TPTP Problem File: SLH0093^1.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/0025_Finite_Fields_Isomorphic/prob_00120_004642__18460772_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2093 ( 189 unt; 814 typ;   0 def)
%            Number of atoms       : 3939 ( 925 equ;   0 cnn)
%            Maximal formula atoms :    8 (   3 avg)
%            Number of connectives : 24222 ( 113   ~;   1   |; 115   &;21872   @)
%                                         (   0 <=>;2121  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   27 (   9 avg)
%            Number of types       :   92 (  91 usr)
%            Number of type conns  : 2399 (2399   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  724 ( 723 usr;  18 con; 0-4 aty)
%            Number of variables   : 3302 ( 457   ^;2768   !;  77   ?;3302   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 13:24:03.236
%------------------------------------------------------------------------------
% Could-be-implicit typings (91)
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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__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
    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_t__Int__Oint,type,
    int: $tType ).

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

% Explicit typings (723)
thf(sy_c_AbelCoset_OA__RCOSETS_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
    a_RCOS5818729163757401641t_unit: partia6652711048488803900t_unit > set_list_set_int > set_set_list_set_int ).

thf(sy_c_AbelCoset_OA__RCOSETS_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_OA__RCOSETS_001tf__a_001tf__b,type,
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thf(sy_c_AbelCoset_Oa__kernel_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    a_kern885551317661704990it_a_b: partia6652711048488803900t_unit > partia2175431115845679010xt_a_b > ( list_set_int > a ) > set_list_set_int ).

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thf(sy_c_AbelCoset_Oa__kernel_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
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thf(sy_c_AbelCoset_Oa__kernel_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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    a_l_co7008843373686234386t_unit: partia2670972154091845814t_unit > list_a > set_list_a > set_list_a ).

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    a_l_co3504123944629134560t_unit: partia4934656038542163276t_unit > set_int > set_set_int > set_set_int ).

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    a_l_co4050093972498570666t_unit: partia3601206958761670294t_unit > set_set_int > set_set_set_int > set_set_set_int ).

thf(sy_c_AbelCoset_Oa__l__coset_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
    a_l_co8760856729277977016t_unit: partia6043505979758434576t_unit > set_a > set_set_a > set_set_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__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
    a_r_co2531912773843275078t_unit: partia7144473111789202860t_unit > set_li2838760105183549752et_int > list_list_set_int > set_li2838760105183549752et_int ).

thf(sy_c_AbelCoset_Oa__r__coset_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
    a_r_co912825291596486358t_unit: partia6652711048488803900t_unit > set_list_set_int > list_set_int > set_list_set_int ).

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__Int__Oint_J_001t__Product____Type__Ounit,type,
    a_r_co692709266861932262t_unit: partia4934656038542163276t_unit > set_set_int > set_int > set_set_int ).

thf(sy_c_AbelCoset_Oa__r__coset_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
    a_r_co7111076282206511392t_unit: partia3352279114339913222t_unit > set_set_list_set_int > set_list_set_int > set_set_list_set_int ).

thf(sy_c_AbelCoset_Oa__r__coset_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
    a_r_co7378703583274672048t_unit: partia3601206958761670294t_unit > set_set_set_int > set_set_int > set_set_set_int ).

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    a_r_co6984235733410409778t_unit: partia6043505979758434576t_unit > set_set_a > set_a > set_set_a ).

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    a_r_coset_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a ).

thf(sy_c_AbelCoset_Oabelian__group__hom_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    abelia8034434169041113351it_a_b: partia6652711048488803900t_unit > partia2175431115845679010xt_a_b > ( list_set_int > a ) > $o ).

thf(sy_c_AbelCoset_Oabelian__subgroup_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
    abelia4826137221796835534t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).

thf(sy_c_AbelCoset_Oadditive__subgroup_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
    additi4441901441605882380t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).

thf(sy_c_AbelCoset_Oadditive__subgroup_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
    additi7073586575563672860t_unit: set_set_int > partia4934656038542163276t_unit > $o ).

thf(sy_c_AbelCoset_Oadditive__subgroup_001tf__a_001tf__b,type,
    additi2834746164131130830up_a_b: set_a > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_AbelCoset_Oset__add_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oset__add_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oset__add_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oset__add_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oset__add_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oset__add_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
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thf(sy_c_Card__Irreducible__Polynomials__Aux_Ogauss__poly_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
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thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    comp_s3539599085587913099et_int: ( set_int > set_int ) > ( set_int > set_int ) > set_int > set_int ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Set__Oset_It__Int__Oint_J,type,
    comp_s4919011123983036881et_int: ( set_int > set_list_set_int ) > ( set_int > set_int ) > set_int > set_list_set_int ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Int__Oint_J_001tf__a_001t__Set__Oset_It__Int__Oint_J,type,
    comp_s6482183999417960411et_int: ( set_int > a ) > ( set_int > set_int ) > set_int > a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Int__Oint_J_001tf__a_001tf__a,type,
    comp_set_int_a_a: ( set_int > a ) > ( a > set_int ) > a > a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
    comp_s3671312503824928807et_int: ( set_list_set_int > set_list_set_int ) > ( list_set_int > set_list_set_int ) > list_set_int > set_list_set_int ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Set__Oset_It__Int__Oint_J,type,
    comp_s2717484327125700759et_int: ( set_list_set_int > set_list_set_int ) > ( set_int > set_list_set_int ) > set_int > set_list_set_int ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001tf__a_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    comp_s3918679199482572123et_int: ( set_list_set_int > a ) > ( set_list_set_int > set_list_set_int ) > set_list_set_int > a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001tf__a_001tf__a,type,
    comp_s10382787549540875nt_a_a: ( set_list_set_int > a ) > ( a > set_list_set_int ) > a > a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
    comp_s1152226898037786081list_a: ( set_list_a > set_list_a ) > ( list_a > set_list_a ) > list_a > set_list_a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001tf__a,type,
    comp_s7504242949508807643st_a_a: ( set_list_a > set_list_a ) > ( a > set_list_a ) > a > set_list_a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    comp_s3581714896423519297et_int: ( set_set_int > set_int ) > ( set_int > set_set_int ) > set_int > set_int ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J_001tf__a,type,
    comp_set_a_set_a_a: ( set_a > set_a ) > ( a > set_a ) > a > set_a ).

thf(sy_c_Fun_Ocomp_001t__Set__Oset_Itf__a_J_001tf__a_001tf__a,type,
    comp_set_a_a_a: ( set_a > a ) > ( a > set_a ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
    comp_a4633020022310225089et_int: ( a > list_set_int ) > ( list_set_int > a ) > list_set_int > list_set_int ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__List__Olist_Itf__a_J_001tf__a,type,
    comp_a_list_a_a: ( a > list_a ) > ( a > a ) > a > list_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
    comp_a7082422283589945889et_int: ( a > set_int ) > ( set_int > a ) > set_int > set_int ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Set__Oset_It__Int__Oint_J_001tf__a,type,
    comp_a_set_int_a: ( a > set_int ) > ( a > a ) > a > set_int ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001tf__a,type,
    comp_a_set_list_a_a: ( a > set_list_a ) > ( a > a ) > a > set_list_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__Set__Oset_Itf__a_J_001tf__a,type,
    comp_a_set_a_a: ( a > set_a ) > ( a > a ) > a > set_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
    comp_a6958603637840890197et_int: ( a > a ) > ( list_set_int > a ) > list_set_int > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__List__Olist_Itf__a_J,type,
    comp_a_a_list_a: ( a > a ) > ( list_a > a ) > list_a > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    comp_a3704501331711576715et_int: ( a > a ) > ( set_list_set_int > a ) > set_list_set_int > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
    comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).

thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
    inj_on_nat_a_a: ( ( nat > a ) > a ) > set_nat_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    inj_on5847296377779290343et_int: ( list_set_int > list_list_set_int ) > set_list_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
    inj_on3170377866336917463et_int: ( list_set_int > list_set_int ) > set_list_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__List__Olist_Itf__a_J,type,
    inj_on1856771975478933727list_a: ( list_set_int > list_a ) > set_list_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
    inj_on144625271008030877et_int: ( list_set_int > set_li2838760105183549752et_int ) > set_list_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001tf__a,type,
    inj_on3061584747197171161_int_a: ( list_set_int > a ) > set_list_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001tf__a,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
    inj_on2182408081467232967et_int: ( set_int > list_set_int ) > set_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    inj_on6725311424131697661et_int: ( set_int > set_list_set_int ) > set_set_int > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Int__Oint_J_001tf__a,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001tf__a,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001tf__a,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001_062_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_It__Int__Oint_J,type,
    inj_on_set_a_set_int: ( set_a > set_int ) > set_set_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_Itf__a_J_001tf__a,type,
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thf(sy_c_Fun_Oinj__on_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_Fun_Oinj__on_001tf__a_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
    inj_on7524766755751376063et_int: ( a > list_set_int ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__List__Olist_Itf__a_J,type,
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thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_It__Int__Oint_J,type,
    inj_on_a_set_int: ( a > set_int ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    inj_on_a_set_list_a: ( a > set_list_a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
    inj_on_a_set_set_int: ( a > set_set_int ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
    inj_on_a_a: ( a > a ) > set_a > $o ).

thf(sy_c_Group_OUnits_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Group_Omonoid_Omult_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    minus_4948633356380776833et_int: set_list_set_int > set_list_set_int > set_list_set_int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    minus_6857623457997529383et_int: set_set_set_int > set_set_set_int > set_set_set_int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_Hilbert__Choice_Oinv__into_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001tf__a,type,
    hilber7965021363618566206_int_a: set_list_set_int > ( list_set_int > a ) > a > list_set_int ).

thf(sy_c_Ideal_Ocgenideal_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Ideal_Ocgenideal_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Ideal_Ocgenideal_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_Ideal_Ocgenideal_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Product____Type__Ounit_J,type,
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thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mtf__a_J,type,
    member1264592267077841462_int_a: ( set_list_set_int > a ) > set_se1925108120559301869_int_a > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    member5068272912271824380list_a: ( set_list_a > set_list_a ) > set_se1917860372504128155list_a > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    member_set_list_a_a: ( set_list_a > a ) > set_set_list_a_a > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    member45303223370921408et_int: ( set_set_int > list_set_int ) > set_se6417573174631468127et_int > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member2198268278526348592et_int: ( set_set_int > set_int ) > set_se7044434731094029775et_int > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    member8939550387393699046et_int: ( set_set_int > set_set_int ) > set_se1350388527137716869et_int > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
    member5110176937812460956et_int: ( set_set_int > set_set_set_int ) > set_se2452231264655218747et_int > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mtf__a_J,type,
    member_set_set_int_a: ( set_set_int > a ) > set_set_set_int_a > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J_Mtf__a_J,type,
    member5392328839176676752_int_a: ( set_set_set_int > a ) > set_se6206509870384177479_int_a > $o ).

thf(sy_c_member_001_062_It__Set__Oset_It__Set__Oset_Itf__a_J_J_Mtf__a_J,type,
    member_set_set_a_a: ( set_set_a > a ) > set_set_set_a_a > $o ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    member1867902588855634794et_int: ( set_a > list_set_int ) > set_se8264765221139854369et_int > $o ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member_set_a_list_a: ( set_a > list_a ) > set_set_a_list_a > $o ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member_set_a_set_int: ( set_a > set_int ) > set_set_a_set_int > $o ).

thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mt__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    member73722476670768732_set_a: ( set_a > set_set_a ) > set_set_a_set_set_a > $o ).

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

thf(sy_c_member_001_062_It__Set__Oset_Itf__a_J_Mtf__a_J,type,
    member_set_a_a: ( set_a > a ) > set_set_a_a > $o ).

thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
    member2665369038410323738et_int: ( a > list_list_set_int ) > set_a_1117089631521071569et_int > $o ).

thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    member6392905276882098698et_int: ( a > list_set_int ) > set_a_list_set_int > $o ).

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

thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_It__Int__Oint_J_J,type,
    member_a_set_int: ( a > set_int ) > set_a_set_int > $o ).

thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
    member8440910143787153472et_int: ( a > set_list_set_int ) > set_a_3643850743625881079et_int > $o ).

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

thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
    member_a_set_set_int: ( a > set_set_int ) > set_a_set_set_int > $o ).

thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_Itf__a_J_J,type,
    member_a_set_a: ( a > set_a ) > set_a_set_a > $o ).

thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
    member_a_a: ( a > a ) > set_a_a > $o ).

thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    member7440485371949894425et_int: list_list_set_int > set_li2838760105183549752et_int > $o ).

thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
    member_list_set_int: list_set_int > set_list_set_int > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: set_int > set_set_int > $o ).

thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
    member4487585027056014399et_int: set_list_set_int > set_set_list_set_int > $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__Int__Oint_J_J,type,
    member_set_set_int: set_set_int > set_set_set_int > $o ).

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

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

thf(sy_v_R,type,
    r: partia2175431115845679010xt_a_b ).

thf(sy_v_p,type,
    p: nat ).

thf(sy_v_x,type,
    x: a ).

thf(sy_v_y____,type,
    y: set_list_set_int ).

% Relevant facts (1278)
thf(fact_0_finite__field__axioms,axiom,
    ring_f9002592259703987221ld_a_b @ r ).

% finite_field_axioms
thf(fact_1_factorial__domain__axioms,axiom,
    ring_f5272581269873410839in_a_b @ r ).

% factorial_domain_axioms
thf(fact_2_p__def,axiom,
    ( p
    = ( ring_char_a_b @ r ) ) ).

% p_def
thf(fact_3_assms_I5_J,axiom,
    member_a @ x @ ( partia707051561876973205xt_a_b @ r ) ).

% assms(5)
thf(fact_4_noetherian__domain__axioms,axiom,
    ring_n4045954140777738665in_a_b @ r ).

% noetherian_domain_axioms
thf(fact_5_zf_Ofactorial__domain__axioms,axiom,
    ring_f5979754897444624531t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.factorial_domain_axioms
thf(fact_6_pzf_Ofactorial__domain__axioms,axiom,
    ring_f3415241477199771139t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.factorial_domain_axioms
thf(fact_7_zf_Onoetherian__domain__axioms,axiom,
    ring_n4948453042392689537t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.noetherian_domain_axioms
thf(fact_8_pzf_Onoetherian__domain__axioms,axiom,
    ring_n3580647943193786865t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.noetherian_domain_axioms
thf(fact_9_zf_Oprincipal__domain__axioms,axiom,
    ring_p2413783099608688842t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.principal_domain_axioms
thf(fact_10_principal__domain__axioms,axiom,
    ring_p8803135361686045600in_a_b @ r ).

% principal_domain_axioms
thf(fact_11_pzf_Onoetherian__ring__axioms,axiom,
    ring_n7202549095058397701t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.noetherian_ring_axioms
thf(fact_12_pzf_Oprincipal__domain__axioms,axiom,
    ring_p1500425057661402810t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.principal_domain_axioms
thf(fact_13_zf_Onoetherian__ring__axioms,axiom,
    ring_n4918880612521764245t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.noetherian_ring_axioms
thf(fact_14_zf_Oonepideal,axiom,
    princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.onepideal
thf(fact_15_h_Oadditive__subgroup__a__kernel,axiom,
    ( additi4441901441605882380t_unit
    @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
    @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% h.additive_subgroup_a_kernel
thf(fact_16_h_Ois__abelian__group__hom,axiom,
    ( abelia8034434169041113351it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
    @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ).

% h.is_abelian_group_hom
thf(fact_17__092_060open_062_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_096_Aa__kernel_A_Ipoly__ring_A_IZFact_A_Iint_Ap_J_J_J_AR_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_060_L_062_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_096_Ay_A_061_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_096_A_Ia__kernel_A_Ipoly__ring_A_IZFact_A_Iint_Ap_J_J_J_AR_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_060_L_062_092_060_094bsub_062poly__ring_A_IZFact_A_Iint_Ap_J_J_092_060_094esub_062_Ay_J_092_060close_062,axiom,
    ( ( set_add_a_b @ r
      @ ( image_list_set_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) )
      @ ( image_list_set_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ y ) )
    = ( image_list_set_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
        @ y ) ) ) ).

% \<open>(\<lambda>p. local.eval (map (char_iso R) p) x) ` a_kernel (poly_ring (ZFact (int p))) R (\<lambda>p. local.eval (map (char_iso R) p) x) <+> (\<lambda>p. local.eval (map (char_iso R) p) x) ` y = (\<lambda>p. local.eval (map (char_iso R) p) x) ` (a_kernel (poly_ring (ZFact (int p))) R (\<lambda>p. local.eval (map (char_iso R) p) x) <+>\<^bsub>poly_ring (ZFact (int p))\<^esub> y)\<close>
thf(fact_18_noetherian__ring__axioms,axiom,
    ring_n3639167112692572309ng_a_b @ r ).

% noetherian_ring_axioms
thf(fact_19_h_Oabelian__subgroup__a__kernel,axiom,
    ( abelia4826137221796835534t_unit
    @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
    @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% h.abelian_subgroup_a_kernel
thf(fact_20_a,axiom,
    ( member6759595599802011116_int_a
    @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
    @ ( ring_h3322734120376767392it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r ) ) ).

% a
thf(fact_21_pzf_Oadd__additive__subgroups,axiom,
    ! [H: set_list_set_int,K: set_list_set_int] :
      ( ( additi4441901441605882380t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( additi4441901441605882380t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
       => ( additi4441901441605882380t_unit @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ K ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.add_additive_subgroups
thf(fact_22_onepideal,axiom,
    principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% onepideal
thf(fact_23_h_Ohom__closed,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( member_a @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% h.hom_closed
thf(fact_24__092_060open_062_092_060zero_062_092_060_094bsub_062poly__ring_A_IZFact_A_Iint_Ap_J_J_092_060_094esub_062_A_092_060in_062_Aa__kernel_A_Ipoly__ring_A_IZFact_A_Iint_Ap_J_J_J_AR_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_092_060close_062,axiom,
    ( member_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
    @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) ).

% \<open>\<zero>\<^bsub>poly_ring (ZFact (int p))\<^esub> \<in> a_kernel (poly_ring (ZFact (int p))) R (\<lambda>p. local.eval (map (char_iso R) p) x)\<close>
thf(fact_25_h_Ozero__closed,axiom,
    member_a @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ).

% h.zero_closed
thf(fact_26__092_060open_062_I_092_060lambda_062p_O_Alocal_Oeval_Ap_Ax_J_A_092_060circ_062_Amap_A_Ichar__iso_AR_J_A_092_060in_062_Aring__hom_A_Ipoly__ring_A_IZFact_A_Iint_Ap_J_J_J_AR_092_060close_062,axiom,
    ( member6759595599802011116_int_a
    @ ( comp_l364663056984283087et_int
      @ ^ [P: list_a] : ( eval_a_b @ r @ P @ x )
      @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) ) )
    @ ( ring_h3322734120376767392it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r ) ) ).

% \<open>(\<lambda>p. local.eval p x) \<circ> map (char_iso R) \<in> ring_hom (poly_ring (ZFact (int p))) R\<close>
thf(fact_27_h_Okernel__is__ideal,axiom,
    ( ideal_3307802554209726317t_unit
    @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
    @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% h.kernel_is_ideal
thf(fact_28_pzf_Osemiring__axioms,axiom,
    semiri2117368926114562431t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.semiring_axioms
thf(fact_29_zf_Ofinite__field__axioms,axiom,
    ring_f302724563095964181t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.finite_field_axioms
thf(fact_30_h_Oring__hom__ring__axioms,axiom,
    ( ring_h6525298532103594462it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
    @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ).

% h.ring_hom_ring_axioms
thf(fact_31_zf_Ocgenideal__is__principalideal,axiom,
    ! [I: set_int] :
      ( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( princi8860937869964495385t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.cgenideal_is_principalideal
thf(fact_32_zf_Ocgenideal__self,axiom,
    ! [I: set_int] :
      ( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( member_set_int @ I @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I ) ) ) ).

% zf.cgenideal_self
thf(fact_33_pzf_Ooneideal,axiom,
    ideal_3307802554209726317t_unit @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.oneideal
thf(fact_34_pzf_Oadd__ideals,axiom,
    ! [I2: set_list_set_int,J: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ideal_3307802554209726317t_unit @ J @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
       => ( ideal_3307802554209726317t_unit @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ J ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.add_ideals
thf(fact_35_pzf_Ozero__closed,axiom,
    member_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% pzf.zero_closed
thf(fact_36_pzf_Oonepideal,axiom,
    princi2884690910141361289t_unit @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.onepideal
thf(fact_37_zf_Ogauss__poly__carr,axiom,
    ! [N: nat] : ( member_list_set_int @ ( card_I2978652876270886323t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ N ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.gauss_poly_carr
thf(fact_38_h_Oset__add__ker__hom_I2_J,axiom,
    ! [I2: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( image_list_set_int_a
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
          @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
              @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
            @ I2 ) )
        = ( image_list_set_int_a
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
          @ I2 ) ) ) ).

% h.set_add_ker_hom(2)
thf(fact_39_h_Oset__add__ker__hom_I1_J,axiom,
    ! [I2: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( image_list_set_int_a
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
          @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2
            @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
              @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) )
        = ( image_list_set_int_a
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
          @ I2 ) ) ) ).

% h.set_add_ker_hom(1)
thf(fact_40_pzf_Ozero__is__prime_I1_J,axiom,
    prime_7199642643890090290t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% pzf.zero_is_prime(1)
thf(fact_41_pzf_Oring__primeE_I1_J,axiom,
    ! [P2: list_set_int] :
      ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
       => ( P2
         != ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.ring_primeE(1)
thf(fact_42_h_Ohomeq__imp__rcos,axiom,
    ! [A: list_set_int,X: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X ) @ x )
            = ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ A ) @ x ) )
         => ( member_list_set_int @ X
            @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
              @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
                @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
              @ A ) ) ) ) ) ).

% h.homeq_imp_rcos
thf(fact_43_h_Orcos__eq__homeq,axiom,
    ! [A: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
          @ A )
        = ( collect_list_set_int
          @ ^ [X2: list_set_int] :
              ( ( member_list_set_int @ X2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
              & ( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X2 ) @ x )
                = ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ A ) @ x ) ) ) ) ) ) ).

% h.rcos_eq_homeq
thf(fact_44_mem__Collect__eq,axiom,
    ! [A: nat > set_int,P3: ( nat > set_int ) > $o] :
      ( ( member_nat_set_int @ A @ ( collect_nat_set_int @ P3 ) )
      = ( P3 @ A ) ) ).

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

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

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

% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
    ! [A: set_set_int,P3: set_set_int > $o] :
      ( ( member_set_set_int @ A @ ( collect_set_set_int @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
    ! [A: set_int,P3: set_int > $o] :
      ( ( member_set_int @ A @ ( collect_set_int @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
    ! [A: set_a,P3: set_a > $o] :
      ( ( member_set_a @ A @ ( collect_set_a @ P3 ) )
      = ( P3 @ A ) ) ).

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

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

% mem_Collect_eq
thf(fact_53_Collect__mem__eq,axiom,
    ! [A2: set_nat_set_int] :
      ( ( collect_nat_set_int
        @ ^ [X2: nat > set_int] : ( member_nat_set_int @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
    ! [A2: set_nat_a] :
      ( ( collect_nat_a
        @ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
    ! [A2: set_a_list_a] :
      ( ( collect_a_list_a
        @ ^ [X2: a > list_a] : ( member_a_list_a @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
    ! [A2: set_a_set_list_a] :
      ( ( collect_a_set_list_a
        @ ^ [X2: a > set_list_a] : ( member_a_set_list_a @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
    ! [A2: set_set_set_int] :
      ( ( collect_set_set_int
        @ ^ [X2: set_set_int] : ( member_set_set_int @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
    ! [A2: set_set_int] :
      ( ( collect_set_int
        @ ^ [X2: set_int] : ( member_set_int @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
    ! [A2: set_set_a] :
      ( ( collect_set_a
        @ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
    ! [A2: set_a] :
      ( ( collect_a
        @ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_62_Collect__cong,axiom,
    ! [P3: set_set_int > $o,Q: set_set_int > $o] :
      ( ! [X3: set_set_int] :
          ( ( P3 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_set_int @ P3 )
        = ( collect_set_set_int @ Q ) ) ) ).

% Collect_cong
thf(fact_63_Collect__cong,axiom,
    ! [P3: set_int > $o,Q: set_int > $o] :
      ( ! [X3: set_int] :
          ( ( P3 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_int @ P3 )
        = ( collect_set_int @ Q ) ) ) ).

% Collect_cong
thf(fact_64_Collect__cong,axiom,
    ! [P3: set_a > $o,Q: set_a > $o] :
      ( ! [X3: set_a] :
          ( ( P3 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_a @ P3 )
        = ( collect_set_a @ Q ) ) ) ).

% Collect_cong
thf(fact_65_Collect__cong,axiom,
    ! [P3: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P3 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P3 )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_66_Collect__cong,axiom,
    ! [P3: a > $o,Q: a > $o] :
      ( ! [X3: a] :
          ( ( P3 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_a @ P3 )
        = ( collect_a @ Q ) ) ) ).

% Collect_cong
thf(fact_67_h_Orcos__imp__homeq,axiom,
    ! [A: list_set_int,X: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ X
          @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
              @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
            @ A ) )
       => ( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X ) @ x )
          = ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ A ) @ x ) ) ) ) ).

% h.rcos_imp_homeq
thf(fact_68_h_Oa__inv__closed,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( member_a @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% h.a_inv_closed
thf(fact_69_h_Oideal__vimage,axiom,
    ! [I2: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ideal_3307802554209726317t_unit
        @ ( collect_list_set_int
          @ ^ [R: list_set_int] :
              ( ( member_list_set_int @ R @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
              & ( member_a @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ R ) @ x ) @ I2 ) ) )
        @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% h.ideal_vimage
thf(fact_70_oneideal,axiom,
    ideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% oneideal
thf(fact_71_add__ideals,axiom,
    ! [I2: set_a,J: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ideal_a_b @ J @ r )
       => ( ideal_a_b @ ( set_add_a_b @ r @ I2 @ J ) @ r ) ) ) ).

% add_ideals
thf(fact_72_pzf_Oa__rcos__zero,axiom,
    ! [I2: set_list_set_int,I: list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member_list_set_int @ I @ I2 )
       => ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ I )
          = I2 ) ) ) ).

% pzf.a_rcos_zero
thf(fact_73_pzf_Oa__r__coset__subset__G,axiom,
    ! [H: set_list_set_int,X: list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ord_le4527193859438875528et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ X ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.a_r_coset_subset_G
thf(fact_74_pzf_Osetadd__subset__G,axiom,
    ! [H: set_list_set_int,K: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ord_le4527193859438875528et_int @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ K ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.setadd_subset_G
thf(fact_75_pzf_Oset__add__comm,axiom,
    ! [I2: set_list_set_int,J: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ord_le4527193859438875528et_int @ J @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ J )
          = ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ J @ I2 ) ) ) ) ).

% pzf.set_add_comm
thf(fact_76_pzf_Oset__add__closed,axiom,
    ! [A2: set_list_set_int,B: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ A2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ord_le4527193859438875528et_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ord_le4527193859438875528et_int @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A2 @ B ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.set_add_closed
thf(fact_77_pzf_Oring__primeE_I3_J,axiom,
    ! [P2: list_set_int] :
      ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
       => ( prime_7199642643890090290t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ).

% pzf.ring_primeE(3)
thf(fact_78_pzf_Oring__primeI,axiom,
    ! [P2: list_set_int] :
      ( ( P2
       != ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( prime_7199642643890090290t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
       => ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ).

% pzf.ring_primeI
thf(fact_79_y__range,axiom,
    ord_le4527193859438875528et_int @ y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% y_range
thf(fact_80_pzf_Oa__setmult__rcos__assoc,axiom,
    ! [H: set_list_set_int,K: set_list_set_int,X: list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ X ) )
            = ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ K ) @ X ) ) ) ) ) ).

% pzf.a_setmult_rcos_assoc
thf(fact_81_pzf_Ominus__minus,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) )
        = X ) ) ).

% pzf.minus_minus
thf(fact_82_pzf_Oadd_Oinv__closed,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( member_list_set_int @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.add.inv_closed
thf(fact_83_pzf_Ominus__zero,axiom,
    ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
    = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.minus_zero
thf(fact_84_pzf_Oadd_Oinv__eq__1__iff,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X )
          = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        = ( X
          = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.add.inv_eq_1_iff
thf(fact_85_pzf_Oa__coset__add__zero,axiom,
    ! [M: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        = M ) ) ).

% pzf.a_coset_add_zero
thf(fact_86_pzf_Oa__rcos__assoc__lcos,axiom,
    ! [H: set_list_set_int,K: set_list_set_int,X: list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ X ) @ K )
            = ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ K ) ) ) ) ) ) ).

% pzf.a_rcos_assoc_lcos
thf(fact_87_pzf_Oideal__is__subalgebra,axiom,
    ! [K: set_list_set_int,I2: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
       => ( embedd3025314616789055568t_unit @ K @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.ideal_is_subalgebra
thf(fact_88_pzf_Oa__lcos__mult__one,axiom,
    ! [M: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ M )
        = M ) ) ).

% pzf.a_lcos_mult_one
thf(fact_89_pzf_Ogenideal__ideal,axiom,
    ! [S: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ S @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ideal_3307802554209726317t_unit @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.genideal_ideal
thf(fact_90_pzf_OIdl__subset__ideal,axiom,
    ! [I2: set_list_set_int,H: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( ord_le4527193859438875528et_int @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H ) @ I2 )
          = ( ord_le4527193859438875528et_int @ H @ I2 ) ) ) ) ).

% pzf.Idl_subset_ideal
thf(fact_91_pzf_Oquotient__eq__iff__same__a__r__cos,axiom,
    ! [I2: set_list_set_int,A: list_set_int,B2: list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( member_list_set_int @ ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B2 ) @ I2 )
            = ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ A )
              = ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ B2 ) ) ) ) ) ) ).

% pzf.quotient_eq_iff_same_a_r_cos
thf(fact_92_pzf_Oa__rcosetsI,axiom,
    ! [H: set_list_set_int,X: list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( member4487585027056014399et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ X ) @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H ) ) ) ) ).

% pzf.a_rcosetsI
thf(fact_93_pzf_Osubalgebra__in__carrier,axiom,
    ! [K: set_list_set_int,V: set_list_set_int] :
      ( ( embedd3025314616789055568t_unit @ K @ V @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ord_le4527193859438875528et_int @ V @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.subalgebra_in_carrier
thf(fact_94_pzf_Ocarrier__is__subalgebra,axiom,
    ! [K: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( embedd3025314616789055568t_unit @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.carrier_is_subalgebra
thf(fact_95_pzf_Ogenideal__minimal,axiom,
    ! [I2: set_list_set_int,S: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ord_le4527193859438875528et_int @ S @ I2 )
       => ( ord_le4527193859438875528et_int @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) @ I2 ) ) ) ).

% pzf.genideal_minimal
thf(fact_96_pzf_Oa__l__coset__subset__G,axiom,
    ! [H: set_list_set_int,X: list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ord_le4527193859438875528et_int @ ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ H ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.a_l_coset_subset_G
thf(fact_97_pzf_Ogenideal__self,axiom,
    ! [S: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ S @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ord_le4527193859438875528et_int @ S @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) ) ) ).

% pzf.genideal_self
thf(fact_98_pzf_Osubset__Idl__subset,axiom,
    ! [I2: set_list_set_int,H: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ord_le4527193859438875528et_int @ H @ I2 )
       => ( ord_le4527193859438875528et_int @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H ) @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) ) ) ) ).

% pzf.subset_Idl_subset
thf(fact_99_pzf_Ominus__closed,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( member_list_set_int @ ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.minus_closed
thf(fact_100_pzf_Or__right__minus__eq,axiom,
    ! [A: list_set_int,B2: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B2 )
            = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          = ( A = B2 ) ) ) ) ).

% pzf.r_right_minus_eq
thf(fact_101_h_Oimg__is__subalgebra,axiom,
    ! [K: set_list_set_int,V: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( embedd3025314616789055568t_unit @ K @ V @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
       => ( embedd9027525575939734154ra_a_b
          @ ( image_list_set_int_a
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
            @ K )
          @ ( image_list_set_int_a
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
            @ V )
          @ r ) ) ) ).

% h.img_is_subalgebra
thf(fact_102_pzf_Onoetherian__ringI,axiom,
    ( ! [I3: set_list_set_int] :
        ( ( ideal_3307802554209726317t_unit @ I3 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
       => ? [A3: set_list_set_int] :
            ( ( ord_le4527193859438875528et_int @ A3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
            & ( finite7825103155802847241et_int @ A3 )
            & ( I3
              = ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A3 ) ) ) )
   => ( ring_n7202549095058397701t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.noetherian_ringI
thf(fact_103_pzf_Ofinetely__gen,axiom,
    ! [I2: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ? [A4: set_list_set_int] :
          ( ( ord_le4527193859438875528et_int @ A4 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          & ( finite7825103155802847241et_int @ A4 )
          & ( I2
            = ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A4 ) ) ) ) ).

% pzf.finetely_gen
thf(fact_104_pzf_Oa__coset__add__inv1,axiom,
    ! [M: set_list_set_int,X: list_set_int,Y: list_set_int] :
      ( ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) ) )
        = M )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
           => ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ X )
              = ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ Y ) ) ) ) ) ) ).

% pzf.a_coset_add_inv1
thf(fact_105_pzf_Oa__coset__add__inv2,axiom,
    ! [M: set_list_set_int,X: list_set_int,Y: list_set_int] :
      ( ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ X )
        = ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ Y ) )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
           => ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) ) )
              = M ) ) ) ) ) ).

% pzf.a_coset_add_inv2
thf(fact_106_pzf_Oline__extension__in__carrier,axiom,
    ! [K: set_list_set_int,A: list_set_int,E: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( ord_le4527193859438875528et_int @ E @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ord_le4527193859438875528et_int @ ( embedd942231993492570031t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ A @ E ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).

% pzf.line_extension_in_carrier
thf(fact_107_pzf_Oup__minus__closed,axiom,
    ! [P2: nat > list_set_int,Q2: nat > list_set_int] :
      ( ( member5167342361288740600et_int @ P2 @ ( up_lis4326185709802647968t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member5167342361288740600et_int @ Q2 @ ( up_lis4326185709802647968t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( member5167342361288740600et_int
          @ ^ [I4: nat] : ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
          @ ( up_lis4326185709802647968t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.up_minus_closed
thf(fact_108_pzf_Oup__a__inv__closed,axiom,
    ! [P2: nat > list_set_int] :
      ( ( member5167342361288740600et_int @ P2 @ ( up_lis4326185709802647968t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( member5167342361288740600et_int
        @ ^ [I4: nat] : ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( P2 @ I4 ) )
        @ ( up_lis4326185709802647968t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.up_a_inv_closed
thf(fact_109_pzf_Oa__lcos__m__assoc,axiom,
    ! [M: set_list_set_int,G: list_set_int,H2: list_set_int] :
      ( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ G @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ G @ ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ M ) )
            = ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ G @ H2 ) @ M ) ) ) ) ) ).

% pzf.a_lcos_m_assoc
thf(fact_110_pzf_Oadd_Or__cancel,axiom,
    ! [A: list_set_int,C: list_set_int,B2: list_set_int] :
      ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C )
        = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 @ C ) )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
           => ( A = B2 ) ) ) ) ) ).

% pzf.add.r_cancel
thf(fact_111_pzf_Oadd_Om__lcomm,axiom,
    ! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ Z ) )
            = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Z ) ) ) ) ) ) ).

% pzf.add.m_lcomm
thf(fact_112_pzf_Oadd_Om__comm,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
          = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X ) ) ) ) ).

% pzf.add.m_comm
thf(fact_113_pzf_Oadd_Om__assoc,axiom,
    ! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ Z )
            = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ Z ) ) ) ) ) ) ).

% pzf.add.m_assoc
thf(fact_114_pzf_Oadd_Ol__cancel,axiom,
    ! [C: list_set_int,A: list_set_int,B2: list_set_int] :
      ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ A )
        = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ B2 ) )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
           => ( A = B2 ) ) ) ) ) ).

% pzf.add.l_cancel
thf(fact_115_pzf_Oup__add__closed,axiom,
    ! [P2: nat > list_set_int,Q2: nat > list_set_int] :
      ( ( member5167342361288740600et_int @ P2 @ ( up_lis4326185709802647968t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member5167342361288740600et_int @ Q2 @ ( up_lis4326185709802647968t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( member5167342361288740600et_int
          @ ^ [I4: nat] : ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
          @ ( up_lis4326185709802647968t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.up_add_closed
thf(fact_116_pzf_Ominus__unique,axiom,
    ! [Y: list_set_int,X: list_set_int,Y2: list_set_int] :
      ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X )
        = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y2 )
          = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
           => ( ( member_list_set_int @ Y2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
             => ( Y = Y2 ) ) ) ) ) ) ).

