J,axiom,
    ! [P: list_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P @ nil_set_list_a )
        = ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P ) ) ) ).

% h.poly_add_zero'(1)
thf(fact_834_h_Oone__closed,axiom,
    member_set_list_a @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ).

% h.one_closed
thf(fact_835_h_Opoly__add__consistent,axiom,
    ! [K: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( poly_a3396581058288038507t_unit
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : K
            @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.poly_add_consistent
thf(fact_836_h_Oline__extension__hom,axiom,
    ! [K: set_list_a,A: list_a,E2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( embedd5951720228509767443t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ K ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ A ) @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ E2 ) )
            = ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E2 ) ) ) ) ) ) ).

% h.line_extension_hom
thf(fact_837_h_Opoly__add__hom_H,axiom,
    ! [P: list_list_a,Q2: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( normal4052021864830707619t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( map_li2657846702790696612list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) ) )
          = ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( map_li2657846702790696612list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ P ) @ ( map_li2657846702790696612list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ Q2 ) ) ) ) ) ).

% h.poly_add_hom'
thf(fact_838_UP_Opoly__mult__var,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( ( P = nil_list_a )
           => ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) )
              = nil_list_a ) )
          & ( ( P != nil_list_a )
           => ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ k ) ) )
              = ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ nil_list_a ) ) ) ) ) ) ) ).

% UP.poly_mult_var
thf(fact_839_h_Oimg__is__subfield_I1_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
         != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ K ) ) ) ).

% h.img_is_subfield(1)
thf(fact_840_zero__not__one,axiom,
    ( ( zero_a_b @ r )
   != ( one_a_ring_ext_a_b @ r ) ) ).

% zero_not_one
thf(fact_841_poly__add__in__carrier,axiom,
    ! [P1: list_a,P22: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P1 @ P22 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% poly_add_in_carrier
thf(fact_842_poly__add__comm,axiom,
    ! [P1: list_a,P22: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ P1 @ P22 )
          = ( poly_add_a_b @ r @ P22 @ P1 ) ) ) ) ).

% poly_add_comm
thf(fact_843_UP_Ozero__not__one,axiom,
    ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) )
   != ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.zero_not_one
thf(fact_844_UP_Osubring__props_I3_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) ) ).

% UP.subring_props(3)
thf(fact_845_poly__add__normalize__aux,axiom,
    ! [P1: list_a,P22: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ P1 @ P22 )
          = ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P22 ) ) ) ) ).

% poly_add_normalize_aux
thf(fact_846_poly__add__normalize_I2_J,axiom,
    ! [P1: list_a,P22: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ P1 @ P22 )
          = ( poly_add_a_b @ r @ P1 @ ( normalize_a_b @ r @ P22 ) ) ) ) ) ).

% poly_add_normalize(2)
thf(fact_847_poly__add__normalize_I3_J,axiom,
    ! [P1: list_a,P22: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ P1 @ P22 )
          = ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ ( normalize_a_b @ r @ P22 ) ) ) ) ) ).

% poly_add_normalize(3)
thf(fact_848_UP_Osubring__props_I2_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) ) ).

% UP.subring_props(2)
thf(fact_849_UP_Osubring__props_I5_J,axiom,
    ! [K: set_list_a,H: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_a @ H @ K )
       => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ H ) @ K ) ) ) ).

% UP.subring_props(5)
thf(fact_850_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_851_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_852_UP_Osubring__props_I1_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.subring_props(1)
thf(fact_853_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_854_UP_Oline__extension__in__carrier,axiom,
    ! [K: set_list_a,A: list_a,E2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).

% UP.line_extension_in_carrier
thf(fact_855_UP_Oline__extension__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd5150658419831591667t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) ) )
        = ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.line_extension_consistent
thf(fact_856_UP_Opoly__add__in__carrier,axiom,
    ! [P1: list_list_a,P22: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P1 @ P22 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.poly_add_in_carrier
thf(fact_857_UP_Opoly__add__comm,axiom,
    ! [P1: list_list_a,P22: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P1 @ P22 )
          = ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P22 @ P1 ) ) ) ) ).

% UP.poly_add_comm
thf(fact_858_UP_Opoly__add__normalize__aux,axiom,
    ! [P1: list_list_a,P22: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P1 @ P22 )
          = ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ P1 ) @ P22 ) ) ) ) ).

% UP.poly_add_normalize_aux
thf(fact_859_UP_Opoly__add__normalize_I2_J,axiom,
    ! [P1: list_list_a,P22: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P1 @ P22 )
          = ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P1 @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ P22 ) ) ) ) ) ).

% UP.poly_add_normalize(2)
thf(fact_860_UP_Opoly__add__normalize_I3_J,axiom,
    ! [P1: list_list_a,P22: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P1 @ P22 )
          = ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ P1 ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ P22 ) ) ) ) ) ).

% UP.poly_add_normalize(3)
thf(fact_861_UP_Omonic__degree__one__root__condition,axiom,
    ! [A: list_a,B: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ A ) @ nil_list_a ) ) @ B )
        = ( A = B ) ) ) ).

% UP.monic_degree_one_root_condition
thf(fact_862_UP_Ois__root__poly__mult__imp__is__root,axiom,
    ! [P: list_list_a,Q2: list_list_a,X2: list_a] :
      ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
       => ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) @ P @ Q2 ) @ X2 )
         => ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ X2 )
            | ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ Q2 @ X2 ) ) ) ) ) ).

% UP.is_root_poly_mult_imp_is_root
thf(fact_863_one__closed,axiom,
    member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% one_closed
thf(fact_864_poly__add__append__zero,axiom,
    ! [P: list_a,Q2: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( poly_add_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
          = ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P @ Q2 ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).

% poly_add_append_zero
thf(fact_865_UP_Opoly__add__zero_H_I1_J,axiom,
    ! [P: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ nil_list_a )
        = ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) ) ) ).

% UP.poly_add_zero'(1)
thf(fact_866_UP_Opoly__add__zero_H_I2_J,axiom,
    ! [P: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a @ P )
        = ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) ) ) ).

% UP.poly_add_zero'(2)
thf(fact_867_UP_Opoly__add__append__zero,axiom,
    ! [P: list_list_a,Q2: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ nil_list_a ) ) @ ( append_list_a @ Q2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ nil_list_a ) ) )
          = ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ k ) @ ( append_list_a @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ nil_list_a ) ) ) ) ) ) ).

% UP.poly_add_append_zero
thf(fact_868_poly__add__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( poly_add_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r ) )
        = ( poly_add_a_b @ r ) ) ) ).

% poly_add_consistent
thf(fact_869_UP_Oone__closed,axiom,
    member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).

% UP.one_closed
thf(fact_870_UP_Opoly__add__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( poly_a7601779127272115787t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) ) )
        = ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.poly_add_consistent
thf(fact_871_h_Ohom__one,axiom,
    ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
    = ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ).

% h.hom_one
thf(fact_872_UP_Ouniv__poly__is__principal,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% UP.univ_poly_is_principal
thf(fact_873_UP_Oexists__unique__long__division,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( Q2 != nil_list_a )
           => ? [X3: produc7709606177366032167list_a] :
                ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 @ X3 )
                & ! [Y5: produc7709606177366032167list_a] :
                    ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 @ Y5 )
                   => ( Y5 = X3 ) ) ) ) ) ) ) ).

% UP.exists_unique_long_division
thf(fact_874_UP_OpprimeE_I1_J,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P )
         => ( P != nil_list_a ) ) ) ) ).

% UP.pprimeE(1)
thf(fact_875_UP_Ouniv__poly__infinite__dimension,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ~ ( embedd2411333406617385593t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ).

% UP.univ_poly_infinite_dimension
thf(fact_876_subring__props_I6_J,axiom,
    ! [K: set_a,H1: a,H22: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ H1 @ K )
       => ( ( member_a @ H22 @ K )
         => ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K ) ) ) ) ).

% subring_props(6)
thf(fact_877_m__assoc,axiom,
    ! [X2: a,Y: a,Z4: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ Z4 )
            = ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z4 ) ) ) ) ) ) ).

% m_assoc
thf(fact_878_m__comm,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
          = ( mult_a_ring_ext_a_b @ r @ Y @ X2 ) ) ) ) ).

% m_comm
thf(fact_879_m__lcomm,axiom,
    ! [X2: a,Y: a,Z4: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z4 ) )
            = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X2 @ Z4 ) ) ) ) ) ) ).

% m_lcomm
thf(fact_880_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_881_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_882_subring__props_I5_J,axiom,
    ! [K: set_a,H: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ H @ K )
       => ( member_a @ ( a_inv_a_b @ r @ H ) @ K ) ) ) ).

% subring_props(5)
thf(fact_883_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_884_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_885_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_886_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_887_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_888_inv__unique,axiom,
    ! [Y: a,X2: a,Y6: a] :
      ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X2 )
        = ( one_a_ring_ext_a_b @ r ) )
     => ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y6 )
          = ( one_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ r ) )
             => ( Y = Y6 ) ) ) ) ) ) ).

% inv_unique
thf(fact_889_one__unique,axiom,
    ! [U: a] :
      ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
              = X3 ) )
       => ( U
          = ( one_a_ring_ext_a_b @ r ) ) ) ) ).

% one_unique
thf(fact_890_l__minus,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ Y )
          = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) ) ) ) ) ).

% l_minus
thf(fact_891_r__minus,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( a_inv_a_b @ r @ Y ) )
          = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) ) ) ) ) ).

% r_minus
thf(fact_892_square__eq__one,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ X2 )
          = ( one_a_ring_ext_a_b @ r ) )
       => ( ( X2
            = ( one_a_ring_ext_a_b @ r ) )
          | ( X2
            = ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).

% square_eq_one
thf(fact_893_univ__poly__subfield__of__consts,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_subfield_of_consts
thf(fact_894_UP_Om__assoc,axiom,
    ! [X2: list_a,Y: list_a,Z4: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Z4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) @ Z4 )
            = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Z4 ) ) ) ) ) ) ).

% UP.m_assoc
thf(fact_895_UP_Om__comm,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y )
          = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X2 ) ) ) ) ).

% UP.m_comm
thf(fact_896_UP_Om__lcomm,axiom,
    ! [X2: list_a,Y: list_a,Z4: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Z4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Z4 ) )
            = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Z4 ) ) ) ) ) ) ).

% UP.m_lcomm
thf(fact_897_UP_Osubring__props_I6_J,axiom,
    ! [K: set_list_a,H1: list_a,H22: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_a @ H1 @ K )
       => ( ( member_list_a @ H22 @ K )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H1 @ H22 ) @ K ) ) ) ) ).

% UP.subring_props(6)
thf(fact_898_h_Osubring__props_I6_J,axiom,
    ! [K: set_set_list_a,H1: set_list_a,H22: set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( member_set_list_a @ H1 @ K )
       => ( ( member_set_list_a @ H22 @ K )
         => ( member_set_list_a @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H1 @ H22 ) @ K ) ) ) ) ).

% h.subring_props(6)
thf(fact_899_is__root__poly__mult__imp__is__root,axiom,
    ! [P: list_a,Q2: list_a,X2: a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Q2 @ ( 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 @ Q2 ) @ X2 )
         => ( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
            | ( polyno4133073214067823460ot_a_b @ r @ Q2 @ X2 ) ) ) ) ) ).

% is_root_poly_mult_imp_is_root
thf(fact_900_const__term__simprules__shell_I2_J,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 ) )
            = ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q2 ) ) ) ) ) ) ).

% const_term_simprules_shell(2)
thf(fact_901_UP_Ointegral,axiom,
    ! [A: list_a,B: list_a] :
      ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( A
              = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
            | ( B
              = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).

% UP.integral
thf(fact_902_UP_Ointegral__iff,axiom,
    ! [A: list_a,B: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          = ( ( A
              = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
            | ( B
              = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).

% UP.integral_iff
thf(fact_903_UP_Om__lcancel,axiom,
    ! [A: list_a,B: list_a,C: list_a] :
      ( ( A
       != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( 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 ) ) )
           => ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ B )
                = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C ) )
              = ( B = C ) ) ) ) ) ) ).

% UP.m_lcancel
thf(fact_904_UP_Om__rcancel,axiom,
    ! [A: list_a,B: list_a,C: list_a] :
      ( ( A
       != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( 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 ) ) )
           => ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ A )
                = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ C @ A ) )
              = ( B = C ) ) ) ) ) ) ).

% UP.m_rcancel
thf(fact_905_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_906_UP_Oinv__unique,axiom,
    ! [Y: list_a,X2: list_a,Y6: list_a] :
      ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y6 )
          = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
             => ( Y = Y6 ) ) ) ) ) ) ).

% UP.inv_unique
thf(fact_907_UP_Oone__unique,axiom,
    ! [U: list_a] :
      ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ! [X3: list_a] :
            ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ U @ X3 )
              = X3 ) )
       => ( U
          = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.one_unique
thf(fact_908_UP_Ol__minus,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) @ Y )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) ) ) ) ) ).

% UP.l_minus
thf(fact_909_UP_Or__minus,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) ) ) ) ) ).

% UP.r_minus
thf(fact_910_h_Om__assoc,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z4: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Z4 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) @ Z4 )
            = ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ Z4 ) ) ) ) ) ) ).

% h.m_assoc
thf(fact_911_h_Osubring__props_I2_J,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ K ) ) ).

% h.subring_props(2)
thf(fact_912_h_Osubring__props_I5_J,axiom,
    ! [K: set_set_list_a,H: set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( member_set_list_a @ H @ K )
       => ( member_set_list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H ) @ K ) ) ) ).

% h.subring_props(5)
thf(fact_913_h_Osubring__props_I3_J,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( member_set_list_a @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ K ) ) ).

% h.subring_props(3)
thf(fact_914_UP_Osquare__eq__one,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ X2 )
          = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( X2
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          | ( X2
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).

% UP.square_eq_one
thf(fact_915_h_Osubring__props_I1_J,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ord_le8877086941679407844list_a @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.subring_props(1)
thf(fact_916_h_Ol__minus,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 ) @ Y )
          = ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) ) ) ) ) ).

% h.l_minus
thf(fact_917_h_Or__minus,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y ) )
          = ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) ) ) ) ) ).

% h.r_minus
thf(fact_918_h_Oinv__unique,axiom,
    ! [Y: set_list_a,X2: set_list_a,Y6: set_list_a] :
      ( ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ X2 )
        = ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y6 )
          = ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( member_set_list_a @ Y6 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
             => ( Y = Y6 ) ) ) ) ) ) ).

% h.inv_unique
thf(fact_919_h_Oone__unique,axiom,
    ! [U: set_list_a] :
      ( ( member_set_list_a @ U @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ! [X3: set_list_a] :
            ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ U @ X3 )
              = X3 ) )
       => ( U
          = ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.one_unique
thf(fact_920_h_Ouniv__poly__subfield__of__consts,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( subfie868023609828841932t_unit @ ( image_3509949574358669579list_a @ ( poly_o3535013565302768950t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ K ) @ ( univ_p863672496597069550t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K ) ) ) ).

% h.univ_poly_subfield_of_consts
thf(fact_921_UP_Oline__extension__smult__closed,axiom,
    ! [K: set_list_a,E2: set_list_a,A: list_a,K5: list_a,U: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ! [K3: list_a,V2: list_a] :
            ( ( member_list_a @ K3 @ K )
           => ( ( member_list_a @ V2 @ E2 )
             => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K3 @ V2 ) @ E2 ) ) )
       => ( ( ord_le8861187494160871172list_a @ E2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( member_list_a @ K5 @ K )
             => ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E2 ) )
               => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K5 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E2 ) ) ) ) ) ) ) ) ).

% UP.line_extension_smult_closed
thf(fact_922_h_Oline__extension__smult__closed,axiom,
    ! [K: set_set_list_a,E2: set_set_list_a,A: set_list_a,K5: set_list_a,U: set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ! [K3: set_list_a,V2: set_list_a] :
            ( ( member_set_list_a @ K3 @ K )
           => ( ( member_set_list_a @ V2 @ E2 )
             => ( member_set_list_a @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K3 @ V2 ) @ E2 ) ) )
       => ( ( ord_le8877086941679407844list_a @ E2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( member_set_list_a @ K5 @ K )
             => ( ( member_set_list_a @ U @ ( embedd5951720228509767443t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ A @ E2 ) )
               => ( member_set_list_a @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K5 @ U ) @ ( embedd5951720228509767443t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ A @ E2 ) ) ) ) ) ) ) ) ).

% h.line_extension_smult_closed
thf(fact_923_UP_Ouniv__poly__subfield__of__consts,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ k ) ) @ K ) @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% UP.univ_poly_subfield_of_consts
thf(fact_924_UP_Oconst__term__simprules__shell_I2_J,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P @ Q2 ) )
            = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ Q2 ) ) ) ) ) ) ).

% UP.const_term_simprules_shell(2)
thf(fact_925_m__closed,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% m_closed
thf(fact_926_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_927_h_Ofactors__mult,axiom,
    ! [Fa: list_set_list_a,A: set_list_a,Fb: list_set_list_a,B: set_list_a] :
      ( ( factor7234795573716641441t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Fa @ A )
     => ( ( factor7234795573716641441t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Fb @ B )
       => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Fa ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Fb ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( factor7234795573716641441t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( append_set_list_a @ Fa @ Fb ) @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A @ B ) ) ) ) ) ) ).

% h.factors_mult
thf(fact_928_h_Oimg__is__subfield_I2_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
         != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( subfie4339374749748326226t_unit @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ K ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.img_is_subfield(2)
thf(fact_929_l__null,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X2 )
        = ( zero_a_b @ r ) ) ) ).

% l_null
thf(fact_930_r__null,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( zero_a_b @ r ) )
        = ( zero_a_b @ r ) ) ) ).

% r_null
thf(fact_931_r__one,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( one_a_ring_ext_a_b @ r ) )
        = X2 ) ) ).

% r_one
thf(fact_932_l__one,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X2 )
        = X2 ) ) ).

% l_one
thf(fact_933_UP_Om__closed,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.m_closed
thf(fact_934_UP_Ol__null,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.l_null
thf(fact_935_UP_Or__null,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.r_null
thf(fact_936_UP_Ol__one,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X2 )
        = X2 ) ) ).

% UP.l_one
thf(fact_937_UP_Or__one,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        = X2 ) ) ).

% UP.r_one
thf(fact_938_h_Om__closed,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( member_set_list_a @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.m_closed
thf(fact_939_h_Ol__null,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ X2 )
        = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.l_null
thf(fact_940_h_Or__null,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.r_null
thf(fact_941_h_Ol__one,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ X2 )
        = X2 ) ) ).

% h.l_one
thf(fact_942_h_Or__one,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = X2 ) ) ).

% h.r_one
thf(fact_943_h_Ohom__mult,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) )
          = ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ X2 ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ Y ) ) ) ) ) ).

% h.hom_mult
thf(fact_944_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_945_exists__unique__long__division,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q2 != nil_a )
           => ? [X3: produc9164743771328383783list_a] :
                ( ( polyno2806191415236617128es_a_b @ r @ P @ Q2 @ X3 )
                & ! [Y5: produc9164743771328383783list_a] :
                    ( ( polyno2806191415236617128es_a_b @ r @ P @ Q2 @ Y5 )
                   => ( Y5 = X3 ) ) ) ) ) ) ) ).

% exists_unique_long_division
thf(fact_946_UP_Olong__division__a__inv_I1_J,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) @ Q2 )
            = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) ) ) ) ) ) ).

% UP.long_division_a_inv(1)
thf(fact_947_UP_Olong__division__zero_I1_J,axiom,
    ! [K: set_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a @ Q2 )
          = nil_list_a ) ) ) ).

% UP.long_division_zero(1)
thf(fact_948_UP_Otelescopic__base__dim_I1_J,axiom,
    ! [K: set_list_a,F2: set_list_a,E2: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ k ) )
       => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ F2 )
         => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ F2 @ E2 )
           => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E2 ) ) ) ) ) ).

% UP.telescopic_base_dim(1)
thf(fact_949_UP_Osum__space__dim_I1_J,axiom,
    ! [K: set_list_a,E2: set_list_a,F2: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E2 )
       => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ F2 )
         => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ E2 @ F2 ) ) ) ) ) ).

% UP.sum_space_dim(1)
thf(fact_950_UP_Ofinite__dimension__imp__subalgebra,axiom,
    ! [K: set_list_a,E2: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E2 )
       => ( embedd1768981623711841426t_unit @ K @ E2 @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.finite_dimension_imp_subalgebra
thf(fact_951_UP_Osubalbegra__incl__imp__finite__dimension,axiom,
    ! [K: set_list_a,E2: set_list_a,V3: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E2 )
       => ( ( embedd1768981623711841426t_unit @ K @ V3 @ ( univ_poly_a_b @ r @ k ) )
         => ( ( ord_le8861187494160871172list_a @ V3 @ E2 )
           => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ V3 ) ) ) ) ) ).

% UP.subalbegra_incl_imp_finite_dimension
thf(fact_952_univ__poly__infinite__dimension,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).

% univ_poly_infinite_dimension
thf(fact_953_UP_Olong__division__closed_I1_J,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( member_list_list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ) ) ).

% UP.long_division_closed(1)
thf(fact_954_h_Oinfinite__dimension__hom,axiom,
    ! [K: set_list_a,E2: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
         != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ E2 )
         => ( ( embedd1768981623711841426t_unit @ K @ E2 @ ( univ_poly_a_b @ r @ k ) )
           => ( ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ E2 )
             => ~ ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ K ) @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ E2 ) ) ) ) ) ) ) ).

% h.infinite_dimension_hom
thf(fact_955_h_Omonoid__cancelI,axiom,
    ( ! [A3: set_list_a,B3: set_list_a,C2: set_list_a] :
        ( ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ C2 @ A3 )
          = ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ C2 @ B3 ) )
       => ( ( member_set_list_a @ A3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( member_set_list_a @ B3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( member_set_list_a @ C2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
             => ( A3 = B3 ) ) ) ) )
   => ( ! [A3: set_list_a,B3: set_list_a,C2: set_list_a] :
          ( ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A3 @ C2 )
            = ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ B3 @ C2 ) )
         => ( ( member_set_list_a @ A3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( member_set_list_a @ B3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
             => ( ( member_set_list_a @ C2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
               => ( A3 = B3 ) ) ) ) )
     => ( monoid5701485383557686215t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.monoid_cancelI
thf(fact_956_h_OsubdomainI,axiom,
    ! [H2: set_set_list_a] :
      ( ( subcri7783154434480317835t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
         != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ! [H12: set_list_a,H23: set_list_a] :
              ( ( member_set_list_a @ H12 @ H2 )
             => ( ( member_set_list_a @ H23 @ H2 )
               => ( ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H12 @ H23 )
                    = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
                 => ( ( H12
                      = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
                    | ( H23
                      = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ) )
         => ( subdom3220114454046903646t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.subdomainI
thf(fact_957_long__division__closed_I1_J,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% long_division_closed(1)
thf(fact_958_long__division__zero_I1_J,axiom,
    ! [K: set_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q2 )
          = nil_a ) ) ) ).

% long_division_zero(1)
thf(fact_959_long__division__a__inv_I1_J,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q2 )
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P @ Q2 ) ) ) ) ) ) ).

% long_division_a_inv(1)
thf(fact_960_h_Otelescopic__base__dim_I1_J,axiom,
    ! [K: set_set_list_a,F2: set_set_list_a,E2: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( subfie4339374749748326226t_unit @ F2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
       => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ F2 )
         => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ F2 @ E2 )
           => ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ E2 ) ) ) ) ) ).

% h.telescopic_base_dim(1)
thf(fact_961_h_Osum__space__dim_I1_J,axiom,
    ! [K: set_set_list_a,E2: set_set_list_a,F2: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ E2 )
       => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ F2 )
         => ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ E2 @ F2 ) ) ) ) ) ).