% pzf.minus_unique
thf(fact_117_pzf_Oadd_Or__inv__ex,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ? [X3: list_set_int] :
          ( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          & ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ X3 )
            = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.add.r_inv_ex
thf(fact_118_pzf_Oadd_Oone__unique,axiom,
    ! [U: list_set_int] :
      ( ( member_list_set_int @ U @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ! [X3: list_set_int] :
            ( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
           => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ U @ X3 )
              = X3 ) )
       => ( U
          = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.add.one_unique
thf(fact_119_pzf_Oadd_Ol__inv__ex,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ? [X3: list_set_int] :
          ( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          & ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X3 @ X )
            = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.add.l_inv_ex
thf(fact_120_pzf_Oadd_Oinv__comm,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
        = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X )
            = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).

% pzf.add.inv_comm
thf(fact_121_pzf_Or__neg2,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ Y ) )
          = Y ) ) ) ).

% pzf.r_neg2
thf(fact_122_pzf_Or__neg1,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) )
          = Y ) ) ) ).

% pzf.r_neg1
thf(fact_123_pzf_Ominus__add,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) )
          = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) ) ) ) ) ).

% pzf.minus_add
thf(fact_124_pzf_Oadd_Oinv__solve__right_H,axiom,
    ! [A: list_set_int,B2: list_set_int,C: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C ) )
              = A )
            = ( B2
              = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C ) ) ) ) ) ) ).

% pzf.add.inv_solve_right'
thf(fact_125_pzf_Oadd_Oinv__solve__right,axiom,
    ! [A: list_set_int,B2: list_set_int,C: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( A
              = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C ) ) )
            = ( B2
              = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C ) ) ) ) ) ) ).

% pzf.add.inv_solve_right
thf(fact_126_pzf_Oadd_Oinv__solve__left_H,axiom,
    ! [A: list_set_int,B2: list_set_int,C: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 ) @ C )
              = A )
            = ( C
              = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 @ A ) ) ) ) ) ) ).

% pzf.add.inv_solve_left'
thf(fact_127_pzf_Oadd_Oinv__solve__left,axiom,
    ! [A: list_set_int,B2: list_set_int,C: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( A
              = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 ) @ C ) )
            = ( C
              = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 @ A ) ) ) ) ) ) ).

% pzf.add.inv_solve_left
thf(fact_128_pzf_Oadd_Oinv__mult__group,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) )
          = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) ) ) ) ).

% pzf.add.inv_mult_group
thf(fact_129_pzf_Oa__transpose__inv,axiom,
    ! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
      ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
        = Z )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
           => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ Z )
              = Y ) ) ) ) ) ).

% pzf.a_transpose_inv
thf(fact_130_pzf_Ominus__eq,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
      = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) ) ) ).

% pzf.minus_eq
thf(fact_131_pzf_Osum__zero__eq__neg,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
            = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( X
            = ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) ) ) ) ) ).

% pzf.sum_zero_eq_neg
thf(fact_132_pzf_Or__neg,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) )
        = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.r_neg
thf(fact_133_pzf_Ominus__equality,axiom,
    ! [Y: list_set_int,X: list_set_int] :
      ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X )
        = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X )
            = Y ) ) ) ) ).

% pzf.minus_equality
thf(fact_134_pzf_Ol__neg,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ X )
        = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.l_neg
thf(fact_135_pzf_Oa__rcosI,axiom,
    ! [H2: list_set_int,H: set_list_set_int,X: list_set_int] :
      ( ( member_list_set_int @ H2 @ H )
     => ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ X ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ X ) ) ) ) ) ).

% pzf.a_rcosI
thf(fact_136_pzf_Oa__coset__add__assoc,axiom,
    ! [M: set_list_set_int,G: list_set_int,H2: list_set_int] :
      ( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ G @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ G ) @ H2 )
            = ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ G @ H2 ) ) ) ) ) ) ).

% pzf.a_coset_add_assoc
thf(fact_137_pzf_Oadd_Oright__cancel,axiom,
    ! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X )
              = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Z @ X ) )
            = ( Y = Z ) ) ) ) ) ).

% pzf.add.right_cancel
thf(fact_138_pzf_Oadd_Om__closed,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.add.m_closed
thf(fact_139_pzf_Or__zero,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        = X ) ) ).

% pzf.r_zero
thf(fact_140_pzf_Ol__zero,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ X )
        = X ) ) ).

% pzf.l_zero
thf(fact_141_pzf_Oadd_Or__cancel__one_H,axiom,
    ! [X: list_set_int,A: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( X
            = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ X ) )
          = ( A
            = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).

% pzf.add.r_cancel_one'
thf(fact_142_pzf_Oadd_Or__cancel__one,axiom,
    ! [X: list_set_int,A: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ X )
            = X )
          = ( A
            = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).

% pzf.add.r_cancel_one
thf(fact_143_pzf_Oadd_Ol__cancel__one_H,axiom,
    ! [X: list_set_int,A: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( X
            = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ A ) )
          = ( A
            = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).

% pzf.add.l_cancel_one'
thf(fact_144_pzf_Oadd_Ol__cancel__one,axiom,
    ! [X: list_set_int,A: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ A )
            = X )
          = ( A
            = ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).

% pzf.add.l_cancel_one
thf(fact_145_pzf_Obound__upD,axiom,
    ! [F: nat > list_set_int] :
      ( ( member5167342361288740600et_int @ F @ ( up_lis4326185709802647968t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ? [N2: nat] : ( bound_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ N2 @ F ) ) ).

% pzf.bound_upD
thf(fact_146_h_Oline__extension__hom,axiom,
    ! [K: set_list_set_int,A: list_set_int,E: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( ord_le4527193859438875528et_int @ E @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( embedd971793762689825387on_a_b @ r
              @ ( image_list_set_int_a
                @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
                @ K )
              @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ A ) @ x )
              @ ( image_list_set_int_a
                @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
                @ E ) )
            = ( image_list_set_int_a
              @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
              @ ( embedd942231993492570031t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ A @ E ) ) ) ) ) ) ).

% h.line_extension_hom
thf(fact_147_pzf_Oadd_Oone__in__subset,axiom,
    ! [H: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( H != bot_bo7344713902834881468et_int )
       => ( ! [X3: list_set_int] :
              ( ( member_list_set_int @ X3 @ H )
             => ( member_list_set_int @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X3 ) @ H ) )
         => ( ! [X3: list_set_int] :
                ( ( member_list_set_int @ X3 @ H )
               => ! [Xa: list_set_int] :
                    ( ( member_list_set_int @ Xa @ H )
                   => ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X3 @ Xa ) @ H ) ) )
           => ( member_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ H ) ) ) ) ) ).

% pzf.add.one_in_subset
thf(fact_148_pzf_Oa__card__cosets__equal,axiom,
    ! [C: set_list_set_int,H: set_list_set_int] :
      ( ( member4487585027056014399et_int @ C @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H ) )
     => ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( finite7825103155802847241et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( finite3141824629951824842et_int @ C )
            = ( finite3141824629951824842et_int @ H ) ) ) ) ) ).

% pzf.a_card_cosets_equal
thf(fact_149_h_Othe__elem__simp,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( the_elem_a
          @ ( image_list_set_int_a
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
            @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
              @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
                @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
              @ X ) ) )
        = ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X ) @ x ) ) ) ).

% h.the_elem_simp
thf(fact_150_pzf_Ocarrier__is__subcring,axiom,
    subcri4185096033226064695t_unit @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.carrier_is_subcring
thf(fact_151_pzf_Ounion__genideal,axiom,
    ! [I2: set_list_set_int,J: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ideal_3307802554209726317t_unit @ J @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
       => ( ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( sup_su6733955740315415124et_int @ I2 @ J ) )
          = ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ J ) ) ) ) ).

% pzf.union_genideal
thf(fact_152_noetherian__ring_Ofinetely__gen,axiom,
    ! [R2: partia3352279114339913222t_unit,I2: set_set_list_set_int] :
      ( ( ring_n1943713544749852751t_unit @ R2 )
     => ( ( ideal_7892184242768911799t_unit @ I2 @ R2 )
       => ? [A4: set_set_list_set_int] :
            ( ( ord_le8104031629355235774et_int @ A4 @ ( partia7954853462395303941t_unit @ R2 ) )
            & ( finite5161535640157284671et_int @ A4 )
            & ( I2
              = ( genide8305543099697017237t_unit @ R2 @ A4 ) ) ) ) ) ).

% noetherian_ring.finetely_gen
thf(fact_153_noetherian__ring_Ofinetely__gen,axiom,
    ! [R2: partia3601206958761670294t_unit,I2: set_set_set_int] :
      ( ( ring_n6121427063012330079t_unit @ R2 )
     => ( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
       => ? [A4: set_set_set_int] :
            ( ( ord_le4317611570275147438et_int @ A4 @ ( partia1981568256555877781t_unit @ R2 ) )
            & ( finite4249678464180374575et_int @ A4 )
            & ( I2
              = ( genide1354039622117027493t_unit @ R2 @ A4 ) ) ) ) ) ).

% noetherian_ring.finetely_gen
thf(fact_154_noetherian__ring_Ofinetely__gen,axiom,
    ! [R2: partia6043505979758434576t_unit,I2: set_set_a] :
      ( ( ring_n5014428767265248323t_unit @ R2 )
     => ( ( ideal_4463284918206690523t_unit @ I2 @ R2 )
       => ? [A4: set_set_a] :
            ( ( ord_le3724670747650509150_set_a @ A4 @ ( partia5907974310037520643t_unit @ R2 ) )
            & ( finite_finite_set_a @ A4 )
            & ( I2
              = ( genide4542190045063241085t_unit @ R2 @ A4 ) ) ) ) ) ).

% noetherian_ring.finetely_gen
thf(fact_155_noetherian__ring_Ofinetely__gen,axiom,
    ! [R2: partia6652711048488803900t_unit,I2: set_list_set_int] :
      ( ( ring_n7202549095058397701t_unit @ R2 )
     => ( ( ideal_3307802554209726317t_unit @ I2 @ R2 )
       => ? [A4: set_list_set_int] :
            ( ( ord_le4527193859438875528et_int @ A4 @ ( partia5243588073521899067t_unit @ R2 ) )
            & ( finite7825103155802847241et_int @ A4 )
            & ( I2
              = ( genide4509635305121348427t_unit @ R2 @ A4 ) ) ) ) ) ).

% noetherian_ring.finetely_gen
thf(fact_156_noetherian__ring_Ofinetely__gen,axiom,
    ! [R2: partia4934656038542163276t_unit,I2: set_set_int] :
      ( ( ring_n4918880612521764245t_unit @ R2 )
     => ( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
       => ? [A4: set_set_int] :
            ( ( ord_le4403425263959731960et_int @ A4 @ ( partia966996272515721803t_unit @ R2 ) )
            & ( finite6197958912794628473et_int @ A4 )
            & ( I2
              = ( genide1545711809618862555t_unit @ R2 @ A4 ) ) ) ) ) ).

% noetherian_ring.finetely_gen
thf(fact_157_noetherian__ring_Ofinetely__gen,axiom,
    ! [R2: partia2175431115845679010xt_a_b,I2: set_a] :
      ( ( ring_n3639167112692572309ng_a_b @ R2 )
     => ( ( ideal_a_b @ I2 @ R2 )
       => ? [A4: set_a] :
            ( ( ord_less_eq_set_a @ A4 @ ( partia707051561876973205xt_a_b @ R2 ) )
            & ( finite_finite_a @ A4 )
            & ( I2
              = ( genideal_a_b @ R2 @ A4 ) ) ) ) ) ).

% noetherian_ring.finetely_gen
thf(fact_158_ring__hom__ring_Oset__add__ker__hom_I1_J,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a,I2: set_list_set_int] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( image_list_set_int_a @ H2 @ ( set_ad2159827153730365080t_unit @ R2 @ I2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) ) )
          = ( image_list_set_int_a @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(1)
thf(fact_159_ring__hom__ring_Oset__add__ker__hom_I1_J,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H2: list_a > set_list_a,I2: set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H2 )
     => ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( image_5464838071766335845list_a @ H2 @ ( set_ad92425877771022410t_unit @ R2 @ I2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H2 ) ) )
          = ( image_5464838071766335845list_a @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(1)
thf(fact_160_ring__hom__ring_Oset__add__ker__hom_I1_J,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,I2: set_list_a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( image_list_a_a @ H2 @ ( set_ad92425877771022410t_unit @ R2 @ I2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) ) )
          = ( image_list_a_a @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(1)
thf(fact_161_ring__hom__ring_Oset__add__ker__hom_I1_J,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6652711048488803900t_unit,H2: set_int > list_set_int,I2: set_set_int] :
      ( ( ring_h615793168808539756t_unit @ R2 @ S @ H2 )
     => ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( image_8094898911679120219et_int @ H2 @ ( set_ad273131178244904872t_unit @ R2 @ I2 @ ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 ) ) )
          = ( image_8094898911679120219et_int @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(1)
thf(fact_162_ring__hom__ring_Oset__add__ker__hom_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int,I2: set_a] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( image_a_list_set_int @ H2 @ ( set_add_a_b @ R2 @ I2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) ) )
          = ( image_a_list_set_int @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(1)
thf(fact_163_ring__hom__ring_Oset__add__ker__hom_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,I2: set_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( image_a_list_a @ H2 @ ( set_add_a_b @ R2 @ I2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) ) )
          = ( image_a_list_a @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(1)
thf(fact_164_ring__hom__ring_Oset__add__ker__hom_I2_J,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a,I2: set_list_set_int] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( image_list_set_int_a @ H2 @ ( set_ad2159827153730365080t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) @ I2 ) )
          = ( image_list_set_int_a @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(2)
thf(fact_165_ring__hom__ring_Oset__add__ker__hom_I2_J,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H2: list_a > set_list_a,I2: set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H2 )
     => ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( image_5464838071766335845list_a @ H2 @ ( set_ad92425877771022410t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H2 ) @ I2 ) )
          = ( image_5464838071766335845list_a @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(2)
thf(fact_166_ring__hom__ring_Oset__add__ker__hom_I2_J,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,I2: set_list_a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( image_list_a_a @ H2 @ ( set_ad92425877771022410t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) @ I2 ) )
          = ( image_list_a_a @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(2)
thf(fact_167_ring__hom__ring_Oset__add__ker__hom_I2_J,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6652711048488803900t_unit,H2: set_int > list_set_int,I2: set_set_int] :
      ( ( ring_h615793168808539756t_unit @ R2 @ S @ H2 )
     => ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( image_8094898911679120219et_int @ H2 @ ( set_ad273131178244904872t_unit @ R2 @ ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 ) @ I2 ) )
          = ( image_8094898911679120219et_int @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(2)
thf(fact_168_ring__hom__ring_Oset__add__ker__hom_I2_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int,I2: set_a] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( image_a_list_set_int @ H2 @ ( set_add_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) @ I2 ) )
          = ( image_a_list_set_int @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(2)
thf(fact_169_ring__hom__ring_Oset__add__ker__hom_I2_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,I2: set_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( image_a_list_a @ H2 @ ( set_add_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) @ I2 ) )
          = ( image_a_list_a @ H2 @ I2 ) ) ) ) ).

% ring_hom_ring.set_add_ker_hom(2)
thf(fact_170_genideal__self,axiom,
    ! [S: set_a] :
      ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).

% genideal_self
thf(fact_171_subset__Idl__subset,axiom,
    ! [I2: set_a,H: set_a] :
      ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ H @ I2 )
       => ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).

% subset_Idl_subset
thf(fact_172_finite__carrier,axiom,
    finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ).

% finite_carrier
thf(fact_173_genideal__minimal,axiom,
    ! [I2: set_a,S: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ord_less_eq_set_a @ S @ I2 )
       => ( ord_less_eq_set_a @ ( genideal_a_b @ r @ S ) @ I2 ) ) ) ).

% genideal_minimal
thf(fact_174_zf_Ogenideal__minimal,axiom,
    ! [I2: set_set_int,S: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( ord_le4403425263959731960et_int @ S @ I2 )
       => ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) @ I2 ) ) ) ).

% zf.genideal_minimal
thf(fact_175_zf_Oadd__ideals,axiom,
    ! [I2: set_set_int,J: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
       => ( ideal_7262958097527394045t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ J ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.add_ideals
thf(fact_176_finetely__gen,axiom,
    ! [I2: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ? [A4: set_a] :
          ( ( ord_less_eq_set_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( finite_finite_a @ A4 )
          & ( I2
            = ( genideal_a_b @ r @ A4 ) ) ) ) ).

% finetely_gen
thf(fact_177_genideal__ideal,axiom,
    ! [S: set_a] :
      ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ideal_a_b @ ( genideal_a_b @ r @ S ) @ r ) ) ).

% genideal_ideal
thf(fact_178_Idl__subset__ideal,axiom,
    ! [I2: set_a,H: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ I2 )
          = ( ord_less_eq_set_a @ H @ I2 ) ) ) ) ).

% Idl_subset_ideal
thf(fact_179_set__add__closed,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% set_add_closed
thf(fact_180_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_181_setadd__subset__G,axiom,
    ! [H: set_a,K: set_a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% setadd_subset_G
thf(fact_182_line__extension__in__carrier,axiom,
    ! [K: set_a,A: a,E: set_a] :
      ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% line_extension_in_carrier
thf(fact_183_carrier__is__subalgebra,axiom,
    ! [K: set_a] :
      ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
     => ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).

% carrier_is_subalgebra
thf(fact_184_subalgebra__in__carrier,axiom,
    ! [K: set_a,V: set_a] :
      ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
     => ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% subalgebra_in_carrier
thf(fact_185_zf_Ofinetely__gen,axiom,
    ! [I2: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ? [A4: set_set_int] :
          ( ( ord_le4403425263959731960et_int @ A4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          & ( finite6197958912794628473et_int @ A4 )
          & ( I2
            = ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A4 ) ) ) ) ).

% zf.finetely_gen
thf(fact_186_zf_OIdl__subset__ideal,axiom,
    ! [I2: set_set_int,H: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H ) @ I2 )
          = ( ord_le4403425263959731960et_int @ H @ I2 ) ) ) ) ).

% zf.Idl_subset_ideal
thf(fact_187_zf_Ogenideal__ideal,axiom,
    ! [S: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ideal_7262958097527394045t_unit @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.genideal_ideal
thf(fact_188_zf_Oset__add__closed,axiom,
    ! [A2: set_set_int,B: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A2 @ B ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.set_add_closed
thf(fact_189_zf_Oset__add__comm,axiom,
    ! [I2: set_set_int,J: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ J )
          = ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ J @ I2 ) ) ) ) ).

% zf.set_add_comm
thf(fact_190_zf_Osetadd__subset__G,axiom,
    ! [H: set_set_int,K: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ K ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.setadd_subset_G
thf(fact_191_zf_Ogenideal__self,axiom,
    ! [S: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ord_le4403425263959731960et_int @ S @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) ) ) ).

% zf.genideal_self
thf(fact_192_zf_Osubset__Idl__subset,axiom,
    ! [I2: set_set_int,H: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ H @ I2 )
       => ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) ) ) ) ).

% zf.subset_Idl_subset
thf(fact_193_zf_Ooneideal,axiom,
    ideal_7262958097527394045t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.oneideal
thf(fact_194_zf_Ofinite__carrier,axiom,
    finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ).

% zf.finite_carrier
thf(fact_195_zf_Ocgenideal__minimal,axiom,
    ! [J: set_set_int,A: set_int] :
      ( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( member_set_int @ A @ J )
       => ( ord_le4403425263959731960et_int @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) @ J ) ) ) ).

% zf.cgenideal_minimal
thf(fact_196_ideal__is__subalgebra,axiom,
    ! [K: set_a,I2: set_a] :
      ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ideal_a_b @ I2 @ r )
       => ( embedd9027525575939734154ra_a_b @ K @ I2 @ r ) ) ) ).

% ideal_is_subalgebra
thf(fact_197_zf_Ocgenideal__ideal,axiom,
    ! [A: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ideal_7262958097527394045t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.cgenideal_ideal
thf(fact_198_zf_Oexists__gen,axiom,
    ! [I2: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ? [X3: set_int] :
          ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          & ( I2
            = ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X3 ) ) ) ) ).

% zf.exists_gen
thf(fact_199_noetherian__ringI,axiom,
    ( ! [I3: set_a] :
        ( ( ideal_a_b @ I3 @ r )
       => ? [A3: set_a] :
            ( ( ord_less_eq_set_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
            & ( finite_finite_a @ A3 )
            & ( I3
              = ( genideal_a_b @ r @ A3 ) ) ) )
   => ( ring_n3639167112692572309ng_a_b @ r ) ) ).

% noetherian_ringI
thf(fact_200_zf_Onoetherian__ringI,axiom,
    ( ! [I3: set_set_int] :
        ( ( ideal_7262958097527394045t_unit @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
       => ? [A3: set_set_int] :
            ( ( ord_le4403425263959731960et_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            & ( finite6197958912794628473et_int @ A3 )
            & ( I3
              = ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A3 ) ) ) )
   => ( ring_n4918880612521764245t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.noetherian_ringI
thf(fact_201_pzf_Ocarrier__not__empty,axiom,
    ( ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
   != bot_bo7344713902834881468et_int ) ).

% pzf.carrier_not_empty
thf(fact_202_pzf_Oadd_Osurj__const__mult,axiom,
    ! [A: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( image_586265162069257835et_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        = ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.add.surj_const_mult
thf(fact_203_ring__hom__ring_Othe__elem__simp,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a,X: list_set_int] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( the_elem_a @ ( image_list_set_int_a @ H2 @ ( a_r_co912825291596486358t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) @ X ) ) )
          = ( H2 @ X ) ) ) ) ).

% ring_hom_ring.the_elem_simp
thf(fact_204_ring__hom__ring_Othe__elem__simp,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H2: list_a > set_list_a,X: list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H2 )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ H2 @ ( a_r_co1577572119920843660t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H2 ) @ X ) ) )
          = ( H2 @ X ) ) ) ) ).

% ring_hom_ring.the_elem_simp
thf(fact_205_ring__hom__ring_Othe__elem__simp,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,X: list_a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( the_elem_a @ ( image_list_a_a @ H2 @ ( a_r_co1577572119920843660t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) @ X ) ) )
          = ( H2 @ X ) ) ) ) ).

% ring_hom_ring.the_elem_simp
thf(fact_206_ring__hom__ring_Othe__elem__simp,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6652711048488803900t_unit,H2: set_int > list_set_int,X: set_int] :
      ( ( ring_h615793168808539756t_unit @ R2 @ S @ H2 )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( the_el6263124322315510589et_int @ ( image_8094898911679120219et_int @ H2 @ ( a_r_co692709266861932262t_unit @ R2 @ ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 ) @ X ) ) )
          = ( H2 @ X ) ) ) ) ).

% ring_hom_ring.the_elem_simp
thf(fact_207_ring__hom__ring_Othe__elem__simp,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int,X: a] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( the_el6263124322315510589et_int @ ( image_a_list_set_int @ H2 @ ( a_r_coset_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) @ X ) ) )
          = ( H2 @ X ) ) ) ) ).

% ring_hom_ring.the_elem_simp
thf(fact_208_ring__hom__ring_Othe__elem__simp,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,X: a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( the_elem_list_a @ ( image_a_list_a @ H2 @ ( a_r_coset_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) @ X ) ) )
          = ( H2 @ X ) ) ) ) ).

% ring_hom_ring.the_elem_simp
thf(fact_209_ring__iso__memE_I1_J,axiom,
    ! [H2: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
      ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R2 @ S ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).

% ring_iso_memE(1)
thf(fact_210_ring__iso__memE_I1_J,axiom,
    ! [H2: set_int > a,R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,X: set_int] :
      ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R2 @ S ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
       => ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).

% ring_iso_memE(1)
thf(fact_211_ring__iso__memE_I1_J,axiom,
    ! [H2: set_a > a,R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,X: set_a] :
      ( ( member_set_a_a @ H2 @ ( ring_i4751279688611836608it_a_b @ R2 @ S ) )
     => ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).

% ring_iso_memE(1)
thf(fact_212_ring__iso__memE_I1_J,axiom,
    ! [H2: a > set_int,R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,X: a] :
      ( ( member_a_set_int @ H2 @ ( ring_i4118987378922720216t_unit @ R2 @ S ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( member_set_int @ ( H2 @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).

% ring_iso_memE(1)
thf(fact_213_ring__iso__memE_I1_J,axiom,
    ! [H2: a > set_a,R2: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,X: a] :
      ( ( member_a_set_a @ H2 @ ( ring_i7849008455817099456t_unit @ R2 @ S ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( member_set_a @ ( H2 @ X ) @ ( partia5907974310037520643t_unit @ S ) ) ) ) ).

% ring_iso_memE(1)
thf(fact_214_ring__iso__memE_I1_J,axiom,
    ! [H2: set_set_int > a,R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,X: set_set_int] :
      ( ( member_set_set_int_a @ H2 @ ( ring_i5461826592427486820it_a_b @ R2 @ S ) )
     => ( ( member_set_set_int @ X @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).

% ring_iso_memE(1)
thf(fact_215_ring__iso__memE_I1_J,axiom,
    ! [H2: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,X: set_int] :
      ( ( member5205197933313416826et_int @ H2 @ ( ring_i7857697894474155344t_unit @ R2 @ S ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
       => ( member_set_int @ ( H2 @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).

% ring_iso_memE(1)
thf(fact_216_ring__iso__memE_I1_J,axiom,
    ! [H2: set_int > set_a,R2: partia4934656038542163276t_unit,S: partia6043505979758434576t_unit,X: set_int] :
      ( ( member_set_int_set_a @ H2 @ ( ring_i5186029982270319944t_unit @ R2 @ S ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
       => ( member_set_a @ ( H2 @ X ) @ ( partia5907974310037520643t_unit @ S ) ) ) ) ).

% ring_iso_memE(1)
thf(fact_217_ring__iso__memE_I1_J,axiom,
    ! [H2: set_a > set_int,R2: partia6043505979758434576t_unit,S: partia4934656038542163276t_unit,X: set_a] :
      ( ( member_set_a_set_int @ H2 @ ( ring_i1340567420592345514t_unit @ R2 @ S ) )
     => ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( member_set_int @ ( H2 @ X ) @ ( partia966996272515721803t_unit @ S ) ) ) ) ).

% ring_iso_memE(1)
thf(fact_218_ring__iso__memE_I1_J,axiom,
    ! [H2: set_a > set_a,R2: partia6043505979758434576t_unit,S: partia6043505979758434576t_unit,X: set_a] :
      ( ( member_set_a_set_a @ H2 @ ( ring_i7821126338072249198t_unit @ R2 @ S ) )
     => ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( member_set_a @ ( H2 @ X ) @ ( partia5907974310037520643t_unit @ S ) ) ) ) ).

% ring_iso_memE(1)
thf(fact_219_ring__iso__set__trans,axiom,
    ! [F: set_list_set_int > set_list_set_int,R2: partia3352279114339913222t_unit,S: partia3352279114339913222t_unit,G: set_list_set_int > a,Q: partia2175431115845679010xt_a_b] :
      ( ( member1822656912387541382et_int @ F @ ( ring_i4301655954165343056t_unit @ R2 @ S ) )
     => ( ( member1264592267077841462_int_a @ G @ ( ring_i1463878111215264340it_a_b @ S @ Q ) )
       => ( member1264592267077841462_int_a @ ( comp_s3918679199482572123et_int @ G @ F ) @ ( ring_i1463878111215264340it_a_b @ R2 @ Q ) ) ) ) ).

% ring_iso_set_trans
thf(fact_220_ring__iso__set__trans,axiom,
    ! [F: set_list_set_int > a,R2: partia3352279114339913222t_unit,S: partia2175431115845679010xt_a_b,G: a > a,Q: partia2175431115845679010xt_a_b] :
      ( ( member1264592267077841462_int_a @ F @ ( ring_i1463878111215264340it_a_b @ R2 @ S ) )
     => ( ( member_a_a @ G @ ( ring_iso_a_b_a_b @ S @ Q ) )
       => ( member1264592267077841462_int_a @ ( comp_a3704501331711576715et_int @ G @ F ) @ ( ring_i1463878111215264340it_a_b @ R2 @ Q ) ) ) ) ).

% ring_iso_set_trans
thf(fact_221_noetherian__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b] :
      ( ( ring_n4045954140777738665in_a_b @ R2 )
     => ( ring_n3639167112692572309ng_a_b @ R2 ) ) ).

% noetherian_domain.axioms(1)
thf(fact_222_noetherian__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia4934656038542163276t_unit] :
      ( ( ring_n4948453042392689537t_unit @ R2 )
     => ( ring_n4918880612521764245t_unit @ R2 ) ) ).

% noetherian_domain.axioms(1)
thf(fact_223_noetherian__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia6652711048488803900t_unit] :
      ( ( ring_n3580647943193786865t_unit @ R2 )
     => ( ring_n7202549095058397701t_unit @ R2 ) ) ).

% noetherian_domain.axioms(1)
thf(fact_224_ring__iso__memE_I3_J,axiom,
    ! [H2: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R2 @ S ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( H2 @ ( add_a_b @ R2 @ X @ Y ) )
            = ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).

% ring_iso_memE(3)
thf(fact_225_ring__iso__memE_I3_J,axiom,
    ! [H2: set_int > a,R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,X: set_int,Y: set_int] :
      ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R2 @ S ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
         => ( ( H2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
            = ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).

% ring_iso_memE(3)
thf(fact_226_ring__iso__memE_I3_J,axiom,
    ! [H2: set_a > a,R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,X: set_a,Y: set_a] :
      ( ( member_set_a_a @ H2 @ ( ring_i4751279688611836608it_a_b @ R2 @ S ) )
     => ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( H2 @ ( add_se3735415688806051380t_unit @ R2 @ X @ Y ) )
            = ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).

% ring_iso_memE(3)
thf(fact_227_ring__iso__memE_I3_J,axiom,
    ! [H2: a > set_int,R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,X: a,Y: a] :
      ( ( member_a_set_int @ H2 @ ( ring_i4118987378922720216t_unit @ R2 @ S ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( H2 @ ( add_a_b @ R2 @ X @ Y ) )
            = ( add_se5859248395121729892t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).

% ring_iso_memE(3)
thf(fact_228_ring__iso__memE_I3_J,axiom,
    ! [H2: list_set_int > a,R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,X: list_set_int,Y: list_set_int] :
      ( ( member6759595599802011116_int_a @ H2 @ ( ring_i4082836582328865034it_a_b @ R2 @ S ) )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ R2 ) )
         => ( ( H2 @ ( add_li6184372983183467092t_unit @ R2 @ X @ Y ) )
            = ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).

% ring_iso_memE(3)
thf(fact_229_ring__iso__memE_I3_J,axiom,
    ! [H2: set_set_int > a,R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,X: set_set_int,Y: set_set_int] :
      ( ( member_set_set_int_a @ H2 @ ( ring_i5461826592427486820it_a_b @ R2 @ S ) )
     => ( ( member_set_set_int @ X @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( member_set_set_int @ Y @ ( partia1981568256555877781t_unit @ R2 ) )
         => ( ( H2 @ ( add_se7096102115476048430t_unit @ R2 @ X @ Y ) )
            = ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).

% ring_iso_memE(3)
thf(fact_230_ring__iso__memE_I3_J,axiom,
    ! [H2: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,X: set_int,Y: set_int] :
      ( ( member5205197933313416826et_int @ H2 @ ( ring_i7857697894474155344t_unit @ R2 @ S ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ R2 ) )
         => ( ( H2 @ ( add_se5859248395121729892t_unit @ R2 @ X @ Y ) )
            = ( add_se5859248395121729892t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).

% ring_iso_memE(3)
thf(fact_231_ring__iso__memE_I3_J,axiom,
    ! [H2: set_a > set_int,R2: partia6043505979758434576t_unit,S: partia4934656038542163276t_unit,X: set_a,Y: set_a] :
      ( ( member_set_a_set_int @ H2 @ ( ring_i1340567420592345514t_unit @ R2 @ S ) )
     => ( ( member_set_a @ X @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( H2 @ ( add_se3735415688806051380t_unit @ R2 @ X @ Y ) )
            = ( add_se5859248395121729892t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).

% ring_iso_memE(3)
thf(fact_232_ring__iso__memE_I3_J,axiom,
    ! [H2: a > list_set_int,R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,X: a,Y: a] :
      ( ( member6392905276882098698et_int @ H2 @ ( ring_i8932118395132445384t_unit @ R2 @ S ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( H2 @ ( add_a_b @ R2 @ X @ Y ) )
            = ( add_li6184372983183467092t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).

% ring_iso_memE(3)
thf(fact_233_ring__iso__memE_I3_J,axiom,
    ! [H2: list_set_int > set_int,R2: partia6652711048488803900t_unit,S: partia4934656038542163276t_unit,X: list_set_int,Y: list_set_int] :
      ( ( member4589863265046936970et_int @ H2 @ ( ring_i372858595358256864t_unit @ R2 @ S ) )
     => ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ R2 ) )
         => ( ( H2 @ ( add_li6184372983183467092t_unit @ R2 @ X @ Y ) )
            = ( add_se5859248395121729892t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).

% ring_iso_memE(3)
thf(fact_234_ring__prime__def,axiom,
    ( ring_r7068619596278734765t_unit
    = ( ^ [R3: partia7144473111789202860t_unit,A5: list_list_set_int] :
          ( ( A5
           != ( zero_l4263514192383796165t_unit @ R3 ) )
          & ( prime_6779480824860900370t_unit @ R3 @ A5 ) ) ) ) ).

% ring_prime_def
thf(fact_235_ring__prime__def,axiom,
    ( ring_r7101923692879464637t_unit
    = ( ^ [R3: partia6652711048488803900t_unit,A5: list_set_int] :
          ( ( A5
           != ( zero_l5318887016005761237t_unit @ R3 ) )
          & ( prime_7199642643890090290t_unit @ R3 @ A5 ) ) ) ) ).

% ring_prime_def
thf(fact_236_ring__prime__def,axiom,
    ( ring_ring_prime_a_b
    = ( ^ [R3: partia2175431115845679010xt_a_b,A5: a] :
          ( ( A5
           != ( zero_a_b @ R3 ) )
          & ( prime_a_ring_ext_a_b @ R3 @ A5 ) ) ) ) ).

% ring_prime_def
thf(fact_237_ring__prime__def,axiom,
    ( ring_r2768735206093983821t_unit
    = ( ^ [R3: partia4934656038542163276t_unit,A5: set_int] :
          ( ( A5
           != ( zero_s6269048424454532197t_unit @ R3 ) )
          & ( prime_2746729220581880658t_unit @ R3 @ A5 ) ) ) ) ).

% ring_prime_def
thf(fact_238_ring__hom__ring_Oideal__vimage,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a,I2: set_a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H2 )
     => ( ( ideal_a_b @ I2 @ S )
       => ( ideal_a_b
          @ ( collect_a
            @ ^ [R: a] :
                ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ R2 ) )
                & ( member_a @ ( H2 @ R ) @ I2 ) ) )
          @ R2 ) ) ) ).

% ring_hom_ring.ideal_vimage
thf(fact_239_ring__hom__ring_Oideal__vimage,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,I2: set_a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( ideal_a_b @ I2 @ S )
       => ( ideal_8896367198367571637t_unit
          @ ( collect_list_a
            @ ^ [R: list_a] :
                ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ R2 ) )
                & ( member_a @ ( H2 @ R ) @ I2 ) ) )
          @ R2 ) ) ) ).

% ring_hom_ring.ideal_vimage
thf(fact_240_ring__hom__ring_Oideal__vimage,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a,I2: set_a] :
      ( ( ring_h5360235766848426350it_a_b @ R2 @ S @ H2 )
     => ( ( ideal_a_b @ I2 @ S )
       => ( ideal_7262958097527394045t_unit
          @ ( collect_set_int
            @ ^ [R: set_int] :
                ( ( member_set_int @ R @ ( partia966996272515721803t_unit @ R2 ) )
                & ( member_a @ ( H2 @ R ) @ I2 ) ) )
          @ R2 ) ) ) ).

% ring_hom_ring.ideal_vimage
thf(fact_241_ring__hom__ring_Oideal__vimage,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,H2: set_a > a,I2: set_a] :
      ( ( ring_h8937780225337603308it_a_b @ R2 @ S @ H2 )
     => ( ( ideal_a_b @ I2 @ S )
       => ( ideal_4463284918206690523t_unit
          @ ( collect_set_a
            @ ^ [R: set_a] :
                ( ( member_set_a @ R @ ( partia5907974310037520643t_unit @ R2 ) )
                & ( member_a @ ( H2 @ R ) @ I2 ) ) )
          @ R2 ) ) ) ).

% ring_hom_ring.ideal_vimage
thf(fact_242_ring__hom__ring_Oideal__vimage,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,H2: a > set_int,I2: set_set_int] :
      ( ( ring_h7266096129245845164t_unit @ R2 @ S @ H2 )
     => ( ( ideal_7262958097527394045t_unit @ I2 @ S )
       => ( ideal_a_b
          @ ( collect_a
            @ ^ [R: a] :
                ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ R2 ) )
                & ( member_set_int @ ( H2 @ R ) @ I2 ) ) )
          @ R2 ) ) ) ).

% ring_hom_ring.ideal_vimage
thf(fact_243_ring__hom__ring_Oideal__vimage,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,H2: a > set_a,I2: set_set_a] :
      ( ( ring_h2812136955688090348t_unit @ R2 @ S @ H2 )
     => ( ( ideal_4463284918206690523t_unit @ I2 @ S )
       => ( ideal_a_b
          @ ( collect_a
            @ ^ [R: a] :
                ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ R2 ) )
                & ( member_set_a @ ( H2 @ R ) @ I2 ) ) )
          @ R2 ) ) ) ).

% ring_hom_ring.ideal_vimage
thf(fact_244_ring__hom__ring_Oideal__vimage,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,I2: set_list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( ideal_8896367198367571637t_unit @ I2 @ S )
       => ( ideal_a_b
          @ ( collect_a
            @ ^ [R: a] :
                ( ( member_a @ R @ ( partia707051561876973205xt_a_b @ R2 ) )
                & ( member_list_a @ ( H2 @ R ) @ I2 ) ) )
          @ R2 ) ) ) ).

% ring_hom_ring.ideal_vimage
thf(fact_245_ring__hom__ring_Oideal__vimage,axiom,
    ! [R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,H2: set_set_int > a,I2: set_a] :
      ( ( ring_h5493655406951491896it_a_b @ R2 @ S @ H2 )
     => ( ( ideal_a_b @ I2 @ S )
       => ( ideal_1992573625310827463t_unit
          @ ( collect_set_set_int
            @ ^ [R: set_set_int] :
                ( ( member_set_set_int @ R @ ( partia1981568256555877781t_unit @ R2 ) )
                & ( member_a @ ( H2 @ R ) @ I2 ) ) )
          @ R2 ) ) ) ).

% ring_hom_ring.ideal_vimage
thf(fact_246_ring__hom__ring_Oideal__vimage,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H2: set_int > set_int,I2: set_set_int] :
      ( ( ring_h98094481560215932t_unit @ R2 @ S @ H2 )
     => ( ( ideal_7262958097527394045t_unit @ I2 @ S )
       => ( ideal_7262958097527394045t_unit
          @ ( collect_set_int
            @ ^ [R: set_int] :
                ( ( member_set_int @ R @ ( partia966996272515721803t_unit @ R2 ) )
                & ( member_set_int @ ( H2 @ R ) @ I2 ) ) )
          @ R2 ) ) ) ).

% ring_hom_ring.ideal_vimage
thf(fact_247_ring__hom__ring_Oideal__vimage,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6043505979758434576t_unit,H2: set_int > set_a,I2: set_set_a] :
      ( ( ring_h5212181533761340444t_unit @ R2 @ S @ H2 )
     => ( ( ideal_4463284918206690523t_unit @ I2 @ S )
       => ( ideal_7262958097527394045t_unit
          @ ( collect_set_int
            @ ^ [R: set_int] :
                ( ( member_set_int @ R @ ( partia966996272515721803t_unit @ R2 ) )
                & ( member_set_a @ ( H2 @ R ) @ I2 ) ) )
          @ R2 ) ) ) ).

% ring_hom_ring.ideal_vimage
thf(fact_248_principal__domain_Oexists__gen,axiom,
    ! [R2: partia2670972154091845814t_unit,I2: set_list_a] :
      ( ( ring_p8098905331641078952t_unit @ R2 )
     => ( ( ideal_8896367198367571637t_unit @ I2 @ R2 )
       => ? [X3: list_a] :
            ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
            & ( I2
              = ( cgenid9131348535277946915t_unit @ R2 @ X3 ) ) ) ) ) ).

% principal_domain.exists_gen
thf(fact_249_principal__domain_Oexists__gen,axiom,
    ! [R2: partia7144473111789202860t_unit,I2: set_li2838760105183549752et_int] :
      ( ( ring_p1105522777314784042t_unit @ R2 )
     => ( ( ideal_1683033260801535581t_unit @ I2 @ R2 )
       => ? [X3: list_list_set_int] :
            ( ( member7440485371949894425et_int @ X3 @ ( partia3237421854401958059t_unit @ R2 ) )
            & ( I2
              = ( cgenid7864102925118505423t_unit @ R2 @ X3 ) ) ) ) ) ).

% principal_domain.exists_gen
thf(fact_250_principal__domain_Oexists__gen,axiom,
    ! [R2: partia3352279114339913222t_unit,I2: set_set_list_set_int] :
      ( ( ring_p4083217851854698756t_unit @ R2 )
     => ( ( ideal_7892184242768911799t_unit @ I2 @ R2 )
       => ? [X3: set_list_set_int] :
            ( ( member4487585027056014399et_int @ X3 @ ( partia7954853462395303941t_unit @ R2 ) )
            & ( I2
              = ( cgenid3413961061359118467t_unit @ R2 @ X3 ) ) ) ) ) ).

% principal_domain.exists_gen
thf(fact_251_principal__domain_Oexists__gen,axiom,
    ! [R2: partia3601206958761670294t_unit,I2: set_set_set_int] :
      ( ( ring_p4005655891433666964t_unit @ R2 )
     => ( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
       => ? [X3: set_set_int] :
            ( ( member_set_set_int @ X3 @ ( partia1981568256555877781t_unit @ R2 ) )
            & ( I2
              = ( cgenid8920438795327392163t_unit @ R2 @ X3 ) ) ) ) ) ).

% principal_domain.exists_gen
thf(fact_252_principal__domain_Oexists__gen,axiom,
    ! [R2: partia6043505979758434576t_unit,I2: set_set_a] :
      ( ( ring_p2862007038493914190t_unit @ R2 )
     => ( ( ideal_4463284918206690523t_unit @ I2 @ R2 )
       => ? [X3: set_a] :
            ( ( member_set_a @ X3 @ ( partia5907974310037520643t_unit @ R2 ) )
            & ( I2
              = ( cgenid6682780793756002467t_unit @ R2 @ X3 ) ) ) ) ) ).

% principal_domain.exists_gen
thf(fact_253_principal__domain_Oexists__gen,axiom,
    ! [R2: partia4934656038542163276t_unit,I2: set_set_int] :
      ( ( ring_p2413783099608688842t_unit @ R2 )
     => ( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ R2 ) )
            & ( I2
              = ( cgenid8502489213727343375t_unit @ R2 @ X3 ) ) ) ) ) ).

% principal_domain.exists_gen
thf(fact_254_principal__domain_Oexists__gen,axiom,
    ! [R2: partia2175431115845679010xt_a_b,I2: set_a] :
      ( ( ring_p8803135361686045600in_a_b @ R2 )
     => ( ( ideal_a_b @ I2 @ R2 )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
            & ( I2
              = ( cgenid547466209912283029xt_a_b @ R2 @ X3 ) ) ) ) ) ).

% principal_domain.exists_gen
thf(fact_255_principal__domain_Oexists__gen,axiom,
    ! [R2: partia6652711048488803900t_unit,I2: set_list_set_int] :
      ( ( ring_p1500425057661402810t_unit @ R2 )
     => ( ( ideal_3307802554209726317t_unit @ I2 @ R2 )
       => ? [X3: list_set_int] :
            ( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ R2 ) )
            & ( I2
              = ( cgenid1447163913751362543t_unit @ R2 @ X3 ) ) ) ) ) ).

% principal_domain.exists_gen
thf(fact_256_a__coset__hom_I2_J,axiom,
    ! [H2: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,I2: set_a,A: a] :
      ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R2 @ S ) )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( image_a_a @ H2 @ ( a_r_coset_a_b @ R2 @ I2 @ A ) )
            = ( a_r_coset_a_b @ S @ ( image_a_a @ H2 @ I2 ) @ ( H2 @ A ) ) ) ) ) ) ).

% a_coset_hom(2)
thf(fact_257_a__coset__hom_I2_J,axiom,
    ! [H2: list_a > a,R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,I2: set_list_a,A: list_a] :
      ( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R2 @ S ) )
     => ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ( image_list_a_a @ H2 @ ( a_r_co1577572119920843660t_unit @ R2 @ I2 @ A ) )
            = ( a_r_coset_a_b @ S @ ( image_list_a_a @ H2 @ I2 ) @ ( H2 @ A ) ) ) ) ) ) ).

% a_coset_hom(2)
thf(fact_258_a__coset__hom_I2_J,axiom,
    ! [H2: set_int > a,R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_int,A: set_int] :
      ( ( member_set_int_a @ H2 @ ( ring_h6431528728137583664it_a_b @ R2 @ S ) )
     => ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
         => ( ( image_set_int_a @ H2 @ ( a_r_co692709266861932262t_unit @ R2 @ I2 @ A ) )
            = ( a_r_coset_a_b @ S @ ( image_set_int_a @ H2 @ I2 ) @ ( H2 @ A ) ) ) ) ) ) ).

% a_coset_hom(2)
thf(fact_259_a__coset__hom_I2_J,axiom,
    ! [H2: set_a > a,R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_a,A: set_a] :
      ( ( member_set_a_a @ H2 @ ( ring_h4811522740288071338it_a_b @ R2 @ S ) )
     => ( ( ord_le3724670747650509150_set_a @ I2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( image_set_a_a @ H2 @ ( a_r_co6984235733410409778t_unit @ R2 @ I2 @ A ) )
            = ( a_r_coset_a_b @ S @ ( image_set_a_a @ H2 @ I2 ) @ ( H2 @ A ) ) ) ) ) ) ).

% a_coset_hom(2)
thf(fact_260_a__coset__hom_I2_J,axiom,
    ! [H2: a > set_int,R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,I2: set_a,A: a] :
      ( ( member_a_set_int @ H2 @ ( ring_h8337389090535002478t_unit @ R2 @ S ) )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( image_a_set_int @ H2 @ ( a_r_coset_a_b @ R2 @ I2 @ A ) )
            = ( a_r_co692709266861932262t_unit @ S @ ( image_a_set_int @ H2 @ I2 ) @ ( H2 @ A ) ) ) ) ) ) ).

% a_coset_hom(2)
thf(fact_261_a__coset__hom_I2_J,axiom,
    ! [H2: a > list_a,R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,I2: set_a,A: a] :
      ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R2 @ S ) )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( image_a_list_a @ H2 @ ( a_r_coset_a_b @ R2 @ I2 @ A ) )
            = ( a_r_co1577572119920843660t_unit @ S @ ( image_a_list_a @ H2 @ I2 ) @ ( H2 @ A ) ) ) ) ) ) ).

% a_coset_hom(2)
thf(fact_262_a__coset__hom_I2_J,axiom,
    ! [H2: list_set_int > a,R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,I2: set_list_set_int,A: list_set_int] :
      ( ( member6759595599802011116_int_a @ H2 @ ( ring_h3322734120376767392it_a_b @ R2 @ S ) )
     => ( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ R2 ) )
         => ( ( image_list_set_int_a @ H2 @ ( a_r_co912825291596486358t_unit @ R2 @ I2 @ A ) )
            = ( a_r_coset_a_b @ S @ ( image_list_set_int_a @ H2 @ I2 ) @ ( H2 @ A ) ) ) ) ) ) ).

% a_coset_hom(2)
thf(fact_263_a__coset__hom_I2_J,axiom,
    ! [H2: list_a > set_int,R2: partia2670972154091845814t_unit,S: partia4934656038542163276t_unit,I2: set_list_a,A: list_a] :
      ( ( member5309957867533224448et_int @ H2 @ ( ring_h5391894047676762854t_unit @ R2 @ S ) )
     => ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ( image_list_a_set_int @ H2 @ ( a_r_co1577572119920843660t_unit @ R2 @ I2 @ A ) )
            = ( a_r_co692709266861932262t_unit @ S @ ( image_list_a_set_int @ H2 @ I2 ) @ ( H2 @ A ) ) ) ) ) ) ).

% a_coset_hom(2)
thf(fact_264_a__coset__hom_I2_J,axiom,
    ! [H2: list_a > list_a,R2: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,I2: set_list_a,A: list_a] :
      ( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R2 @ S ) )
     => ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ( image_list_a_list_a @ H2 @ ( a_r_co1577572119920843660t_unit @ R2 @ I2 @ A ) )
            = ( a_r_co1577572119920843660t_unit @ S @ ( image_list_a_list_a @ H2 @ I2 ) @ ( H2 @ A ) ) ) ) ) ) ).

% a_coset_hom(2)
thf(fact_265_a__coset__hom_I2_J,axiom,
    ! [H2: set_set_int > a,R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_set_int,A: set_set_int] :
      ( ( member_set_set_int_a @ H2 @ ( ring_h5866496596994661370it_a_b @ R2 @ S ) )
     => ( ( ord_le4317611570275147438et_int @ I2 @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( member_set_set_int @ A @ ( partia1981568256555877781t_unit @ R2 ) )
         => ( ( image_set_set_int_a @ H2 @ ( a_r_co7378703583274672048t_unit @ R2 @ I2 @ A ) )
            = ( a_r_coset_a_b @ S @ ( image_set_set_int_a @ H2 @ I2 ) @ ( H2 @ A ) ) ) ) ) ) ).

% a_coset_hom(2)
thf(fact_266_set__add__hom,axiom,
    ! [H2: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,I2: set_a,J: set_a] :
      ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R2 @ S ) )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( image_a_a @ H2 @ ( set_add_a_b @ R2 @ I2 @ J ) )
            = ( set_add_a_b @ S @ ( image_a_a @ H2 @ I2 ) @ ( image_a_a @ H2 @ J ) ) ) ) ) ) ).

% set_add_hom
thf(fact_267_set__add__hom,axiom,
    ! [H2: set_int > a,R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_int,J: set_set_int] :
      ( ( member_set_int_a @ H2 @ ( ring_h6431528728137583664it_a_b @ R2 @ S ) )
     => ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ R2 ) )
         => ( ( image_set_int_a @ H2 @ ( set_ad273131178244904872t_unit @ R2 @ I2 @ J ) )
            = ( set_add_a_b @ S @ ( image_set_int_a @ H2 @ I2 ) @ ( image_set_int_a @ H2 @ J ) ) ) ) ) ) ).

% set_add_hom
thf(fact_268_set__add__hom,axiom,
    ! [H2: set_a > a,R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_a,J: set_set_a] :
      ( ( member_set_a_a @ H2 @ ( ring_h4811522740288071338it_a_b @ R2 @ S ) )
     => ( ( ord_le3724670747650509150_set_a @ I2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( image_set_a_a @ H2 @ ( set_ad3113708954360070384t_unit @ R2 @ I2 @ J ) )
            = ( set_add_a_b @ S @ ( image_set_a_a @ H2 @ I2 ) @ ( image_set_a_a @ H2 @ J ) ) ) ) ) ) ).

% set_add_hom
thf(fact_269_set__add__hom,axiom,
    ! [H2: a > list_a,R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,I2: set_a,J: set_a] :
      ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R2 @ S ) )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( image_a_list_a @ H2 @ ( set_add_a_b @ R2 @ I2 @ J ) )
            = ( set_ad92425877771022410t_unit @ S @ ( image_a_list_a @ H2 @ I2 ) @ ( image_a_list_a @ H2 @ J ) ) ) ) ) ) ).

% set_add_hom
thf(fact_270_set__add__hom,axiom,
    ! [H2: a > set_int,R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,I2: set_a,J: set_a] :
      ( ( member_a_set_int @ H2 @ ( ring_h8337389090535002478t_unit @ R2 @ S ) )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( image_a_set_int @ H2 @ ( set_add_a_b @ R2 @ I2 @ J ) )
            = ( set_ad273131178244904872t_unit @ S @ ( image_a_set_int @ H2 @ I2 ) @ ( image_a_set_int @ H2 @ J ) ) ) ) ) ) ).

% set_add_hom
thf(fact_271_set__add__hom,axiom,
    ! [H2: list_set_int > a,R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,I2: set_list_set_int,J: set_list_set_int] :
      ( ( member6759595599802011116_int_a @ H2 @ ( ring_h3322734120376767392it_a_b @ R2 @ S ) )
     => ( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( ord_le4527193859438875528et_int @ J @ ( partia5243588073521899067t_unit @ R2 ) )
         => ( ( image_list_set_int_a @ H2 @ ( set_ad2159827153730365080t_unit @ R2 @ I2 @ J ) )
            = ( set_add_a_b @ S @ ( image_list_set_int_a @ H2 @ I2 ) @ ( image_list_set_int_a @ H2 @ J ) ) ) ) ) ) ).

% set_add_hom
thf(fact_272_set__add__hom,axiom,
    ! [H2: set_set_int > a,R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_set_int,J: set_set_set_int] :
      ( ( member_set_set_int_a @ H2 @ ( ring_h5866496596994661370it_a_b @ R2 @ S ) )
     => ( ( ord_le4317611570275147438et_int @ I2 @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( ord_le4317611570275147438et_int @ J @ ( partia1981568256555877781t_unit @ R2 ) )
         => ( ( image_set_set_int_a @ H2 @ ( set_ad1658175378941472882t_unit @ R2 @ I2 @ J ) )
            = ( set_add_a_b @ S @ ( image_set_set_int_a @ H2 @ I2 ) @ ( image_set_set_int_a @ H2 @ J ) ) ) ) ) ) ).

% set_add_hom
thf(fact_273_set__add__hom,axiom,
    ! [H2: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,I2: set_set_int,J: set_set_int] :
      ( ( member5205197933313416826et_int @ H2 @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
     => ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ R2 ) )
         => ( ( image_524474410958335435et_int @ H2 @ ( set_ad273131178244904872t_unit @ R2 @ I2 @ J ) )
            = ( set_ad273131178244904872t_unit @ S @ ( image_524474410958335435et_int @ H2 @ I2 ) @ ( image_524474410958335435et_int @ H2 @ J ) ) ) ) ) ) ).

% set_add_hom
thf(fact_274_set__add__hom,axiom,
    ! [H2: set_a > set_a,R2: partia6043505979758434576t_unit,S: partia6043505979758434576t_unit,I2: set_set_a,J: set_set_a] :
      ( ( member_set_a_set_a @ H2 @ ( ring_h4754363464466234712t_unit @ R2 @ S ) )
     => ( ( ord_le3724670747650509150_set_a @ I2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( image_set_a_set_a @ H2 @ ( set_ad3113708954360070384t_unit @ R2 @ I2 @ J ) )
            = ( set_ad3113708954360070384t_unit @ S @ ( image_set_a_set_a @ H2 @ I2 ) @ ( image_set_a_set_a @ H2 @ J ) ) ) ) ) ) ).

% set_add_hom
thf(fact_275_set__add__hom,axiom,
    ! [H2: set_a > set_int,R2: partia6043505979758434576t_unit,S: partia4934656038542163276t_unit,I2: set_set_a,J: set_set_a] :
      ( ( member_set_a_set_int @ H2 @ ( ring_h4827446719030694208t_unit @ R2 @ S ) )
     => ( ( ord_le3724670747650509150_set_a @ I2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( image_set_a_set_int @ H2 @ ( set_ad3113708954360070384t_unit @ R2 @ I2 @ J ) )
            = ( set_ad273131178244904872t_unit @ S @ ( image_set_a_set_int @ H2 @ I2 ) @ ( image_set_a_set_int @ H2 @ J ) ) ) ) ) ) ).

% set_add_hom
thf(fact_276_a__coset__hom_I1_J,axiom,
    ! [H2: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,I2: set_a,A: a] :
      ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R2 @ S ) )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( image_a_a @ H2 @ ( a_l_coset_a_b @ R2 @ A @ I2 ) )
            = ( a_l_coset_a_b @ S @ ( H2 @ A ) @ ( image_a_a @ H2 @ I2 ) ) ) ) ) ) ).

% a_coset_hom(1)
thf(fact_277_a__coset__hom_I1_J,axiom,
    ! [H2: set_int > a,R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_int,A: set_int] :
      ( ( member_set_int_a @ H2 @ ( ring_h6431528728137583664it_a_b @ R2 @ S ) )
     => ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
         => ( ( image_set_int_a @ H2 @ ( a_l_co3504123944629134560t_unit @ R2 @ A @ I2 ) )
            = ( a_l_coset_a_b @ S @ ( H2 @ A ) @ ( image_set_int_a @ H2 @ I2 ) ) ) ) ) ) ).

% a_coset_hom(1)
thf(fact_278_a__coset__hom_I1_J,axiom,
    ! [H2: set_a > a,R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_a,A: set_a] :
      ( ( member_set_a_a @ H2 @ ( ring_h4811522740288071338it_a_b @ R2 @ S ) )
     => ( ( ord_le3724670747650509150_set_a @ I2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( image_set_a_a @ H2 @ ( a_l_co8760856729277977016t_unit @ R2 @ A @ I2 ) )
            = ( a_l_coset_a_b @ S @ ( H2 @ A ) @ ( image_set_a_a @ H2 @ I2 ) ) ) ) ) ) ).

% a_coset_hom(1)
thf(fact_279_a__coset__hom_I1_J,axiom,
    ! [H2: a > list_a,R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,I2: set_a,A: a] :
      ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R2 @ S ) )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( image_a_list_a @ H2 @ ( a_l_coset_a_b @ R2 @ A @ I2 ) )
            = ( a_l_co7008843373686234386t_unit @ S @ ( H2 @ A ) @ ( image_a_list_a @ H2 @ I2 ) ) ) ) ) ) ).

% a_coset_hom(1)
thf(fact_280_a__coset__hom_I1_J,axiom,
    ! [H2: a > set_int,R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,I2: set_a,A: a] :
      ( ( member_a_set_int @ H2 @ ( ring_h8337389090535002478t_unit @ R2 @ S ) )
     => ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( image_a_set_int @ H2 @ ( a_l_coset_a_b @ R2 @ A @ I2 ) )
            = ( a_l_co3504123944629134560t_unit @ S @ ( H2 @ A ) @ ( image_a_set_int @ H2 @ I2 ) ) ) ) ) ) ).

% a_coset_hom(1)
thf(fact_281_a__coset__hom_I1_J,axiom,
    ! [H2: list_set_int > a,R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,I2: set_list_set_int,A: list_set_int] :
      ( ( member6759595599802011116_int_a @ H2 @ ( ring_h3322734120376767392it_a_b @ R2 @ S ) )
     => ( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ R2 ) )
         => ( ( image_list_set_int_a @ H2 @ ( a_l_co143929110759192528t_unit @ R2 @ A @ I2 ) )
            = ( a_l_coset_a_b @ S @ ( H2 @ A ) @ ( image_list_set_int_a @ H2 @ I2 ) ) ) ) ) ) ).

% a_coset_hom(1)
thf(fact_282_a__coset__hom_I1_J,axiom,
    ! [H2: set_set_int > a,R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,I2: set_set_set_int,A: set_set_int] :
      ( ( member_set_set_int_a @ H2 @ ( ring_h5866496596994661370it_a_b @ R2 @ S ) )
     => ( ( ord_le4317611570275147438et_int @ I2 @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( member_set_set_int @ A @ ( partia1981568256555877781t_unit @ R2 ) )
         => ( ( image_set_set_int_a @ H2 @ ( a_l_co4050093972498570666t_unit @ R2 @ A @ I2 ) )
            = ( a_l_coset_a_b @ S @ ( H2 @ A ) @ ( image_set_set_int_a @ H2 @ I2 ) ) ) ) ) ) ).

% a_coset_hom(1)
thf(fact_283_a__coset__hom_I1_J,axiom,
    ! [H2: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,I2: set_set_int,A: set_int] :
      ( ( member5205197933313416826et_int @ H2 @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
     => ( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
         => ( ( image_524474410958335435et_int @ H2 @ ( a_l_co3504123944629134560t_unit @ R2 @ A @ I2 ) )
            = ( a_l_co3504123944629134560t_unit @ S @ ( H2 @ A ) @ ( image_524474410958335435et_int @ H2 @ I2 ) ) ) ) ) ) ).

% a_coset_hom(1)
thf(fact_284_a__coset__hom_I1_J,axiom,
    ! [H2: set_a > set_a,R2: partia6043505979758434576t_unit,S: partia6043505979758434576t_unit,I2: set_set_a,A: set_a] :
      ( ( member_set_a_set_a @ H2 @ ( ring_h4754363464466234712t_unit @ R2 @ S ) )
     => ( ( ord_le3724670747650509150_set_a @ I2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( image_set_a_set_a @ H2 @ ( a_l_co8760856729277977016t_unit @ R2 @ A @ I2 ) )
            = ( a_l_co8760856729277977016t_unit @ S @ ( H2 @ A ) @ ( image_set_a_set_a @ H2 @ I2 ) ) ) ) ) ) ).

% a_coset_hom(1)
thf(fact_285_a__coset__hom_I1_J,axiom,
    ! [H2: set_a > set_int,R2: partia6043505979758434576t_unit,S: partia4934656038542163276t_unit,I2: set_set_a,A: set_a] :
      ( ( member_set_a_set_int @ H2 @ ( ring_h4827446719030694208t_unit @ R2 @ S ) )
     => ( ( ord_le3724670747650509150_set_a @ I2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( image_set_a_set_int @ H2 @ ( a_l_co8760856729277977016t_unit @ R2 @ A @ I2 ) )
            = ( a_l_co3504123944629134560t_unit @ S @ ( H2 @ A ) @ ( image_set_a_set_int @ H2 @ I2 ) ) ) ) ) ) ).

% a_coset_hom(1)
thf(fact_286_zf_Oideal__sum__iff__gcd,axiom,
    ! [A: set_int,B2: set_int,D: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ D @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ D )
              = ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 ) ) )
            = ( isgcd_4636411027072998995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ D @ A @ B2 ) ) ) ) ) ).

% zf.ideal_sum_iff_gcd
thf(fact_287_h_Othe__elem__surj,axiom,
    ( ( image_5898968216206821007_int_a
      @ ^ [X4: set_list_set_int] :
          ( the_elem_a
          @ ( image_list_set_int_a
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
            @ X4 ) )
      @ ( partia7954853462395303941t_unit
        @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) ) )
    = ( image_list_set_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% h.the_elem_surj
thf(fact_288_h_Othe__elem__inj,axiom,
    ! [X5: set_list_set_int,Y3: set_list_set_int] :
      ( ( member4487585027056014399et_int @ X5
        @ ( partia7954853462395303941t_unit
          @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
              @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) ) )
     => ( ( member4487585027056014399et_int @ Y3
          @ ( partia7954853462395303941t_unit
            @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
              @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
                @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) ) )
       => ( ( ( the_elem_a
              @ ( image_list_set_int_a
                @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
                @ X5 ) )
            = ( the_elem_a
              @ ( image_list_set_int_a
                @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
                @ Y3 ) ) )
         => ( X5 = Y3 ) ) ) ) ).

% h.the_elem_inj
thf(fact_289_h_Oquot__mem,axiom,
    ! [X5: set_list_set_int] :
      ( ( member4487585027056014399et_int @ X5
        @ ( partia7954853462395303941t_unit
          @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
              @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) ) )
     => ? [X3: list_set_int] :
          ( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          & ( X5
            = ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
              @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
                @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
              @ X3 ) ) ) ) ).

% h.quot_mem
thf(fact_290_mem__upI,axiom,
    ! [F: nat > list_list_set_int,R2: partia7144473111789202860t_unit] :
      ( ! [N2: nat] : ( member7440485371949894425et_int @ ( F @ N2 ) @ ( partia3237421854401958059t_unit @ R2 ) )
     => ( ? [N3: nat] : ( bound_704489707291117315et_int @ ( zero_l4263514192383796165t_unit @ R2 ) @ N3 @ F )
       => ( member3828452275266370824et_int @ F @ ( up_lis1317569024804092688t_unit @ R2 ) ) ) ) ).

% mem_upI
thf(fact_291_mem__upI,axiom,
    ! [F: nat > list_set_int,R2: partia6652711048488803900t_unit] :
      ( ! [N2: nat] : ( member_list_set_int @ ( F @ N2 ) @ ( partia5243588073521899067t_unit @ R2 ) )
     => ( ? [N3: nat] : ( bound_list_set_int @ ( zero_l5318887016005761237t_unit @ R2 ) @ N3 @ F )
       => ( member5167342361288740600et_int @ F @ ( up_lis4326185709802647968t_unit @ R2 ) ) ) ) ).

% mem_upI
thf(fact_292_mem__upI,axiom,
    ! [F: nat > set_list_set_int,R2: partia3352279114339913222t_unit] :
      ( ! [N2: nat] : ( member4487585027056014399et_int @ ( F @ N2 ) @ ( partia7954853462395303941t_unit @ R2 ) )
     => ( ? [N3: nat] : ( bound_3086541607571825961et_int @ ( zero_s8971087499367319327t_unit @ R2 ) @ N3 @ F )
       => ( member5182700891607430446et_int @ F @ ( up_set8076327455042720746t_unit @ R2 ) ) ) ) ).

% mem_upI
thf(fact_293_mem__upI,axiom,
    ! [F: nat > set_set_int,R2: partia3601206958761670294t_unit] :
      ( ! [N2: nat] : ( member_set_set_int @ ( F @ N2 ) @ ( partia1981568256555877781t_unit @ R2 ) )
     => ( ? [N3: nat] : ( bound_set_set_int @ ( zero_s2946895028292161839t_unit @ R2 ) @ N3 @ F )
       => ( member5812803276464394782et_int @ F @ ( up_set7246145586214403962t_unit @ R2 ) ) ) ) ).

% mem_upI
thf(fact_294_mem__upI,axiom,
    ! [F: nat > set_int,R2: partia4934656038542163276t_unit] :
      ( ! [N2: nat] : ( member_set_int @ ( F @ N2 ) @ ( partia966996272515721803t_unit @ R2 ) )
     => ( ? [N3: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ N3 @ F )
       => ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R2 ) ) ) ) ).

% mem_upI
thf(fact_295_mem__upI,axiom,
    ! [F: nat > set_a,R2: partia6043505979758434576t_unit] :
      ( ! [N2: nat] : ( member_set_a @ ( F @ N2 ) @ ( partia5907974310037520643t_unit @ R2 ) )
     => ( ? [N3: nat] : ( bound_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ N3 @ F )
       => ( member_nat_set_a @ F @ ( up_set6103515193466504168t_unit @ R2 ) ) ) ) ).

% mem_upI
thf(fact_296_mem__upI,axiom,
    ! [F: nat > a,R2: partia2175431115845679010xt_a_b] :
      ( ! [N2: nat] : ( member_a @ ( F @ N2 ) @ ( partia707051561876973205xt_a_b @ R2 ) )
     => ( ? [N3: nat] : ( bound_a @ ( zero_a_b @ R2 ) @ N3 @ F )
       => ( member_nat_a @ F @ ( up_a_b @ R2 ) ) ) ) ).

% mem_upI
thf(fact_297_zf_Obezout__identity,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 ) )
          = ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( somegc3856857245158089085t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B2 ) ) ) ) ) ).

% zf.bezout_identity
thf(fact_298_zf_Ofinite__number__of__roots,axiom,
    ! [P2: list_set_int] :
      ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( finite6197958912794628473et_int @ ( collect_set_int @ ( polyno6587735422648734726t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) ) ) ) ).

% zf.finite_number_of_roots
thf(fact_299_pzf_Oset__add__zero,axiom,
    ! [A2: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ A2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ A2 )
        = A2 ) ) ).

% pzf.set_add_zero
thf(fact_300_pzf_OIdl__subset__ideal_H,axiom,
    ! [A: list_set_int,B2: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( ord_le4527193859438875528et_int @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int ) ) @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ B2 @ bot_bo7344713902834881468et_int ) ) )
          = ( member_list_set_int @ A @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ B2 @ bot_bo7344713902834881468et_int ) ) ) ) ) ) ).

% pzf.Idl_subset_ideal'
thf(fact_301_bound__upD,axiom,
    ! [F: nat > a] :
      ( ( member_nat_a @ F @ ( up_a_b @ r ) )
     => ? [N2: nat] : ( bound_a @ ( zero_a_b @ r ) @ N2 @ F ) ) ).

% bound_upD
thf(fact_302_ring__primeI,axiom,
    ! [P2: a] :
      ( ( P2
       != ( zero_a_b @ r ) )
     => ( ( prime_a_ring_ext_a_b @ r @ P2 )
       => ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).

% ring_primeI
thf(fact_303_zero__is__prime_I1_J,axiom,
    prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).

% zero_is_prime(1)
thf(fact_304_up__add__closed,axiom,
    ! [P2: nat > a,Q2: nat > a] :
      ( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
     => ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
       => ( member_nat_a
          @ ^ [I4: nat] : ( add_a_b @ r @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
          @ ( up_a_b @ r ) ) ) ) ).

% up_add_closed
thf(fact_305_local_Ominus__unique,axiom,
    ! [Y: a,X: a,Y2: a] :
      ( ( ( add_a_b @ r @ Y @ X )
        = ( zero_a_b @ r ) )
     => ( ( ( add_a_b @ r @ X @ Y2 )
          = ( zero_a_b @ r ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
             => ( Y = Y2 ) ) ) ) ) ) ).

% local.minus_unique
thf(fact_306_add_Or__inv__ex,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( add_a_b @ r @ X @ X3 )
            = ( zero_a_b @ r ) ) ) ) ).

% add.r_inv_ex
thf(fact_307_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_308_add_Ol__inv__ex,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( add_a_b @ r @ X3 @ X )
            = ( zero_a_b @ r ) ) ) ) ).

% add.l_inv_ex
thf(fact_309_add_Oinv__comm,axiom,
    ! [X: a,Y: a] :
      ( ( ( add_a_b @ r @ X @ Y )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( add_a_b @ r @ Y @ X )
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.inv_comm
thf(fact_310_ring__primeE_I1_J,axiom,
    ! [P2: a] :
      ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_ring_prime_a_b @ r @ P2 )
       => ( P2
         != ( zero_a_b @ r ) ) ) ) ).

% ring_primeE(1)
thf(fact_311_ring__primeE_I3_J,axiom,
    ! [P2: a] :
      ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_ring_prime_a_b @ r @ P2 )
       => ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).

% ring_primeE(3)
thf(fact_312_a__lcomm,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
            = ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).

% a_lcomm
thf(fact_313_a__comm,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( add_a_b @ r @ X @ Y )
          = ( add_a_b @ r @ Y @ X ) ) ) ) ).

% a_comm
thf(fact_314_a__assoc,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
            = ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).

% a_assoc
thf(fact_315_add_Or__cancel,axiom,
    ! [A: a,C: a,B2: a] :
      ( ( ( add_a_b @ r @ A @ C )
        = ( add_a_b @ r @ B2 @ C ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
           => ( A = B2 ) ) ) ) ) ).

% add.r_cancel
thf(fact_316_add_Ol__cancel,axiom,
    ! [C: a,A: a,B2: a] :
      ( ( ( add_a_b @ r @ C @ A )
        = ( add_a_b @ r @ C @ B2 ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
           => ( A = B2 ) ) ) ) ) ).

% add.l_cancel
thf(fact_317_cgenideal__self,axiom,
    ! [I: a] :
      ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).

% cgenideal_self
thf(fact_318_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 ) )
     => ( ! [R4: a] :
            ( ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( F @ R4 )
              = ( G @ R4 ) ) )
       => ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ r @ S ) ) ) ) ).

% ring_hom_restrict
thf(fact_319_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 ) )
     => ( ! [R4: a] :
            ( ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( F @ R4 )
              = ( G @ R4 ) ) )
       => ( member_a_set_list_a @ G @ ( ring_h6109298854714515236t_unit @ r @ S ) ) ) ) ).

% ring_hom_restrict
thf(fact_320_a__rcos__zero,axiom,
    ! [I2: set_a,I: a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( member_a @ I @ I2 )
       => ( ( a_r_coset_a_b @ r @ I2 @ I )
          = I2 ) ) ) ).

% a_rcos_zero
thf(fact_321_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_322_zf_Obound__upD,axiom,
    ! [F: nat > set_int] :
      ( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ? [N2: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ N2 @ F ) ) ).

% zf.bound_upD
thf(fact_323_zf_Oring__primeI,axiom,
    ! [P2: set_int] :
      ( ( P2
       != ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( prime_2746729220581880658t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 )
       => ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) ) ) ).

% zf.ring_primeI
thf(fact_324_zf_Ozero__is__prime_I1_J,axiom,
    prime_2746729220581880658t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ).

% zf.zero_is_prime(1)
thf(fact_325_a__coset__add__assoc,axiom,
    ! [M: set_a,G: a,H2: a] :
      ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( a_r_coset_a_b @ r @ ( a_r_coset_a_b @ r @ M @ G ) @ H2 )
            = ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ G @ H2 ) ) ) ) ) ) ).