% h.sum_space_dim(1)
thf(fact_962_h_OsubcringI,axiom,
    ! [H2: set_set_list_a] :
      ( ( subrin5643252653130547402t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ! [H12: set_list_a,H23: set_list_a] :
            ( ( member_set_list_a @ H12 @ H2 )
           => ( ( member_set_list_a @ H23 @ H2 )
             => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H12 @ H23 )
                = ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H23 @ H12 ) ) ) )
       => ( subcri7783154434480317835t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.subcringI
thf(fact_963_h_Ofinite__dimension__imp__subalgebra,axiom,
    ! [K: set_set_list_a,E2: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ E2 )
       => ( embedd3586486045337765042t_unit @ K @ E2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.finite_dimension_imp_subalgebra
thf(fact_964_h_Osubalbegra__incl__imp__finite__dimension,axiom,
    ! [K: set_set_list_a,E2: set_set_list_a,V3: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ E2 )
       => ( ( embedd3586486045337765042t_unit @ K @ V3 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
         => ( ( ord_le8877086941679407844list_a @ V3 @ E2 )
           => ( embedd3118038802735549087t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ V3 ) ) ) ) ) ).

% h.subalbegra_incl_imp_finite_dimension
thf(fact_965_h_Oeval__append__aux,axiom,
    ! [P: list_set_list_a,B: set_list_a,A: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( append_set_list_a @ P @ ( cons_set_list_a @ B @ nil_set_list_a ) ) @ A )
            = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P @ A ) @ A ) @ B ) ) ) ) ) ).

% h.eval_append_aux
thf(fact_966_UP_Orupture__eq__0__iff,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) @ Q2 )
              = ( zero_s2920163772466840039t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P ) ) )
            = ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) ) ) ) ) ).

% UP.rupture_eq_0_iff
thf(fact_967_UP_Ozero__pdivides__zero,axiom,
    polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a @ nil_list_a ).

% UP.zero_pdivides_zero
thf(fact_968_UP_Ozero__pdivides,axiom,
    ! [P: list_list_a] :
      ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a @ P )
      = ( P = nil_list_a ) ) ).

% UP.zero_pdivides
thf(fact_969_h_Oadd_Or__cancel,axiom,
    ! [A: set_list_a,C: set_list_a,B: set_list_a] :
      ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A @ C )
        = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ B @ C ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( A = B ) ) ) ) ) ).

% h.add.r_cancel
thf(fact_970_h_Oadd_Om__lcomm,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z4: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Z4 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ Z4 ) )
            = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Z4 ) ) ) ) ) ) ).

% h.add.m_lcomm
thf(fact_971_h_Oadd_Om__comm,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y )
          = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ X2 ) ) ) ) ).

% h.add.m_comm
thf(fact_972_h_Oadd_Om__assoc,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z4: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Z4 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) @ Z4 )
            = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ Z4 ) ) ) ) ) ) ).

% h.add.m_assoc
thf(fact_973_h_Oadd_Ol__cancel,axiom,
    ! [C: set_list_a,A: set_list_a,B: set_list_a] :
      ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ C @ A )
        = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ C @ B ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( A = B ) ) ) ) ) ).

% h.add.l_cancel
thf(fact_974_h_Osubring__props_I7_J,axiom,
    ! [K: set_set_list_a,H1: set_list_a,H22: set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( ( member_set_list_a @ H1 @ K )
       => ( ( member_set_list_a @ H22 @ K )
         => ( member_set_list_a @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H1 @ H22 ) @ K ) ) ) ) ).

% h.subring_props(7)
thf(fact_975_h_Ominus__unique,axiom,
    ! [Y: set_list_a,X2: set_list_a,Y6: set_list_a] :
      ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ X2 )
        = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y6 )
          = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( member_set_list_a @ Y6 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
             => ( Y = Y6 ) ) ) ) ) ) ).

% h.minus_unique
thf(fact_976_h_Oadd_Or__inv__ex,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ? [X3: set_list_a] :
          ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
          & ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ X3 )
            = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.add.r_inv_ex
thf(fact_977_h_Oadd_Oone__unique,axiom,
    ! [U: set_list_a] :
      ( ( member_set_list_a @ U @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ! [X3: set_list_a] :
            ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ U @ X3 )
              = X3 ) )
       => ( U
          = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.add.one_unique
thf(fact_978_h_Oadd_Ol__inv__ex,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ? [X3: set_list_a] :
          ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
          & ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X3 @ X2 )
            = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.add.l_inv_ex
thf(fact_979_h_Oadd_Oinv__comm,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y )
        = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ X2 )
            = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ) ).

% h.add.inv_comm
thf(fact_980_h_Or__distr,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z4: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Z4 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Z4 @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) )
            = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Z4 @ X2 ) @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Z4 @ Y ) ) ) ) ) ) ).

% h.r_distr
thf(fact_981_h_Ol__distr,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z4: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Z4 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) @ Z4 )
            = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Z4 ) @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ Z4 ) ) ) ) ) ) ).

% h.l_distr
thf(fact_982_h_Or__neg2,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 ) @ Y ) )
          = Y ) ) ) ).

% h.r_neg2
thf(fact_983_h_Or__neg1,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 ) @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) )
          = Y ) ) ) ).

% h.r_neg1
thf(fact_984_h_Ominus__add,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) )
          = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y ) ) ) ) ) ).

% h.minus_add
thf(fact_985_h_Oadd_Oinv__solve__right_H,axiom,
    ! [A: set_list_a,B: set_list_a,C: set_list_a] :
      ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ B @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ C ) )
              = A )
            = ( B
              = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A @ C ) ) ) ) ) ) ).

% h.add.inv_solve_right'
thf(fact_986_h_Oadd_Oinv__solve__right,axiom,
    ! [A: set_list_a,B: set_list_a,C: set_list_a] :
      ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( A
              = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ B @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ C ) ) )
            = ( B
              = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A @ C ) ) ) ) ) ) ).

% h.add.inv_solve_right
thf(fact_987_h_Oadd_Oinv__solve__left_H,axiom,
    ! [A: set_list_a,B: set_list_a,C: set_list_a] :
      ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ B ) @ C )
              = A )
            = ( C
              = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ B @ A ) ) ) ) ) ) ).

% h.add.inv_solve_left'
thf(fact_988_h_Oadd_Oinv__solve__left,axiom,
    ! [A: set_list_a,B: set_list_a,C: set_list_a] :
      ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( A
              = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ B ) @ C ) )
            = ( C
              = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ B @ A ) ) ) ) ) ) ).

% h.add.inv_solve_left
thf(fact_989_h_Oadd_Oinv__mult__group,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) )
          = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 ) ) ) ) ) ).

% h.add.inv_mult_group
thf(fact_990_h_Oa__transpose__inv,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z4: set_list_a] :
      ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y )
        = Z4 )
     => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( member_set_list_a @ Z4 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 ) @ Z4 )
              = Y ) ) ) ) ) ).

% h.a_transpose_inv
thf(fact_991_h_Oline__extension__mem__iff,axiom,
    ! [U: set_list_a,K: set_set_list_a,A: set_list_a,E2: set_set_list_a] :
      ( ( member_set_list_a @ U @ ( embedd5951720228509767443t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ K @ A @ E2 ) )
      = ( ? [X: set_list_a] :
            ( ( member_set_list_a @ X @ K )
            & ? [Y4: set_list_a] :
                ( ( member_set_list_a @ Y4 @ E2 )
                & ( U
                  = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X @ A ) @ Y4 ) ) ) ) ) ) ).

% h.line_extension_mem_iff
thf(fact_992_h_Oadd_Oinj__on__multc,axiom,
    ! [C: set_list_a] :
      ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( inj_on727569427812895985list_a
        @ ^ [X: set_list_a] : ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X @ C )
        @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.add.inj_on_multc
thf(fact_993_h_Oadd_Oinj__on__cmult,axiom,
    ! [C: set_list_a] :
      ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( inj_on727569427812895985list_a @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ C ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.add.inj_on_cmult
thf(fact_994_UP_Opdivides__zero,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ nil_list_a ) ) ) ).

% UP.pdivides_zero
thf(fact_995_h_Or__neg,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 ) )
        = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.r_neg
thf(fact_996_h_Ominus__equality,axiom,
    ! [Y: set_list_a,X2: set_list_a] :
      ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ X2 )
        = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 )
            = Y ) ) ) ) ).

% h.minus_equality
thf(fact_997_h_Ol__neg,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 ) @ X2 )
        = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.l_neg
thf(fact_998_h_Oa__rcosI,axiom,
    ! [H: set_list_a,H2: set_set_list_a,X2: set_list_a] :
      ( ( member_set_list_a @ H @ H2 )
     => ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( member_set_list_a @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H @ X2 ) @ ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 @ X2 ) ) ) ) ) ).

% h.a_rcosI
thf(fact_999_h_Oa__coset__add__assoc,axiom,
    ! [M: set_set_list_a,G: set_list_a,H: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ M @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ G @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ H @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ M @ G ) @ H )
            = ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ M @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ G @ H ) ) ) ) ) ) ).

% h.a_coset_add_assoc
thf(fact_1000_h_Oadd_Osurj__const__mult,axiom,
    ! [A: set_list_a] :
      ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( image_5749939591322298757list_a @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.add.surj_const_mult
thf(fact_1001_h_Oa__lcos__m__assoc,axiom,
    ! [M: set_set_list_a,G: set_list_a,H: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ M @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ G @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ H @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( a_l_co4135707798524667186t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ G @ ( a_l_co4135707798524667186t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H @ M ) )
            = ( a_l_co4135707798524667186t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ G @ H ) @ M ) ) ) ) ) ).

% h.a_lcos_m_assoc
thf(fact_1002_h_Oadd_Oinj__on__g,axiom,
    ! [H2: set_set_list_a,A: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( inj_on727569427812895985list_a
          @ ^ [Y4: set_list_a] : ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y4 @ A )
          @ H2 ) ) ) ).

% h.add.inj_on_g
thf(fact_1003_UP_Opdivides__imp__root__sharing,axiom,
    ! [P: list_list_a,Q2: list_list_a,A: list_a] :
      ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ A )
              = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ Q2 @ A )
              = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).

% UP.pdivides_imp_root_sharing
thf(fact_1004_h_Oa__coset__add__inv2,axiom,
    ! [M: set_set_list_a,X2: set_list_a,Y: set_list_a] :
      ( ( ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ M @ X2 )
        = ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ M @ Y ) )
     => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( ord_le8877086941679407844list_a @ M @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ M @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y ) ) )
              = M ) ) ) ) ) ).

% h.a_coset_add_inv2
thf(fact_1005_h_Oa__coset__add__inv1,axiom,
    ! [M: set_set_list_a,X2: set_list_a,Y: set_list_a] :
      ( ( ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ M @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y ) ) )
        = M )
     => ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( ord_le8877086941679407844list_a @ M @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
           => ( ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ M @ X2 )
              = ( a_r_co866426763593333676t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ M @ Y ) ) ) ) ) ) ).

% h.a_coset_add_inv1
thf(fact_1006_UP_OpprimeE_I3_J,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a,R4: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P )
         => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
           => ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
             => ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ Q2 @ R4 ) )
               => ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 )
                  | ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ R4 ) ) ) ) ) ) ) ) ).

% UP.pprimeE(3)
thf(fact_1007_h_Oeval__poly__add,axiom,
    ! [P: list_set_list_a,Q2: list_set_list_a,A: set_list_a] :
      ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ P ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ ( set_set_list_a2 @ Q2 ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( poly_a3396581058288038507t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P @ Q2 ) @ A )
            = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ P @ A ) @ ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Q2 @ A ) ) ) ) ) ) ).

% h.eval_poly_add
thf(fact_1008_UP_Opdivides__imp__is__root,axiom,
    ! [P: list_list_a,X2: list_a] :
      ( ( P != nil_list_a )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) @ nil_list_a ) ) @ P )
         => ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ X2 ) ) ) ) ).