% a_coset_add_assoc
thf(fact_326_a__rcosI,axiom,
    ! [H2: a,H: set_a,X: a] :
      ( ( member_a @ H2 @ H )
     => ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
         => ( member_a @ ( add_a_b @ r @ H2 @ X ) @ ( a_r_coset_a_b @ r @ H @ X ) ) ) ) ) ).

% a_rcosI
thf(fact_327_a__lcos__m__assoc,axiom,
    ! [M: set_a,G: a,H2: a] :
      ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H2 @ M ) )
            = ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H2 ) @ M ) ) ) ) ) ).

% a_lcos_m_assoc
thf(fact_328_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_329_a__r__coset__subset__G,axiom,
    ! [H: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% a_r_coset_subset_G
thf(fact_330_a__l__coset__subset__G,axiom,
    ! [H: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% a_l_coset_subset_G
thf(fact_331_exists__gen,axiom,
    ! [I2: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ? [X3: a] :
          ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( I2
            = ( cgenid547466209912283029xt_a_b @ r @ X3 ) ) ) ) ).

% exists_gen
thf(fact_332_cgenideal__ideal,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ r ) ) ).

% cgenideal_ideal
thf(fact_333_cgenideal__minimal,axiom,
    ! [J: set_a,A: a] :
      ( ( ideal_a_b @ J @ r )
     => ( ( member_a @ A @ J )
       => ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ J ) ) ) ).

% cgenideal_minimal
thf(fact_334_zf_Oup__add__closed,axiom,
    ! [P2: nat > set_int,Q2: nat > set_int] :
      ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_nat_set_int @ Q2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( member_nat_set_int
          @ ^ [I4: nat] : ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
          @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.up_add_closed
thf(fact_335_zf_Ominus__unique,axiom,
    ! [Y: set_int,X: set_int,Y2: set_int] :
      ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X )
        = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y2 )
          = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
           => ( ( member_set_int @ Y2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
             => ( Y = Y2 ) ) ) ) ) ) ).

% zf.minus_unique
thf(fact_336_zf_Oadd_Or__inv__ex,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ? [X3: set_int] :
          ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          & ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ X3 )
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.add.r_inv_ex
thf(fact_337_zf_Oadd_Oone__unique,axiom,
    ! [U: set_int] :
      ( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ! [X3: set_int] :
            ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
           => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ U @ X3 )
              = X3 ) )
       => ( U
          = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.add.one_unique
thf(fact_338_zf_Oadd_Ol__inv__ex,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ? [X3: set_int] :
          ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          & ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X3 @ X )
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.add.l_inv_ex
thf(fact_339_zf_Oadd_Oinv__comm,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
        = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X )
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% zf.add.inv_comm
thf(fact_340_zf_Oring__primeE_I1_J,axiom,
    ! [P2: set_int] :
      ( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 )
       => ( P2
         != ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.ring_primeE(1)
thf(fact_341_zf_Oring__primeE_I3_J,axiom,
    ! [P2: set_int] :
      ( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 )
       => ( prime_2746729220581880658t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) ) ) ).

% zf.ring_primeE(3)
thf(fact_342_zf_Oadd_Or__cancel,axiom,
    ! [A: set_int,C: set_int,B2: set_int] :
      ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C )
        = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 @ C ) )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
           => ( A = B2 ) ) ) ) ) ).

% zf.add.r_cancel
thf(fact_343_zf_Oadd_Om__lcomm,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ Z ) )
            = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Z ) ) ) ) ) ) ).

% zf.add.m_lcomm
thf(fact_344_zf_Oadd_Om__comm,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
          = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X ) ) ) ) ).

% zf.add.m_comm
thf(fact_345_zf_Oadd_Om__assoc,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) @ Z )
            = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ Z ) ) ) ) ) ) ).

% zf.add.m_assoc
thf(fact_346_zf_Oadd_Ol__cancel,axiom,
    ! [C: set_int,A: set_int,B2: set_int] :
      ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ A )
        = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ B2 ) )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
           => ( A = B2 ) ) ) ) ) ).

% zf.add.l_cancel
thf(fact_347_zf_Oring__hom__restrict,axiom,
    ! [F: set_int > list_set_int,S: partia6652711048488803900t_unit,G: set_int > list_set_int] :
      ( ( member7759640485799516682et_int @ F @ ( ring_h8178984891965966890t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ! [R4: set_int] :
            ( ( member_set_int @ R4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
           => ( ( F @ R4 )
              = ( G @ R4 ) ) )
       => ( member7759640485799516682et_int @ G @ ( ring_h8178984891965966890t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) ) ) ) ).

% zf.ring_hom_restrict
thf(fact_348_zf_Oring__hom__restrict,axiom,
    ! [F: set_int > set_list_set_int,S: partia3352279114339913222t_unit,G: set_int > set_list_set_int] :
      ( ( member7361235547688761408et_int @ F @ ( ring_h2330097277019473012t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ! [R4: set_int] :
            ( ( member_set_int @ R4 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
           => ( ( F @ R4 )
              = ( G @ R4 ) ) )
       => ( member7361235547688761408et_int @ G @ ( ring_h2330097277019473012t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) ) ) ) ).

% zf.ring_hom_restrict
thf(fact_349_cgenideal__is__principalideal,axiom,
    ! [I: a] :
      ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
     => ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).

% cgenideal_is_principalideal
thf(fact_350_zf_Oa__rcos__zero,axiom,
    ! [I2: set_set_int,I: set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( member_set_int @ I @ I2 )
       => ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ I )
          = I2 ) ) ) ).

% zf.a_rcos_zero
thf(fact_351_zf_Oadd_Osurj__const__mult,axiom,
    ! [A: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( image_524474410958335435et_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        = ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.add.surj_const_mult
thf(fact_352_a__rcos__assoc__lcos,axiom,
    ! [H: set_a,K: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( set_add_a_b @ r @ ( a_r_coset_a_b @ r @ H @ X ) @ K )
            = ( set_add_a_b @ r @ H @ ( a_l_coset_a_b @ r @ X @ K ) ) ) ) ) ) ).

% a_rcos_assoc_lcos
thf(fact_353_a__setmult__rcos__assoc,axiom,
    ! [H: set_a,K: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( set_add_a_b @ r @ H @ ( a_r_coset_a_b @ r @ K @ X ) )
            = ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H @ K ) @ X ) ) ) ) ) ).

% a_setmult_rcos_assoc
thf(fact_354_zf_Oa__rcosI,axiom,
    ! [H2: set_int,H: set_set_int,X: set_int] :
      ( ( member_set_int @ H2 @ H )
     => ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ X ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ X ) ) ) ) ) ).

% zf.a_rcosI
thf(fact_355_zf_Oa__coset__add__assoc,axiom,
    ! [M: set_set_int,G: set_int,H2: set_int] :
      ( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ G @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ G ) @ H2 )
            = ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ G @ H2 ) ) ) ) ) ) ).

% zf.a_coset_add_assoc
thf(fact_356_zf_Oa__lcos__m__assoc,axiom,
    ! [M: set_set_int,G: set_int,H2: set_int] :
      ( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ G @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ G @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ M ) )
            = ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ G @ H2 ) @ M ) ) ) ) ) ).

% zf.a_lcos_m_assoc
thf(fact_357_zf_Oa__lcos__mult__one,axiom,
    ! [M: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ M )
        = M ) ) ).

% zf.a_lcos_mult_one
thf(fact_358_zf_Oa__r__coset__subset__G,axiom,
    ! [H: set_set_int,X: set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ord_le4403425263959731960et_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.a_r_coset_subset_G
thf(fact_359_zf_Oa__l__coset__subset__G,axiom,
    ! [H: set_set_int,X: set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ord_le4403425263959731960et_int @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ H ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.a_l_coset_subset_G
thf(fact_360_pzf_Ocgenideal__self,axiom,
    ! [I: list_set_int] :
      ( ( member_list_set_int @ I @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( member_list_set_int @ I @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I ) ) ) ).

% pzf.cgenideal_self
thf(fact_361_zf_Oa__rcos__assoc__lcos,axiom,
    ! [H: set_set_int,K: set_set_int,X: set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ X ) @ K )
            = ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ K ) ) ) ) ) ) ).

% zf.a_rcos_assoc_lcos
thf(fact_362_zf_Oa__setmult__rcos__assoc,axiom,
    ! [H: set_set_int,K: set_set_int,X: set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K @ X ) )
            = ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ K ) @ X ) ) ) ) ) ).

% zf.a_setmult_rcos_assoc
thf(fact_363_pzf_Oexists__gen,axiom,
    ! [I2: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ? [X3: list_set_int] :
          ( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          & ( I2
            = ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X3 ) ) ) ) ).

% pzf.exists_gen
thf(fact_364_pzf_Ocgenideal__ideal,axiom,
    ! [A: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ideal_3307802554209726317t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.cgenideal_ideal
thf(fact_365_pzf_Ocgenideal__minimal,axiom,
    ! [J: set_list_set_int,A: list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ J @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member_list_set_int @ A @ J )
       => ( ord_le4527193859438875528et_int @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) @ J ) ) ) ).

% pzf.cgenideal_minimal
thf(fact_366_pzf_Ocgenideal__is__principalideal,axiom,
    ! [I: list_set_int] :
      ( ( member_list_set_int @ I @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( princi2884690910141361289t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.cgenideal_is_principalideal
thf(fact_367_pzf_Oring__ideal__imp__quot__ideal,axiom,
    ! [I2: set_list_set_int,J: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ideal_3307802554209726317t_unit @ J @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
       => ( ideal_7892184242768911799t_unit @ ( image_8713293071697434529et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) @ J ) @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) ) ) ) ).

% pzf.ring_ideal_imp_quot_ideal
thf(fact_368_pzf_Ozeroideal,axiom,
    ideal_3307802554209726317t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.zeroideal
thf(fact_369_pzf_Ocgenideal__eq__genideal,axiom,
    ! [I: list_set_int] :
      ( ( member_list_set_int @ I @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I )
        = ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ I @ bot_bo7344713902834881468et_int ) ) ) ) ).

% pzf.cgenideal_eq_genideal
thf(fact_370_pzf_Ogenideal__self_H,axiom,
    ! [I: list_set_int] :
      ( ( member_list_set_int @ I @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( member_list_set_int @ I @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ I @ bot_bo7344713902834881468et_int ) ) ) ) ).

% pzf.genideal_self'
thf(fact_371_pzf_Ogenideal__zero,axiom,
    ( ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) )
    = ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) ).

% pzf.genideal_zero
thf(fact_372_r__zero,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
        = X ) ) ).

% r_zero
thf(fact_373_l__zero,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
        = X ) ) ).

% l_zero
thf(fact_374_add_Or__cancel__one_H,axiom,
    ! [X: a,A: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( X
            = ( add_a_b @ r @ A @ X ) )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.r_cancel_one'
thf(fact_375_add_Or__cancel__one,axiom,
    ! [X: a,A: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ A @ X )
            = X )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.r_cancel_one
thf(fact_376_add_Ol__cancel__one_H,axiom,
    ! [X: a,A: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( X
            = ( add_a_b @ r @ X @ A ) )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.l_cancel_one'
thf(fact_377_add_Ol__cancel__one,axiom,
    ! [X: a,A: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ X @ A )
            = X )
          = ( A
            = ( zero_a_b @ r ) ) ) ) ) ).

% add.l_cancel_one
thf(fact_378_a__closed,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% a_closed
thf(fact_379_local_Oadd_Oright__cancel,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ Y @ X )
              = ( add_a_b @ r @ Z @ X ) )
            = ( Y = Z ) ) ) ) ) ).

% local.add.right_cancel
thf(fact_380_zero__closed,axiom,
    member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% zero_closed
thf(fact_381_pzf_Oquot__quot__hom,axiom,
    ! [I2: set_list_set_int,J: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ideal_3307802554209726317t_unit @ J @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
       => ( ( ord_le4527193859438875528et_int @ I2 @ J )
         => ( member1822656912387541382et_int @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ J ) @ ( ring_h5575170385439490106t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ J ) ) ) ) ) ) ).

% pzf.quot_quot_hom
thf(fact_382_pzf_Ozeropideal,axiom,
    princi2884690910141361289t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.zeropideal
thf(fact_383_pzf_Oquot__carr,axiom,
    ! [I2: set_list_set_int,Y: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member4487585027056014399et_int @ Y @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) ) )
       => ( ord_le4527193859438875528et_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.quot_carr
thf(fact_384_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_385_zf_Or__zero,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        = X ) ) ).

% zf.r_zero
thf(fact_386_zf_Ol__zero,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ X )
        = X ) ) ).

% zf.l_zero
thf(fact_387_zf_Oadd_Or__cancel__one_H,axiom,
    ! [X: set_int,A: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( X
            = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ X ) )
          = ( A
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% zf.add.r_cancel_one'
thf(fact_388_zf_Oadd_Or__cancel__one,axiom,
    ! [X: set_int,A: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ X )
            = X )
          = ( A
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% zf.add.r_cancel_one
thf(fact_389_zf_Oadd_Ol__cancel__one_H,axiom,
    ! [X: set_int,A: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( X
            = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ A ) )
          = ( A
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% zf.add.l_cancel_one'
thf(fact_390_zf_Oadd_Ol__cancel__one,axiom,
    ! [X: set_int,A: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ A )
            = X )
          = ( A
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% zf.add.l_cancel_one
thf(fact_391_zf_Oadd_Oright__cancel,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X )
              = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Z @ X ) )
            = ( Y = Z ) ) ) ) ) ).

% zf.add.right_cancel
thf(fact_392_zf_Oadd_Om__closed,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.add.m_closed
thf(fact_393_zf_Ozero__closed,axiom,
    member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ).

% zf.zero_closed
thf(fact_394_h_Othe__elem__hom,axiom,
    ( member1264592267077841462_int_a
    @ ^ [X4: set_list_set_int] :
        ( the_elem_a
        @ ( image_list_set_int_a
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
          @ X4 ) )
    @ ( ring_h6043350348530273514it_a_b
      @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) )
      @ r ) ) ).

% h.the_elem_hom
thf(fact_395_h_OFactRing__iso__set,axiom,
    ( ( ( image_list_set_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
      = ( partia707051561876973205xt_a_b @ r ) )
   => ( member1264592267077841462_int_a
      @ ^ [X4: set_list_set_int] :
          ( the_elem_a
          @ ( image_list_set_int_a
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
            @ X4 ) )
      @ ( ring_i1463878111215264340it_a_b
        @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) )
        @ r ) ) ) ).

% h.FactRing_iso_set
thf(fact_396_zf_Oa__coset__add__zero,axiom,
    ! [M: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        = M ) ) ).

% zf.a_coset_add_zero
thf(fact_397_h_Okernel__zero,axiom,
    ! [I: list_set_int,R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( member_list_set_int @ I
        @ ( a_kern885551317661704990it_a_b @ R2 @ S
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) )
     => ( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ I ) @ x )
        = ( zero_a_b @ S ) ) ) ).

% h.kernel_zero
thf(fact_398_h_Ohom__zero,axiom,
    ( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ x )
    = ( zero_a_b @ r ) ) ).

% h.hom_zero
thf(fact_399_h_Ohom__add,axiom,
    ! [X: list_set_int,Y: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) ) @ x )
          = ( add_a_b @ r @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X ) @ x ) @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ Y ) @ x ) ) ) ) ) ).

% h.hom_add
thf(fact_400_pzf_OFactRing__zeroideal_I2_J,axiom,
    is_rin8255480188913837456t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) ).

% pzf.FactRing_zeroideal(2)
thf(fact_401_pzf_OFactRing__zeroideal_I1_J,axiom,
    is_rin7318007858330770044t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.FactRing_zeroideal(1)
thf(fact_402_h_OFactRing__iso,axiom,
    ( ( ( image_list_set_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
      = ( partia707051561876973205xt_a_b @ r ) )
   => ( is_rin39764828050323806it_a_b
      @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) )
      @ r ) ) ).

% h.FactRing_iso
thf(fact_403_ideal_Orcos__ring__hom,axiom,
    ! [I2: set_set_list_set_int,R2: partia3352279114339913222t_unit] :
      ( ( ideal_7892184242768911799t_unit @ I2 @ R2 )
     => ( member3351879598113499324et_int @ ( a_r_co7111076282206511392t_unit @ R2 @ I2 ) @ ( ring_h5324936237077678468t_unit @ R2 @ ( factRi8810600641151685442t_unit @ R2 @ I2 ) ) ) ) ).

% ideal.rcos_ring_hom
thf(fact_404_ideal_Orcos__ring__hom,axiom,
    ! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
      ( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
     => ( member5110176937812460956et_int @ ( a_r_co7378703583274672048t_unit @ R2 @ I2 ) @ ( ring_h7502273849718182020t_unit @ R2 @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) ) ) ) ).

% ideal.rcos_ring_hom
thf(fact_405_ideal_Orcos__ring__hom,axiom,
    ! [I2: set_set_a,R2: partia6043505979758434576t_unit] :
      ( ( ideal_4463284918206690523t_unit @ I2 @ R2 )
     => ( member73722476670768732_set_a @ ( a_r_co6984235733410409778t_unit @ R2 @ I2 ) @ ( ring_h8231902973048596472t_unit @ R2 @ ( factRi9077435707053964944t_unit @ R2 @ I2 ) ) ) ) ).

% ideal.rcos_ring_hom
thf(fact_406_ideal_Orcos__ring__hom,axiom,
    ! [I2: set_list_set_int,R2: partia6652711048488803900t_unit] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ R2 )
     => ( member5554658343060950864et_int @ ( a_r_co912825291596486358t_unit @ R2 @ I2 ) @ ( ring_h7954335661761407236t_unit @ R2 @ ( factRi8995599497114439928t_unit @ R2 @ I2 ) ) ) ) ).

% ideal.rcos_ring_hom
thf(fact_407_ideal_Orcos__ring__hom,axiom,
    ! [I2: set_a,R2: partia2175431115845679010xt_a_b] :
      ( ( ideal_a_b @ I2 @ R2 )
     => ( member_a_set_a @ ( a_r_coset_a_b @ R2 @ I2 ) @ ( ring_h7909251507493334186t_unit @ R2 @ ( factRing_a_b @ R2 @ I2 ) ) ) ) ).

% ideal.rcos_ring_hom
thf(fact_408_ideal_Orcos__ring__hom,axiom,
    ! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
     => ( member1123395007370249776et_int @ ( a_r_co692709266861932262t_unit @ R2 @ I2 ) @ ( ring_h2644251554410904836t_unit @ R2 @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) ) ) ) ).

% ideal.rcos_ring_hom
thf(fact_409_ideal_Orcos__ring__hom,axiom,
    ! [I2: set_list_a,R2: partia2670972154091845814t_unit] :
      ( ( ideal_8896367198367571637t_unit @ I2 @ R2 )
     => ( member4263473470251683292list_a @ ( a_r_co1577572119920843660t_unit @ R2 @ I2 ) @ ( ring_h6188449271506562988t_unit @ R2 @ ( factRi3329376332477095402t_unit @ R2 @ I2 ) ) ) ) ).

% ideal.rcos_ring_hom
thf(fact_410_ideal_Orcos__ring__hom,axiom,
    ! [I2: set_li2838760105183549752et_int,R2: partia7144473111789202860t_unit] :
      ( ( ideal_1683033260801535581t_unit @ I2 @ R2 )
     => ( member8798303191263431536et_int @ ( a_r_co2531912773843275078t_unit @ R2 @ I2 ) @ ( ring_h2819912158977254660t_unit @ R2 @ ( factRi1079867023605223016t_unit @ R2 @ I2 ) ) ) ) ).

% ideal.rcos_ring_hom
thf(fact_411_ideal_Orcos__ring__hom__ring,axiom,
    ! [I2: set_set_list_set_int,R2: partia3352279114339913222t_unit] :
      ( ( ideal_7892184242768911799t_unit @ I2 @ R2 )
     => ( ring_h7916722715803227334t_unit @ R2 @ ( factRi8810600641151685442t_unit @ R2 @ I2 ) @ ( a_r_co7111076282206511392t_unit @ R2 @ I2 ) ) ) ).

% ideal.rcos_ring_hom_ring
thf(fact_412_ideal_Orcos__ring__hom__ring,axiom,
    ! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
      ( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
     => ( ring_h8406274701427011014t_unit @ R2 @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) @ ( a_r_co7378703583274672048t_unit @ R2 @ I2 ) ) ) ).

% ideal.rcos_ring_hom_ring
thf(fact_413_ideal_Orcos__ring__hom__ring,axiom,
    ! [I2: set_set_a,R2: partia6043505979758434576t_unit] :
      ( ( ideal_4463284918206690523t_unit @ I2 @ R2 )
     => ( ring_h3283350608632093754t_unit @ R2 @ ( factRi9077435707053964944t_unit @ R2 @ I2 ) @ ( a_r_co6984235733410409778t_unit @ R2 @ I2 ) ) ) ).

% ideal.rcos_ring_hom_ring
thf(fact_414_ideal_Orcos__ring__hom__ring,axiom,
    ! [I2: set_list_set_int,R2: partia6652711048488803900t_unit] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ R2 )
     => ( ring_h4009149326251492422t_unit @ R2 @ ( factRi8995599497114439928t_unit @ R2 @ I2 ) @ ( a_r_co912825291596486358t_unit @ R2 @ I2 ) ) ) ).

% ideal.rcos_ring_hom_ring
thf(fact_415_ideal_Orcos__ring__hom__ring,axiom,
    ! [I2: set_a,R2: partia2175431115845679010xt_a_b] :
      ( ( ideal_a_b @ I2 @ R2 )
     => ( ring_h2812136955688090348t_unit @ R2 @ ( factRing_a_b @ R2 @ I2 ) @ ( a_r_coset_a_b @ R2 @ I2 ) ) ) ).

% ideal.rcos_ring_hom_ring
thf(fact_416_ideal_Orcos__ring__hom__ring,axiom,
    ! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
     => ( ring_h2557103153054820934t_unit @ R2 @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) @ ( a_r_co692709266861932262t_unit @ R2 @ I2 ) ) ) ).

% ideal.rcos_ring_hom_ring
thf(fact_417_ideal_Orcos__ring__hom__ring,axiom,
    ! [I2: set_li2838760105183549752et_int,R2: partia7144473111789202860t_unit] :
      ( ( ideal_1683033260801535581t_unit @ I2 @ R2 )
     => ( ring_h5795583995102309958t_unit @ R2 @ ( factRi1079867023605223016t_unit @ R2 @ I2 ) @ ( a_r_co2531912773843275078t_unit @ R2 @ I2 ) ) ) ).

% ideal.rcos_ring_hom_ring
thf(fact_418_ideal_Orcos__ring__hom__ring,axiom,
    ! [I2: set_list_a,R2: partia2670972154091845814t_unit] :
      ( ( ideal_8896367198367571637t_unit @ I2 @ R2 )
     => ( ring_h6719293541701509614t_unit @ R2 @ ( factRi3329376332477095402t_unit @ R2 @ I2 ) @ ( a_r_co1577572119920843660t_unit @ R2 @ I2 ) ) ) ).

% ideal.rcos_ring_hom_ring
thf(fact_419_ring__hom__ring_Othe__elem__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,H2: a > set_a] :
      ( ( ring_h2812136955688090348t_unit @ R2 @ S @ H2 )
     => ( member_set_a_set_a
        @ ^ [X4: set_a] : ( the_elem_set_a @ ( image_a_set_a @ H2 @ X4 ) )
        @ ( ring_h4754363464466234712t_unit @ ( factRing_a_b @ R2 @ ( a_kern6814816853359880620t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.the_elem_hom
thf(fact_420_ring__hom__ring_Othe__elem__hom,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( member1264592267077841462_int_a
        @ ^ [X4: set_list_set_int] : ( the_elem_a @ ( image_list_set_int_a @ H2 @ X4 ) )
        @ ( ring_h6043350348530273514it_a_b @ ( factRi8995599497114439928t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.the_elem_hom
thf(fact_421_ring__hom__ring_Othe__elem__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H2: list_a > set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H2 )
     => ( member5068272912271824380list_a
        @ ^ [X4: set_list_a] : ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ H2 @ X4 ) )
        @ ( ring_h6076331213207892940t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.the_elem_hom
thf(fact_422_ring__hom__ring_Othe__elem__hom,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( member_set_list_a_a
        @ ^ [X4: set_list_a] : ( the_elem_a @ ( image_list_a_a @ H2 @ X4 ) )
        @ ( ring_h8906680420194085028it_a_b @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.the_elem_hom
thf(fact_423_ring__hom__ring_Othe__elem__hom,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6652711048488803900t_unit,H2: set_int > list_set_int] :
      ( ( ring_h615793168808539756t_unit @ R2 @ S @ H2 )
     => ( member45303223370921408et_int
        @ ^ [X4: set_set_int] : ( the_el6263124322315510589et_int @ ( image_8094898911679120219et_int @ H2 @ X4 ) )
        @ ( ring_h288618570656695008t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.the_elem_hom
thf(fact_424_ring__hom__ring_Othe__elem__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( member1867902588855634794et_int
        @ ^ [X4: set_a] : ( the_el6263124322315510589et_int @ ( image_a_list_set_int @ H2 @ X4 ) )
        @ ( ring_h112604941399733296t_unit @ ( factRing_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.the_elem_hom
thf(fact_425_ring__hom__ring_Othe__elem__hom,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( member_set_a_list_a
        @ ^ [X4: set_a] : ( the_elem_list_a @ ( image_a_list_a @ H2 @ X4 ) )
        @ ( ring_h6465547548369483954t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.the_elem_hom
thf(fact_426_ring__hom__ring_Othe__elem__wf_H,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H2: list_a > set_list_a,X5: set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H2 )
     => ( ( member_set_list_a @ X5 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: list_a] :
            ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
            & ( ( image_5464838071766335845list_a @ H2 @ X5 )
              = ( insert_set_list_a @ ( H2 @ X3 ) @ bot_bo3186585308812441520list_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf'
thf(fact_427_ring__hom__ring_Othe__elem__wf_H,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,X5: set_list_a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( member_set_list_a @ X5 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) ) ) )
       => ? [X3: list_a] :
            ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
            & ( ( image_list_a_a @ H2 @ X5 )
              = ( insert_a @ ( H2 @ X3 ) @ bot_bot_set_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf'
thf(fact_428_ring__hom__ring_Othe__elem__wf_H,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a,X5: set_list_set_int] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( member4487585027056014399et_int @ X5 @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) ) ) )
       => ? [X3: list_set_int] :
            ( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ R2 ) )
            & ( ( image_list_set_int_a @ H2 @ X5 )
              = ( insert_a @ ( H2 @ X3 ) @ bot_bot_set_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf'
thf(fact_429_ring__hom__ring_Othe__elem__wf_H,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6652711048488803900t_unit,H2: set_int > list_set_int,X5: set_set_int] :
      ( ( ring_h615793168808539756t_unit @ R2 @ S @ H2 )
     => ( ( member_set_set_int @ X5 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ R2 ) )
            & ( ( image_8094898911679120219et_int @ H2 @ X5 )
              = ( insert_list_set_int @ ( H2 @ X3 ) @ bot_bo7344713902834881468et_int ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf'
thf(fact_430_ring__hom__ring_Othe__elem__wf_H,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int,X5: set_a] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
            & ( ( image_a_list_set_int @ H2 @ X5 )
              = ( insert_list_set_int @ ( H2 @ X3 ) @ bot_bo7344713902834881468et_int ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf'
thf(fact_431_ring__hom__ring_Othe__elem__wf_H,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,X5: set_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
            & ( ( image_a_list_a @ H2 @ X5 )
              = ( insert_list_a @ ( H2 @ X3 ) @ bot_bot_set_list_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf'
thf(fact_432_ring__hom__ring_Othe__elem__wf,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a,X5: set_a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kernel_a_b_a_b @ R2 @ S @ H2 ) ) ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ S ) )
            & ( ( image_a_a @ H2 @ X5 )
              = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf
thf(fact_433_ring__hom__ring_Othe__elem__wf,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,X5: set_list_a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( member_set_list_a @ X5 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) ) ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ S ) )
            & ( ( image_list_a_a @ H2 @ X5 )
              = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf
thf(fact_434_ring__hom__ring_Othe__elem__wf,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a,X5: set_set_int] :
      ( ( ring_h5360235766848426350it_a_b @ R2 @ S @ H2 )
     => ( ( member_set_set_int @ X5 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern189768852501166it_a_b @ R2 @ S @ H2 ) ) ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ S ) )
            & ( ( image_set_int_a @ H2 @ X5 )
              = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf
thf(fact_435_ring__hom__ring_Othe__elem__wf,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,X5: set_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: list_a] :
            ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ S ) )
            & ( ( image_a_list_a @ H2 @ X5 )
              = ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf
thf(fact_436_ring__hom__ring_Othe__elem__wf,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,H2: a > set_int,X5: set_a] :
      ( ( ring_h7266096129245845164t_unit @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern1906050131249919980t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ S ) )
            & ( ( image_a_set_int @ H2 @ X5 )
              = ( insert_set_int @ X3 @ bot_bot_set_set_int ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf
thf(fact_437_ring__hom__ring_Othe__elem__wf,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,H2: a > set_a,X5: set_a] :
      ( ( ring_h2812136955688090348t_unit @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern6814816853359880620t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: set_a] :
            ( ( member_set_a @ X3 @ ( partia5907974310037520643t_unit @ S ) )
            & ( ( image_a_set_a @ H2 @ X5 )
              = ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf
thf(fact_438_ring__hom__ring_Othe__elem__wf,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a,X5: set_list_set_int] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( member4487585027056014399et_int @ X5 @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) ) ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ S ) )
            & ( ( image_list_set_int_a @ H2 @ X5 )
              = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf
thf(fact_439_ring__hom__ring_Othe__elem__wf,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H2: set_int > set_int,X5: set_set_int] :
      ( ( ring_h98094481560215932t_unit @ R2 @ S @ H2 )
     => ( ( member_set_set_int @ X5 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern4456122948206014524t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ S ) )
            & ( ( image_524474410958335435et_int @ H2 @ X5 )
              = ( insert_set_int @ X3 @ bot_bot_set_set_int ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf
thf(fact_440_ring__hom__ring_Othe__elem__wf,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6043505979758434576t_unit,H2: set_int > set_a,X5: set_set_int] :
      ( ( ring_h5212181533761340444t_unit @ R2 @ S @ H2 )
     => ( ( member_set_set_int @ X5 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern888292444459869532t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: set_a] :
            ( ( member_set_a @ X3 @ ( partia5907974310037520643t_unit @ S ) )
            & ( ( image_set_int_set_a @ H2 @ X5 )
              = ( insert_set_a @ X3 @ bot_bot_set_set_a ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf
thf(fact_441_ring__hom__ring_Othe__elem__wf,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int,X5: set_a] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: list_set_int] :
            ( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ S ) )
            & ( ( image_a_list_set_int @ H2 @ X5 )
              = ( insert_list_set_int @ X3 @ bot_bo7344713902834881468et_int ) ) ) ) ) ).

% ring_hom_ring.the_elem_wf
thf(fact_442_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H2 )
     => ( ( ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member_set_a_a
          @ ^ [X4: set_a] : ( the_elem_a @ ( image_a_a @ H2 @ X4 ) )
          @ ( ring_i4751279688611836608it_a_b @ ( factRing_a_b @ R2 @ ( a_kernel_a_b_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_443_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member_set_list_a_a
          @ ^ [X4: set_list_a] : ( the_elem_a @ ( image_list_a_a @ H2 @ X4 ) )
          @ ( ring_i8122894263081988538it_a_b @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_444_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
      ( ( ring_h5360235766848426350it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member_set_set_int_a
          @ ^ [X4: set_set_int] : ( the_elem_a @ ( image_set_int_a @ H2 @ X4 ) )
          @ ( ring_i5461826592427486820it_a_b @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern189768852501166it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_445_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,H2: set_a > a] :
      ( ( ring_h8937780225337603308it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_set_a_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member_set_set_a_a
          @ ^ [X4: set_set_a] : ( the_elem_a @ ( image_set_a_a @ H2 @ X4 ) )
          @ ( ring_i7178773322895389024it_a_b @ ( factRi9077435707053964944t_unit @ R2 @ ( a_kern3717088086154617772it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_446_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia5361259788508890537t_unit @ S ) )
       => ( member_set_a_list_a
          @ ^ [X4: set_a] : ( the_elem_list_a @ ( image_a_list_a @ H2 @ X4 ) )
          @ ( ring_i2253880877871059912t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_447_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,H2: a > set_int] :
      ( ( ring_h7266096129245845164t_unit @ R2 @ S @ H2 )
     => ( ( ( image_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia966996272515721803t_unit @ S ) )
       => ( member_set_a_set_int
          @ ^ [X4: set_a] : ( the_elem_set_int @ ( image_a_set_int @ H2 @ X4 ) )
          @ ( ring_i1340567420592345514t_unit @ ( factRing_a_b @ R2 @ ( a_kern1906050131249919980t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_448_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,H2: a > set_a] :
      ( ( ring_h2812136955688090348t_unit @ R2 @ S @ H2 )
     => ( ( ( image_a_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia5907974310037520643t_unit @ S ) )
       => ( member_set_a_set_a
          @ ^ [X4: set_a] : ( the_elem_set_a @ ( image_a_set_a @ H2 @ X4 ) )
          @ ( ring_i7821126338072249198t_unit @ ( factRing_a_b @ R2 @ ( a_kern6814816853359880620t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_449_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_list_set_int_a @ H2 @ ( partia5243588073521899067t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member1264592267077841462_int_a
          @ ^ [X4: set_list_set_int] : ( the_elem_a @ ( image_list_set_int_a @ H2 @ X4 ) )
          @ ( ring_i1463878111215264340it_a_b @ ( factRi8995599497114439928t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_450_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,H2: set_set_int > a] :
      ( ( ring_h5493655406951491896it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_set_set_int_a @ H2 @ ( partia1981568256555877781t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( member5392328839176676752_int_a
          @ ^ [X4: set_set_set_int] : ( the_elem_a @ ( image_set_set_int_a @ H2 @ X4 ) )
          @ ( ring_i1231480633883639086it_a_b @ ( factRi2699865449404500690t_unit @ R2 @ ( a_kern1569243063284222584it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_451_ring__hom__ring_OFactRing__iso__set,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H2: set_int > set_int] :
      ( ( ring_h98094481560215932t_unit @ R2 @ S @ H2 )
     => ( ( ( image_524474410958335435et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
          = ( partia966996272515721803t_unit @ S ) )
       => ( member2198268278526348592et_int
          @ ^ [X4: set_set_int] : ( the_elem_set_int @ ( image_524474410958335435et_int @ H2 @ X4 ) )
          @ ( ring_i4798011830457088262t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern4456122948206014524t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set
thf(fact_452_principal__domain_Obezout__identity,axiom,
    ! [R2: partia2670972154091845814t_unit,A: list_a,B2: list_a] :
      ( ( ring_p8098905331641078952t_unit @ R2 )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ( set_ad92425877771022410t_unit @ R2 @ ( cgenid9131348535277946915t_unit @ R2 @ A ) @ ( cgenid9131348535277946915t_unit @ R2 @ B2 ) )
            = ( cgenid9131348535277946915t_unit @ R2 @ ( somegc2556875419254249873t_unit @ R2 @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.bezout_identity
thf(fact_453_principal__domain_Obezout__identity,axiom,
    ! [R2: partia7144473111789202860t_unit,A: list_list_set_int,B2: list_list_set_int] :
      ( ( ring_p1105522777314784042t_unit @ R2 )
     => ( ( member7440485371949894425et_int @ A @ ( partia3237421854401958059t_unit @ R2 ) )
       => ( ( member7440485371949894425et_int @ B2 @ ( partia3237421854401958059t_unit @ R2 ) )
         => ( ( set_ad945020892712637448t_unit @ R2 @ ( cgenid7864102925118505423t_unit @ R2 @ A ) @ ( cgenid7864102925118505423t_unit @ R2 @ B2 ) )
            = ( cgenid7864102925118505423t_unit @ R2 @ ( somegc8859314853175125053t_unit @ R2 @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.bezout_identity
thf(fact_454_principal__domain_Obezout__identity,axiom,
    ! [R2: partia3352279114339913222t_unit,A: set_list_set_int,B2: set_list_set_int] :
      ( ( ring_p4083217851854698756t_unit @ R2 )
     => ( ( member4487585027056014399et_int @ A @ ( partia7954853462395303941t_unit @ R2 ) )
       => ( ( member4487585027056014399et_int @ B2 @ ( partia7954853462395303941t_unit @ R2 ) )
         => ( ( set_ad4103617194169821410t_unit @ R2 @ ( cgenid3413961061359118467t_unit @ R2 @ A ) @ ( cgenid3413961061359118467t_unit @ R2 @ B2 ) )
            = ( cgenid3413961061359118467t_unit @ R2 @ ( somegc3010301331330988273t_unit @ R2 @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.bezout_identity
thf(fact_455_principal__domain_Obezout__identity,axiom,
    ! [R2: partia3601206958761670294t_unit,A: set_set_int,B2: set_set_int] :
      ( ( ring_p4005655891433666964t_unit @ R2 )
     => ( ( member_set_set_int @ A @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( member_set_set_int @ B2 @ ( partia1981568256555877781t_unit @ R2 ) )
         => ( ( set_ad1658175378941472882t_unit @ R2 @ ( cgenid8920438795327392163t_unit @ R2 @ A ) @ ( cgenid8920438795327392163t_unit @ R2 @ B2 ) )
            = ( cgenid8920438795327392163t_unit @ R2 @ ( somegc6715311040209614865t_unit @ R2 @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.bezout_identity
thf(fact_456_principal__domain_Obezout__identity,axiom,
    ! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a] :
      ( ( ring_p2862007038493914190t_unit @ R2 )
     => ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( set_ad3113708954360070384t_unit @ R2 @ ( cgenid6682780793756002467t_unit @ R2 @ A ) @ ( cgenid6682780793756002467t_unit @ R2 @ B2 ) )
            = ( cgenid6682780793756002467t_unit @ R2 @ ( somegc5022794125158426641t_unit @ R2 @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.bezout_identity
thf(fact_457_principal__domain_Obezout__identity,axiom,
    ! [R2: partia4934656038542163276t_unit,A: set_int,B2: set_int] :
      ( ( ring_p2413783099608688842t_unit @ R2 )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ R2 ) )
         => ( ( set_ad273131178244904872t_unit @ R2 @ ( cgenid8502489213727343375t_unit @ R2 @ A ) @ ( cgenid8502489213727343375t_unit @ R2 @ B2 ) )
            = ( cgenid8502489213727343375t_unit @ R2 @ ( somegc3856857245158089085t_unit @ R2 @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.bezout_identity
thf(fact_458_principal__domain_Obezout__identity,axiom,
    ! [R2: partia2175431115845679010xt_a_b,A: a,B2: a] :
      ( ( ring_p8803135361686045600in_a_b @ R2 )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( set_add_a_b @ R2 @ ( cgenid547466209912283029xt_a_b @ R2 @ A ) @ ( cgenid547466209912283029xt_a_b @ R2 @ B2 ) )
            = ( cgenid547466209912283029xt_a_b @ R2 @ ( somegc1600592057159103747xt_a_b @ R2 @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.bezout_identity
thf(fact_459_principal__domain_Obezout__identity,axiom,
    ! [R2: partia6652711048488803900t_unit,A: list_set_int,B2: list_set_int] :
      ( ( ring_p1500425057661402810t_unit @ R2 )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ R2 ) )
         => ( ( set_ad2159827153730365080t_unit @ R2 @ ( cgenid1447163913751362543t_unit @ R2 @ A ) @ ( cgenid1447163913751362543t_unit @ R2 @ B2 ) )
            = ( cgenid1447163913751362543t_unit @ R2 @ ( somegc4977764851446611037t_unit @ R2 @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.bezout_identity
thf(fact_460_ring__hom__ring_Otrivial__hom__iff,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H2 )
     => ( ( ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
        = ( ( a_kernel_a_b_a_b @ R2 @ S @ H2 )
          = ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_hom_iff
thf(fact_461_ring__hom__ring_Otrivial__hom__iff,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
          = ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
        = ( ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 )
          = ( partia5361259788508890537t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_hom_iff
thf(fact_462_ring__hom__ring_Otrivial__hom__iff,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
      ( ( ring_h5360235766848426350it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
          = ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
        = ( ( a_kern189768852501166it_a_b @ R2 @ S @ H2 )
          = ( partia966996272515721803t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_hom_iff
thf(fact_463_ring__hom__ring_Otrivial__hom__iff,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,H2: set_a > a] :
      ( ( ring_h8937780225337603308it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_set_a_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
          = ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
        = ( ( a_kern3717088086154617772it_a_b @ R2 @ S @ H2 )
          = ( partia5907974310037520643t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_hom_iff
thf(fact_464_ring__hom__ring_Otrivial__hom__iff,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ S ) @ bot_bot_set_list_a ) )
        = ( ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 )
          = ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_hom_iff
thf(fact_465_ring__hom__ring_Otrivial__hom__iff,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,H2: a > set_int] :
      ( ( ring_h7266096129245845164t_unit @ R2 @ S @ H2 )
     => ( ( ( image_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
        = ( ( a_kern1906050131249919980t_unit @ R2 @ S @ H2 )
          = ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_hom_iff
thf(fact_466_ring__hom__ring_Otrivial__hom__iff,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_list_set_int_a @ H2 @ ( partia5243588073521899067t_unit @ R2 ) )
          = ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
        = ( ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 )
          = ( partia5243588073521899067t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_hom_iff
thf(fact_467_ring__hom__ring_Otrivial__hom__iff,axiom,
    ! [R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,H2: set_set_int > a] :
      ( ( ring_h5493655406951491896it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_set_set_int_a @ H2 @ ( partia1981568256555877781t_unit @ R2 ) )
          = ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
        = ( ( a_kern1569243063284222584it_a_b @ R2 @ S @ H2 )
          = ( partia1981568256555877781t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_hom_iff
thf(fact_468_ring__hom__ring_Otrivial__hom__iff,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H2: set_int > set_int] :
      ( ( ring_h98094481560215932t_unit @ R2 @ S @ H2 )
     => ( ( ( image_524474410958335435et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
          = ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
        = ( ( a_kern4456122948206014524t_unit @ R2 @ S @ H2 )
          = ( partia966996272515721803t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_hom_iff
thf(fact_469_ring__hom__ring_Otrivial__hom__iff,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia4934656038542163276t_unit,H2: set_a > set_int] :
      ( ( ring_h1366718972083366014t_unit @ R2 @ S @ H2 )
     => ( ( ( image_set_a_set_int @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
          = ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
        = ( ( a_kern6266201919636670910t_unit @ R2 @ S @ H2 )
          = ( partia5907974310037520643t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_hom_iff
thf(fact_470_ring__hom__ring_Oquot__mem,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H2: list_a > set_list_a,X5: set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H2 )
     => ( ( member_set_list_a @ X5 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: list_a] :
            ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
            & ( X5
              = ( a_r_co1577572119920843660t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H2 ) @ X3 ) ) ) ) ) ).

% ring_hom_ring.quot_mem
thf(fact_471_ring__hom__ring_Oquot__mem,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,X5: set_list_a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( member_set_list_a @ X5 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) ) ) )
       => ? [X3: list_a] :
            ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R2 ) )
            & ( X5
              = ( a_r_co1577572119920843660t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) @ X3 ) ) ) ) ) ).

% ring_hom_ring.quot_mem
thf(fact_472_ring__hom__ring_Oquot__mem,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a,X5: set_list_set_int] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( member4487585027056014399et_int @ X5 @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) ) ) )
       => ? [X3: list_set_int] :
            ( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ R2 ) )
            & ( X5
              = ( a_r_co912825291596486358t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) @ X3 ) ) ) ) ) ).

% ring_hom_ring.quot_mem
thf(fact_473_ring__hom__ring_Oquot__mem,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6652711048488803900t_unit,H2: set_int > list_set_int,X5: set_set_int] :
      ( ( ring_h615793168808539756t_unit @ R2 @ S @ H2 )
     => ( ( member_set_set_int @ X5 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: set_int] :
            ( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ R2 ) )
            & ( X5
              = ( a_r_co692709266861932262t_unit @ R2 @ ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 ) @ X3 ) ) ) ) ) ).

% ring_hom_ring.quot_mem
thf(fact_474_ring__hom__ring_Oquot__mem,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int,X5: set_a] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
            & ( X5
              = ( a_r_coset_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) @ X3 ) ) ) ) ) ).

% ring_hom_ring.quot_mem
thf(fact_475_ring__hom__ring_Oquot__mem,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,X5: set_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) ) ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R2 ) )
            & ( X5
              = ( a_r_coset_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) @ X3 ) ) ) ) ) ).

% ring_hom_ring.quot_mem
thf(fact_476_ring__hom__ring_Othe__elem__inj,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H2: list_a > set_list_a,X5: set_list_a,Y3: set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H2 )
     => ( ( member_set_list_a @ X5 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H2 ) ) ) )
       => ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H2 ) ) ) )
         => ( ( ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ H2 @ X5 ) )
              = ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ H2 @ Y3 ) ) )
           => ( X5 = Y3 ) ) ) ) ) ).

% ring_hom_ring.the_elem_inj
thf(fact_477_ring__hom__ring_Othe__elem__inj,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a,X5: set_list_a,Y3: set_list_a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( member_set_list_a @ X5 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) ) ) )
       => ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) ) ) )
         => ( ( ( the_elem_a @ ( image_list_a_a @ H2 @ X5 ) )
              = ( the_elem_a @ ( image_list_a_a @ H2 @ Y3 ) ) )
           => ( X5 = Y3 ) ) ) ) ) ).

% ring_hom_ring.the_elem_inj
thf(fact_478_ring__hom__ring_Othe__elem__inj,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a,X5: set_list_set_int,Y3: set_list_set_int] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( member4487585027056014399et_int @ X5 @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) ) ) )
       => ( ( member4487585027056014399et_int @ Y3 @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) ) ) )
         => ( ( ( the_elem_a @ ( image_list_set_int_a @ H2 @ X5 ) )
              = ( the_elem_a @ ( image_list_set_int_a @ H2 @ Y3 ) ) )
           => ( X5 = Y3 ) ) ) ) ) ).

% ring_hom_ring.the_elem_inj
thf(fact_479_ring__hom__ring_Othe__elem__inj,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6652711048488803900t_unit,H2: set_int > list_set_int,X5: set_set_int,Y3: set_set_int] :
      ( ( ring_h615793168808539756t_unit @ R2 @ S @ H2 )
     => ( ( member_set_set_int @ X5 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 ) ) ) )
       => ( ( member_set_set_int @ Y3 @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 ) ) ) )
         => ( ( ( the_el6263124322315510589et_int @ ( image_8094898911679120219et_int @ H2 @ X5 ) )
              = ( the_el6263124322315510589et_int @ ( image_8094898911679120219et_int @ H2 @ Y3 ) ) )
           => ( X5 = Y3 ) ) ) ) ) ).

% ring_hom_ring.the_elem_inj
thf(fact_480_ring__hom__ring_Othe__elem__inj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int,X5: set_a,Y3: set_a] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) ) ) )
       => ( ( member_set_a @ Y3 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) ) ) )
         => ( ( ( the_el6263124322315510589et_int @ ( image_a_list_set_int @ H2 @ X5 ) )
              = ( the_el6263124322315510589et_int @ ( image_a_list_set_int @ H2 @ Y3 ) ) )
           => ( X5 = Y3 ) ) ) ) ) ).

% ring_hom_ring.the_elem_inj
thf(fact_481_ring__hom__ring_Othe__elem__inj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a,X5: set_a,Y3: set_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( member_set_a @ X5 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) ) ) )
       => ( ( member_set_a @ Y3 @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) ) ) )
         => ( ( ( the_elem_list_a @ ( image_a_list_a @ H2 @ X5 ) )
              = ( the_elem_list_a @ ( image_a_list_a @ H2 @ Y3 ) ) )
           => ( X5 = Y3 ) ) ) ) ) ).

% ring_hom_ring.the_elem_inj
thf(fact_482_principal__domain_Oideal__sum__iff__gcd,axiom,
    ! [R2: partia2670972154091845814t_unit,A: list_a,B2: list_a,D: list_a] :
      ( ( ring_p8098905331641078952t_unit @ R2 )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R2 ) )
         => ( ( member_list_a @ D @ ( partia5361259788508890537t_unit @ R2 ) )
           => ( ( ( cgenid9131348535277946915t_unit @ R2 @ D )
                = ( set_ad92425877771022410t_unit @ R2 @ ( cgenid9131348535277946915t_unit @ R2 @ A ) @ ( cgenid9131348535277946915t_unit @ R2 @ B2 ) ) )
              = ( isgcd_1118609098697428327t_unit @ R2 @ D @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.ideal_sum_iff_gcd
thf(fact_483_principal__domain_Oideal__sum__iff__gcd,axiom,
    ! [R2: partia7144473111789202860t_unit,A: list_list_set_int,B2: list_list_set_int,D: list_list_set_int] :
      ( ( ring_p1105522777314784042t_unit @ R2 )
     => ( ( member7440485371949894425et_int @ A @ ( partia3237421854401958059t_unit @ R2 ) )
       => ( ( member7440485371949894425et_int @ B2 @ ( partia3237421854401958059t_unit @ R2 ) )
         => ( ( member7440485371949894425et_int @ D @ ( partia3237421854401958059t_unit @ R2 ) )
           => ( ( ( cgenid7864102925118505423t_unit @ R2 @ D )
                = ( set_ad945020892712637448t_unit @ R2 @ ( cgenid7864102925118505423t_unit @ R2 @ A ) @ ( cgenid7864102925118505423t_unit @ R2 @ B2 ) ) )
              = ( isgcd_1310364588453500179t_unit @ R2 @ D @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.ideal_sum_iff_gcd
thf(fact_484_principal__domain_Oideal__sum__iff__gcd,axiom,
    ! [R2: partia3352279114339913222t_unit,A: set_list_set_int,B2: set_list_set_int,D: set_list_set_int] :
      ( ( ring_p4083217851854698756t_unit @ R2 )
     => ( ( member4487585027056014399et_int @ A @ ( partia7954853462395303941t_unit @ R2 ) )
       => ( ( member4487585027056014399et_int @ B2 @ ( partia7954853462395303941t_unit @ R2 ) )
         => ( ( member4487585027056014399et_int @ D @ ( partia7954853462395303941t_unit @ R2 ) )
           => ( ( ( cgenid3413961061359118467t_unit @ R2 @ D )
                = ( set_ad4103617194169821410t_unit @ R2 @ ( cgenid3413961061359118467t_unit @ R2 @ A ) @ ( cgenid3413961061359118467t_unit @ R2 @ B2 ) ) )
              = ( isgcd_8007709855726583751t_unit @ R2 @ D @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.ideal_sum_iff_gcd
thf(fact_485_principal__domain_Oideal__sum__iff__gcd,axiom,
    ! [R2: partia3601206958761670294t_unit,A: set_set_int,B2: set_set_int,D: set_set_int] :
      ( ( ring_p4005655891433666964t_unit @ R2 )
     => ( ( member_set_set_int @ A @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( member_set_set_int @ B2 @ ( partia1981568256555877781t_unit @ R2 ) )
         => ( ( member_set_set_int @ D @ ( partia1981568256555877781t_unit @ R2 ) )
           => ( ( ( cgenid8920438795327392163t_unit @ R2 @ D )
                = ( set_ad1658175378941472882t_unit @ R2 @ ( cgenid8920438795327392163t_unit @ R2 @ A ) @ ( cgenid8920438795327392163t_unit @ R2 @ B2 ) ) )
              = ( isgcd_6085643619309520103t_unit @ R2 @ D @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.ideal_sum_iff_gcd
thf(fact_486_principal__domain_Oideal__sum__iff__gcd,axiom,
    ! [R2: partia6043505979758434576t_unit,A: set_a,B2: set_a,D: set_a] :
      ( ( ring_p2862007038493914190t_unit @ R2 )
     => ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R2 ) )
         => ( ( member_set_a @ D @ ( partia5907974310037520643t_unit @ R2 ) )
           => ( ( ( cgenid6682780793756002467t_unit @ R2 @ D )
                = ( set_ad3113708954360070384t_unit @ R2 @ ( cgenid6682780793756002467t_unit @ R2 @ A ) @ ( cgenid6682780793756002467t_unit @ R2 @ B2 ) ) )
              = ( isgcd_8277756548069700071t_unit @ R2 @ D @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.ideal_sum_iff_gcd
thf(fact_487_principal__domain_Oideal__sum__iff__gcd,axiom,
    ! [R2: partia4934656038542163276t_unit,A: set_int,B2: set_int,D: set_int] :
      ( ( ring_p2413783099608688842t_unit @ R2 )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ R2 ) )
         => ( ( member_set_int @ D @ ( partia966996272515721803t_unit @ R2 ) )
           => ( ( ( cgenid8502489213727343375t_unit @ R2 @ D )
                = ( set_ad273131178244904872t_unit @ R2 @ ( cgenid8502489213727343375t_unit @ R2 @ A ) @ ( cgenid8502489213727343375t_unit @ R2 @ B2 ) ) )
              = ( isgcd_4636411027072998995t_unit @ R2 @ D @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.ideal_sum_iff_gcd
thf(fact_488_principal__domain_Oideal__sum__iff__gcd,axiom,
    ! [R2: partia2175431115845679010xt_a_b,A: a,B2: a,D: a] :
      ( ( ring_p8803135361686045600in_a_b @ R2 )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R2 ) )
         => ( ( member_a @ D @ ( partia707051561876973205xt_a_b @ R2 ) )
           => ( ( ( cgenid547466209912283029xt_a_b @ R2 @ D )
                = ( set_add_a_b @ R2 @ ( cgenid547466209912283029xt_a_b @ R2 @ A ) @ ( cgenid547466209912283029xt_a_b @ R2 @ B2 ) ) )
              = ( isgcd_a_ring_ext_a_b @ R2 @ D @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.ideal_sum_iff_gcd
thf(fact_489_principal__domain_Oideal__sum__iff__gcd,axiom,
    ! [R2: partia6652711048488803900t_unit,A: list_set_int,B2: list_set_int,D: list_set_int] :
      ( ( ring_p1500425057661402810t_unit @ R2 )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ R2 ) )
         => ( ( member_list_set_int @ D @ ( partia5243588073521899067t_unit @ R2 ) )
           => ( ( ( cgenid1447163913751362543t_unit @ R2 @ D )
                = ( set_ad2159827153730365080t_unit @ R2 @ ( cgenid1447163913751362543t_unit @ R2 @ A ) @ ( cgenid1447163913751362543t_unit @ R2 @ B2 ) ) )
              = ( isgcd_1196993390438183731t_unit @ R2 @ D @ A @ B2 ) ) ) ) ) ) ).

% principal_domain.ideal_sum_iff_gcd
thf(fact_490_mem__upD,axiom,
    ! [F: nat > list_set_int,R2: partia6652711048488803900t_unit,N: nat] :
      ( ( member5167342361288740600et_int @ F @ ( up_lis4326185709802647968t_unit @ R2 ) )
     => ( member_list_set_int @ ( F @ N ) @ ( partia5243588073521899067t_unit @ R2 ) ) ) ).

% mem_upD
thf(fact_491_mem__upD,axiom,
    ! [F: nat > set_list_set_int,R2: partia3352279114339913222t_unit,N: nat] :
      ( ( member5182700891607430446et_int @ F @ ( up_set8076327455042720746t_unit @ R2 ) )
     => ( member4487585027056014399et_int @ ( F @ N ) @ ( partia7954853462395303941t_unit @ R2 ) ) ) ).

% mem_upD
thf(fact_492_mem__upD,axiom,
    ! [F: nat > set_set_int,R2: partia3601206958761670294t_unit,N: nat] :
      ( ( member5812803276464394782et_int @ F @ ( up_set7246145586214403962t_unit @ R2 ) )
     => ( member_set_set_int @ ( F @ N ) @ ( partia1981568256555877781t_unit @ R2 ) ) ) ).

% mem_upD
thf(fact_493_mem__upD,axiom,
    ! [F: nat > set_int,R2: partia4934656038542163276t_unit,N: nat] :
      ( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R2 ) )
     => ( member_set_int @ ( F @ N ) @ ( partia966996272515721803t_unit @ R2 ) ) ) ).

% mem_upD
thf(fact_494_mem__upD,axiom,
    ! [F: nat > set_a,R2: partia6043505979758434576t_unit,N: nat] :
      ( ( member_nat_set_a @ F @ ( up_set6103515193466504168t_unit @ R2 ) )
     => ( member_set_a @ ( F @ N ) @ ( partia5907974310037520643t_unit @ R2 ) ) ) ).

% mem_upD
thf(fact_495_mem__upD,axiom,
    ! [F: nat > a,R2: partia2175431115845679010xt_a_b,N: nat] :
      ( ( member_nat_a @ F @ ( up_a_b @ R2 ) )
     => ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ).

% mem_upD
thf(fact_496_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H2: list_a > set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H2 )
     => ( ( image_5749939591322298757list_a
          @ ^ [X4: set_list_a] : ( the_elem_set_list_a @ ( image_5464838071766335845list_a @ H2 @ X4 ) )
          @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern570848875181920430t_unit @ R2 @ S @ H2 ) ) ) )
        = ( image_5464838071766335845list_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_497_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( image_set_list_a_a
          @ ^ [X4: set_list_a] : ( the_elem_a @ ( image_list_a_a @ H2 @ X4 ) )
          @ ( partia141011252114345353t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) ) ) )
        = ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_498_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( image_5898968216206821007_int_a
          @ ^ [X4: set_list_set_int] : ( the_elem_a @ ( image_list_set_int_a @ H2 @ X4 ) )
          @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) ) ) )
        = ( image_list_set_int_a @ H2 @ ( partia5243588073521899067t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_499_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6652711048488803900t_unit,H2: set_int > list_set_int] :
      ( ( ring_h615793168808539756t_unit @ R2 @ S @ H2 )
     => ( ( image_5007591269197315857et_int
          @ ^ [X4: set_set_int] : ( the_el6263124322315510589et_int @ ( image_8094898911679120219et_int @ H2 @ X4 ) )
          @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 ) ) ) )
        = ( image_8094898911679120219et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_500_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( image_2108778832603955851et_int
          @ ^ [X4: set_a] : ( the_el6263124322315510589et_int @ ( image_a_list_set_int @ H2 @ X4 ) )
          @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) ) ) )
        = ( image_a_list_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_501_ring__hom__ring_Othe__elem__surj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( image_set_a_list_a
          @ ^ [X4: set_a] : ( the_elem_list_a @ ( image_a_list_a @ H2 @ X4 ) )
          @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) ) ) )
        = ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ).

% ring_hom_ring.the_elem_surj
thf(fact_502_pzf_Oideal__sum__iff__gcd,axiom,
    ! [A: list_set_int,B2: list_set_int,D: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( member_list_set_int @ D @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ D )
              = ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 ) ) )
            = ( isgcd_1196993390438183731t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ D @ A @ B2 ) ) ) ) ) ).

% pzf.ideal_sum_iff_gcd
thf(fact_503_ideal__sum__iff__gcd,axiom,
    ! [A: a,B2: a,D: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ D @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( cgenid547466209912283029xt_a_b @ r @ D )
              = ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B2 ) ) )
            = ( isgcd_a_ring_ext_a_b @ r @ D @ A @ B2 ) ) ) ) ) ).

% ideal_sum_iff_gcd
thf(fact_504_pzf_Obezout__identity,axiom,
    ! [A: list_set_int,B2: list_set_int] :
      ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 ) )
          = ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( somegc4977764851446611037t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B2 ) ) ) ) ) ).

% pzf.bezout_identity
thf(fact_505_h_Otrivial__ker__imp__inj,axiom,
    ( ( ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
      = ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) )
   => ( inj_on3061584747197171161_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% h.trivial_ker_imp_inj
thf(fact_506_h_Oinj__iff__trivial__ker,axiom,
    ( ( inj_on3061584747197171161_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
    = ( ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
      = ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) ) ).

% h.inj_iff_trivial_ker
thf(fact_507_h_OFactRing__iso__set__aux,axiom,
    ( member1264592267077841462_int_a
    @ ^ [X4: set_list_set_int] :
        ( the_elem_a
        @ ( image_list_set_int_a
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
          @ X4 ) )
    @ ( ring_i1463878111215264340it_a_b
      @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) )
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] :
            ( image_list_set_int_a
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
            @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        @ r ) ) ) ).

% h.FactRing_iso_set_aux
thf(fact_508_pzf_Ozeroprimeideal,axiom,
    primei7054412349171268410t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.zeroprimeideal
thf(fact_509_zf_Oring__irreducibleE_I1_J,axiom,
    ! [R5: set_int] :
      ( ( member_set_int @ R5 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ring_r5206707500676108452t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ R5 )
       => ( R5
         != ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.ring_irreducibleE(1)
thf(fact_510_carrier__not__empty,axiom,
    ( ( partia707051561876973205xt_a_b @ r )
   != bot_bot_set_a ) ).

% carrier_not_empty
thf(fact_511_zeroideal,axiom,
    ideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).

% zeroideal
thf(fact_512_genideal__self_H,axiom,
    ! [I: a] :
      ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).

% genideal_self'
thf(fact_513_genideal__zero,axiom,
    ( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
    = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).

% genideal_zero
thf(fact_514_ring__ideal__imp__quot__ideal,axiom,
    ! [I2: set_a,J: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ideal_a_b @ J @ r )
       => ( ideal_4463284918206690523t_unit @ ( image_a_set_a @ ( a_r_coset_a_b @ r @ I2 ) @ J ) @ ( factRing_a_b @ r @ I2 ) ) ) ) ).

% ring_ideal_imp_quot_ideal
thf(fact_515_zf_Ocarrier__not__empty,axiom,
    ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
   != bot_bot_set_set_int ) ).

% zf.carrier_not_empty
thf(fact_516_zeropideal,axiom,
    principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).

% zeropideal
thf(fact_517_quot__carr,axiom,
    ! [I2: set_a,Y: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
       => ( ord_less_eq_set_a @ Y @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% quot_carr
thf(fact_518_Idl__subset__ideal_H,axiom,
    ! [A: a,B2: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
          = ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ) ) ).

% Idl_subset_ideal'
thf(fact_519_bezout__identity,axiom,
    ! [A: a,B2: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B2 ) )
          = ( cgenid547466209912283029xt_a_b @ r @ ( somegc1600592057159103747xt_a_b @ r @ A @ B2 ) ) ) ) ) ).

% bezout_identity
thf(fact_520_all__ideals,axiom,
    ( ( collect_set_a
      @ ^ [I5: set_a] : ( ideal_a_b @ I5 @ r ) )
    = ( insert_set_a @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia707051561876973205xt_a_b @ r ) @ bot_bot_set_set_a ) ) ) ).

% all_ideals
thf(fact_521_cgenideal__eq__genideal,axiom,
    ! [I: a] :
      ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( cgenid547466209912283029xt_a_b @ r @ I )
        = ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).

% cgenideal_eq_genideal
thf(fact_522_quot__quot__hom,axiom,
    ! [I2: set_a,J: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ideal_a_b @ J @ r )
       => ( ( ord_less_eq_set_a @ I2 @ J )
         => ( member_set_a_set_a @ ( set_add_a_b @ r @ J ) @ ( ring_h4754363464466234712t_unit @ ( factRing_a_b @ r @ I2 ) @ ( factRing_a_b @ r @ J ) ) ) ) ) ) ).

% quot_quot_hom
thf(fact_523_zf_Ozeroideal,axiom,
    ideal_7262958097527394045t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.zeroideal
thf(fact_524_zf_Ogenideal__self_H,axiom,
    ! [I: set_int] :
      ( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( member_set_int @ I @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ).

% zf.genideal_self'
thf(fact_525_zf_Ogenideal__zero,axiom,
    ( ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) )
    = ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) ).

% zf.genideal_zero
thf(fact_526_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_527_zf_Oring__ideal__imp__quot__ideal,axiom,
    ! [I2: set_set_int,J: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
       => ( ideal_1992573625310827463t_unit @ ( image_1010086626112315521et_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) @ J ) @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) ) ) ) ).

% zf.ring_ideal_imp_quot_ideal
thf(fact_528_zf_Ozeropideal,axiom,
    princi8860937869964495385t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.zeropideal
thf(fact_529_zf_Oquot__carr,axiom,
    ! [I2: set_set_int,Y: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( member_set_set_int @ Y @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) ) )
       => ( ord_le4403425263959731960et_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.quot_carr
thf(fact_530_zf_OIdl__subset__ideal_H,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ B2 @ bot_bot_set_set_int ) ) )
          = ( member_set_int @ A @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ B2 @ bot_bot_set_set_int ) ) ) ) ) ) ).

% zf.Idl_subset_ideal'
thf(fact_531_zf_Oall__ideals,axiom,
    ( ( collect_set_set_int
      @ ^ [I5: set_set_int] : ( ideal_7262958097527394045t_unit @ I5 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
    = ( insert_set_set_int @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( insert_set_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bo2384636101374064866et_int ) ) ) ).

% zf.all_ideals
thf(fact_532_zf_Ocgenideal__eq__genideal,axiom,
    ! [I: set_int] :
      ( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I )
        = ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ).

% zf.cgenideal_eq_genideal
thf(fact_533_zf_Oprimeness__condition,axiom,
    ! [P2: set_int] :
      ( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ring_r5206707500676108452t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 )
        = ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) ) ) ).

% zf.primeness_condition
thf(fact_534_zf_Oquot__quot__hom,axiom,
    ! [I2: set_set_int,J: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
       => ( ( ord_le4403425263959731960et_int @ I2 @ J )
         => ( member8939550387393699046et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ J ) @ ( ring_h3060642073748304058t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ J ) ) ) ) ) ) ).

% zf.quot_quot_hom
thf(fact_535_zf_Oset__add__zero,axiom,
    ! [A2: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ A2 )
        = A2 ) ) ).

% zf.set_add_zero
thf(fact_536__092_060open_062_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_096_Ay_A_061_A_123_092_060zero_062_125_A_060_L_062_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_096_Ay_092_060close_062,axiom,
    ( ( image_list_set_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ y )
    = ( set_add_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a )
      @ ( image_list_set_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ y ) ) ) ).

% \<open>(\<lambda>p. local.eval (map (char_iso R) p) x) ` y = {\<zero>} <+> (\<lambda>p. local.eval (map (char_iso R) p) x) ` y\<close>
thf(fact_537_c,axiom,
    ( ( image_list_set_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) )
    = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).

% c
thf(fact_538_h_Otrivial__hom__iff,axiom,
    ( ( ( image_list_set_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
      = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
    = ( ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
      = ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% h.trivial_hom_iff
thf(fact_539__092_060open_062_123_092_060zero_062_125_A_092_060subseteq_062_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_096_Aa__kernel_A_Ipoly__ring_A_IZFact_A_Iint_Ap_J_J_J_AR_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_092_060close_062,axiom,
    ( ord_less_eq_set_a @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a )
    @ ( image_list_set_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) ) ).

% \<open>{\<zero>} \<subseteq> (\<lambda>p. local.eval (map (char_iso R) p) x) ` a_kernel (poly_ring (ZFact (int p))) R (\<lambda>p. local.eval (map (char_iso R) p) x)\<close>
thf(fact_540__092_060open_062_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_096_Aa__kernel_A_Ipoly__ring_A_IZFact_A_Iint_Ap_J_J_J_AR_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_092_060subseteq_062_A_123_092_060zero_062_125_092_060close_062,axiom,
    ( ord_less_eq_set_a
    @ ( image_list_set_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) )
    @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).

% \<open>(\<lambda>p. local.eval (map (char_iso R) p) x) ` a_kernel (poly_ring (ZFact (int p))) R (\<lambda>p. local.eval (map (char_iso R) p) x) \<subseteq> {\<zero>}\<close>
thf(fact_541_h_Othe__elem__wf_H,axiom,
    ! [X5: set_list_set_int] :
      ( ( member4487585027056014399et_int @ X5
        @ ( partia7954853462395303941t_unit
          @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
              @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) ) )
     => ? [X3: list_set_int] :
          ( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          & ( ( image_list_set_int_a
              @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
              @ X5 )
            = ( insert_a @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X3 ) @ x ) @ bot_bot_set_a ) ) ) ) ).

% h.the_elem_wf'
thf(fact_542_h_Othe__elem__wf,axiom,
    ! [X5: set_list_set_int] :
      ( ( member4487585027056014399et_int @ X5
        @ ( partia7954853462395303941t_unit
          @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
              @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( image_list_set_int_a
              @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
              @ X5 )
            = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).

% h.the_elem_wf
thf(fact_543__092_060open_062_123_092_060zero_062_125_A_060_L_062_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_096_Ay_A_061_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_096_Aa__kernel_A_Ipoly__ring_A_IZFact_A_Iint_Ap_J_J_J_AR_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_060_L_062_A_I_092_060lambda_062p_O_Alocal_Oeval_A_Imap_A_Ichar__iso_AR_J_Ap_J_Ax_J_A_096_Ay_092_060close_062,axiom,
    ( ( set_add_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a )
      @ ( image_list_set_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ y ) )
    = ( set_add_a_b @ r
      @ ( image_list_set_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) )
      @ ( image_list_set_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ y ) ) ) ).

% \<open>{\<zero>} <+> (\<lambda>p. local.eval (map (char_iso R) p) x) ` y = (\<lambda>p. local.eval (map (char_iso R) p) x) ` a_kernel (poly_ring (ZFact (int p))) R (\<lambda>p. local.eval (map (char_iso R) p) x) <+> (\<lambda>p. local.eval (map (char_iso R) p) x) ` y\<close>
thf(fact_544_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_545_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_546_is__ring__iso__def,axiom,
    ( is_rin39764828050323806it_a_b
    = ( ^ [R3: partia3352279114339913222t_unit,S2: partia2175431115845679010xt_a_b] :
          ( ( ring_i1463878111215264340it_a_b @ R3 @ S2 )
         != bot_bo831862522269835097_int_a ) ) ) ).

% is_ring_iso_def
thf(fact_547_is__ring__iso__def,axiom,
    ( is_rin1729965262454186108t_unit
    = ( ^ [R3: partia3601206958761670294t_unit,S2: partia4934656038542163276t_unit] :
          ( ( ring_i4798011830457088262t_unit @ R3 @ S2 )
         != bot_bo7126174074431328867et_int ) ) ) ).

% is_ring_iso_def
thf(fact_548_is__ring__iso__def,axiom,
    ( is_rin4964436081320486288t_unit
    = ( ^ [R3: partia4934656038542163276t_unit,S2: partia3601206958761670294t_unit] :
          ( ( ring_i8032482649323388442t_unit @ R3 @ S2 )
         != bot_bo3844255703400516451et_int ) ) ) ).

% is_ring_iso_def
thf(fact_549_is__ring__iso__def,axiom,
    ( is_rin6001486760346555702it_a_b
    = ( ^ [R3: partia6043505979758434576t_unit,S2: partia2175431115845679010xt_a_b] :
          ( ( ring_i4751279688611836608it_a_b @ R3 @ S2 )
         != bot_bot_set_set_a_a ) ) ) ).

% is_ring_iso_def
thf(fact_550_is__ring__iso__def,axiom,
    ( is_rin9099215527551818550t_unit
    = ( ^ [R3: partia2175431115845679010xt_a_b,S2: partia6043505979758434576t_unit] :
          ( ( ring_i7849008455817099456t_unit @ R3 @ S2 )
         != bot_bot_set_a_set_a ) ) ) ).

% is_ring_iso_def
thf(fact_551_ring__iso__same__card,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
      ( ( is_ring_iso_a_b_a_b @ R2 @ S )
     => ( ( finite_card_a @ ( partia707051561876973205xt_a_b @ R2 ) )
        = ( finite_card_a @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).

% ring_iso_same_card
thf(fact_552_ring__iso__same__card,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( is_rin8239664008114016932it_a_b @ R2 @ S )
     => ( ( finite_card_set_int @ ( partia966996272515721803t_unit @ R2 ) )
        = ( finite_card_a @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).

% ring_iso_same_card
thf(fact_553_ring__iso__same__card,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( is_rin6001486760346555702it_a_b @ R2 @ S )
     => ( ( finite_card_set_a @ ( partia5907974310037520643t_unit @ R2 ) )
        = ( finite_card_a @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).

% ring_iso_same_card
thf(fact_554_ring__iso__same__card,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit] :
      ( ( is_rin922152333656659938t_unit @ R2 @ S )
     => ( ( finite_card_a @ ( partia707051561876973205xt_a_b @ R2 ) )
        = ( finite_card_set_int @ ( partia966996272515721803t_unit @ S ) ) ) ) ).

% ring_iso_same_card
thf(fact_555_ring__iso__same__card,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit] :
      ( ( is_rin9099215527551818550t_unit @ R2 @ S )
     => ( ( finite_card_a @ ( partia707051561876973205xt_a_b @ R2 ) )
        = ( finite_card_set_a @ ( partia5907974310037520643t_unit @ S ) ) ) ) ).

% ring_iso_same_card
thf(fact_556_ring__iso__same__card,axiom,
    ! [R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( is_rin1503616481324995182it_a_b @ R2 @ S )
     => ( ( finite7882580182802147440et_int @ ( partia1981568256555877781t_unit @ R2 ) )
        = ( finite_card_a @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).

% ring_iso_same_card
thf(fact_557_ring__iso__same__card,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit] :
      ( ( is_rin6292643797401928902t_unit @ R2 @ S )
     => ( ( finite_card_set_int @ ( partia966996272515721803t_unit @ R2 ) )
        = ( finite_card_set_int @ ( partia966996272515721803t_unit @ S ) ) ) ) ).