% UP.pdivides_imp_is_root
thf(fact_1009_h_OsubringI,axiom,
    ! [H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ H2 )
       => ( ! [H3: set_list_a] :
              ( ( member_set_list_a @ H3 @ H2 )
             => ( member_set_list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H3 ) @ H2 ) )
         => ( ! [H12: set_list_a,H23: set_list_a] :
                ( ( member_set_list_a @ H12 @ H2 )
               => ( ( member_set_list_a @ H23 @ H2 )
                 => ( member_set_list_a @ ( mult_s7802724872828879953t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H12 @ H23 ) @ H2 ) ) )
           => ( ! [H12: set_list_a,H23: set_list_a] :
                  ( ( member_set_list_a @ H12 @ H2 )
                 => ( ( member_set_list_a @ H23 @ H2 )
                   => ( member_set_list_a @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H12 @ H23 ) @ H2 ) ) )
             => ( subrin5643252653130547402t_unit @ H2 @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ) ) ).

% h.subringI
thf(fact_1010_UP_Ois__root__imp__pdivides,axiom,
    ! [P: list_list_a,X2: list_a] :
      ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) )
     => ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ X2 )
       => ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) @ nil_list_a ) ) @ P ) ) ) ).

% UP.is_root_imp_pdivides
thf(fact_1011_h_Oadd_Oright__cancel,axiom,
    ! [X2: set_list_a,Y: set_list_a,Z4: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( member_set_list_a @ Z4 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
         => ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y @ X2 )
              = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Z4 @ X2 ) )
            = ( Y = Z4 ) ) ) ) ) ).

% h.add.right_cancel
thf(fact_1012_h_Oadd_Om__closed,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( member_set_list_a @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.add.m_closed
thf(fact_1013_h_Or__zero,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
        = X2 ) ) ).

% h.r_zero
thf(fact_1014_h_Ol__zero,axiom,
    ! [X2: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ X2 )
        = X2 ) ) ).

% h.l_zero
thf(fact_1015_h_Oadd_Or__cancel__one_H,axiom,
    ! [X2: set_list_a,A: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( X2
            = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A @ X2 ) )
          = ( A
            = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ) ).

% h.add.r_cancel_one'
thf(fact_1016_h_Oadd_Or__cancel__one,axiom,
    ! [X2: set_list_a,A: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A @ X2 )
            = X2 )
          = ( A
            = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ) ).

% h.add.r_cancel_one
thf(fact_1017_h_Oadd_Ol__cancel__one_H,axiom,
    ! [X2: set_list_a,A: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( X2
            = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ A ) )
          = ( A
            = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ) ).

% h.add.l_cancel_one'
thf(fact_1018_h_Oadd_Ol__cancel__one,axiom,
    ! [X2: set_list_a,A: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ A )
            = X2 )
          = ( A
            = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ) ).

% h.add.l_cancel_one
thf(fact_1019_h_Ominus__eq,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( a_minu2642007939804611572t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y )
      = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ Y ) ) ) ).

% h.minus_eq
thf(fact_1020_UP_Orupture__carrier__as__pmod__image,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( image_551801017575455717list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) )
            @ ( image_5446147394702115205list_a
              @ ^ [Q3: list_list_a] : ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ Q3 @ P )
              @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) )
          = ( partia3317168157747563407t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ P ) ) ) ) ) ).

% UP.rupture_carrier_as_pmod_image
thf(fact_1021_UP_Oeval__append__aux,axiom,
    ! [P: list_list_a,B: list_a,A: list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ ( append_list_a @ P @ ( cons_list_a @ B @ nil_list_a ) ) @ A )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ A ) @ A ) @ B ) ) ) ) ) ).

% UP.eval_append_aux
thf(fact_1022_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_1023_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_1024_a__assoc,axiom,
    ! [X2: a,Y: a,Z4: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( add_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) @ Z4 )
            = ( add_a_b @ r @ X2 @ ( add_a_b @ r @ Y @ Z4 ) ) ) ) ) ) ).

% a_assoc
thf(fact_1025_a__comm,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( add_a_b @ r @ X2 @ Y )
          = ( add_a_b @ r @ Y @ X2 ) ) ) ) ).

% a_comm
thf(fact_1026_a__lcomm,axiom,
    ! [X2: a,Y: a,Z4: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( add_a_b @ r @ X2 @ ( add_a_b @ r @ Y @ Z4 ) )
            = ( add_a_b @ r @ Y @ ( add_a_b @ r @ X2 @ Z4 ) ) ) ) ) ) ).

% a_lcomm
thf(fact_1027_carrier__is__subcring,axiom,
    subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% carrier_is_subcring
thf(fact_1028_zero__pdivides,axiom,
    ! [P: list_a] :
      ( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
      = ( P = nil_a ) ) ).

% zero_pdivides
thf(fact_1029_zero__pdivides__zero,axiom,
    polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).

% zero_pdivides_zero
thf(fact_1030_subring__props_I7_J,axiom,
    ! [K: set_a,H1: a,H22: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ H1 @ K )
       => ( ( member_a @ H22 @ K )
         => ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K ) ) ) ) ).

% subring_props(7)
thf(fact_1031_subcringI_H,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( subcring_a_b @ H2 @ r ) ) ).

% subcringI'
thf(fact_1032_add_Osurj__const__mult,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( image_a_a @ ( add_a_b @ r @ A ) @ ( partia707051561876973205xt_a_b @ r ) )
        = ( partia707051561876973205xt_a_b @ r ) ) ) ).

% add.surj_const_mult
thf(fact_1033_add_Oinv__comm,axiom,
    ! [X2: a,Y: a] :
      ( ( ( add_a_b @ r @ X2 @ Y )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( add_a_b @ r @ Y @ X2 )
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.inv_comm
thf(fact_1034_add_Ol__inv__ex,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( add_a_b @ r @ X3 @ X2 )
            = ( zero_a_b @ r ) ) ) ) ).

% add.l_inv_ex
thf(fact_1035_add_Oone__unique,axiom,
    ! [U: a] :
      ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( add_a_b @ r @ U @ X3 )
              = X3 ) )
       => ( U
          = ( zero_a_b @ r ) ) ) ) ).

% add.one_unique
thf(fact_1036_add_Or__inv__ex,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( add_a_b @ r @ X2 @ X3 )
            = ( zero_a_b @ r ) ) ) ) ).

% add.r_inv_ex
thf(fact_1037_local_Ominus__unique,axiom,
    ! [Y: a,X2: a,Y6: a] :
      ( ( ( add_a_b @ r @ Y @ X2 )
        = ( zero_a_b @ r ) )
     => ( ( ( add_a_b @ r @ X2 @ Y6 )
          = ( zero_a_b @ r ) )
       => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ Y6 @ ( partia707051561876973205xt_a_b @ r ) )
             => ( Y = Y6 ) ) ) ) ) ) ).

% local.minus_unique
thf(fact_1038_l__distr,axiom,
    ! [X2: a,Y: a,Z4: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) @ Z4 )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Z4 ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z4 ) ) ) ) ) ) ).

% l_distr
thf(fact_1039_r__distr,axiom,
    ! [X2: a,Y: a,Z4: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ Z4 @ ( add_a_b @ r @ X2 @ Y ) )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z4 @ X2 ) @ ( mult_a_ring_ext_a_b @ r @ Z4 @ Y ) ) ) ) ) ) ).

% r_distr
thf(fact_1040_add_Oinv__mult__group,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) )
          = ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X2 ) ) ) ) ) ).

% add.inv_mult_group
thf(fact_1041_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_1042_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_1043_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_1044_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_1045_a__transpose__inv,axiom,
    ! [X2: a,Y: a,Z4: a] :
      ( ( ( add_a_b @ r @ X2 @ Y )
        = Z4 )
     => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ Z4 )
              = Y ) ) ) ) ) ).

% a_transpose_inv
thf(fact_1046_local_Ominus__add,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) )
          = ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).

% local.minus_add
thf(fact_1047_r__neg1,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ ( add_a_b @ r @ X2 @ Y ) )
          = Y ) ) ) ).

% r_neg1
thf(fact_1048_r__neg2,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( add_a_b @ r @ X2 @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ Y ) )
          = Y ) ) ) ).

% r_neg2
thf(fact_1049_subcringI,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( ! [H12: a,H23: a] :
            ( ( member_a @ H12 @ H2 )
           => ( ( member_a @ H23 @ H2 )
             => ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
                = ( mult_a_ring_ext_a_b @ r @ H23 @ H12 ) ) ) )
       => ( subcring_a_b @ H2 @ r ) ) ) ).

% subcringI
thf(fact_1050_add_Oinj__on__cmult,axiom,
    ! [C: a] :
      ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
     => ( inj_on_a_a @ ( add_a_b @ r @ C ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% add.inj_on_cmult
thf(fact_1051_add_Oinj__on__multc,axiom,
    ! [C: a] :
      ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
     => ( inj_on_a_a
        @ ^ [X: a] : ( add_a_b @ r @ X @ C )
        @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% add.inj_on_multc
thf(fact_1052_l__neg,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ X2 )
        = ( zero_a_b @ r ) ) ) ).

% l_neg
thf(fact_1053_minus__equality,axiom,
    ! [Y: a,X2: a] :
      ( ( ( add_a_b @ r @ Y @ X2 )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( a_inv_a_b @ r @ X2 )
            = Y ) ) ) ) ).

% minus_equality
thf(fact_1054_r__neg,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ X2 @ ( a_inv_a_b @ r @ X2 ) )
        = ( zero_a_b @ r ) ) ) ).

% r_neg
thf(fact_1055_sum__zero__eq__neg,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ X2 @ Y )
            = ( zero_a_b @ r ) )
         => ( X2
            = ( a_inv_a_b @ r @ Y ) ) ) ) ) ).

% sum_zero_eq_neg
thf(fact_1056_a__coset__add__assoc,axiom,
    ! [M: set_a,G: a,H: a] :
      ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( a_r_coset_a_b @ r @ ( a_r_coset_a_b @ r @ M @ G ) @ H )
            = ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ G @ H ) ) ) ) ) ) ).

% a_coset_add_assoc
thf(fact_1057_a__rcosI,axiom,
    ! [H: a,H2: set_a,X2: a] :
      ( ( member_a @ H @ H2 )
     => ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( member_a @ ( add_a_b @ r @ H @ X2 ) @ ( a_r_coset_a_b @ r @ H2 @ X2 ) ) ) ) ) ).

% a_rcosI
thf(fact_1058_UP_Ocarrier__is__subcring,axiom,
    subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( univ_poly_a_b @ r @ k ) ).

% UP.carrier_is_subcring
thf(fact_1059_a__lcos__m__assoc,axiom,
    ! [M: set_a,G: a,H: a] :
      ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H @ M ) )
            = ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H ) @ M ) ) ) ) ) ).

% a_lcos_m_assoc
thf(fact_1060_UP_OsubcringI_H,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.subcringI'
thf(fact_1061_add_Oinj__on__g,axiom,
    ! [H2: set_a,A: a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( inj_on_a_a
          @ ^ [Y4: a] : ( add_a_b @ r @ Y4 @ A )
          @ H2 ) ) ) ).

% add.inj_on_g
thf(fact_1062_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_1063_UP_Oadd_Ol__cancel,axiom,
    ! [C: list_a,A: list_a,B: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ C @ A )
        = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ C @ B ) )
     => ( ( 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 ) ) )
           => ( A = B ) ) ) ) ) ).

% UP.add.l_cancel
thf(fact_1064_UP_Oadd_Om__assoc,axiom,
    ! [X2: list_a,Y: list_a,Z4: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Z4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) @ Z4 )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Z4 ) ) ) ) ) ) ).

% UP.add.m_assoc
thf(fact_1065_UP_Oadd_Om__comm,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y )
          = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X2 ) ) ) ) ).

% UP.add.m_comm
thf(fact_1066_UP_Oadd_Om__lcomm,axiom,
    ! [X2: list_a,Y: list_a,Z4: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Z4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Z4 ) )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Z4 ) ) ) ) ) ) ).

% UP.add.m_lcomm
thf(fact_1067_UP_Oadd_Or__cancel,axiom,
    ! [A: list_a,C: list_a,B: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C )
        = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ C ) )
     => ( ( 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 ) ) )
           => ( A = B ) ) ) ) ) ).

% UP.add.r_cancel
thf(fact_1068_a__coset__add__inv2,axiom,
    ! [M: set_a,X2: a,Y: a] :
      ( ( ( a_r_coset_a_b @ r @ M @ X2 )
        = ( a_r_coset_a_b @ r @ M @ Y ) )
     => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ X2 @ ( a_inv_a_b @ r @ Y ) ) )
              = M ) ) ) ) ) ).