% ring_iso_same_card
thf(fact_558_ring__iso__same__card,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6043505979758434576t_unit] :
      ( ( is_rin3768964980678483794t_unit @ R2 @ S )
     => ( ( finite_card_set_int @ ( partia966996272515721803t_unit @ R2 ) )
        = ( finite_card_set_a @ ( partia5907974310037520643t_unit @ S ) ) ) ) ).

% ring_iso_same_card
thf(fact_559_ring__iso__same__card,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia4934656038542163276t_unit] :
      ( ( is_rin9146874455855285172t_unit @ R2 @ S )
     => ( ( finite_card_set_a @ ( partia5907974310037520643t_unit @ R2 ) )
        = ( finite_card_set_int @ ( partia966996272515721803t_unit @ S ) ) ) ) ).

% ring_iso_same_card
thf(fact_560_ring__iso__same__card,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia6043505979758434576t_unit] :
      ( ( is_rin3089442944380665828t_unit @ R2 @ S )
     => ( ( finite_card_set_a @ ( partia5907974310037520643t_unit @ R2 ) )
        = ( finite_card_set_a @ ( partia5907974310037520643t_unit @ S ) ) ) ) ).

% ring_iso_same_card
thf(fact_561_ring__hom__ring_Oinj__iff__trivial__ker,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( inj_on3061584747197171161_int_a @ H2 @ ( partia5243588073521899067t_unit @ R2 ) )
        = ( ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 )
          = ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ R2 ) @ bot_bo7344713902834881468et_int ) ) ) ) ).

% ring_hom_ring.inj_iff_trivial_ker
thf(fact_562_ring__hom__ring_Oinj__iff__trivial__ker,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H2: list_a > set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H2 )
     => ( ( inj_on1264545500884751569list_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
        = ( ( a_kern570848875181920430t_unit @ R2 @ S @ H2 )
          = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ bot_bot_set_list_a ) ) ) ) ).

% ring_hom_ring.inj_iff_trivial_ker
thf(fact_563_ring__hom__ring_Oinj__iff__trivial__ker,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( inj_on_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
        = ( ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 )
          = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ bot_bot_set_list_a ) ) ) ) ).

% ring_hom_ring.inj_iff_trivial_ker
thf(fact_564_ring__hom__ring_Oinj__iff__trivial__ker,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6652711048488803900t_unit,H2: set_int > list_set_int] :
      ( ( ring_h615793168808539756t_unit @ R2 @ S @ H2 )
     => ( ( inj_on2182408081467232967et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
        = ( ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 )
          = ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ) ).

% ring_hom_ring.inj_iff_trivial_ker
thf(fact_565_ring__hom__ring_Oinj__iff__trivial__ker,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( inj_on7524766755751376063et_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
        = ( ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 )
          = ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) ) ) ).

% ring_hom_ring.inj_iff_trivial_ker
thf(fact_566_ring__hom__ring_Oinj__iff__trivial__ker,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( inj_on_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
        = ( ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 )
          = ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) ) ) ).

% ring_hom_ring.inj_iff_trivial_ker
thf(fact_567_ring__hom__ring_Otrivial__ker__imp__inj,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 )
          = ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ R2 ) @ bot_bo7344713902834881468et_int ) )
       => ( inj_on3061584747197171161_int_a @ H2 @ ( partia5243588073521899067t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_ker_imp_inj
thf(fact_568_ring__hom__ring_Otrivial__ker__imp__inj,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H2: list_a > set_list_a] :
      ( ( ring_h6719293541701509614t_unit @ R2 @ S @ H2 )
     => ( ( ( a_kern570848875181920430t_unit @ R2 @ S @ H2 )
          = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ bot_bot_set_list_a ) )
       => ( inj_on1264545500884751569list_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_ker_imp_inj
thf(fact_569_ring__hom__ring_Otrivial__ker__imp__inj,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 )
          = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ bot_bot_set_list_a ) )
       => ( inj_on_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_ker_imp_inj
thf(fact_570_ring__hom__ring_Otrivial__ker__imp__inj,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6652711048488803900t_unit,H2: set_int > list_set_int] :
      ( ( ring_h615793168808539756t_unit @ R2 @ S @ H2 )
     => ( ( ( a_kern8808281968458474796t_unit @ R2 @ S @ H2 )
          = ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) )
       => ( inj_on2182408081467232967et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_ker_imp_inj
thf(fact_571_ring__hom__ring_Otrivial__ker__imp__inj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 )
          = ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) )
       => ( inj_on7524766755751376063et_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_ker_imp_inj
thf(fact_572_ring__hom__ring_Otrivial__ker__imp__inj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 )
          = ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) )
       => ( inj_on_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ).

% ring_hom_ring.trivial_ker_imp_inj
thf(fact_573_principal__domain_Oprimeness__condition,axiom,
    ! [R2: partia3352279114339913222t_unit,P2: set_list_set_int] :
      ( ( ring_p4083217851854698756t_unit @ R2 )
     => ( ( member4487585027056014399et_int @ P2 @ ( partia7954853462395303941t_unit @ R2 ) )
       => ( ( ring_r8396396158236165086t_unit @ R2 @ P2 )
          = ( ring_r1442117713053202183t_unit @ R2 @ P2 ) ) ) ) ).

% principal_domain.primeness_condition
thf(fact_574_principal__domain_Oprimeness__condition,axiom,
    ! [R2: partia3601206958761670294t_unit,P2: set_set_int] :
      ( ( ring_p4005655891433666964t_unit @ R2 )
     => ( ( member_set_set_int @ P2 @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( ring_r5146550843102935406t_unit @ R2 @ P2 )
          = ( ring_r8238242118552278295t_unit @ R2 @ P2 ) ) ) ) ).

% principal_domain.primeness_condition
thf(fact_575_principal__domain_Oprimeness__condition,axiom,
    ! [R2: partia6043505979758434576t_unit,P2: set_a] :
      ( ( ring_p2862007038493914190t_unit @ R2 )
     => ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( ring_r7790391342995787508t_unit @ R2 @ P2 )
          = ( ring_r6795642478576035723t_unit @ R2 @ P2 ) ) ) ) ).

% principal_domain.primeness_condition
thf(fact_576_principal__domain_Oprimeness__condition,axiom,
    ! [R2: partia4934656038542163276t_unit,P2: set_int] :
      ( ( ring_p2413783099608688842t_unit @ R2 )
     => ( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( ring_r5206707500676108452t_unit @ R2 @ P2 )
          = ( ring_r2768735206093983821t_unit @ R2 @ P2 ) ) ) ) ).

% principal_domain.primeness_condition
thf(fact_577_principal__domain_Oprimeness__condition,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P2: a] :
      ( ( ring_p8803135361686045600in_a_b @ R2 )
     => ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ring_r999134135267193926le_a_b @ R2 @ P2 )
          = ( ring_ring_prime_a_b @ R2 @ P2 ) ) ) ) ).

% principal_domain.primeness_condition
thf(fact_578_principal__domain_Oprimeness__condition,axiom,
    ! [R2: partia6652711048488803900t_unit,P2: list_set_int] :
      ( ( ring_p1500425057661402810t_unit @ R2 )
     => ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( ring_r7480558297017414036t_unit @ R2 @ P2 )
          = ( ring_r7101923692879464637t_unit @ R2 @ P2 ) ) ) ) ).

% principal_domain.primeness_condition
thf(fact_579_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H2 )
     => ( member_set_a_a
        @ ^ [X4: set_a] : ( the_elem_a @ ( image_a_a @ H2 @ X4 ) )
        @ ( ring_i4751279688611836608it_a_b @ ( factRing_a_b @ R2 @ ( a_kernel_a_b_a_b @ R2 @ S @ H2 ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_580_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( member_set_a_list_a
        @ ^ [X4: set_a] : ( the_elem_list_a @ ( image_a_list_a @ H2 @ X4 ) )
        @ ( ring_i2253880877871059912t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) )
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_581_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( member_set_list_a_a
        @ ^ [X4: set_list_a] : ( the_elem_a @ ( image_list_a_a @ H2 @ X4 ) )
        @ ( ring_i8122894263081988538it_a_b @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_582_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
      ( ( ring_h5360235766848426350it_a_b @ R2 @ S @ H2 )
     => ( member_set_set_int_a
        @ ^ [X4: set_set_int] : ( the_elem_a @ ( image_set_int_a @ H2 @ X4 ) )
        @ ( ring_i5461826592427486820it_a_b @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern189768852501166it_a_b @ R2 @ S @ H2 ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_583_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,H2: set_a > a] :
      ( ( ring_h8937780225337603308it_a_b @ R2 @ S @ H2 )
     => ( member_set_set_a_a
        @ ^ [X4: set_set_a] : ( the_elem_a @ ( image_set_a_a @ H2 @ X4 ) )
        @ ( ring_i7178773322895389024it_a_b @ ( factRi9077435707053964944t_unit @ R2 @ ( a_kern3717088086154617772it_a_b @ R2 @ S @ H2 ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_set_a_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_584_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,H2: a > set_int] :
      ( ( ring_h7266096129245845164t_unit @ R2 @ S @ H2 )
     => ( member_set_a_set_int
        @ ^ [X4: set_a] : ( the_elem_set_int @ ( image_a_set_int @ H2 @ X4 ) )
        @ ( ring_i1340567420592345514t_unit @ ( factRing_a_b @ R2 @ ( a_kern1906050131249919980t_unit @ R2 @ S @ H2 ) )
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : ( image_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_585_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( member1264592267077841462_int_a
        @ ^ [X4: set_list_set_int] : ( the_elem_a @ ( image_list_set_int_a @ H2 @ X4 ) )
        @ ( ring_i1463878111215264340it_a_b @ ( factRi8995599497114439928t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_list_set_int_a @ H2 @ ( partia5243588073521899067t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_586_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,H2: set_set_int > a] :
      ( ( ring_h5493655406951491896it_a_b @ R2 @ S @ H2 )
     => ( member5392328839176676752_int_a
        @ ^ [X4: set_set_set_int] : ( the_elem_a @ ( image_set_set_int_a @ H2 @ X4 ) )
        @ ( ring_i1231480633883639086it_a_b @ ( factRi2699865449404500690t_unit @ R2 @ ( a_kern1569243063284222584it_a_b @ R2 @ S @ H2 ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_set_set_int_a @ H2 @ ( partia1981568256555877781t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_587_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( member1867902588855634794et_int
        @ ^ [X4: set_a] : ( the_el6263124322315510589et_int @ ( image_a_list_set_int @ H2 @ X4 ) )
        @ ( ring_i5263253608767469466t_unit @ ( factRing_a_b @ R2 @ ( a_kern5734833130465285340t_unit @ R2 @ S @ H2 ) )
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : ( image_a_list_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_588_ring__hom__ring_OFactRing__iso__set__aux,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H2: set_int > set_int] :
      ( ( ring_h98094481560215932t_unit @ R2 @ S @ H2 )
     => ( member2198268278526348592et_int
        @ ^ [X4: set_set_int] : ( the_elem_set_int @ ( image_524474410958335435et_int @ H2 @ X4 ) )
        @ ( ring_i4798011830457088262t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern4456122948206014524t_unit @ R2 @ S @ H2 ) )
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : ( image_524474410958335435et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.FactRing_iso_set_aux
thf(fact_589_ring__hom__ring_OFactRing__iso,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H2 )
     => ( ( ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( is_rin6001486760346555702it_a_b @ ( factRing_a_b @ R2 @ ( a_kernel_a_b_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.FactRing_iso
thf(fact_590_ring__hom__ring_OFactRing__iso,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( is_rin5597148638330396976it_a_b @ ( factRi3329376332477095402t_unit @ R2 @ ( a_kern7116238624728830086it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.FactRing_iso
thf(fact_591_ring__hom__ring_OFactRing__iso,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
      ( ( ring_h5360235766848426350it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( is_rin1503616481324995182it_a_b @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern189768852501166it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.FactRing_iso
thf(fact_592_ring__hom__ring_OFactRing__iso,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,H2: set_a > a] :
      ( ( ring_h8937780225337603308it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_set_a_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( is_rin8432218625136655318it_a_b @ ( factRi9077435707053964944t_unit @ R2 @ ( a_kern3717088086154617772it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.FactRing_iso
thf(fact_593_ring__hom__ring_OFactRing__iso,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia5361259788508890537t_unit @ S ) )
       => ( is_rin1441393272211633470t_unit @ ( factRing_a_b @ R2 @ ( a_kern4625283579065039622t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.FactRing_iso
thf(fact_594_ring__hom__ring_OFactRing__iso,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,H2: a > set_int] :
      ( ( ring_h7266096129245845164t_unit @ R2 @ S @ H2 )
     => ( ( ( image_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia966996272515721803t_unit @ S ) )
       => ( is_rin9146874455855285172t_unit @ ( factRing_a_b @ R2 @ ( a_kern1906050131249919980t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.FactRing_iso
thf(fact_595_ring__hom__ring_OFactRing__iso,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,H2: a > set_a] :
      ( ( ring_h2812136955688090348t_unit @ R2 @ S @ H2 )
     => ( ( ( image_a_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          = ( partia5907974310037520643t_unit @ S ) )
       => ( is_rin3089442944380665828t_unit @ ( factRing_a_b @ R2 @ ( a_kern6814816853359880620t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.FactRing_iso
thf(fact_596_ring__hom__ring_OFactRing__iso,axiom,
    ! [R2: partia6652711048488803900t_unit,S: partia2175431115845679010xt_a_b,H2: list_set_int > a] :
      ( ( ring_h6525298532103594462it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_list_set_int_a @ H2 @ ( partia5243588073521899067t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( is_rin39764828050323806it_a_b @ ( factRi8995599497114439928t_unit @ R2 @ ( a_kern885551317661704990it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.FactRing_iso
thf(fact_597_ring__hom__ring_OFactRing__iso,axiom,
    ! [R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,H2: set_set_int > a] :
      ( ( ring_h5493655406951491896it_a_b @ R2 @ S @ H2 )
     => ( ( ( image_set_set_int_a @ H2 @ ( partia1981568256555877781t_unit @ R2 ) )
          = ( partia707051561876973205xt_a_b @ S ) )
       => ( is_rin4724037964917864760it_a_b @ ( factRi2699865449404500690t_unit @ R2 @ ( a_kern1569243063284222584it_a_b @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.FactRing_iso
thf(fact_598_ring__hom__ring_OFactRing__iso,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H2: set_int > set_int] :
      ( ( ring_h98094481560215932t_unit @ R2 @ S @ H2 )
     => ( ( ( image_524474410958335435et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
          = ( partia966996272515721803t_unit @ S ) )
       => ( is_rin1729965262454186108t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( a_kern4456122948206014524t_unit @ R2 @ S @ H2 ) ) @ S ) ) ) ).

% ring_hom_ring.FactRing_iso
thf(fact_599_zf_OFactRing__zeroideal_I2_J,axiom,
    is_rin4964436081320486288t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) ).

% zf.FactRing_zeroideal(2)
thf(fact_600_zf_OFactRing__zeroideal_I1_J,axiom,
    is_rin1729965262454186108t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.FactRing_zeroideal(1)
thf(fact_601_pzf_Omaximalideal__prime,axiom,
    ! [I2: set_list_set_int] :
      ( ( maxima7372504373286735342t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( primei7054412349171268410t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.maximalideal_prime
thf(fact_602_zf_Oirreducible__imp__maximalideal,axiom,
    ! [P2: set_int] :
      ( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ring_r5206707500676108452t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 )
       => ( maxima6262477034536100350t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.irreducible_imp_maximalideal
thf(fact_603_h_Ochar__consistent,axiom,
    ( ( inj_on3061584747197171161_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
   => ( ( ring_c4550445564046854380t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
      = ( ring_char_a_b @ r ) ) ) ).

% h.char_consistent
thf(fact_604_zf_Oquotient__eq__iff__same__a__r__cos,axiom,
    ! [I2: set_set_int,A: set_int,B2: set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( member_set_int @ ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B2 ) @ I2 )
            = ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ A )
              = ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ B2 ) ) ) ) ) ) ).

% zf.quotient_eq_iff_same_a_r_cos
thf(fact_605_ring__irreducibleE_I1_J,axiom,
    ! [R5: a] :
      ( ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_r999134135267193926le_a_b @ r @ R5 )
       => ( R5
         != ( zero_a_b @ r ) ) ) ) ).

% ring_irreducibleE(1)
thf(fact_606_add_Oinj__on__multc,axiom,
    ! [C: a] :
      ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
     => ( inj_on_a_a
        @ ^ [X2: a] : ( add_a_b @ r @ X2 @ C )
        @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% add.inj_on_multc
thf(fact_607_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_608_primeness__condition,axiom,
    ! [P2: a] :
      ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_r999134135267193926le_a_b @ r @ P2 )
        = ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).

% primeness_condition
thf(fact_609_add_Oinj__on__g,axiom,
    ! [H: set_a,A: a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( inj_on_a_a
          @ ^ [Y4: a] : ( add_a_b @ r @ Y4 @ A )
          @ H ) ) ) ).

% add.inj_on_g
thf(fact_610_zf_Oadd_Oinj__on__multc,axiom,
    ! [C: set_int] :
      ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( inj_on6435365835345961783et_int
        @ ^ [X2: set_int] : ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X2 @ C )
        @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.add.inj_on_multc
thf(fact_611_zf_Oadd_Oinj__on__cmult,axiom,
    ! [C: set_int] :
      ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( inj_on6435365835345961783et_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.add.inj_on_cmult
thf(fact_612_zf_Oup__minus__closed,axiom,
    ! [P2: nat > set_int,Q2: nat > set_int] :
      ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_nat_set_int @ Q2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( member_nat_set_int
          @ ^ [I4: nat] : ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
          @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.up_minus_closed
thf(fact_613_zf_Oadd_Oinj__on__g,axiom,
    ! [H: set_set_int,A: set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( inj_on6435365835345961783et_int
          @ ^ [Y4: set_int] : ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y4 @ A )
          @ H ) ) ) ).

% zf.add.inj_on_g
thf(fact_614_pzf_Oring__irreducibleE_I1_J,axiom,
    ! [R5: list_set_int] :
      ( ( member_list_set_int @ R5 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ R5 )
       => ( R5
         != ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.ring_irreducibleE(1)
thf(fact_615_pzf_Oadd_Oinv__inj,axiom,
    inj_on3170377866336917463et_int @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% pzf.add.inv_inj
thf(fact_616_pzf_Oprimeness__condition,axiom,
    ! [P2: list_set_int] :
      ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
        = ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ).

% pzf.primeness_condition
thf(fact_617_pzf_Oadd_Oinj__on__multc,axiom,
    ! [C: list_set_int] :
      ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( inj_on3170377866336917463et_int
        @ ^ [X2: list_set_int] : ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X2 @ C )
        @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.add.inj_on_multc
thf(fact_618_pzf_Oadd_Oinj__on__cmult,axiom,
    ! [C: list_set_int] :
      ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( inj_on3170377866336917463et_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.add.inj_on_cmult
thf(fact_619_zf_Ozeromaximalideal,axiom,
    maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.zeromaximalideal
thf(fact_620_pzf_Oadd_Oinj__on__g,axiom,
    ! [H: set_list_set_int,A: list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( inj_on3170377866336917463et_int
          @ ^ [Y4: list_set_int] : ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y4 @ A )
          @ H ) ) ) ).

% pzf.add.inj_on_g
thf(fact_621_pzf_Oirreducible__imp__maximalideal,axiom,
    ! [P2: list_set_int] :
      ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
       => ( maxima7372504373286735342t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.irreducible_imp_maximalideal
thf(fact_622_zf_Ominus__closed,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( member_set_int @ ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.minus_closed
thf(fact_623_zf_Or__right__minus__eq,axiom,
    ! [A: set_int,B2: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B2 )
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          = ( A = B2 ) ) ) ) ).

% zf.r_right_minus_eq
thf(fact_624_principal__domain_Oirreducible__imp__maximalideal,axiom,
    ! [R2: partia2670972154091845814t_unit,P2: list_a] :
      ( ( ring_p8098905331641078952t_unit @ R2 )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R2 ) )
       => ( ( ring_r932985474545269838t_unit @ R2 @ P2 )
         => ( maxima6585700282301356660t_unit @ ( cgenid9131348535277946915t_unit @ R2 @ P2 ) @ R2 ) ) ) ) ).

% principal_domain.irreducible_imp_maximalideal
thf(fact_625_principal__domain_Oirreducible__imp__maximalideal,axiom,
    ! [R2: partia7144473111789202860t_unit,P2: list_list_set_int] :
      ( ( ring_p1105522777314784042t_unit @ R2 )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ R2 ) )
       => ( ( ring_r5263416876369801476t_unit @ R2 @ P2 )
         => ( maxima4593186045562755166t_unit @ ( cgenid7864102925118505423t_unit @ R2 @ P2 ) @ R2 ) ) ) ) ).

% principal_domain.irreducible_imp_maximalideal
thf(fact_626_principal__domain_Oirreducible__imp__maximalideal,axiom,
    ! [R2: partia3352279114339913222t_unit,P2: set_list_set_int] :
      ( ( ring_p4083217851854698756t_unit @ R2 )
     => ( ( member4487585027056014399et_int @ P2 @ ( partia7954853462395303941t_unit @ R2 ) )
       => ( ( ring_r8396396158236165086t_unit @ R2 @ P2 )
         => ( maxima3691383912927508536t_unit @ ( cgenid3413961061359118467t_unit @ R2 @ P2 ) @ R2 ) ) ) ) ).

% principal_domain.irreducible_imp_maximalideal
thf(fact_627_principal__domain_Oirreducible__imp__maximalideal,axiom,
    ! [R2: partia3601206958761670294t_unit,P2: set_set_int] :
      ( ( ring_p4005655891433666964t_unit @ R2 )
     => ( ( member_set_set_int @ P2 @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( ring_r5146550843102935406t_unit @ R2 @ P2 )
         => ( maxima8339209307728405192t_unit @ ( cgenid8920438795327392163t_unit @ R2 @ P2 ) @ R2 ) ) ) ) ).

% principal_domain.irreducible_imp_maximalideal
thf(fact_628_principal__domain_Oirreducible__imp__maximalideal,axiom,
    ! [R2: partia6043505979758434576t_unit,P2: set_a] :
      ( ( ring_p2862007038493914190t_unit @ R2 )
     => ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( ring_r7790391342995787508t_unit @ R2 @ P2 )
         => ( maxima2253313296322093082t_unit @ ( cgenid6682780793756002467t_unit @ R2 @ P2 ) @ R2 ) ) ) ) ).

% principal_domain.irreducible_imp_maximalideal
thf(fact_629_principal__domain_Oirreducible__imp__maximalideal,axiom,
    ! [R2: partia4934656038542163276t_unit,P2: set_int] :
      ( ( ring_p2413783099608688842t_unit @ R2 )
     => ( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( ring_r5206707500676108452t_unit @ R2 @ P2 )
         => ( maxima6262477034536100350t_unit @ ( cgenid8502489213727343375t_unit @ R2 @ P2 ) @ R2 ) ) ) ) ).

% principal_domain.irreducible_imp_maximalideal
thf(fact_630_principal__domain_Oirreducible__imp__maximalideal,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P2: a] :
      ( ( ring_p8803135361686045600in_a_b @ R2 )
     => ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ring_r999134135267193926le_a_b @ R2 @ P2 )
         => ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R2 @ P2 ) @ R2 ) ) ) ) ).

% principal_domain.irreducible_imp_maximalideal
thf(fact_631_principal__domain_Oirreducible__imp__maximalideal,axiom,
    ! [R2: partia6652711048488803900t_unit,P2: list_set_int] :
      ( ( ring_p1500425057661402810t_unit @ R2 )
     => ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( ring_r7480558297017414036t_unit @ R2 @ P2 )
         => ( maxima7372504373286735342t_unit @ ( cgenid1447163913751362543t_unit @ R2 @ P2 ) @ R2 ) ) ) ) ).

% principal_domain.irreducible_imp_maximalideal
thf(fact_632_zeromaximalideal,axiom,
    maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).

% zeromaximalideal
thf(fact_633_h_Onon__trivial__field__hom__imp__inj,axiom,
    ( ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
   => ( ( ( image_list_set_int_a
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
          @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       != ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
     => ( inj_on3061584747197171161_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% h.non_trivial_field_hom_imp_inj
thf(fact_634_h_Oimg__is__ring,axiom,
    ( ring_a_b
    @ ( partia8674076737563717228xt_a_b
      @ ^ [Uu: set_a] :
          ( image_list_set_int_a
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
          @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
      @ r ) ) ).

% h.img_is_ring
thf(fact_635_h_Oimg__is__domain,axiom,
    ( ( domain_a_b @ r )
   => ( domain_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] :
            ( image_list_set_int_a
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
            @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        @ r ) ) ) ).

% h.img_is_domain
thf(fact_636_pzf_Oprimeideal__iff__prime,axiom,
    ! [P2: list_set_int] :
      ( ( member_list_set_int @ P2 @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
     => ( ( primei7054412349171268410t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
        = ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ).

% pzf.primeideal_iff_prime
thf(fact_637_domain__axioms,axiom,
    domain_a_b @ r ).

% domain_axioms
thf(fact_638_local_Oring__axioms,axiom,
    ring_a_b @ r ).

% local.ring_axioms
thf(fact_639_irreducible__imp__maximalideal,axiom,
    ! [P2: a] :
      ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_r999134135267193926le_a_b @ r @ P2 )
       => ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r ) ) ) ).

% irreducible_imp_maximalideal
thf(fact_640_pzf_Ozeromaximalideal__eq__field,axiom,
    ( ( maxima7372504373286735342t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
    = ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.zeromaximalideal_eq_field
thf(fact_641_pzf_Ozeromaximalideal__fieldI,axiom,
    ( ( maxima7372504373286735342t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
   => ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.zeromaximalideal_fieldI
thf(fact_642_pzf_Otrivialideals__eq__field,axiom,
    ( ( ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     != ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) )
   => ( ( ( collec8150298066296752253et_int
          @ ^ [I5: set_list_set_int] : ( ideal_3307802554209726317t_unit @ I5 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        = ( insert4179525362955487576et_int @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( insert4179525362955487576et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo4503415735588907506et_int ) ) )
      = ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.trivialideals_eq_field
thf(fact_643_pzf_Otrivialideals__fieldI,axiom,
    ( ( ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     != ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) )
   => ( ( ( collec8150298066296752253et_int
          @ ^ [I5: set_list_set_int] : ( ideal_3307802554209726317t_unit @ I5 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        = ( insert4179525362955487576et_int @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( insert4179525362955487576et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo4503415735588907506et_int ) ) )
     => ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.trivialideals_fieldI
thf(fact_644_noetherian__ring_Oaxioms_I1_J,axiom,
    ! [R2: partia2670972154091845814t_unit] :
      ( ( ring_n5188127996776581661t_unit @ R2 )
     => ( ring_l6212528067271185461t_unit @ R2 ) ) ).

% noetherian_ring.axioms(1)
thf(fact_645_noetherian__ring_Oaxioms_I1_J,axiom,
    ! [R2: partia7144473111789202860t_unit] :
      ( ( ring_n8807844084639296757t_unit @ R2 )
     => ( ring_l6429277945453189341t_unit @ R2 ) ) ).

% noetherian_ring.axioms(1)
thf(fact_646_noetherian__ring_Oaxioms_I1_J,axiom,
    ! [R2: partia6652711048488803900t_unit] :
      ( ( ring_n7202549095058397701t_unit @ R2 )
     => ( ring_l5834914389771393517t_unit @ R2 ) ) ).

% noetherian_ring.axioms(1)
thf(fact_647_noetherian__ring_Oaxioms_I1_J,axiom,
    ! [R2: partia4934656038542163276t_unit] :
      ( ( ring_n4918880612521764245t_unit @ R2 )
     => ( ring_s5316885176909347197t_unit @ R2 ) ) ).

% noetherian_ring.axioms(1)
thf(fact_648_noetherian__ring_Oaxioms_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b] :
      ( ( ring_n3639167112692572309ng_a_b @ R2 )
     => ( ring_a_b @ R2 ) ) ).

% noetherian_ring.axioms(1)
thf(fact_649_principal__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia3352279114339913222t_unit] :
      ( ( ring_p4083217851854698756t_unit @ R2 )
     => ( domain6977320662053078051t_unit @ R2 ) ) ).

% principal_domain.axioms(1)
thf(fact_650_principal__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia3601206958761670294t_unit] :
      ( ( ring_p4005655891433666964t_unit @ R2 )
     => ( domain2593739924244766771t_unit @ R2 ) ) ).

% principal_domain.axioms(1)
thf(fact_651_principal__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia6043505979758434576t_unit] :
      ( ( ring_p2862007038493914190t_unit @ R2 )
     => ( domain4236798911309298543t_unit @ R2 ) ) ).

% principal_domain.axioms(1)
thf(fact_652_principal__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia4934656038542163276t_unit] :
      ( ( ring_p2413783099608688842t_unit @ R2 )
     => ( domain6183376680155302761t_unit @ R2 ) ) ).

% principal_domain.axioms(1)
thf(fact_653_principal__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b] :
      ( ( ring_p8803135361686045600in_a_b @ R2 )
     => ( domain_a_b @ R2 ) ) ).

% principal_domain.axioms(1)
thf(fact_654_principal__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia6652711048488803900t_unit] :
      ( ( ring_p1500425057661402810t_unit @ R2 )
     => ( domain1631227219952682457t_unit @ R2 ) ) ).

% principal_domain.axioms(1)
thf(fact_655_noetherian__domain_Oaxioms_I2_J,axiom,
    ! [R2: partia3352279114339913222t_unit] :
      ( ( ring_n132514764384409659t_unit @ R2 )
     => ( domain6977320662053078051t_unit @ R2 ) ) ).

% noetherian_domain.axioms(2)
thf(fact_656_noetherian__domain_Oaxioms_I2_J,axiom,
    ! [R2: partia3601206958761670294t_unit] :
      ( ( ring_n3961374629531770955t_unit @ R2 )
     => ( domain2593739924244766771t_unit @ R2 ) ) ).

% noetherian_domain.axioms(2)
thf(fact_657_noetherian__domain_Oaxioms_I2_J,axiom,
    ! [R2: partia6043505979758434576t_unit] :
      ( ( ring_n3212398840814694743t_unit @ R2 )
     => ( domain4236798911309298543t_unit @ R2 ) ) ).

% noetherian_domain.axioms(2)
thf(fact_658_noetherian__domain_Oaxioms_I2_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b] :
      ( ( ring_n4045954140777738665in_a_b @ R2 )
     => ( domain_a_b @ R2 ) ) ).

% noetherian_domain.axioms(2)
thf(fact_659_noetherian__domain_Oaxioms_I2_J,axiom,
    ! [R2: partia4934656038542163276t_unit] :
      ( ( ring_n4948453042392689537t_unit @ R2 )
     => ( domain6183376680155302761t_unit @ R2 ) ) ).

% noetherian_domain.axioms(2)
thf(fact_660_noetherian__domain_Oaxioms_I2_J,axiom,
    ! [R2: partia6652711048488803900t_unit] :
      ( ( ring_n3580647943193786865t_unit @ R2 )
     => ( domain1631227219952682457t_unit @ R2 ) ) ).

% noetherian_domain.axioms(2)
thf(fact_661_factorial__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia3352279114339913222t_unit] :
      ( ( ring_f2444916658485529677t_unit @ R2 )
     => ( domain6977320662053078051t_unit @ R2 ) ) ).

% factorial_domain.axioms(1)
thf(fact_662_factorial__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia3601206958761670294t_unit] :
      ( ( ring_f8772991872198083933t_unit @ R2 )
     => ( domain2593739924244766771t_unit @ R2 ) ) ).

% factorial_domain.axioms(1)
thf(fact_663_factorial__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia6043505979758434576t_unit] :
      ( ( ring_f6820247627256571077t_unit @ R2 )
     => ( domain4236798911309298543t_unit @ R2 ) ) ).

% factorial_domain.axioms(1)
thf(fact_664_factorial__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b] :
      ( ( ring_f5272581269873410839in_a_b @ R2 )
     => ( domain_a_b @ R2 ) ) ).

% factorial_domain.axioms(1)
thf(fact_665_factorial__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia4934656038542163276t_unit] :
      ( ( ring_f5979754897444624531t_unit @ R2 )
     => ( domain6183376680155302761t_unit @ R2 ) ) ).

% factorial_domain.axioms(1)
thf(fact_666_factorial__domain_Oaxioms_I1_J,axiom,
    ! [R2: partia6652711048488803900t_unit] :
      ( ( ring_f3415241477199771139t_unit @ R2 )
     => ( domain1631227219952682457t_unit @ R2 ) ) ).

% factorial_domain.axioms(1)
thf(fact_667_ring_Oring__hom__restrict,axiom,
    ! [R2: partia4934656038542163276t_unit,F: set_int > list_set_int,S: partia6652711048488803900t_unit,G: set_int > list_set_int] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( member7759640485799516682et_int @ F @ ( ring_h8178984891965966890t_unit @ R2 @ S ) )
       => ( ! [R4: set_int] :
              ( ( member_set_int @ R4 @ ( partia966996272515721803t_unit @ R2 ) )
             => ( ( F @ R4 )
                = ( G @ R4 ) ) )
         => ( member7759640485799516682et_int @ G @ ( ring_h8178984891965966890t_unit @ R2 @ S ) ) ) ) ) ).

% ring.ring_hom_restrict
thf(fact_668_ring_Oring__hom__restrict,axiom,
    ! [R2: partia4934656038542163276t_unit,F: set_int > set_list_set_int,S: partia3352279114339913222t_unit,G: set_int > set_list_set_int] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( member7361235547688761408et_int @ F @ ( ring_h2330097277019473012t_unit @ R2 @ S ) )
       => ( ! [R4: set_int] :
              ( ( member_set_int @ R4 @ ( partia966996272515721803t_unit @ R2 ) )
             => ( ( F @ R4 )
                = ( G @ R4 ) ) )
         => ( member7361235547688761408et_int @ G @ ( ring_h2330097277019473012t_unit @ R2 @ S ) ) ) ) ) ).

% ring.ring_hom_restrict
thf(fact_669_ring_Oring__hom__restrict,axiom,
    ! [R2: partia6043505979758434576t_unit,F: set_a > set_a,S: partia6043505979758434576t_unit,G: set_a > set_a] :
      ( ( ring_s5549108798478166619t_unit @ R2 )
     => ( ( member_set_a_set_a @ F @ ( ring_h4754363464466234712t_unit @ R2 @ S ) )
       => ( ! [R4: set_a] :
              ( ( member_set_a @ R4 @ ( partia5907974310037520643t_unit @ R2 ) )
             => ( ( F @ R4 )
                = ( G @ R4 ) ) )
         => ( member_set_a_set_a @ G @ ( ring_h4754363464466234712t_unit @ R2 @ S ) ) ) ) ) ).

% ring.ring_hom_restrict
thf(fact_670_ring_Oring__hom__restrict,axiom,
    ! [R2: partia2175431115845679010xt_a_b,F: a > list_a,S: partia2670972154091845814t_unit,G: a > list_a] :
      ( ( ring_a_b @ R2 )
     => ( ( member_a_list_a @ F @ ( ring_h405018892823518980t_unit @ R2 @ S ) )
       => ( ! [R4: a] :
              ( ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ R2 ) )
             => ( ( F @ R4 )
                = ( G @ R4 ) ) )
         => ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ R2 @ S ) ) ) ) ) ).

% ring.ring_hom_restrict
thf(fact_671_ring_Oring__hom__restrict,axiom,
    ! [R2: partia2175431115845679010xt_a_b,F: a > set_list_a,S: partia7496981018696276118t_unit,G: a > set_list_a] :
      ( ( ring_a_b @ R2 )
     => ( ( member_a_set_list_a @ F @ ( ring_h6109298854714515236t_unit @ R2 @ S ) )
       => ( ! [R4: a] :
              ( ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ R2 ) )
             => ( ( F @ R4 )
                = ( G @ R4 ) ) )
         => ( member_a_set_list_a @ G @ ( ring_h6109298854714515236t_unit @ R2 @ S ) ) ) ) ) ).

% ring.ring_hom_restrict
thf(fact_672_ring_Oring__iso__restrict,axiom,
    ! [R2: partia3352279114339913222t_unit,F: set_list_set_int > a,S: partia2175431115845679010xt_a_b,G: set_list_set_int > a] :
      ( ( ring_s3147478420139132983t_unit @ R2 )
     => ( ( member1264592267077841462_int_a @ F @ ( ring_i1463878111215264340it_a_b @ R2 @ S ) )
       => ( ! [R4: set_list_set_int] :
              ( ( member4487585027056014399et_int @ R4 @ ( partia7954853462395303941t_unit @ R2 ) )
             => ( ( F @ R4 )
                = ( G @ R4 ) ) )
         => ( member1264592267077841462_int_a @ G @ ( ring_i1463878111215264340it_a_b @ R2 @ S ) ) ) ) ) ).