% a_coset_add_inv2
thf(fact_1069_a__coset__add__inv1,axiom,
    ! [M: set_a,X2: a,Y: a] :
      ( ( ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ X2 @ ( a_inv_a_b @ r @ Y ) ) )
        = M )
     => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( a_r_coset_a_b @ r @ M @ X2 )
              = ( a_r_coset_a_b @ r @ M @ Y ) ) ) ) ) ) ).

% a_coset_add_inv1
thf(fact_1070_UP_Osubring__props_I7_J,axiom,
    ! [K: set_list_a,H1: list_a,H22: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_a @ H1 @ K )
       => ( ( member_list_a @ H22 @ K )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ H1 @ H22 ) @ K ) ) ) ) ).

% UP.subring_props(7)
thf(fact_1071_UP_OsubcringI,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( ! [H12: list_a,H23: list_a] :
            ( ( member_list_a @ H12 @ H2 )
           => ( ( member_list_a @ H23 @ H2 )
             => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H12 @ H23 )
                = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H23 @ H12 ) ) ) )
       => ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.subcringI
thf(fact_1072_UP_Oadd_Osurj__const__mult,axiom,
    ! [A: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( image_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        = ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.add.surj_const_mult
thf(fact_1073_subdomainI,axiom,
    ! [H2: set_a] :
      ( ( subcring_a_b @ H2 @ r )
     => ( ( ( one_a_ring_ext_a_b @ r )
         != ( zero_a_b @ r ) )
       => ( ! [H12: a,H23: a] :
              ( ( member_a @ H12 @ H2 )
             => ( ( member_a @ H23 @ H2 )
               => ( ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
                    = ( zero_a_b @ r ) )
                 => ( ( H12
                      = ( zero_a_b @ r ) )
                    | ( H23
                      = ( zero_a_b @ r ) ) ) ) ) )
         => ( subdomain_a_b @ H2 @ r ) ) ) ) ).

% subdomainI
thf(fact_1074_pdivides__imp__root__sharing,axiom,
    ! [P: list_a,Q2: list_a,A: a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( eval_a_b @ r @ P @ A )
              = ( zero_a_b @ r ) )
           => ( ( eval_a_b @ r @ Q2 @ A )
              = ( zero_a_b @ r ) ) ) ) ) ) ).

% pdivides_imp_root_sharing
thf(fact_1075_const__term__simprules__shell_I3_J,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 ) )
            = ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q2 ) ) ) ) ) ) ).

% const_term_simprules_shell(3)
thf(fact_1076_UP_Oadd_Oinv__comm,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).

% UP.add.inv_comm
thf(fact_1077_UP_Oadd_Ol__inv__ex,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ? [X3: list_a] :
          ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          & ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X3 @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.add.l_inv_ex
thf(fact_1078_UP_Oadd_Oone__unique,axiom,
    ! [U: list_a] :
      ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ! [X3: list_a] :
            ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ U @ X3 )
              = X3 ) )
       => ( U
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.add.one_unique
thf(fact_1079_UP_Oadd_Or__inv__ex,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ? [X3: list_a] :
          ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          & ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ X3 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.add.r_inv_ex
thf(fact_1080_UP_Ominus__unique,axiom,
    ! [Y: list_a,X2: list_a,Y6: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y6 )
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
             => ( Y = Y6 ) ) ) ) ) ) ).

% UP.minus_unique
thf(fact_1081_UP_Ol__distr,axiom,
    ! [X2: list_a,Y: list_a,Z4: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Z4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) @ Z4 )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Z4 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ Z4 ) ) ) ) ) ) ).

% UP.l_distr
thf(fact_1082_UP_Or__distr,axiom,
    ! [X2: list_a,Y: list_a,Z4: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Z4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Z4 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Z4 @ X2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ Z4 @ Y ) ) ) ) ) ) ).

% UP.r_distr
thf(fact_1083_long__division__add_I1_J,axiom,
    ! [K: set_a,A: list_a,B: list_a,Q2: 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q2 )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ A @ Q2 ) @ ( polynomial_pdiv_a_b @ r @ B @ Q2 ) ) ) ) ) ) ) ).

% long_division_add(1)
thf(fact_1084_eval__poly__add,axiom,
    ! [P: list_a,Q2: list_a,A: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q2 ) @ A )
            = ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q2 @ A ) ) ) ) ) ) ).

% eval_poly_add
thf(fact_1085_UP_Oa__transpose__inv,axiom,
    ! [X2: list_a,Y: list_a,Z4: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y )
        = Z4 )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( member_list_a @ Z4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) @ Z4 )
              = Y ) ) ) ) ) ).

% UP.a_transpose_inv
thf(fact_1086_UP_Oadd_Oinv__mult__group,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) )
          = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) ) ) ) ) ).

% UP.add.inv_mult_group
thf(fact_1087_UP_Oadd_Oinv__solve__left,axiom,
    ! [A: list_a,B: list_a,C: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( A
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ B ) @ C ) )
            = ( C
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ A ) ) ) ) ) ) ).

% UP.add.inv_solve_left
thf(fact_1088_UP_Oadd_Oinv__solve__left_H,axiom,
    ! [A: list_a,B: list_a,C: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ B ) @ C )
              = A )
            = ( C
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ A ) ) ) ) ) ) ).

% UP.add.inv_solve_left'
thf(fact_1089_UP_Oadd_Oinv__solve__right,axiom,
    ! [A: list_a,B: list_a,C: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( A
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ C ) ) )
            = ( B
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C ) ) ) ) ) ) ).

% UP.add.inv_solve_right
thf(fact_1090_UP_Oadd_Oinv__solve__right_H,axiom,
    ! [A: list_a,B: list_a,C: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ C ) )
              = A )
            = ( B
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ C ) ) ) ) ) ) ).

% UP.add.inv_solve_right'
thf(fact_1091_UP_Ominus__add,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) )
          = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) ) ) ) ) ).

% UP.minus_add
thf(fact_1092_UP_Or__neg1,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) )
          = Y ) ) ) ).

% UP.r_neg1
thf(fact_1093_UP_Or__neg2,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) @ Y ) )
          = Y ) ) ) ).

% UP.r_neg2
thf(fact_1094_pprimeE_I3_J,axiom,
    ! [K: set_a,P: list_a,Q2: list_a,R4: 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 @ 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 ) ) ) ) ) ) ) ) ).

% pprimeE(3)
thf(fact_1095_const__term__simprules_I3_J,axiom,
    ! [P: list_a,Q2: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( const_term_a_b @ r @ ( poly_add_a_b @ r @ P @ Q2 ) )
          = ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q2 ) ) ) ) ) ).

% const_term_simprules(3)
thf(fact_1096_UP_Oline__extension__mem__iff,axiom,
    ! [U: list_a,K: set_list_a,A: list_a,E2: set_list_a] :
      ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ A @ E2 ) )
      = ( ? [X: list_a] :
            ( ( member_list_a @ X @ K )
            & ? [Y4: list_a] :
                ( ( member_list_a @ Y4 @ E2 )
                & ( U
                  = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ A ) @ Y4 ) ) ) ) ) ) ).

% UP.line_extension_mem_iff
thf(fact_1097_UP_Ominus__eq,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y )
      = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) ) ) ).

% UP.minus_eq
thf(fact_1098_UP_Oadd_Oinj__on__cmult,axiom,
    ! [C: list_a] :
      ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( inj_on_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ C ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.add.inj_on_cmult
thf(fact_1099_UP_Oadd_Oinj__on__multc,axiom,
    ! [C: list_a] :
      ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( inj_on_list_a_list_a
        @ ^ [X: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X @ C )
        @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.add.inj_on_multc
thf(fact_1100_subringI,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H2 )
       => ( ! [H3: a] :
              ( ( member_a @ H3 @ H2 )
             => ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H2 ) )
         => ( ! [H12: a,H23: a] :
                ( ( member_a @ H12 @ H2 )
               => ( ( member_a @ H23 @ H2 )
                 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ H2 ) ) )
           => ( ! [H12: a,H23: a] :
                  ( ( member_a @ H12 @ H2 )
                 => ( ( member_a @ H23 @ H2 )
                   => ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ H2 ) ) )
             => ( subring_a_b @ H2 @ r ) ) ) ) ) ) ).

% subringI
thf(fact_1101_UP_Oa__coset__add__assoc,axiom,
    ! [M: set_list_a,G: list_a,H: list_a] :
      ( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ G ) @ H )
            = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ G @ H ) ) ) ) ) ) ).

% UP.a_coset_add_assoc
thf(fact_1102_UP_Oa__rcosI,axiom,
    ! [H: list_a,H2: set_list_a,X2: list_a] :
      ( ( member_list_a @ H @ H2 )
     => ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ H @ X2 ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ H2 @ X2 ) ) ) ) ) ).

% UP.a_rcosI
thf(fact_1103_UP_Ol__neg,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) @ X2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.l_neg
thf(fact_1104_UP_Ominus__equality,axiom,
    ! [Y: list_a,X2: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 )
            = Y ) ) ) ) ).

% UP.minus_equality
thf(fact_1105_UP_Or__neg,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 ) )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.r_neg
thf(fact_1106_UP_Osum__zero__eq__neg,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( X2
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) ) ) ) ) ).

% UP.sum_zero_eq_neg
thf(fact_1107_pdivides__imp__is__root,axiom,
    ! [P: list_a,X2: a] :
      ( ( P != nil_a )
     => ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X2 ) @ nil_a ) ) @ P )
         => ( polyno4133073214067823460ot_a_b @ r @ P @ X2 ) ) ) ) ).

% pdivides_imp_is_root
thf(fact_1108_UP_Oa__lcos__m__assoc,axiom,
    ! [M: set_list_a,G: list_a,H: list_a] :
      ( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ H @ M ) )
            = ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ G @ H ) @ M ) ) ) ) ) ).

% UP.a_lcos_m_assoc
thf(fact_1109_UP_Olong__division__add_I2_J,axiom,
    ! [K: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
           => ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ A @ B ) @ Q2 )
              = ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ Q2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ Q2 ) ) ) ) ) ) ) ).

% UP.long_division_add(2)
thf(fact_1110_UP_Olong__division__add__iff,axiom,
    ! [K: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
           => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
             => ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ Q2 )
                  = ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ Q2 ) )
                = ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ A @ C ) @ Q2 )
                  = ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ).

% UP.long_division_add_iff
thf(fact_1111_UP_Olong__division__closed_I2_J,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( member_list_list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ) ) ).

% UP.long_division_closed(2)
thf(fact_1112_UP_Olong__division__add_I1_J,axiom,
    ! [K: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
           => ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ A @ B ) @ Q2 )
              = ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ Q2 ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ Q2 ) ) ) ) ) ) ) ).

% UP.long_division_add(1)
thf(fact_1113_rupture__eq__0__iff,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q2 )
              = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) ) )
            = ( polyno5814909790663948098es_a_b @ r @ P @ Q2 ) ) ) ) ) ).

% rupture_eq_0_iff
thf(fact_1114_UP_OsubdomainI,axiom,
    ! [H2: set_list_a] :
      ( ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
         != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ! [H12: list_a,H23: list_a] :
              ( ( member_list_a @ H12 @ H2 )
             => ( ( member_list_a @ H23 @ H2 )
               => ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H12 @ H23 )
                    = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
                 => ( ( H12
                      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
                    | ( H23
                      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) )
         => ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.subdomainI
thf(fact_1115_UP_Oadd_Oinj__on__g,axiom,
    ! [H2: set_list_a,A: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( inj_on_list_a_list_a
          @ ^ [Y4: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y4 @ A )
          @ H2 ) ) ) ).

% UP.add.inj_on_g
thf(fact_1116_UP_Oa__coset__add__inv1,axiom,
    ! [M: set_list_a,X2: list_a,Y: list_a] :
      ( ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) ) )
        = M )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ X2 )
              = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ Y ) ) ) ) ) ) ).

% UP.a_coset_add_inv1
thf(fact_1117_UP_Oa__coset__add__inv2,axiom,
    ! [M: set_list_a,X2: list_a,Y: list_a] :
      ( ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ X2 )
        = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ Y ) )
     => ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ M @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ Y ) ) )
              = M ) ) ) ) ) ).

% UP.a_coset_add_inv2
thf(fact_1118_UP_Oconst__term__simprules__shell_I3_J,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P @ Q2 ) )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ Q2 ) ) ) ) ) ) ).

% UP.const_term_simprules_shell(3)
thf(fact_1119_UP_Olong__division__zero_I2_J,axiom,
    ! [K: set_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ nil_list_a @ Q2 )
          = nil_list_a ) ) ) ).

% UP.long_division_zero(2)
thf(fact_1120_UP_Olong__division__a__inv_I2_J,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) @ Q2 )
            = ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) ) ) ) ) ) ).

% UP.long_division_a_inv(2)
thf(fact_1121_local_Oadd_Oright__cancel,axiom,
    ! [X2: a,Y: a,Z4: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ Y @ X2 )
              = ( add_a_b @ r @ Z4 @ X2 ) )
            = ( Y = Z4 ) ) ) ) ) ).

% local.add.right_cancel
thf(fact_1122_a__closed,axiom,
    ! [X2: a,Y: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( add_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% a_closed
thf(fact_1123_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_1124_is__root__imp__pdivides,axiom,
    ! [P: list_a,X2: a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
       => ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X2 ) @ nil_a ) ) @ P ) ) ) ).

% is_root_imp_pdivides
thf(fact_1125_UP_Oeval__poly__add,axiom,
    ! [P: list_list_a,Q2: list_list_a,A: list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) @ A )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ A ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ k ) @ Q2 @ A ) ) ) ) ) ) ).

% UP.eval_poly_add
thf(fact_1126_UP_Oconst__term__simprules_I3_J,axiom,
    ! [P: list_list_a,Q2: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) )
          = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ k ) @ Q2 ) ) ) ) ) ).

% UP.const_term_simprules(3)
thf(fact_1127_UP_Opmod__zero__iff__pdivides,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 )
              = nil_list_a )
            = ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ Q2 @ P ) ) ) ) ) ).

% UP.pmod_zero_iff_pdivides
thf(fact_1128_UP_Opdiv__pmod,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( P
            = ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ Q2 @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) ) ) ) ) ) ).

% UP.pdiv_pmod
thf(fact_1129_UP_Orupture__surj__composed__with__pmod,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) @ Q2 )
            = ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ Q2 @ P ) ) ) ) ) ) ).

% UP.rupture_surj_composed_with_pmod
thf(fact_1130_UP_OsubringI,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) @ H2 )
       => ( ! [H3: list_a] :
              ( ( member_list_a @ H3 @ H2 )
             => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ H3 ) @ H2 ) )
         => ( ! [H12: list_a,H23: list_a] :
                ( ( member_list_a @ H12 @ H2 )
               => ( ( member_list_a @ H23 @ H2 )
                 => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ H12 @ H23 ) @ H2 ) ) )
           => ( ! [H12: list_a,H23: list_a] :
                  ( ( member_list_a @ H12 @ H2 )
                 => ( ( member_list_a @ H23 @ H2 )
                   => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ H12 @ H23 ) @ H2 ) ) )
             => ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ) ).

% UP.subringI
thf(fact_1131_UP_Osame__pmod__iff__pdivides,axiom,
    ! [K: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
           => ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ Q2 )
                = ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ B @ Q2 ) )
              = ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ k ) @ Q2 @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ A @ B ) ) ) ) ) ) ) ).

% UP.same_pmod_iff_pdivides
thf(fact_1132_UP_Orupture__surj__inj__on,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( inj_on3733059618185315153list_a @ ( a_r_co7333016316668015878t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) @ P ) )
          @ ( image_5446147394702115205list_a
            @ ^ [Q3: list_list_a] : ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ Q3 @ P )
            @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ) ) ) ).

% UP.rupture_surj_inj_on
thf(fact_1133_add_Ol__cancel__one,axiom,
    ! [X2: a,A: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ X2 @ A )
            = X2 )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.l_cancel_one
thf(fact_1134_add_Ol__cancel__one_H,axiom,
    ! [X2: a,A: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( X2
            = ( add_a_b @ r @ X2 @ A ) )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.l_cancel_one'
thf(fact_1135_add_Or__cancel__one,axiom,
    ! [X2: a,A: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ A @ X2 )
            = X2 )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.r_cancel_one
thf(fact_1136_add_Or__cancel__one_H,axiom,
    ! [X2: a,A: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( X2
            = ( add_a_b @ r @ A @ X2 ) )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.r_cancel_one'
thf(fact_1137_l__zero,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X2 )
        = X2 ) ) ).

% l_zero
thf(fact_1138_r__zero,axiom,
    ! [X2: a] :
      ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ X2 @ ( zero_a_b @ r ) )
        = X2 ) ) ).

% r_zero
thf(fact_1139_UP_Oadd_Om__closed,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.add.m_closed
thf(fact_1140_UP_Oadd_Oright__cancel,axiom,
    ! [X2: list_a,Y: list_a,Z4: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ Z4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Y @ X2 )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ Z4 @ X2 ) )
            = ( Y = Z4 ) ) ) ) ) ).

% UP.add.right_cancel
thf(fact_1141_UP_Oadd_Ol__cancel__one,axiom,
    ! [X2: list_a,A: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ A )
            = X2 )
          = ( A
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).

% UP.add.l_cancel_one
thf(fact_1142_UP_Oadd_Ol__cancel__one_H,axiom,
    ! [X2: list_a,A: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( X2
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ A ) )
          = ( A
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).

% UP.add.l_cancel_one'
thf(fact_1143_UP_Oadd_Or__cancel__one,axiom,
    ! [X2: list_a,A: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ X2 )
            = X2 )
          = ( A
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).

% UP.add.r_cancel_one
thf(fact_1144_UP_Oadd_Or__cancel__one_H,axiom,
    ! [X2: list_a,A: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( X2
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ A @ X2 ) )
          = ( A
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ) ).

% UP.add.r_cancel_one'
thf(fact_1145_UP_Ol__zero,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ X2 )
        = X2 ) ) ).

% UP.l_zero
thf(fact_1146_UP_Or__zero,axiom,
    ! [X2: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
        = X2 ) ) ).

% UP.r_zero
thf(fact_1147_h_Ominus__closed,axiom,
    ! [X2: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( member_set_list_a @ ( a_minu2642007939804611572t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X2 @ Y ) @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ) ).

% h.minus_closed
thf(fact_1148_h_Or__right__minus__eq,axiom,
    ! [A: set_list_a,B: set_list_a] :
      ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( ( a_minu2642007939804611572t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ A @ B )
            = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
          = ( A = B ) ) ) ) ).

% h.r_right_minus_eq
thf(fact_1149_h_Ohom__add,axiom,
    ! [X2: list_a,Y: list_a] :
      ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X2 @ Y ) )
          = ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ X2 ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ Y ) ) ) ) ) ).

% h.hom_add
thf(fact_1150_UP_Omonoid__cancelI,axiom,
    ( ! [A3: list_a,B3: list_a,C2: list_a] :
        ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ C2 @ A3 )
          = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ C2 @ B3 ) )
       => ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
             => ( A3 = B3 ) ) ) ) )
   => ( ! [A3: list_a,B3: list_a,C2: list_a] :
          ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A3 @ C2 )
            = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ B3 @ C2 ) )
         => ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
             => ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
               => ( A3 = B3 ) ) ) ) )
     => ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.monoid_cancelI
thf(fact_1151_h_Oadd_Oone__in__subset,axiom,
    ! [H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ H2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( H2 != bot_bo3186585308812441520list_a )
       => ( ! [X3: set_list_a] :
              ( ( member_set_list_a @ X3 @ H2 )
             => ( member_set_list_a @ ( a_inv_5715216516650856053t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X3 ) @ H2 ) )
         => ( ! [X3: set_list_a] :
                ( ( member_set_list_a @ X3 @ H2 )
               => ! [Xa2: set_list_a] :
                    ( ( member_set_list_a @ Xa2 @ H2 )
                   => ( member_set_list_a @ ( add_se2486902527185523630t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ X3 @ Xa2 ) @ H2 ) ) )
           => ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ H2 ) ) ) ) ) ).

% h.add.one_in_subset
thf(fact_1152_UP_Olong__divisionE,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( Q2 != nil_list_a )
           => ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) ) ) ) ) ) ) ).

% UP.long_divisionE
thf(fact_1153_UP_Opoly__add_Ocases,axiom,
    ! [X2: produc7709606177366032167list_a] :
      ~ ! [P12: list_list_a,P23: list_list_a] :
          ( X2
         != ( produc8696003437204565271list_a @ P12 @ P23 ) ) ).

% UP.poly_add.cases
thf(fact_1154_UP_Opoly__mult_Ocases,axiom,
    ! [X2: produc7709606177366032167list_a] :
      ( ! [P23: list_list_a] :
          ( X2
         != ( produc8696003437204565271list_a @ nil_list_a @ P23 ) )
     => ~ ! [V2: list_a,Va: list_list_a,P23: list_list_a] :
            ( X2
           != ( produc8696003437204565271list_a @ ( cons_list_a @ V2 @ Va ) @ P23 ) ) ) ).

% UP.poly_mult.cases
thf(fact_1155_UP_Ocombine_Ocases,axiom,
    ! [X2: produc7709606177366032167list_a] :
      ( ! [K3: list_a,Ks: list_list_a,U2: list_a,Us: list_list_a] :
          ( X2
         != ( produc8696003437204565271list_a @ ( cons_list_a @ K3 @ Ks ) @ ( cons_list_a @ U2 @ Us ) ) )
     => ( ! [Us: list_list_a] :
            ( X2
           != ( produc8696003437204565271list_a @ nil_list_a @ Us ) )
       => ~ ! [Ks: list_list_a] :
              ( X2
             != ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).

% UP.combine.cases
thf(fact_1156_long__division__closed_I2_J,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% long_division_closed(2)
thf(fact_1157_long__division__add_I2_J,axiom,
    ! [K: set_a,A: list_a,B: list_a,Q2: 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q2 )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ A @ Q2 ) @ ( polynomial_pmod_a_b @ r @ B @ Q2 ) ) ) ) ) ) ) ).

% long_division_add(2)
thf(fact_1158_long__division__add__iff,axiom,
    ! [K: set_a,A: list_a,B: list_a,C: list_a,Q2: 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
             => ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
                  = ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
                = ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ C ) @ Q2 )
                  = ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ).

% long_division_add_iff
thf(fact_1159_long__division__zero_I2_J,axiom,
    ! [K: set_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( polynomial_pmod_a_b @ r @ nil_a @ Q2 )
          = nil_a ) ) ) ).

% long_division_zero(2)
thf(fact_1160_long__division__a__inv_I2_J,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q2 )
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P @ Q2 ) ) ) ) ) ) ).

% long_division_a_inv(2)
thf(fact_1161_rupture__surj__composed__with__pmod,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q2 )
            = ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ ( polynomial_pmod_a_b @ r @ Q2 @ P ) ) ) ) ) ) ).

% rupture_surj_composed_with_pmod
thf(fact_1162_pmod__zero__iff__pdivides,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( ( polynomial_pmod_a_b @ r @ P @ Q2 )
              = nil_a )
            = ( polyno5814909790663948098es_a_b @ r @ Q2 @ P ) ) ) ) ) ).

% pmod_zero_iff_pdivides
thf(fact_1163_same__pmod__iff__pdivides,axiom,
    ! [K: set_a,A: list_a,B: list_a,Q2: 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 @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
                = ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
              = ( polyno5814909790663948098es_a_b @ r @ Q2 @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) ) ) ) ) ) ) ).

% same_pmod_iff_pdivides
thf(fact_1164_h_Ocarrier__not__empty,axiom,
    ( ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
   != bot_bo3186585308812441520list_a ) ).

% h.carrier_not_empty
thf(fact_1165_h_Osubring__props_I4_J,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( K != bot_bo3186585308812441520list_a ) ) ).