% ring.ring_iso_restrict
thf(fact_673_domain_Ozero__is__prime_I1_J,axiom,
    ! [R2: partia3352279114339913222t_unit] :
      ( ( domain6977320662053078051t_unit @ R2 )
     => ( prime_713070722189034950t_unit @ R2 @ ( zero_s8971087499367319327t_unit @ R2 ) ) ) ).

% domain.zero_is_prime(1)
thf(fact_674_domain_Ozero__is__prime_I1_J,axiom,
    ! [R2: partia3601206958761670294t_unit] :
      ( ( domain2593739924244766771t_unit @ R2 )
     => ( prime_7344002144422207974t_unit @ R2 @ ( zero_s2946895028292161839t_unit @ R2 ) ) ) ).

% domain.zero_is_prime(1)
thf(fact_675_domain_Ozero__is__prime_I1_J,axiom,
    ! [R2: partia6043505979758434576t_unit] :
      ( ( domain4236798911309298543t_unit @ R2 )
     => ( prime_4522187476880896870t_unit @ R2 @ ( zero_s2174465271003423091t_unit @ R2 ) ) ) ).

% domain.zero_is_prime(1)
thf(fact_676_domain_Ozero__is__prime_I1_J,axiom,
    ! [R2: partia7144473111789202860t_unit] :
      ( ( domain1893962698156629705t_unit @ R2 )
     => ( prime_6779480824860900370t_unit @ R2 @ ( zero_l4263514192383796165t_unit @ R2 ) ) ) ).

% domain.zero_is_prime(1)
thf(fact_677_domain_Ozero__is__prime_I1_J,axiom,
    ! [R2: partia6652711048488803900t_unit] :
      ( ( domain1631227219952682457t_unit @ R2 )
     => ( prime_7199642643890090290t_unit @ R2 @ ( zero_l5318887016005761237t_unit @ R2 ) ) ) ).

% domain.zero_is_prime(1)
thf(fact_678_domain_Ozero__is__prime_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b] :
      ( ( domain_a_b @ R2 )
     => ( prime_a_ring_ext_a_b @ R2 @ ( zero_a_b @ R2 ) ) ) ).

% domain.zero_is_prime(1)
thf(fact_679_domain_Ozero__is__prime_I1_J,axiom,
    ! [R2: partia4934656038542163276t_unit] :
      ( ( domain6183376680155302761t_unit @ R2 )
     => ( prime_2746729220581880658t_unit @ R2 @ ( zero_s6269048424454532197t_unit @ R2 ) ) ) ).

% domain.zero_is_prime(1)
thf(fact_680_ring_Oup__add__closed,axiom,
    ! [R2: partia2670972154091845814t_unit,P2: nat > list_a,Q2: nat > list_a] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ R2 ) )
       => ( ( member_nat_list_a @ Q2 @ ( up_lis8464167429055313730t_unit @ R2 ) )
         => ( member_nat_list_a
            @ ^ [I4: nat] : ( add_li7652885771158616974t_unit @ R2 @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
            @ ( up_lis8464167429055313730t_unit @ R2 ) ) ) ) ) ).

% ring.up_add_closed
thf(fact_681_ring_Oup__add__closed,axiom,
    ! [R2: partia6652711048488803900t_unit,P2: nat > list_set_int,Q2: nat > list_set_int] :
      ( ( ring_l5834914389771393517t_unit @ R2 )
     => ( ( member5167342361288740600et_int @ P2 @ ( up_lis4326185709802647968t_unit @ R2 ) )
       => ( ( member5167342361288740600et_int @ Q2 @ ( up_lis4326185709802647968t_unit @ R2 ) )
         => ( member5167342361288740600et_int
            @ ^ [I4: nat] : ( add_li6184372983183467092t_unit @ R2 @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
            @ ( up_lis4326185709802647968t_unit @ R2 ) ) ) ) ) ).

% ring.up_add_closed
thf(fact_682_ring_Oup__add__closed,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P2: nat > a,Q2: nat > a] :
      ( ( ring_a_b @ R2 )
     => ( ( member_nat_a @ P2 @ ( up_a_b @ R2 ) )
       => ( ( member_nat_a @ Q2 @ ( up_a_b @ R2 ) )
         => ( member_nat_a
            @ ^ [I4: nat] : ( add_a_b @ R2 @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
            @ ( up_a_b @ R2 ) ) ) ) ) ).

% ring.up_add_closed
thf(fact_683_ring_Oup__add__closed,axiom,
    ! [R2: partia4934656038542163276t_unit,P2: nat > set_int,Q2: nat > set_int] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ R2 ) )
       => ( ( member_nat_set_int @ Q2 @ ( up_set1168727741560211120t_unit @ R2 ) )
         => ( member_nat_set_int
            @ ^ [I4: nat] : ( add_se5859248395121729892t_unit @ R2 @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
            @ ( up_set1168727741560211120t_unit @ R2 ) ) ) ) ) ).

% ring.up_add_closed
thf(fact_684_ring_Oup__add__closed,axiom,
    ! [R2: partia7144473111789202860t_unit,P2: nat > list_list_set_int,Q2: nat > list_list_set_int] :
      ( ( ring_l6429277945453189341t_unit @ R2 )
     => ( ( member3828452275266370824et_int @ P2 @ ( up_lis1317569024804092688t_unit @ R2 ) )
       => ( ( member3828452275266370824et_int @ Q2 @ ( up_lis1317569024804092688t_unit @ R2 ) )
         => ( member3828452275266370824et_int
            @ ^ [I4: nat] : ( add_li3203230194967740868t_unit @ R2 @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
            @ ( up_lis1317569024804092688t_unit @ R2 ) ) ) ) ) ).

% ring.up_add_closed
thf(fact_685_ring_Oup__a__inv__closed,axiom,
    ! [R2: partia2670972154091845814t_unit,P2: nat > list_a] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ R2 ) )
       => ( member_nat_list_a
          @ ^ [I4: nat] : ( a_inv_8944721093294617173t_unit @ R2 @ ( P2 @ I4 ) )
          @ ( up_lis8464167429055313730t_unit @ R2 ) ) ) ) ).

% ring.up_a_inv_closed
thf(fact_686_ring_Oup__a__inv__closed,axiom,
    ! [R2: partia7144473111789202860t_unit,P2: nat > list_list_set_int] :
      ( ( ring_l6429277945453189341t_unit @ R2 )
     => ( ( member3828452275266370824et_int @ P2 @ ( up_lis1317569024804092688t_unit @ R2 ) )
       => ( member3828452275266370824et_int
          @ ^ [I4: nat] : ( a_inv_9003328026414236349t_unit @ R2 @ ( P2 @ I4 ) )
          @ ( up_lis1317569024804092688t_unit @ R2 ) ) ) ) ).

% ring.up_a_inv_closed
thf(fact_687_ring_Oup__a__inv__closed,axiom,
    ! [R2: partia6652711048488803900t_unit,P2: nat > list_set_int] :
      ( ( ring_l5834914389771393517t_unit @ R2 )
     => ( ( member5167342361288740600et_int @ P2 @ ( up_lis4326185709802647968t_unit @ R2 ) )
       => ( member5167342361288740600et_int
          @ ^ [I4: nat] : ( a_inv_7608866662416757709t_unit @ R2 @ ( P2 @ I4 ) )
          @ ( up_lis4326185709802647968t_unit @ R2 ) ) ) ) ).

% ring.up_a_inv_closed
thf(fact_688_ring_Oup__a__inv__closed,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P2: nat > a] :
      ( ( ring_a_b @ R2 )
     => ( ( member_nat_a @ P2 @ ( up_a_b @ R2 ) )
       => ( member_nat_a
          @ ^ [I4: nat] : ( a_inv_a_b @ R2 @ ( P2 @ I4 ) )
          @ ( up_a_b @ R2 ) ) ) ) ).

% ring.up_a_inv_closed
thf(fact_689_ring_Oup__a__inv__closed,axiom,
    ! [R2: partia4934656038542163276t_unit,P2: nat > set_int] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ R2 ) )
       => ( member_nat_set_int
          @ ^ [I4: nat] : ( a_inv_5951419416477254493t_unit @ R2 @ ( P2 @ I4 ) )
          @ ( up_set1168727741560211120t_unit @ R2 ) ) ) ) ).

% ring.up_a_inv_closed
thf(fact_690_ring_Oup__minus__closed,axiom,
    ! [R2: partia6652711048488803900t_unit,P2: nat > list_set_int,Q2: nat > list_set_int] :
      ( ( ring_l5834914389771393517t_unit @ R2 )
     => ( ( member5167342361288740600et_int @ P2 @ ( up_lis4326185709802647968t_unit @ R2 ) )
       => ( ( member5167342361288740600et_int @ Q2 @ ( up_lis4326185709802647968t_unit @ R2 ) )
         => ( member5167342361288740600et_int
            @ ^ [I4: nat] : ( a_minu3289003089969826958t_unit @ R2 @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
            @ ( up_lis4326185709802647968t_unit @ R2 ) ) ) ) ) ).

% ring.up_minus_closed
thf(fact_691_ring_Oup__minus__closed,axiom,
    ! [R2: partia4934656038542163276t_unit,P2: nat > set_int,Q2: nat > set_int] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ R2 ) )
       => ( ( member_nat_set_int @ Q2 @ ( up_set1168727741560211120t_unit @ R2 ) )
         => ( member_nat_set_int
            @ ^ [I4: nat] : ( a_minu5974516859897376926t_unit @ R2 @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
            @ ( up_set1168727741560211120t_unit @ R2 ) ) ) ) ) ).

% ring.up_minus_closed
thf(fact_692_ring_Oup__minus__closed,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P2: nat > a,Q2: nat > a] :
      ( ( ring_a_b @ R2 )
     => ( ( member_nat_a @ P2 @ ( up_a_b @ R2 ) )
       => ( ( member_nat_a @ Q2 @ ( up_a_b @ R2 ) )
         => ( member_nat_a
            @ ^ [I4: nat] : ( a_minus_a_b @ R2 @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
            @ ( up_a_b @ R2 ) ) ) ) ) ).

% ring.up_minus_closed
thf(fact_693_ring_Oup__minus__closed,axiom,
    ! [R2: partia2670972154091845814t_unit,P2: nat > list_a,Q2: nat > list_a] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ R2 ) )
       => ( ( member_nat_list_a @ Q2 @ ( up_lis8464167429055313730t_unit @ R2 ) )
         => ( member_nat_list_a
            @ ^ [I4: nat] : ( a_minu3984020753470702548t_unit @ R2 @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
            @ ( up_lis8464167429055313730t_unit @ R2 ) ) ) ) ) ).

% ring.up_minus_closed
thf(fact_694_ring_Oup__minus__closed,axiom,
    ! [R2: partia7144473111789202860t_unit,P2: nat > list_list_set_int,Q2: nat > list_list_set_int] :
      ( ( ring_l6429277945453189341t_unit @ R2 )
     => ( ( member3828452275266370824et_int @ P2 @ ( up_lis1317569024804092688t_unit @ R2 ) )
       => ( ( member3828452275266370824et_int @ Q2 @ ( up_lis1317569024804092688t_unit @ R2 ) )
         => ( member3828452275266370824et_int
            @ ^ [I4: nat] : ( a_minu1053138715659470590t_unit @ R2 @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
            @ ( up_lis1317569024804092688t_unit @ R2 ) ) ) ) ) ).

% ring.up_minus_closed
thf(fact_695_noetherian__domain__def,axiom,
    ( ring_n132514764384409659t_unit
    = ( ^ [R3: partia3352279114339913222t_unit] :
          ( ( ring_n1943713544749852751t_unit @ R3 )
          & ( domain6977320662053078051t_unit @ R3 ) ) ) ) ).

% noetherian_domain_def
thf(fact_696_noetherian__domain__def,axiom,
    ( ring_n3961374629531770955t_unit
    = ( ^ [R3: partia3601206958761670294t_unit] :
          ( ( ring_n6121427063012330079t_unit @ R3 )
          & ( domain2593739924244766771t_unit @ R3 ) ) ) ) ).

% noetherian_domain_def
thf(fact_697_noetherian__domain__def,axiom,
    ( ring_n3212398840814694743t_unit
    = ( ^ [R3: partia6043505979758434576t_unit] :
          ( ( ring_n5014428767265248323t_unit @ R3 )
          & ( domain4236798911309298543t_unit @ R3 ) ) ) ) ).

% noetherian_domain_def
thf(fact_698_noetherian__domain__def,axiom,
    ( ring_n4045954140777738665in_a_b
    = ( ^ [R3: partia2175431115845679010xt_a_b] :
          ( ( ring_n3639167112692572309ng_a_b @ R3 )
          & ( domain_a_b @ R3 ) ) ) ) ).

% noetherian_domain_def
thf(fact_699_noetherian__domain__def,axiom,
    ( ring_n4948453042392689537t_unit
    = ( ^ [R3: partia4934656038542163276t_unit] :
          ( ( ring_n4918880612521764245t_unit @ R3 )
          & ( domain6183376680155302761t_unit @ R3 ) ) ) ) ).

% noetherian_domain_def
thf(fact_700_noetherian__domain__def,axiom,
    ( ring_n3580647943193786865t_unit
    = ( ^ [R3: partia6652711048488803900t_unit] :
          ( ( ring_n7202549095058397701t_unit @ R3 )
          & ( domain1631227219952682457t_unit @ R3 ) ) ) ) ).

% noetherian_domain_def
thf(fact_701_noetherian__domain_Ointro,axiom,
    ! [R2: partia3352279114339913222t_unit] :
      ( ( ring_n1943713544749852751t_unit @ R2 )
     => ( ( domain6977320662053078051t_unit @ R2 )
       => ( ring_n132514764384409659t_unit @ R2 ) ) ) ).

% noetherian_domain.intro
thf(fact_702_noetherian__domain_Ointro,axiom,
    ! [R2: partia3601206958761670294t_unit] :
      ( ( ring_n6121427063012330079t_unit @ R2 )
     => ( ( domain2593739924244766771t_unit @ R2 )
       => ( ring_n3961374629531770955t_unit @ R2 ) ) ) ).

% noetherian_domain.intro
thf(fact_703_noetherian__domain_Ointro,axiom,
    ! [R2: partia6043505979758434576t_unit] :
      ( ( ring_n5014428767265248323t_unit @ R2 )
     => ( ( domain4236798911309298543t_unit @ R2 )
       => ( ring_n3212398840814694743t_unit @ R2 ) ) ) ).

% noetherian_domain.intro
thf(fact_704_noetherian__domain_Ointro,axiom,
    ! [R2: partia2175431115845679010xt_a_b] :
      ( ( ring_n3639167112692572309ng_a_b @ R2 )
     => ( ( domain_a_b @ R2 )
       => ( ring_n4045954140777738665in_a_b @ R2 ) ) ) ).

% noetherian_domain.intro
thf(fact_705_noetherian__domain_Ointro,axiom,
    ! [R2: partia4934656038542163276t_unit] :
      ( ( ring_n4918880612521764245t_unit @ R2 )
     => ( ( domain6183376680155302761t_unit @ R2 )
       => ( ring_n4948453042392689537t_unit @ R2 ) ) ) ).

% noetherian_domain.intro
thf(fact_706_noetherian__domain_Ointro,axiom,
    ! [R2: partia6652711048488803900t_unit] :
      ( ( ring_n7202549095058397701t_unit @ R2 )
     => ( ( domain1631227219952682457t_unit @ R2 )
       => ( ring_n3580647943193786865t_unit @ R2 ) ) ) ).

% noetherian_domain.intro
thf(fact_707_non__trivial__field__hom__is__inj,axiom,
    ! [H2: a > a,R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
      ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R2 @ S ) )
     => ( ( field_a_b @ R2 )
       => ( ( field_a_b @ S )
         => ( inj_on_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).

% non_trivial_field_hom_is_inj
thf(fact_708_non__trivial__field__hom__is__inj,axiom,
    ! [H2: set_int > a,R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( member_set_int_a @ H2 @ ( ring_h6431528728137583664it_a_b @ R2 @ S ) )
     => ( ( field_5943785737635511755t_unit @ R2 )
       => ( ( field_a_b @ S )
         => ( inj_on_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).

% non_trivial_field_hom_is_inj
thf(fact_709_non__trivial__field__hom__is__inj,axiom,
    ! [H2: set_a > a,R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( member_set_a_a @ H2 @ ( ring_h4811522740288071338it_a_b @ R2 @ S ) )
     => ( ( field_6045675692312731021t_unit @ R2 )
       => ( ( field_a_b @ S )
         => ( inj_on_set_a_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) ) ) ) ) ).

% non_trivial_field_hom_is_inj
thf(fact_710_non__trivial__field__hom__is__inj,axiom,
    ! [H2: a > set_int,R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit] :
      ( ( member_a_set_int @ H2 @ ( ring_h8337389090535002478t_unit @ R2 @ S ) )
     => ( ( field_a_b @ R2 )
       => ( ( field_5943785737635511755t_unit @ S )
         => ( inj_on_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).

% non_trivial_field_hom_is_inj
thf(fact_711_non__trivial__field__hom__is__inj,axiom,
    ! [H2: a > set_a,R2: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit] :
      ( ( member_a_set_a @ H2 @ ( ring_h7909251507493334186t_unit @ R2 @ S ) )
     => ( ( field_a_b @ R2 )
       => ( ( field_6045675692312731021t_unit @ S )
         => ( inj_on_a_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).

% non_trivial_field_hom_is_inj
thf(fact_712_non__trivial__field__hom__is__inj,axiom,
    ! [H2: a > list_a,R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit] :
      ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R2 @ S ) )
     => ( ( field_a_b @ R2 )
       => ( ( field_6388047844668329575t_unit @ S )
         => ( inj_on_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).

% non_trivial_field_hom_is_inj
thf(fact_713_non__trivial__field__hom__is__inj,axiom,
    ! [H2: set_set_int > a,R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b] :
      ( ( member_set_set_int_a @ H2 @ ( ring_h5866496596994661370it_a_b @ R2 @ S ) )
     => ( ( field_4969826516285655701t_unit @ R2 )
       => ( ( field_a_b @ S )
         => ( inj_on_set_set_int_a @ H2 @ ( partia1981568256555877781t_unit @ R2 ) ) ) ) ) ).

% non_trivial_field_hom_is_inj
thf(fact_714_non__trivial__field__hom__is__inj,axiom,
    ! [H2: set_int > set_int,R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit] :
      ( ( member5205197933313416826et_int @ H2 @ ( ring_h3404898052528352314t_unit @ R2 @ S ) )
     => ( ( field_5943785737635511755t_unit @ R2 )
       => ( ( field_5943785737635511755t_unit @ S )
         => ( inj_on6435365835345961783et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).

% non_trivial_field_hom_is_inj
thf(fact_715_non__trivial__field__hom__is__inj,axiom,
    ! [H2: set_int > set_a,R2: partia4934656038542163276t_unit,S: partia6043505979758434576t_unit] :
      ( ( member_set_int_set_a @ H2 @ ( ring_h8672909280708668638t_unit @ R2 @ S ) )
     => ( ( field_5943785737635511755t_unit @ R2 )
       => ( ( field_6045675692312731021t_unit @ S )
         => ( inj_on_set_int_set_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).

% non_trivial_field_hom_is_inj
thf(fact_716_non__trivial__field__hom__is__inj,axiom,
    ! [H2: set_a > set_int,R2: partia6043505979758434576t_unit,S: partia4934656038542163276t_unit] :
      ( ( member_set_a_set_int @ H2 @ ( ring_h4827446719030694208t_unit @ R2 @ S ) )
     => ( ( field_6045675692312731021t_unit @ R2 )
       => ( ( field_5943785737635511755t_unit @ S )
         => ( inj_on_set_a_set_int @ H2 @ ( partia5907974310037520643t_unit @ R2 ) ) ) ) ) ).

% non_trivial_field_hom_is_inj
thf(fact_717_domain_Oring__irreducibleE_I1_J,axiom,
    ! [R2: partia7144473111789202860t_unit,R5: list_list_set_int] :
      ( ( domain1893962698156629705t_unit @ R2 )
     => ( ( member7440485371949894425et_int @ R5 @ ( partia3237421854401958059t_unit @ R2 ) )
       => ( ( ring_r5263416876369801476t_unit @ R2 @ R5 )
         => ( R5
           != ( zero_l4263514192383796165t_unit @ R2 ) ) ) ) ) ).

% domain.ring_irreducibleE(1)
thf(fact_718_domain_Oring__irreducibleE_I1_J,axiom,
    ! [R2: partia3352279114339913222t_unit,R5: set_list_set_int] :
      ( ( domain6977320662053078051t_unit @ R2 )
     => ( ( member4487585027056014399et_int @ R5 @ ( partia7954853462395303941t_unit @ R2 ) )
       => ( ( ring_r8396396158236165086t_unit @ R2 @ R5 )
         => ( R5
           != ( zero_s8971087499367319327t_unit @ R2 ) ) ) ) ) ).

% domain.ring_irreducibleE(1)
thf(fact_719_domain_Oring__irreducibleE_I1_J,axiom,
    ! [R2: partia3601206958761670294t_unit,R5: set_set_int] :
      ( ( domain2593739924244766771t_unit @ R2 )
     => ( ( member_set_set_int @ R5 @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( ring_r5146550843102935406t_unit @ R2 @ R5 )
         => ( R5
           != ( zero_s2946895028292161839t_unit @ R2 ) ) ) ) ) ).

% domain.ring_irreducibleE(1)
thf(fact_720_domain_Oring__irreducibleE_I1_J,axiom,
    ! [R2: partia6043505979758434576t_unit,R5: set_a] :
      ( ( domain4236798911309298543t_unit @ R2 )
     => ( ( member_set_a @ R5 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( ring_r7790391342995787508t_unit @ R2 @ R5 )
         => ( R5
           != ( zero_s2174465271003423091t_unit @ R2 ) ) ) ) ) ).

% domain.ring_irreducibleE(1)
thf(fact_721_domain_Oring__irreducibleE_I1_J,axiom,
    ! [R2: partia4934656038542163276t_unit,R5: set_int] :
      ( ( domain6183376680155302761t_unit @ R2 )
     => ( ( member_set_int @ R5 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( ring_r5206707500676108452t_unit @ R2 @ R5 )
         => ( R5
           != ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ) ).

% domain.ring_irreducibleE(1)
thf(fact_722_domain_Oring__irreducibleE_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,R5: a] :
      ( ( domain_a_b @ R2 )
     => ( ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ring_r999134135267193926le_a_b @ R2 @ R5 )
         => ( R5
           != ( zero_a_b @ R2 ) ) ) ) ) ).

% domain.ring_irreducibleE(1)
thf(fact_723_domain_Oring__irreducibleE_I1_J,axiom,
    ! [R2: partia6652711048488803900t_unit,R5: list_set_int] :
      ( ( domain1631227219952682457t_unit @ R2 )
     => ( ( member_list_set_int @ R5 @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( ring_r7480558297017414036t_unit @ R2 @ R5 )
         => ( R5
           != ( zero_l5318887016005761237t_unit @ R2 ) ) ) ) ) ).

% domain.ring_irreducibleE(1)
thf(fact_724_domain_Oring__primeE_I1_J,axiom,
    ! [R2: partia7144473111789202860t_unit,P2: list_list_set_int] :
      ( ( domain1893962698156629705t_unit @ R2 )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ R2 ) )
       => ( ( ring_r7068619596278734765t_unit @ R2 @ P2 )
         => ( P2
           != ( zero_l4263514192383796165t_unit @ R2 ) ) ) ) ) ).

% domain.ring_primeE(1)
thf(fact_725_domain_Oring__primeE_I1_J,axiom,
    ! [R2: partia6652711048488803900t_unit,P2: list_set_int] :
      ( ( domain1631227219952682457t_unit @ R2 )
     => ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( ring_r7101923692879464637t_unit @ R2 @ P2 )
         => ( P2
           != ( zero_l5318887016005761237t_unit @ R2 ) ) ) ) ) ).

% domain.ring_primeE(1)
thf(fact_726_domain_Oring__primeE_I1_J,axiom,
    ! [R2: partia3352279114339913222t_unit,P2: set_list_set_int] :
      ( ( domain6977320662053078051t_unit @ R2 )
     => ( ( member4487585027056014399et_int @ P2 @ ( partia7954853462395303941t_unit @ R2 ) )
       => ( ( ring_r1442117713053202183t_unit @ R2 @ P2 )
         => ( P2
           != ( zero_s8971087499367319327t_unit @ R2 ) ) ) ) ) ).

% domain.ring_primeE(1)
thf(fact_727_domain_Oring__primeE_I1_J,axiom,
    ! [R2: partia3601206958761670294t_unit,P2: set_set_int] :
      ( ( domain2593739924244766771t_unit @ R2 )
     => ( ( member_set_set_int @ P2 @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( ring_r8238242118552278295t_unit @ R2 @ P2 )
         => ( P2
           != ( zero_s2946895028292161839t_unit @ R2 ) ) ) ) ) ).

% domain.ring_primeE(1)
thf(fact_728_domain_Oring__primeE_I1_J,axiom,
    ! [R2: partia4934656038542163276t_unit,P2: set_int] :
      ( ( domain6183376680155302761t_unit @ R2 )
     => ( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( ring_r2768735206093983821t_unit @ R2 @ P2 )
         => ( P2
           != ( zero_s6269048424454532197t_unit @ R2 ) ) ) ) ) ).

% domain.ring_primeE(1)
thf(fact_729_domain_Oring__primeE_I1_J,axiom,
    ! [R2: partia6043505979758434576t_unit,P2: set_a] :
      ( ( domain4236798911309298543t_unit @ R2 )
     => ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( ring_r6795642478576035723t_unit @ R2 @ P2 )
         => ( P2
           != ( zero_s2174465271003423091t_unit @ R2 ) ) ) ) ) ).

% domain.ring_primeE(1)
thf(fact_730_domain_Oring__primeE_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P2: a] :
      ( ( domain_a_b @ R2 )
     => ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ring_ring_prime_a_b @ R2 @ P2 )
         => ( P2
           != ( zero_a_b @ R2 ) ) ) ) ) ).

% domain.ring_primeE(1)
thf(fact_731_ring_Obound__upD,axiom,
    ! [R2: partia2670972154091845814t_unit,F: nat > list_a] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R2 ) )
       => ? [N2: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ N2 @ F ) ) ) ).

% ring.bound_upD
thf(fact_732_ring_Obound__upD,axiom,
    ! [R2: partia7144473111789202860t_unit,F: nat > list_list_set_int] :
      ( ( ring_l6429277945453189341t_unit @ R2 )
     => ( ( member3828452275266370824et_int @ F @ ( up_lis1317569024804092688t_unit @ R2 ) )
       => ? [N2: nat] : ( bound_704489707291117315et_int @ ( zero_l4263514192383796165t_unit @ R2 ) @ N2 @ F ) ) ) ).

% ring.bound_upD
thf(fact_733_ring_Obound__upD,axiom,
    ! [R2: partia6652711048488803900t_unit,F: nat > list_set_int] :
      ( ( ring_l5834914389771393517t_unit @ R2 )
     => ( ( member5167342361288740600et_int @ F @ ( up_lis4326185709802647968t_unit @ R2 ) )
       => ? [N2: nat] : ( bound_list_set_int @ ( zero_l5318887016005761237t_unit @ R2 ) @ N2 @ F ) ) ) ).

% ring.bound_upD
thf(fact_734_ring_Obound__upD,axiom,
    ! [R2: partia2175431115845679010xt_a_b,F: nat > a] :
      ( ( ring_a_b @ R2 )
     => ( ( member_nat_a @ F @ ( up_a_b @ R2 ) )
       => ? [N2: nat] : ( bound_a @ ( zero_a_b @ R2 ) @ N2 @ F ) ) ) ).

% ring.bound_upD
thf(fact_735_ring_Obound__upD,axiom,
    ! [R2: partia4934656038542163276t_unit,F: nat > set_int] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( member_nat_set_int @ F @ ( up_set1168727741560211120t_unit @ R2 ) )
       => ? [N2: nat] : ( bound_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ N2 @ F ) ) ) ).

% ring.bound_upD
thf(fact_736_domain_Oring__primeE_I3_J,axiom,
    ! [R2: partia6652711048488803900t_unit,P2: list_set_int] :
      ( ( domain1631227219952682457t_unit @ R2 )
     => ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ R2 ) )
       => ( ( ring_r7101923692879464637t_unit @ R2 @ P2 )
         => ( prime_7199642643890090290t_unit @ R2 @ P2 ) ) ) ) ).

% domain.ring_primeE(3)
thf(fact_737_domain_Oring__primeE_I3_J,axiom,
    ! [R2: partia3352279114339913222t_unit,P2: set_list_set_int] :
      ( ( domain6977320662053078051t_unit @ R2 )
     => ( ( member4487585027056014399et_int @ P2 @ ( partia7954853462395303941t_unit @ R2 ) )
       => ( ( ring_r1442117713053202183t_unit @ R2 @ P2 )
         => ( prime_713070722189034950t_unit @ R2 @ P2 ) ) ) ) ).

% domain.ring_primeE(3)
thf(fact_738_domain_Oring__primeE_I3_J,axiom,
    ! [R2: partia3601206958761670294t_unit,P2: set_set_int] :
      ( ( domain2593739924244766771t_unit @ R2 )
     => ( ( member_set_set_int @ P2 @ ( partia1981568256555877781t_unit @ R2 ) )
       => ( ( ring_r8238242118552278295t_unit @ R2 @ P2 )
         => ( prime_7344002144422207974t_unit @ R2 @ P2 ) ) ) ) ).

% domain.ring_primeE(3)
thf(fact_739_domain_Oring__primeE_I3_J,axiom,
    ! [R2: partia4934656038542163276t_unit,P2: set_int] :
      ( ( domain6183376680155302761t_unit @ R2 )
     => ( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ R2 ) )
       => ( ( ring_r2768735206093983821t_unit @ R2 @ P2 )
         => ( prime_2746729220581880658t_unit @ R2 @ P2 ) ) ) ) ).

% domain.ring_primeE(3)
thf(fact_740_domain_Oring__primeE_I3_J,axiom,
    ! [R2: partia6043505979758434576t_unit,P2: set_a] :
      ( ( domain4236798911309298543t_unit @ R2 )
     => ( ( member_set_a @ P2 @ ( partia5907974310037520643t_unit @ R2 ) )
       => ( ( ring_r6795642478576035723t_unit @ R2 @ P2 )
         => ( prime_4522187476880896870t_unit @ R2 @ P2 ) ) ) ) ).

% domain.ring_primeE(3)
thf(fact_741_domain_Oring__primeE_I3_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P2: a] :
      ( ( domain_a_b @ R2 )
     => ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R2 ) )
       => ( ( ring_ring_prime_a_b @ R2 @ P2 )
         => ( prime_a_ring_ext_a_b @ R2 @ P2 ) ) ) ) ).

% domain.ring_primeE(3)
thf(fact_742_ring_Oring__primeI,axiom,
    ! [R2: partia2670972154091845814t_unit,P2: list_a] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( P2
         != ( zero_l4142658623432671053t_unit @ R2 ) )
       => ( ( prime_2011924034616061926t_unit @ R2 @ P2 )
         => ( ring_r6430282645014804837t_unit @ R2 @ P2 ) ) ) ) ).

% ring.ring_primeI
thf(fact_743_ring_Oring__primeI,axiom,
    ! [R2: partia7144473111789202860t_unit,P2: list_list_set_int] :
      ( ( ring_l6429277945453189341t_unit @ R2 )
     => ( ( P2
         != ( zero_l4263514192383796165t_unit @ R2 ) )
       => ( ( prime_6779480824860900370t_unit @ R2 @ P2 )
         => ( ring_r7068619596278734765t_unit @ R2 @ P2 ) ) ) ) ).

% ring.ring_primeI
thf(fact_744_ring_Oring__primeI,axiom,
    ! [R2: partia6652711048488803900t_unit,P2: list_set_int] :
      ( ( ring_l5834914389771393517t_unit @ R2 )
     => ( ( P2
         != ( zero_l5318887016005761237t_unit @ R2 ) )
       => ( ( prime_7199642643890090290t_unit @ R2 @ P2 )
         => ( ring_r7101923692879464637t_unit @ R2 @ P2 ) ) ) ) ).

% ring.ring_primeI
thf(fact_745_ring_Oring__primeI,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P2: a] :
      ( ( ring_a_b @ R2 )
     => ( ( P2
         != ( zero_a_b @ R2 ) )
       => ( ( prime_a_ring_ext_a_b @ R2 @ P2 )
         => ( ring_ring_prime_a_b @ R2 @ P2 ) ) ) ) ).

% ring.ring_primeI
thf(fact_746_ring_Oring__primeI,axiom,
    ! [R2: partia4934656038542163276t_unit,P2: set_int] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( P2
         != ( zero_s6269048424454532197t_unit @ R2 ) )
       => ( ( prime_2746729220581880658t_unit @ R2 @ P2 )
         => ( ring_r2768735206093983821t_unit @ R2 @ P2 ) ) ) ) ).

% ring.ring_primeI
thf(fact_747_domain_Oprimeideal__iff__prime,axiom,
    ! [R2: partia2670972154091845814t_unit,P2: list_a] :
      ( ( domain6553523120543210313t_unit @ R2 )
     => ( ( member_list_a @ P2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R2 ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ bot_bot_set_list_a ) ) )
       => ( ( primei6309817859076077608t_unit @ ( cgenid9131348535277946915t_unit @ R2 @ P2 ) @ R2 )
          = ( ring_r6430282645014804837t_unit @ R2 @ P2 ) ) ) ) ).

% domain.primeideal_iff_prime
thf(fact_748_domain_Oprimeideal__iff__prime,axiom,
    ! [R2: partia7144473111789202860t_unit,P2: list_list_set_int] :
      ( ( domain1893962698156629705t_unit @ R2 )
     => ( ( member7440485371949894425et_int @ P2 @ ( minus_678714269029007121et_int @ ( partia3237421854401958059t_unit @ R2 ) @ ( insert5982949434426573746et_int @ ( zero_l4263514192383796165t_unit @ R2 ) @ bot_bo8087486759604424652et_int ) ) )
       => ( ( primei8613712984254312362t_unit @ ( cgenid7864102925118505423t_unit @ R2 @ P2 ) @ R2 )
          = ( ring_r7068619596278734765t_unit @ R2 @ P2 ) ) ) ) ).

% domain.primeideal_iff_prime
thf(fact_749_domain_Oprimeideal__iff__prime,axiom,
    ! [R2: partia6652711048488803900t_unit,P2: list_set_int] :
      ( ( domain1631227219952682457t_unit @ R2 )
     => ( ( member_list_set_int @ P2 @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ R2 ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ R2 ) @ bot_bo7344713902834881468et_int ) ) )
       => ( ( primei7054412349171268410t_unit @ ( cgenid1447163913751362543t_unit @ R2 @ P2 ) @ R2 )
          = ( ring_r7101923692879464637t_unit @ R2 @ P2 ) ) ) ) ).

% domain.primeideal_iff_prime
thf(fact_750_domain_Oprimeideal__iff__prime,axiom,
    ! [R2: partia3352279114339913222t_unit,P2: set_list_set_int] :
      ( ( domain6977320662053078051t_unit @ R2 )
     => ( ( member4487585027056014399et_int @ P2 @ ( minus_8524384805321567159et_int @ ( partia7954853462395303941t_unit @ R2 ) @ ( insert4179525362955487576et_int @ ( zero_s8971087499367319327t_unit @ R2 ) @ bot_bo4503415735588907506et_int ) ) )
       => ( ( primei4609795502683059588t_unit @ ( cgenid3413961061359118467t_unit @ R2 @ P2 ) @ R2 )
          = ( ring_r1442117713053202183t_unit @ R2 @ P2 ) ) ) ) ).

% domain.primeideal_iff_prime
thf(fact_751_domain_Oprimeideal__iff__prime,axiom,
    ! [R2: partia3601206958761670294t_unit,P2: set_set_int] :
      ( ( domain2593739924244766771t_unit @ R2 )
     => ( ( member_set_set_int @ P2 @ ( minus_6857623457997529383et_int @ ( partia1981568256555877781t_unit @ R2 ) @ ( insert_set_set_int @ ( zero_s2946895028292161839t_unit @ R2 ) @ bot_bo2384636101374064866et_int ) ) )
       => ( ( primei8868987738197308948t_unit @ ( cgenid8920438795327392163t_unit @ R2 @ P2 ) @ R2 )
          = ( ring_r8238242118552278295t_unit @ R2 @ P2 ) ) ) ) ).