% h.subring_props(4)
thf(fact_1166_pdiv__pmod,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( 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 ) @ Q2 @ ( polynomial_pdiv_a_b @ r @ P @ Q2 ) ) @ ( polynomial_pmod_a_b @ r @ P @ Q2 ) ) ) ) ) ) ).

% pdiv_pmod
thf(fact_1167_rupture__surj__inj__on,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
          @ ( image_list_a_list_a
            @ ^ [Q3: list_a] : ( polynomial_pmod_a_b @ r @ Q3 @ P )
            @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% rupture_surj_inj_on
thf(fact_1168_UP_Oexists__long__division,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( Q2 != nil_list_a )
           => ~ ! [B3: list_list_a] :
                  ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
                 => ! [R: list_list_a] :
                      ( ( member_list_list_a @ R @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
                     => ~ ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 @ ( produc8696003437204565271list_a @ B3 @ R ) ) ) ) ) ) ) ) ).

% UP.exists_long_division
thf(fact_1169_rupture__carrier__as__pmod__image,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
            @ ( image_list_a_list_a
              @ ^ [Q3: list_a] : ( polynomial_pmod_a_b @ r @ Q3 @ P )
              @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) )
          = ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) ) ) ) ) ).

% rupture_carrier_as_pmod_image
thf(fact_1170_UP_Olong__divisionI,axiom,
    ! [K: set_list_a,P: list_list_a,Q2: list_list_a,B: list_list_a,R4: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
       => ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) )
         => ( ( Q2 != nil_list_a )
           => ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 @ ( produc8696003437204565271list_a @ B @ R4 ) )
             => ( ( produc8696003437204565271list_a @ B @ R4 )
                = ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ k ) @ P @ Q2 ) ) ) ) ) ) ) ) ).

% UP.long_divisionI
thf(fact_1171_h_Othe__elem__wf_H,axiom,
    ! [X4: set_list_a] :
      ( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) ) )
     => ? [X3: list_a] :
          ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
          & ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ X4 )
            = ( insert_set_list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ X3 ) @ bot_bo3186585308812441520list_a ) ) ) ) ).

% h.the_elem_wf'
thf(fact_1172_h_Othe__elem__wf,axiom,
    ! [X4: set_list_a] :
      ( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) ) ) ) )
     => ? [X3: set_list_a] :
          ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
          & ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ X4 )
            = ( insert_set_list_a @ X3 @ bot_bo3186585308812441520list_a ) ) ) ) ).

% h.the_elem_wf
thf(fact_1173_poly__mult_Ocases,axiom,
    ! [X2: produc9164743771328383783list_a] :
      ( ! [P23: list_a] :
          ( X2
         != ( produc6837034575241423639list_a @ nil_a @ P23 ) )
     => ~ ! [V2: a,Va: list_a,P23: list_a] :
            ( X2
           != ( produc6837034575241423639list_a @ ( cons_a @ V2 @ Va ) @ P23 ) ) ) ).

% poly_mult.cases
thf(fact_1174_combine_Ocases,axiom,
    ! [X2: produc9164743771328383783list_a] :
      ( ! [K3: a,Ks: list_a,U2: a,Us: list_a] :
          ( X2
         != ( produc6837034575241423639list_a @ ( cons_a @ K3 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
     => ( ! [Us: list_a] :
            ( X2
           != ( produc6837034575241423639list_a @ nil_a @ Us ) )
       => ~ ! [Ks: list_a] :
              ( X2
             != ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).

% combine.cases
thf(fact_1175_h_Opoly__mult_Ocases,axiom,
    ! [X2: produc76396112498735911list_a] :
      ( ! [P23: list_set_list_a] :
          ( X2
         != ( produc8430489356965245207list_a @ nil_set_list_a @ P23 ) )
     => ~ ! [V2: set_list_a,Va: list_set_list_a,P23: list_set_list_a] :
            ( X2
           != ( produc8430489356965245207list_a @ ( cons_set_list_a @ V2 @ Va ) @ P23 ) ) ) ).

% h.poly_mult.cases
thf(fact_1176_h_Ocombine_Ocases,axiom,
    ! [X2: produc76396112498735911list_a] :
      ( ! [K3: set_list_a,Ks: list_set_list_a,U2: set_list_a,Us: list_set_list_a] :
          ( X2
         != ( produc8430489356965245207list_a @ ( cons_set_list_a @ K3 @ Ks ) @ ( cons_set_list_a @ U2 @ Us ) ) )
     => ( ! [Us: list_set_list_a] :
            ( X2
           != ( produc8430489356965245207list_a @ nil_set_list_a @ Us ) )
       => ~ ! [Ks: list_set_list_a] :
              ( X2
             != ( produc8430489356965245207list_a @ Ks @ nil_set_list_a ) ) ) ) ).

% h.combine.cases
thf(fact_1177_carrier__not__empty,axiom,
    ( ( partia707051561876973205xt_a_b @ r )
   != bot_bot_set_a ) ).

% carrier_not_empty
thf(fact_1178_subring__props_I4_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( K != bot_bot_set_a ) ) ).

% subring_props(4)
thf(fact_1179_UP_Ocarrier__not__empty,axiom,
    ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) )
   != bot_bot_set_list_a ) ).

% UP.carrier_not_empty
thf(fact_1180_UP_Osubring__props_I4_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( K != bot_bot_set_list_a ) ) ).

% UP.subring_props(4)
thf(fact_1181_add_Oone__in__subset,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( H2 != bot_bot_set_a )
       => ( ! [X3: a] :
              ( ( member_a @ X3 @ H2 )
             => ( member_a @ ( a_inv_a_b @ r @ X3 ) @ H2 ) )
         => ( ! [X3: a] :
                ( ( member_a @ X3 @ H2 )
               => ! [Xa2: a] :
                    ( ( member_a @ Xa2 @ H2 )
                   => ( member_a @ ( add_a_b @ r @ X3 @ Xa2 ) @ H2 ) ) )
           => ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).

% add.one_in_subset
thf(fact_1182_exists__long__division,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q2 != nil_a )
           => ~ ! [B3: list_a] :
                  ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                 => ! [R: list_a] :
                      ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                     => ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q2 @ ( produc6837034575241423639list_a @ B3 @ R ) ) ) ) ) ) ) ) ).

% exists_long_division
thf(fact_1183_h_OFactRing__zeroideal_I2_J,axiom,
    is_rin3234129319156366968t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( factRi5994137953073989130t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) ) ).

% h.FactRing_zeroideal(2)
thf(fact_1184_h_OFactRing__zeroideal_I1_J,axiom,
    is_rin3921434485553078904t_unit @ ( factRi5994137953073989130t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ).

% h.FactRing_zeroideal(1)
thf(fact_1185_UP_Oadd_Oone__in__subset,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( H2 != bot_bot_set_list_a )
       => ( ! [X3: list_a] :
              ( ( member_list_a @ X3 @ H2 )
             => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ X3 ) @ H2 ) )
         => ( ! [X3: list_a] :
                ( ( member_list_a @ X3 @ H2 )
               => ! [Xa2: list_a] :
                    ( ( member_list_a @ Xa2 @ H2 )
                   => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ X3 @ Xa2 ) @ H2 ) ) )
           => ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ H2 ) ) ) ) ) ).

% UP.add.one_in_subset
thf(fact_1186_long__divisionE,axiom,
    ! [K: set_a,P: list_a,Q2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q2 != nil_a )
           => ( polyno2806191415236617128es_a_b @ r @ P @ Q2 @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P @ Q2 ) ) ) ) ) ) ) ).

% long_divisionE
thf(fact_1187_long__divisionI,axiom,
    ! [K: set_a,P: list_a,Q2: list_a,B: list_a,R4: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q2 != nil_a )
           => ( ( polyno2806191415236617128es_a_b @ r @ P @ Q2 @ ( produc6837034575241423639list_a @ B @ R4 ) )
             => ( ( produc6837034575241423639list_a @ B @ R4 )
                = ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P @ Q2 ) ) ) ) ) ) ) ) ).

% long_divisionI
thf(fact_1188_h_Ocarrier__one__not__zero,axiom,
    ( ( ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     != ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) )
    = ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.carrier_one_not_zero
thf(fact_1189_h_Ocarrier__one__zero,axiom,
    ( ( ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
      = ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) )
    = ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
      = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.carrier_one_zero
thf(fact_1190_h_Oone__zeroD,axiom,
    ( ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
      = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
   => ( ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
      = ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) ) ) ).

% h.one_zeroD
thf(fact_1191_h_Oone__zeroI,axiom,
    ( ( ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
      = ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) )
   => ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
      = ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.one_zeroI
thf(fact_1192_h_Ozeropideal,axiom,
    princi5566405997746891961t_unit @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ).

% h.zeropideal
thf(fact_1193_h_Oset__add__zero,axiom,
    ! [A2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ A2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( set_ad2573391977335003242t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) @ A2 )
        = A2 ) ) ).

% h.set_add_zero
thf(fact_1194_h_Otrivial__hom__iff,axiom,
    ( ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
      = ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) )
    = ( ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) )
      = ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% h.trivial_hom_iff
thf(fact_1195_h_Onon__trivial__field__hom__imp__inj,axiom,
    ( ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) )
   => ( ( ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       != ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) )
     => ( inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% h.non_trivial_field_hom_imp_inj
thf(fact_1196_h_Ogenideal__one,axiom,
    ( ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( insert_set_list_a @ ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) )
    = ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ).

% h.genideal_one
thf(fact_1197_poly__add_Ocases,axiom,
    ! [X2: produc9164743771328383783list_a] :
      ~ ! [P12: list_a,P23: list_a] :
          ( X2
         != ( produc6837034575241423639list_a @ P12 @ P23 ) ) ).

% poly_add.cases
thf(fact_1198_h_Opoly__add_Ocases,axiom,
    ! [X2: produc76396112498735911list_a] :
      ~ ! [P12: list_set_list_a,P23: list_set_list_a] :
          ( X2
         != ( produc8430489356965245207list_a @ P12 @ P23 ) ) ).

% h.poly_add.cases
thf(fact_1199_FactRing__zeroideal_I2_J,axiom,
    is_rin9099215527551818550t_unit @ r @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).

% FactRing_zeroideal(2)
thf(fact_1200_FactRing__zeroideal_I1_J,axiom,
    is_rin6001486760346555702it_a_b @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) @ r ).

% FactRing_zeroideal(1)
thf(fact_1201_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_1202_zeropideal,axiom,
    principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).

% zeropideal
thf(fact_1203_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_1204_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_1205_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_1206_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_1207_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_1208_UP_OFactRing__zeroideal_I2_J,axiom,
    is_rin2993610189962786360t_unit @ ( univ_poly_a_b @ r @ k ) @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) ).

% UP.FactRing_zeroideal(2)
thf(fact_1209_UP_OFactRing__zeroideal_I1_J,axiom,
    is_rin4843644836746533432t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) @ ( univ_poly_a_b @ r @ k ) ).

% UP.FactRing_zeroideal(1)
thf(fact_1210_UP_Ozeropideal,axiom,
    princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) ).

% UP.zeropideal
thf(fact_1211_UP_Osubfield__iff_I2_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( field_6388047844668329575t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : K
          @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.subfield_iff(2)
thf(fact_1212_UP_Oone__zeroI,axiom,
    ( ( ( 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 ) )
   => ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.one_zeroI
thf(fact_1213_UP_Oone__zeroD,axiom,
    ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
   => ( ( 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 ) ) ) ).

% UP.one_zeroD
thf(fact_1214_UP_Ocarrier__one__zero,axiom,
    ( ( ( 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 ) )
    = ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.carrier_one_zero
thf(fact_1215_UP_Ocarrier__one__not__zero,axiom,
    ( ( ( 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 ) )
    = ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) )
     != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.carrier_one_not_zero
thf(fact_1216_h_Osubset__Idl__subset,axiom,
    ! [I2: set_set_list_a,H2: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ I2 @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ H2 @ I2 )
       => ( ord_le8877086941679407844list_a @ ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ H2 ) @ ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ I2 ) ) ) ) ).

% h.subset_Idl_subset
thf(fact_1217_h_Ogenideal__self,axiom,
    ! [S: set_set_list_a] :
      ( ( ord_le8877086941679407844list_a @ S @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ord_le8877086941679407844list_a @ S @ ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ S ) ) ) ).

% h.genideal_self
thf(fact_1218_UP_Oset__add__zero,axiom,
    ! [A2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ A2 )
        = A2 ) ) ).

% UP.set_add_zero
thf(fact_1219_UP_Ocring__fieldI2,axiom,
    ( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) )
     != ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) )
   => ( ! [A3: list_a] :
          ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( A3
             != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ? [X6: list_a] :
                ( ( member_list_a @ X6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
                & ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ A3 @ X6 )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) )
     => ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.cring_fieldI2
thf(fact_1220_h_Ogenideal__self_H,axiom,
    ! [I: set_list_a] :
      ( ( member_set_list_a @ I @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( member_set_list_a @ I @ ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( insert_set_list_a @ I @ bot_bo3186585308812441520list_a ) ) ) ) ).

% h.genideal_self'
thf(fact_1221_h_Ogenideal__zero,axiom,
    ( ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) )
    = ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) @ bot_bo3186585308812441520list_a ) ) ).

% h.genideal_zero
thf(fact_1222_UP_Osubfield__iff_I1_J,axiom,
    ! [K: set_list_a] :
      ( ( field_6388047844668329575t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : K
          @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.subfield_iff(1)
thf(fact_1223_h_OIdl__subset__ideal_H,axiom,
    ! [A: set_list_a,B: set_list_a] :
      ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( ( ord_le8877086941679407844list_a @ ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( insert_set_list_a @ A @ bot_bo3186585308812441520list_a ) ) @ ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( insert_set_list_a @ B @ bot_bo3186585308812441520list_a ) ) )
          = ( member_set_list_a @ A @ ( genide4187322989772540535t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( insert_set_list_a @ B @ bot_bo3186585308812441520list_a ) ) ) ) ) ) ).

% h.Idl_subset_ideal'
thf(fact_1224_h_Otrivial__ker__imp__inj,axiom,
    ( ( ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) )
      = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) )
   => ( inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% h.trivial_ker_imp_inj
thf(fact_1225_h_Oinj__iff__trivial__ker,axiom,
    ( ( inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
    = ( ( a_kern570848875181920430t_unit @ ( univ_poly_a_b @ r @ k ) @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) )
      = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) ) ).

% h.inj_iff_trivial_ker
thf(fact_1226_UP_Ozeromaximalideal__eq__field,axiom,
    ( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) )
    = ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.zeromaximalideal_eq_field
thf(fact_1227_UP_Ozeromaximalideal__fieldI,axiom,
    ( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) )
   => ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.zeromaximalideal_fieldI
thf(fact_1228_subfield__iff_I2_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( field_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : K
          @ r ) ) ) ).

% subfield_iff(2)
thf(fact_1229_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 ) )
           => ? [X6: a] :
                ( ( member_a @ X6 @ ( partia707051561876973205xt_a_b @ r ) )
                & ( ( mult_a_ring_ext_a_b @ r @ A3 @ X6 )
                  = ( one_a_ring_ext_a_b @ r ) ) ) ) )
     => ( field_a_b @ r ) ) ) ).

% cring_fieldI2
thf(fact_1230_UP_Ouniv__poly__not__field,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ~ ( field_1861437471013600865t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ k ) @ K ) ) ) ).

% UP.univ_poly_not_field
thf(fact_1231_subfield__iff_I1_J,axiom,
    ! [K: set_a] :
      ( ( field_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : K
          @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( subfield_a_b @ K @ r ) ) ) ).

% subfield_iff(1)
thf(fact_1232_h_Osubfield__iff_I2_J,axiom,
    ! [K: set_set_list_a] :
      ( ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) )
     => ( field_26233345952514695t_unit
        @ ( partia2314006361284536288t_unit
          @ ^ [Uu: set_set_list_a] : K
          @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.subfield_iff(2)
thf(fact_1233_h_Osubfield__iff_I1_J,axiom,
    ! [K: set_set_list_a] :
      ( ( field_26233345952514695t_unit
        @ ( partia2314006361284536288t_unit
          @ ^ [Uu: set_set_list_a] : K
          @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
     => ( ( ord_le8877086941679407844list_a @ K @ ( partia141011252114345353t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) )
       => ( subfie4339374749748326226t_unit @ K @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) ) ) ) ).

% h.subfield_iff(1)
thf(fact_1234_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_1235_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_1236_h_Oinj__on__Span__iff__trivial__ker,axiom,
    ! [K: set_list_a,Us2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( inj_on1264545500884751569list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) )
          = ( ( a_kern570848875181920430t_unit
              @ ( partia9041243232023819264t_unit
                @ ^ [Uu: set_list_a] : ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 )
                @ ( univ_poly_a_b @ r @ k ) )
              @ ( polyno5459750281392823787re_a_b @ r @ k @ p )
              @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) )
            = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) ) ) ) ).

% h.inj_on_Span_iff_trivial_ker
thf(fact_1237_UP_Odomain__eq__zeroprimeideal,axiom,
    ( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) )
    = ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.domain_eq_zeroprimeideal
thf(fact_1238_UP_Omaximalideal__prime,axiom,
    ! [I2: set_list_a] :
      ( ( maxima6585700282301356660t_unit @ I2 @ ( univ_poly_a_b @ r @ k ) )
     => ( primei6309817859076077608t_unit @ I2 @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.maximalideal_prime
thf(fact_1239_UP_OSpan__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd4402942584324845940t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ k ) ) )
        = ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.Span_consistent
thf(fact_1240_UP_OSpan__in__carrier,axiom,
    ! [K: set_list_a,Us2: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.Span_in_carrier
thf(fact_1241_UP_Ozeroprimeideal,axiom,
    primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) ).

% UP.zeroprimeideal
thf(fact_1242_UP_OSpan__subgroup__props_I1_J,axiom,
    ! [K: set_list_a,Us2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ) ) ).

% UP.Span_subgroup_props(1)
thf(fact_1243_UP_OSpan__base__incl,axiom,
    ! [K: set_list_a,Us2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) ) ) ) ).

% UP.Span_base_incl
thf(fact_1244_UP_OSpan__same__set,axiom,
    ! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ( set_list_a2 @ Us2 )
            = ( set_list_a2 @ Vs ) )
         => ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 )
            = ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Vs ) ) ) ) ) ).

% UP.Span_same_set
thf(fact_1245_UP_Omono__Span__sublist,axiom,
    ! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( set_list_a2 @ Vs ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Vs ) ) ) ) ) ).

% UP.mono_Span_sublist
thf(fact_1246_UP_Omono__Span__subset,axiom,
    ! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Vs ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Vs ) ) ) ) ) ).

% UP.mono_Span_subset
thf(fact_1247_UP_OSpan__subalgebraI,axiom,
    ! [K: set_list_a,E2: set_list_a,Us2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd1768981623711841426t_unit @ K @ E2 @ ( univ_poly_a_b @ r @ k ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ E2 )
         => ( ! [V4: set_list_a] :
                ( ( embedd1768981623711841426t_unit @ K @ V4 @ ( univ_poly_a_b @ r @ k ) )
               => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ V4 )
                 => ( ord_le8861187494160871172list_a @ E2 @ V4 ) ) )
           => ( E2
              = ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) ) ) ) ) ) ).

% UP.Span_subalgebraI
thf(fact_1248_UP_Osubalgebra__Span__incl,axiom,
    ! [K: set_list_a,V3: set_list_a,Us2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( embedd1768981623711841426t_unit @ K @ V3 @ ( univ_poly_a_b @ r @ k ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ V3 )
         => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) @ V3 ) ) ) ) ).

% UP.subalgebra_Span_incl
thf(fact_1249_UP_Ozeroprimeideal__domainI,axiom,
    ( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ k ) )
   => ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ k ) ) ) ).

% UP.zeroprimeideal_domainI
thf(fact_1250_UP_OSpan__subgroup__props_I2_J,axiom,
    ! [K: set_list_a,Us2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) ) ) ) ).

% UP.Span_subgroup_props(2)
thf(fact_1251_UP_OSpan__subgroup__props_I3_J,axiom,
    ! [K: set_list_a,Us2: list_list_a,V1: list_a,V22: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ V1 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) )
         => ( ( member_list_a @ V22 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) )
           => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ k ) @ V1 @ V22 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) ) ) ) ) ) ).

% UP.Span_subgroup_props(3)
thf(fact_1252_UP_OSpan__smult__closed,axiom,
    ! [K: set_list_a,Us2: list_list_a,K5: list_a,V: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ K5 @ K )
         => ( ( member_list_a @ V @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) )
           => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K5 @ V ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) ) ) ) ) ) ).

% UP.Span_smult_closed
thf(fact_1253_UP_Omono__Span,axiom,
    ! [K: set_list_a,Us2: list_list_a,U: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( cons_list_a @ U @ Us2 ) ) ) ) ) ) ).

% UP.mono_Span
thf(fact_1254_UP_OSpan__subgroup__props_I4_J,axiom,
    ! [K: set_list_a,Us2: list_list_a,V: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ V @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) )
         => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ k ) @ V ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) ) ) ) ) ).

% UP.Span_subgroup_props(4)
thf(fact_1255_UP_Omono__Span__append_I2_J,axiom,
    ! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( append_list_a @ Vs @ Us2 ) ) ) ) ) ) ).

% UP.mono_Span_append(2)
thf(fact_1256_UP_Omono__Span__append_I1_J,axiom,
    ! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( append_list_a @ Us2 @ Vs ) ) ) ) ) ) ).

% UP.mono_Span_append(1)
thf(fact_1257_UP_OSpan__finite__dimension,axiom,
    ! [K: set_list_a,Us2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) ) ) ) ).

% UP.Span_finite_dimension
thf(fact_1258_UP_OSpan__is__subalgebra,axiom,
    ! [K: set_list_a,Us2: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( embedd1768981623711841426t_unit @ K @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) @ ( univ_poly_a_b @ r @ k ) ) ) ) ).

% UP.Span_is_subalgebra
thf(fact_1259_UP_OSpan__append__eq__set__add,axiom,
    ! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
         => ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ ( append_list_a @ Us2 @ Vs ) )
            = ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ k ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Vs ) ) ) ) ) ) ).

% UP.Span_append_eq_set_add
thf(fact_1260_UP_OSpan__m__inv__simprule,axiom,
    ! [K: set_list_a,Us2: list_list_a,K5: list_a,A: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ k ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
       => ( ( member_list_a @ K5 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ k ) ) @ bot_bot_set_list_a ) ) )
         => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) )
           => ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ k ) @ K5 @ A ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) )
             => ( member_list_a @ A @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ k ) @ K @ Us2 ) ) ) ) ) ) ) ).

% UP.Span_m_inv_simprule
thf(fact_1261_maximalideal__prime,axiom,
    ! [I2: set_a] :
      ( ( maximalideal_a_b @ I2 @ r )
     => ( primeideal_a_b @ I2 @ r ) ) ).

% maximalideal_prime
thf(fact_1262_Span__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( embedded_Span_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r ) )
        = ( embedded_Span_a_b @ r ) ) ) ).

% Span_consistent
thf(fact_1263_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_1264_zeroprimeideal,axiom,
    primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).

% zeroprimeideal
thf(fact_1265_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_1266_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_1267_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_1268_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_1269_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_1270_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_1271_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_1272_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_1273_Span__subgroup__props_I3_J,axiom,
    ! [K: set_a,Us2: list_a,V1: a,V22: 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 @ V22 @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
           => ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ).

% Span_subgroup_props(3)
thf(fact_1274_Span__smult__closed,axiom,
    ! [K: set_a,Us2: list_a,K5: a,V: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ K5 @ K )
         => ( ( member_a @ V @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
           => ( member_a @ ( mult_a_ring_ext_a_b @ r @ K5 @ V ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ).

% Span_smult_closed
thf(fact_1275_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

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ q )
    = ( eval_s5133945360527818456t_unit @ ( polyno5459750281392823787re_a_b @ r @ k @ p ) @ ( map_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) ) @ ( poly_of_const_a_b @ r ) ) @ q ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ k ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ k ) @ p ) @ ( var_a_b @ r ) ) ) ) ).

%------------------------------------------------------------------------------