% domain.primeideal_iff_prime
thf(fact_752_domain_Oprimeideal__iff__prime,axiom,
    ! [R2: partia4934656038542163276t_unit,P2: set_int] :
      ( ( domain6183376680155302761t_unit @ R2 )
     => ( ( member_set_int @ P2 @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ R2 ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
       => ( ( primei350866878734230858t_unit @ ( cgenid8502489213727343375t_unit @ R2 @ P2 ) @ R2 )
          = ( ring_r2768735206093983821t_unit @ R2 @ P2 ) ) ) ) ).

% domain.primeideal_iff_prime
thf(fact_753_domain_Oprimeideal__iff__prime,axiom,
    ! [R2: partia6043505979758434576t_unit,P2: set_a] :
      ( ( domain4236798911309298543t_unit @ R2 )
     => ( ( member_set_a @ P2 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
       => ( ( primei7645216761534224334t_unit @ ( cgenid6682780793756002467t_unit @ R2 @ P2 ) @ R2 )
          = ( ring_r6795642478576035723t_unit @ R2 @ P2 ) ) ) ) ).

% domain.primeideal_iff_prime
thf(fact_754_domain_Oprimeideal__iff__prime,axiom,
    ! [R2: partia2175431115845679010xt_a_b,P2: a] :
      ( ( domain_a_b @ R2 )
     => ( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
       => ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ R2 @ P2 ) @ R2 )
          = ( ring_ring_prime_a_b @ R2 @ P2 ) ) ) ) ).

% domain.primeideal_iff_prime
thf(fact_755_ring__hom__ring_Oimg__is__domain,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H2 )
     => ( ( domain_a_b @ S )
       => ( domain_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.img_is_domain
thf(fact_756_ring__hom__ring_Oimg__is__domain,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6043505979758434576t_unit,H2: a > set_a] :
      ( ( ring_h2812136955688090348t_unit @ R2 @ S @ H2 )
     => ( ( domain4236798911309298543t_unit @ S )
       => ( domain4236798911309298543t_unit
          @ ( partia7874839658741621210t_unit
            @ ^ [Uu: set_set_a] : ( image_a_set_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.img_is_domain
thf(fact_757_ring__hom__ring_Oimg__is__domain,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( domain6553523120543210313t_unit @ S )
       => ( domain6553523120543210313t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.img_is_domain
thf(fact_758_ring__hom__ring_Oimg__is__domain,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( domain_a_b @ S )
       => ( domain_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.img_is_domain
thf(fact_759_ring__hom__ring_Oimg__is__domain,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
      ( ( ring_h5360235766848426350it_a_b @ R2 @ S @ H2 )
     => ( ( domain_a_b @ S )
       => ( domain_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.img_is_domain
thf(fact_760_ring__hom__ring_Oimg__is__domain,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,H2: set_a > a] :
      ( ( ring_h8937780225337603308it_a_b @ R2 @ S @ H2 )
     => ( ( domain_a_b @ S )
       => ( domain_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_set_a_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.img_is_domain
thf(fact_761_ring__hom__ring_Oimg__is__domain,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,H2: a > set_int] :
      ( ( ring_h7266096129245845164t_unit @ R2 @ S @ H2 )
     => ( ( domain6183376680155302761t_unit @ S )
       => ( domain6183376680155302761t_unit
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : ( image_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.img_is_domain
thf(fact_762_ring__hom__ring_Oimg__is__domain,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia6043505979758434576t_unit,H2: set_int > set_a] :
      ( ( ring_h5212181533761340444t_unit @ R2 @ S @ H2 )
     => ( ( domain4236798911309298543t_unit @ S )
       => ( domain4236798911309298543t_unit
          @ ( partia7874839658741621210t_unit
            @ ^ [Uu: set_set_a] : ( image_set_int_set_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.img_is_domain
thf(fact_763_ring__hom__ring_Oimg__is__domain,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia6043505979758434576t_unit,H2: set_a > set_a] :
      ( ( ring_h8071708631462443418t_unit @ R2 @ S @ H2 )
     => ( ( domain4236798911309298543t_unit @ S )
       => ( domain4236798911309298543t_unit
          @ ( partia7874839658741621210t_unit
            @ ^ [Uu: set_set_a] : ( image_set_a_set_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.img_is_domain
thf(fact_764_ring__hom__ring_Oimg__is__domain,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia3601206958761670294t_unit,H2: a > set_set_int] :
      ( ( ring_h2226868092072877942t_unit @ R2 @ S @ H2 )
     => ( ( domain2593739924244766771t_unit @ S )
       => ( domain2593739924244766771t_unit
          @ ( partia8762098237279093118t_unit
            @ ^ [Uu: set_set_set_int] : ( image_a_set_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring_hom_ring.img_is_domain
thf(fact_765_ring__hom__ring_Oimg__is__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H2 )
     => ( ring_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          @ S ) ) ) ).

% ring_hom_ring.img_is_ring
thf(fact_766_ring__hom__ring_Oimg__is__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ring_l6212528067271185461t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          @ S ) ) ) ).

% ring_hom_ring.img_is_ring
thf(fact_767_ring__hom__ring_Oimg__is__ring,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ring_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
          @ S ) ) ) ).

% ring_hom_ring.img_is_ring
thf(fact_768_ring__hom__ring_Oimg__is__ring,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
      ( ( ring_h5360235766848426350it_a_b @ R2 @ S @ H2 )
     => ( ring_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
          @ S ) ) ) ).

% ring_hom_ring.img_is_ring
thf(fact_769_ring__hom__ring_Oimg__is__ring,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,H2: set_a > a] :
      ( ( ring_h8937780225337603308it_a_b @ R2 @ S @ H2 )
     => ( ring_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : ( image_set_a_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
          @ S ) ) ) ).

% ring_hom_ring.img_is_ring
thf(fact_770_ring__hom__ring_Oimg__is__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,H2: a > set_int] :
      ( ( ring_h7266096129245845164t_unit @ R2 @ S @ H2 )
     => ( ring_s5316885176909347197t_unit
        @ ( partia2914870419229397556t_unit
          @ ^ [Uu: set_set_int] : ( image_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          @ S ) ) ) ).

% ring_hom_ring.img_is_ring
thf(fact_771_ring__hom__ring_Oimg__is__ring,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2670972154091845814t_unit,H2: set_int > list_a] :
      ( ( ring_h3064273570435554678t_unit @ R2 @ S @ H2 )
     => ( ring_l6212528067271185461t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : ( image_set_int_list_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
          @ S ) ) ) ).

% ring_hom_ring.img_is_ring
thf(fact_772_ring__hom__ring_Oimg__is__ring,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia2670972154091845814t_unit,H2: set_a > list_a] :
      ( ( ring_h6190114719687439348t_unit @ R2 @ S @ H2 )
     => ( ring_l6212528067271185461t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : ( image_set_a_list_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
          @ S ) ) ) ).

% ring_hom_ring.img_is_ring
thf(fact_773_ring__hom__ring_Oimg__is__ring,axiom,
    ! [R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,H2: set_set_int > a] :
      ( ( ring_h5493655406951491896it_a_b @ R2 @ S @ H2 )
     => ( ring_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : ( image_set_set_int_a @ H2 @ ( partia1981568256555877781t_unit @ R2 ) )
          @ S ) ) ) ).

% ring_hom_ring.img_is_ring
thf(fact_774_ring__hom__ring_Oimg__is__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ring_l5834914389771393517t_unit
        @ ( partia488849630590579876t_unit
          @ ^ [Uu: set_list_set_int] : ( image_a_list_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
          @ S ) ) ) ).

% ring_hom_ring.img_is_ring
thf(fact_775_ring_Oring__ideal__imp__quot__ideal,axiom,
    ! [R2: partia3352279114339913222t_unit,I2: set_set_list_set_int,J: set_set_list_set_int] :
      ( ( ring_s3147478420139132983t_unit @ R2 )
     => ( ( ideal_7892184242768911799t_unit @ I2 @ R2 )
       => ( ( ideal_7892184242768911799t_unit @ J @ R2 )
         => ( ideal_4170570365519098625t_unit @ ( image_4421660998065691661et_int @ ( a_r_co7111076282206511392t_unit @ R2 @ I2 ) @ J ) @ ( factRi8810600641151685442t_unit @ R2 @ I2 ) ) ) ) ) ).

% ring.ring_ideal_imp_quot_ideal
thf(fact_776_ring_Oring__ideal__imp__quot__ideal,axiom,
    ! [R2: partia3601206958761670294t_unit,I2: set_set_set_int,J: set_set_set_int] :
      ( ( ring_s5321399439445376583t_unit @ R2 )
     => ( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
       => ( ( ideal_1992573625310827463t_unit @ J @ R2 )
         => ( ideal_6096881031695155089t_unit @ ( image_7888150899047434989et_int @ ( a_r_co7378703583274672048t_unit @ R2 @ I2 ) @ J ) @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) ) ) ) ) ).

% ring.ring_ideal_imp_quot_ideal
thf(fact_777_ring_Oring__ideal__imp__quot__ideal,axiom,
    ! [R2: partia6043505979758434576t_unit,I2: set_set_a,J: set_set_a] :
      ( ( ring_s5549108798478166619t_unit @ R2 )
     => ( ( ideal_4463284918206690523t_unit @ I2 @ R2 )
       => ( ( ideal_4463284918206690523t_unit @ J @ R2 )
         => ( ideal_8746426582677721979t_unit @ ( image_4955109552351689957_set_a @ ( a_r_co6984235733410409778t_unit @ R2 @ I2 ) @ J ) @ ( factRi9077435707053964944t_unit @ R2 @ I2 ) ) ) ) ) ).

% ring.ring_ideal_imp_quot_ideal
thf(fact_778_ring_Oring__ideal__imp__quot__ideal,axiom,
    ! [R2: partia6652711048488803900t_unit,I2: set_list_set_int,J: set_list_set_int] :
      ( ( ring_l5834914389771393517t_unit @ R2 )
     => ( ( ideal_3307802554209726317t_unit @ I2 @ R2 )
       => ( ( ideal_3307802554209726317t_unit @ J @ R2 )
         => ( ideal_7892184242768911799t_unit @ ( image_8713293071697434529et_int @ ( a_r_co912825291596486358t_unit @ R2 @ I2 ) @ J ) @ ( factRi8995599497114439928t_unit @ R2 @ I2 ) ) ) ) ) ).

% ring.ring_ideal_imp_quot_ideal
thf(fact_779_ring_Oring__ideal__imp__quot__ideal,axiom,
    ! [R2: partia2175431115845679010xt_a_b,I2: set_a,J: set_a] :
      ( ( ring_a_b @ R2 )
     => ( ( ideal_a_b @ I2 @ R2 )
       => ( ( ideal_a_b @ J @ R2 )
         => ( ideal_4463284918206690523t_unit @ ( image_a_set_a @ ( a_r_coset_a_b @ R2 @ I2 ) @ J ) @ ( factRing_a_b @ R2 @ I2 ) ) ) ) ) ).

% ring.ring_ideal_imp_quot_ideal
thf(fact_780_ring_Oring__ideal__imp__quot__ideal,axiom,
    ! [R2: partia4934656038542163276t_unit,I2: set_set_int,J: set_set_int] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
       => ( ( ideal_7262958097527394045t_unit @ J @ R2 )
         => ( ideal_1992573625310827463t_unit @ ( image_1010086626112315521et_int @ ( a_r_co692709266861932262t_unit @ R2 @ I2 ) @ J ) @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) ) ) ) ) ).

% ring.ring_ideal_imp_quot_ideal
thf(fact_781_ring_Oring__ideal__imp__quot__ideal,axiom,
    ! [R2: partia2670972154091845814t_unit,I2: set_list_a,J: set_list_a] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( ideal_8896367198367571637t_unit @ I2 @ R2 )
       => ( ( ideal_8896367198367571637t_unit @ J @ R2 )
         => ( ideal_5294671857925479125t_unit @ ( image_5464838071766335845list_a @ ( a_r_co1577572119920843660t_unit @ R2 @ I2 ) @ J ) @ ( factRi3329376332477095402t_unit @ R2 @ I2 ) ) ) ) ) ).

% ring.ring_ideal_imp_quot_ideal
thf(fact_782_ring_Oring__ideal__imp__quot__ideal,axiom,
    ! [R2: partia7144473111789202860t_unit,I2: set_li2838760105183549752et_int,J: set_li2838760105183549752et_int] :
      ( ( ring_l6429277945453189341t_unit @ R2 )
     => ( ( ideal_1683033260801535581t_unit @ I2 @ R2 )
       => ( ( ideal_1683033260801535581t_unit @ J @ R2 )
         => ( ideal_3419581485515834919t_unit @ ( image_5431800880458187201et_int @ ( a_r_co2531912773843275078t_unit @ R2 @ I2 ) @ J ) @ ( factRi1079867023605223016t_unit @ R2 @ I2 ) ) ) ) ) ).

% ring.ring_ideal_imp_quot_ideal
thf(fact_783_ring_OFactRing__zeroideal_I1_J,axiom,
    ! [R2: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( is_rin4843644836746533432t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ bot_bot_set_list_a ) ) @ R2 ) ) ).

% ring.FactRing_zeroideal(1)
thf(fact_784_ring_OFactRing__zeroideal_I1_J,axiom,
    ! [R2: partia7144473111789202860t_unit] :
      ( ( ring_l6429277945453189341t_unit @ R2 )
     => ( is_rin3223429467781506172t_unit @ ( factRi1079867023605223016t_unit @ R2 @ ( insert5982949434426573746et_int @ ( zero_l4263514192383796165t_unit @ R2 ) @ bot_bo8087486759604424652et_int ) ) @ R2 ) ) ).

% ring.FactRing_zeroideal(1)
thf(fact_785_ring_OFactRing__zeroideal_I1_J,axiom,
    ! [R2: partia6652711048488803900t_unit] :
      ( ( ring_l5834914389771393517t_unit @ R2 )
     => ( is_rin7318007858330770044t_unit @ ( factRi8995599497114439928t_unit @ R2 @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ R2 ) @ bot_bo7344713902834881468et_int ) ) @ R2 ) ) ).

% ring.FactRing_zeroideal(1)
thf(fact_786_ring_OFactRing__zeroideal_I1_J,axiom,
    ! [R2: partia4934656038542163276t_unit] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( is_rin1729965262454186108t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) @ R2 ) ) ).

% ring.FactRing_zeroideal(1)
thf(fact_787_ring_OFactRing__zeroideal_I1_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R2 )
     => ( is_rin6001486760346555702it_a_b @ ( factRing_a_b @ R2 @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) @ R2 ) ) ).

% ring.FactRing_zeroideal(1)
thf(fact_788_ring_OFactRing__zeroideal_I2_J,axiom,
    ! [R2: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( is_rin2993610189962786360t_unit @ R2 @ ( factRi3329376332477095402t_unit @ R2 @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ bot_bot_set_list_a ) ) ) ) ).

% ring.FactRing_zeroideal(2)
thf(fact_789_ring_OFactRing__zeroideal_I2_J,axiom,
    ! [R2: partia7144473111789202860t_unit] :
      ( ( ring_l6429277945453189341t_unit @ R2 )
     => ( is_rin5671707885937472912t_unit @ R2 @ ( factRi1079867023605223016t_unit @ R2 @ ( insert5982949434426573746et_int @ ( zero_l4263514192383796165t_unit @ R2 ) @ bot_bo8087486759604424652et_int ) ) ) ) ).

% ring.FactRing_zeroideal(2)
thf(fact_790_ring_OFactRing__zeroideal_I2_J,axiom,
    ! [R2: partia6652711048488803900t_unit] :
      ( ( ring_l5834914389771393517t_unit @ R2 )
     => ( is_rin8255480188913837456t_unit @ R2 @ ( factRi8995599497114439928t_unit @ R2 @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ R2 ) @ bot_bo7344713902834881468et_int ) ) ) ) ).

% ring.FactRing_zeroideal(2)
thf(fact_791_ring_OFactRing__zeroideal_I2_J,axiom,
    ! [R2: partia4934656038542163276t_unit] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( is_rin4964436081320486288t_unit @ R2 @ ( factRi3149420076008518152t_unit @ R2 @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) ) ) ).

% ring.FactRing_zeroideal(2)
thf(fact_792_ring_OFactRing__zeroideal_I2_J,axiom,
    ! [R2: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R2 )
     => ( is_rin9099215527551818550t_unit @ R2 @ ( factRing_a_b @ R2 @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) ) ) ).

% ring.FactRing_zeroideal(2)
thf(fact_793_ring_Onoetherian__ringI,axiom,
    ! [R2: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ! [I3: set_list_a] :
            ( ( ideal_8896367198367571637t_unit @ I3 @ R2 )
           => ? [A3: set_list_a] :
                ( ( ord_le8861187494160871172list_a @ A3 @ ( partia5361259788508890537t_unit @ R2 ) )
                & ( finite_finite_list_a @ A3 )
                & ( I3
                  = ( genide3243992037924705879t_unit @ R2 @ A3 ) ) ) )
       => ( ring_n5188127996776581661t_unit @ R2 ) ) ) ).

% ring.noetherian_ringI
thf(fact_794_ring_Onoetherian__ringI,axiom,
    ! [R2: partia7144473111789202860t_unit] :
      ( ( ring_l6429277945453189341t_unit @ R2 )
     => ( ! [I3: set_li2838760105183549752et_int] :
            ( ( ideal_1683033260801535581t_unit @ I3 @ R2 )
           => ? [A3: set_li2838760105183549752et_int] :
                ( ( ord_le6773583430429161880et_int @ A3 @ ( partia3237421854401958059t_unit @ R2 ) )
                & ( finite9197753355890630681et_int @ A3 )
                & ( I3
                  = ( genide35381894626161467t_unit @ R2 @ A3 ) ) ) )
       => ( ring_n8807844084639296757t_unit @ R2 ) ) ) ).

% ring.noetherian_ringI
thf(fact_795_ring_Onoetherian__ringI,axiom,
    ! [R2: partia3352279114339913222t_unit] :
      ( ( ring_s3147478420139132983t_unit @ R2 )
     => ( ! [I3: set_set_list_set_int] :
            ( ( ideal_7892184242768911799t_unit @ I3 @ R2 )
           => ? [A3: set_set_list_set_int] :
                ( ( ord_le8104031629355235774et_int @ A3 @ ( partia7954853462395303941t_unit @ R2 ) )
                & ( finite5161535640157284671et_int @ A3 )
                & ( I3
                  = ( genide8305543099697017237t_unit @ R2 @ A3 ) ) ) )
       => ( ring_n1943713544749852751t_unit @ R2 ) ) ) ).

% ring.noetherian_ringI
thf(fact_796_ring_Onoetherian__ringI,axiom,
    ! [R2: partia3601206958761670294t_unit] :
      ( ( ring_s5321399439445376583t_unit @ R2 )
     => ( ! [I3: set_set_set_int] :
            ( ( ideal_1992573625310827463t_unit @ I3 @ R2 )
           => ? [A3: set_set_set_int] :
                ( ( ord_le4317611570275147438et_int @ A3 @ ( partia1981568256555877781t_unit @ R2 ) )
                & ( finite4249678464180374575et_int @ A3 )
                & ( I3
                  = ( genide1354039622117027493t_unit @ R2 @ A3 ) ) ) )
       => ( ring_n6121427063012330079t_unit @ R2 ) ) ) ).

% ring.noetherian_ringI
thf(fact_797_ring_Onoetherian__ringI,axiom,
    ! [R2: partia6043505979758434576t_unit] :
      ( ( ring_s5549108798478166619t_unit @ R2 )
     => ( ! [I3: set_set_a] :
            ( ( ideal_4463284918206690523t_unit @ I3 @ R2 )
           => ? [A3: set_set_a] :
                ( ( ord_le3724670747650509150_set_a @ A3 @ ( partia5907974310037520643t_unit @ R2 ) )
                & ( finite_finite_set_a @ A3 )
                & ( I3
                  = ( genide4542190045063241085t_unit @ R2 @ A3 ) ) ) )
       => ( ring_n5014428767265248323t_unit @ R2 ) ) ) ).

% ring.noetherian_ringI
thf(fact_798_ring_Onoetherian__ringI,axiom,
    ! [R2: partia6652711048488803900t_unit] :
      ( ( ring_l5834914389771393517t_unit @ R2 )
     => ( ! [I3: set_list_set_int] :
            ( ( ideal_3307802554209726317t_unit @ I3 @ R2 )
           => ? [A3: set_list_set_int] :
                ( ( ord_le4527193859438875528et_int @ A3 @ ( partia5243588073521899067t_unit @ R2 ) )
                & ( finite7825103155802847241et_int @ A3 )
                & ( I3
                  = ( genide4509635305121348427t_unit @ R2 @ A3 ) ) ) )
       => ( ring_n7202549095058397701t_unit @ R2 ) ) ) ).

% ring.noetherian_ringI
thf(fact_799_ring_Onoetherian__ringI,axiom,
    ! [R2: partia4934656038542163276t_unit] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ! [I3: set_set_int] :
            ( ( ideal_7262958097527394045t_unit @ I3 @ R2 )
           => ? [A3: set_set_int] :
                ( ( ord_le4403425263959731960et_int @ A3 @ ( partia966996272515721803t_unit @ R2 ) )
                & ( finite6197958912794628473et_int @ A3 )
                & ( I3
                  = ( genide1545711809618862555t_unit @ R2 @ A3 ) ) ) )
       => ( ring_n4918880612521764245t_unit @ R2 ) ) ) ).

% ring.noetherian_ringI
thf(fact_800_ring_Onoetherian__ringI,axiom,
    ! [R2: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R2 )
     => ( ! [I3: set_a] :
            ( ( ideal_a_b @ I3 @ R2 )
           => ? [A3: set_a] :
                ( ( ord_less_eq_set_a @ A3 @ ( partia707051561876973205xt_a_b @ R2 ) )
                & ( finite_finite_a @ A3 )
                & ( I3
                  = ( genideal_a_b @ R2 @ A3 ) ) ) )
       => ( ring_n3639167112692572309ng_a_b @ R2 ) ) ) ).

% ring.noetherian_ringI
thf(fact_801_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H2: a > a] :
      ( ( ring_h4024360765257340990_b_a_b @ R2 @ S @ H2 )
     => ( ( field_a_b @ R2 )
       => ( ( ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
           != ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
         => ( inj_on_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).

% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_802_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
    ! [R2: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H2: list_a > a] :
      ( ( ring_h7848885096329822662it_a_b @ R2 @ S @ H2 )
     => ( ( field_6388047844668329575t_unit @ R2 )
       => ( ( ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
           != ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
         => ( inj_on_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) ) ) ) ) ).

% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_803_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia2175431115845679010xt_a_b,H2: set_int > a] :
      ( ( ring_h5360235766848426350it_a_b @ R2 @ S @ H2 )
     => ( ( field_5943785737635511755t_unit @ R2 )
       => ( ( ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
           != ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
         => ( inj_on_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).

% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_804_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia2175431115845679010xt_a_b,H2: set_a > a] :
      ( ( ring_h8937780225337603308it_a_b @ R2 @ S @ H2 )
     => ( ( field_6045675692312731021t_unit @ R2 )
       => ( ( ( image_set_a_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
           != ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
         => ( inj_on_set_a_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) ) ) ) ) ).

% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_805_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H2: a > list_a] :
      ( ( ring_h5357930050666032198t_unit @ R2 @ S @ H2 )
     => ( ( field_a_b @ R2 )
       => ( ( ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
           != ( insert_list_a @ ( zero_l4142658623432671053t_unit @ S ) @ bot_bot_set_list_a ) )
         => ( inj_on_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).

% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_806_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia4934656038542163276t_unit,H2: a > set_int] :
      ( ( ring_h7266096129245845164t_unit @ R2 @ S @ H2 )
     => ( ( field_a_b @ R2 )
       => ( ( ( image_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
           != ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
         => ( inj_on_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).

% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_807_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
    ! [R2: partia3601206958761670294t_unit,S: partia2175431115845679010xt_a_b,H2: set_set_int > a] :
      ( ( ring_h5493655406951491896it_a_b @ R2 @ S @ H2 )
     => ( ( field_4969826516285655701t_unit @ R2 )
       => ( ( ( image_set_set_int_a @ H2 @ ( partia1981568256555877781t_unit @ R2 ) )
           != ( insert_a @ ( zero_a_b @ S ) @ bot_bot_set_a ) )
         => ( inj_on_set_set_int_a @ H2 @ ( partia1981568256555877781t_unit @ R2 ) ) ) ) ) ).

% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_808_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
    ! [R2: partia4934656038542163276t_unit,S: partia4934656038542163276t_unit,H2: set_int > set_int] :
      ( ( ring_h98094481560215932t_unit @ R2 @ S @ H2 )
     => ( ( field_5943785737635511755t_unit @ R2 )
       => ( ( ( image_524474410958335435et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
           != ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
         => ( inj_on6435365835345961783et_int @ H2 @ ( partia966996272515721803t_unit @ R2 ) ) ) ) ) ).

% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_809_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
    ! [R2: partia6043505979758434576t_unit,S: partia4934656038542163276t_unit,H2: set_a > set_int] :
      ( ( ring_h1366718972083366014t_unit @ R2 @ S @ H2 )
     => ( ( field_6045675692312731021t_unit @ R2 )
       => ( ( ( image_set_a_set_int @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
           != ( insert_set_int @ ( zero_s6269048424454532197t_unit @ S ) @ bot_bot_set_set_int ) )
         => ( inj_on_set_a_set_int @ H2 @ ( partia5907974310037520643t_unit @ R2 ) ) ) ) ) ).

% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_810_ring__hom__ring_Onon__trivial__field__hom__imp__inj,axiom,
    ! [R2: partia2175431115845679010xt_a_b,S: partia6652711048488803900t_unit,H2: a > list_set_int] :
      ( ( ring_h2151208308052399004t_unit @ R2 @ S @ H2 )
     => ( ( field_a_b @ R2 )
       => ( ( ( image_a_list_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
           != ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ S ) @ bot_bo7344713902834881468et_int ) )
         => ( inj_on7524766755751376063et_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) ) ) ) ) ).

% ring_hom_ring.non_trivial_field_hom_imp_inj
thf(fact_811_inj__on__insert,axiom,
    ! [F: a > a,A: a,A2: set_a] :
      ( ( inj_on_a_a @ F @ ( insert_a @ A @ A2 ) )
      = ( ( inj_on_a_a @ F @ A2 )
        & ~ ( member_a @ ( F @ A ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_812_inj__on__insert,axiom,
    ! [F: set_a > a,A: set_a,A2: set_set_a] :
      ( ( inj_on_set_a_a @ F @ ( insert_set_a @ A @ A2 ) )
      = ( ( inj_on_set_a_a @ F @ A2 )
        & ~ ( member_a @ ( F @ A ) @ ( image_set_a_a @ F @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_813_inj__on__insert,axiom,
    ! [F: a > set_a,A: a,A2: set_a] :
      ( ( inj_on_a_set_a @ F @ ( insert_a @ A @ A2 ) )
      = ( ( inj_on_a_set_a @ F @ A2 )
        & ~ ( member_set_a @ ( F @ A ) @ ( image_a_set_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_814_inj__on__insert,axiom,
    ! [F: set_int > a,A: set_int,A2: set_set_int] :
      ( ( inj_on_set_int_a @ F @ ( insert_set_int @ A @ A2 ) )
      = ( ( inj_on_set_int_a @ F @ A2 )
        & ~ ( member_a @ ( F @ A ) @ ( image_set_int_a @ F @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_815_inj__on__insert,axiom,
    ! [F: set_set_int > a,A: set_set_int,A2: set_set_set_int] :
      ( ( inj_on_set_set_int_a @ F @ ( insert_set_set_int @ A @ A2 ) )
      = ( ( inj_on_set_set_int_a @ F @ A2 )
        & ~ ( member_a @ ( F @ A ) @ ( image_set_set_int_a @ F @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_816_inj__on__insert,axiom,
    ! [F: list_set_int > a,A: list_set_int,A2: set_list_set_int] :
      ( ( inj_on3061584747197171161_int_a @ F @ ( insert_list_set_int @ A @ A2 ) )
      = ( ( inj_on3061584747197171161_int_a @ F @ A2 )
        & ~ ( member_a @ ( F @ A ) @ ( image_list_set_int_a @ F @ ( minus_4948633356380776833et_int @ A2 @ ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_817_inj__on__insert,axiom,
    ! [F: a > nat > a,A: a,A2: set_a] :
      ( ( inj_on_a_nat_a @ F @ ( insert_a @ A @ A2 ) )
      = ( ( inj_on_a_nat_a @ F @ A2 )
        & ~ ( member_nat_a @ ( F @ A ) @ ( image_a_nat_a @ F @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_818_inj__on__insert,axiom,
    ! [F: set_int > set_int,A: set_int,A2: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ ( insert_set_int @ A @ A2 ) )
      = ( ( inj_on6435365835345961783et_int @ F @ A2 )
        & ~ ( member_set_int @ ( F @ A ) @ ( image_524474410958335435et_int @ F @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_819_inj__on__insert,axiom,
    ! [F: set_list_set_int > a,A: set_list_set_int,A2: set_set_list_set_int] :
      ( ( inj_on5590897182042520099_int_a @ F @ ( insert4179525362955487576et_int @ A @ A2 ) )
      = ( ( inj_on5590897182042520099_int_a @ F @ A2 )
        & ~ ( member_a @ ( F @ A ) @ ( image_5898968216206821007_int_a @ F @ ( minus_8524384805321567159et_int @ A2 @ ( insert4179525362955487576et_int @ A @ bot_bo4503415735588907506et_int ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_820_inj__on__insert,axiom,
    ! [F: set_a > nat > a,A: set_a,A2: set_set_a] :
      ( ( inj_on_set_a_nat_a @ F @ ( insert_set_a @ A @ A2 ) )
      = ( ( inj_on_set_a_nat_a @ F @ A2 )
        & ~ ( member_nat_a @ ( F @ A ) @ ( image_set_a_nat_a @ F @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_821_pzf_Oring__hom__imp__img__ring,axiom,
    ! [H2: list_set_int > list_a,S: partia2670972154091845814t_unit] :
      ( ( member7247483931175078386list_a @ H2 @ ( ring_h2345411983690065064t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( ring_l6212528067271185461t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : ( image_1311554247746015051list_a @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
            @ S ) ) ) ) ).

% pzf.ring_hom_imp_img_ring
thf(fact_822_pzf_Oring__hom__imp__img__ring,axiom,
    ! [H2: list_set_int > list_list_set_int,S: partia7144473111789202860t_unit] :
      ( ( member5915730702165157418et_int @ H2 @ ( ring_h7978285497051570986t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( ring_l6429277945453189341t_unit
        @ ( zero_u4006987073238113436t_unit
          @ ^ [Uu: list_list_set_int] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ ( partia2558897084348286356t_unit
            @ ^ [Uu: set_li2838760105183549752et_int] : ( image_2046306602859724155et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
            @ S ) ) ) ) ).

% pzf.ring_hom_imp_img_ring
thf(fact_823_pzf_Oring__hom__imp__img__ring,axiom,
    ! [H2: list_set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( member6759595599802011116_int_a @ H2 @ ( ring_h3322734120376767392it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( ring_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_list_set_int_a @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
            @ S ) ) ) ) ).

% pzf.ring_hom_imp_img_ring
thf(fact_824_pzf_Oring__hom__imp__img__ring,axiom,
    ! [H2: list_set_int > list_set_int,S: partia6652711048488803900t_unit] :
      ( ( member5734324632954958106et_int @ H2 @ ( ring_h5872882829082111674t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( ring_l5834914389771393517t_unit
        @ ( zero_u3668689415746537772t_unit
          @ ^ [Uu: list_set_int] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : ( image_586265162069257835et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
            @ S ) ) ) ) ).

% pzf.ring_hom_imp_img_ring
thf(fact_825_pzf_Oring__hom__imp__img__ring,axiom,
    ! [H2: list_set_int > set_int,S: partia4934656038542163276t_unit] :
      ( ( member4589863265046936970et_int @ H2 @ ( ring_h1343335901776435402t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( ring_s5316885176909347197t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : ( image_6270612139435598043et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
            @ S ) ) ) ) ).

% pzf.ring_hom_imp_img_ring
thf(fact_826_zf_Oring__hom__imp__img__ring,axiom,
    ! [H2: set_int > list_a,S: partia2670972154091845814t_unit] :
      ( ( member6804298160596253954list_a @ H2 @ ( ring_h8006428831458247992t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_l6212528067271185461t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : ( image_set_int_list_a @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ S ) ) ) ) ).

% zf.ring_hom_imp_img_ring
thf(fact_827_zf_Oring__hom__imp__img__ring,axiom,
    ! [H2: set_int > list_list_set_int,S: partia7144473111789202860t_unit] :
      ( ( member1521596788047370010et_int @ H2 @ ( ring_h4623143327427544730t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_l6429277945453189341t_unit
        @ ( zero_u4006987073238113436t_unit
          @ ^ [Uu: list_list_set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( partia2558897084348286356t_unit
            @ ^ [Uu: set_li2838760105183549752et_int] : ( image_4305541057067063915et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ S ) ) ) ) ).

% zf.ring_hom_imp_img_ring
thf(fact_828_zf_Oring__hom__imp__img__ring,axiom,
    ! [H2: set_int > set_list_set_int,S: partia3352279114339913222t_unit] :
      ( ( member7361235547688761408et_int @ H2 @ ( ring_h2330097277019473012t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_s3147478420139132983t_unit
        @ ( zero_u393940103915928438t_unit
          @ ^ [Uu: set_list_set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( partia6871244519201348206t_unit
            @ ^ [Uu: set_set_list_set_int] : ( image_4742829542267907217et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ S ) ) ) ) ).

% zf.ring_hom_imp_img_ring
thf(fact_829_zf_Oring__hom__imp__img__ring,axiom,
    ! [H2: set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_set_int_a @ H2 @ ( ring_h6431528728137583664it_a_b @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ S ) ) ) ) ).

% zf.ring_hom_imp_img_ring
thf(fact_830_zf_Oring__hom__imp__img__ring,axiom,
    ! [H2: set_int > list_set_int,S: partia6652711048488803900t_unit] :
      ( ( member7759640485799516682et_int @ H2 @ ( ring_h8178984891965966890t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_l5834914389771393517t_unit
        @ ( zero_u3668689415746537772t_unit
          @ ^ [Uu: list_set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : ( image_8094898911679120219et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ S ) ) ) ) ).

% zf.ring_hom_imp_img_ring
thf(fact_831_zf_Oring__hom__imp__img__ring,axiom,
    ! [H2: set_int > set_int,S: partia4934656038542163276t_unit] :
      ( ( member5205197933313416826et_int @ H2 @ ( ring_h3404898052528352314t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_s5316885176909347197t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : ( image_524474410958335435et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
            @ S ) ) ) ) ).

% zf.ring_hom_imp_img_ring
thf(fact_832_h_Oinj__on__domain,axiom,
    ( ( inj_on3061584747197171161_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
   => ( ( domain_a_b @ r )
     => ( domain1631227219952682457t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% h.inj_on_domain
thf(fact_833_zf_Oa__rcosetsI,axiom,
    ! [H: set_set_int,X: set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ X ) @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H ) ) ) ) ).

% zf.a_rcosetsI
thf(fact_834_local_Ofield__axioms,axiom,
    field_a_b @ r ).

% local.field_axioms
thf(fact_835_minus__eq,axiom,
    ! [X: a,Y: a] :
      ( ( a_minus_a_b @ r @ X @ Y )
      = ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).

% minus_eq
thf(fact_836_maximalideal__prime,axiom,
    ! [I2: set_a] :
      ( ( maximalideal_a_b @ I2 @ r )
     => ( primeideal_a_b @ I2 @ r ) ) ).

% maximalideal_prime
thf(fact_837_up__a__inv__closed,axiom,
    ! [P2: nat > a] :
      ( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
     => ( member_nat_a
        @ ^ [I4: nat] : ( a_inv_a_b @ r @ ( P2 @ I4 ) )
        @ ( up_a_b @ r ) ) ) ).

% up_a_inv_closed
thf(fact_838_up__minus__closed,axiom,
    ! [P2: nat > a,Q2: nat > a] :
      ( ( member_nat_a @ P2 @ ( up_a_b @ r ) )
     => ( ( member_nat_a @ Q2 @ ( up_a_b @ r ) )
       => ( member_nat_a
          @ ^ [I4: nat] : ( a_minus_a_b @ r @ ( P2 @ I4 ) @ ( Q2 @ I4 ) )
          @ ( up_a_b @ r ) ) ) ) ).

% up_minus_closed
thf(fact_839_zf_Ofield__axioms,axiom,
    field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.field_axioms
thf(fact_840_zf_Odomain__axioms,axiom,
    domain6183376680155302761t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.domain_axioms
thf(fact_841_zf_Oring__axioms,axiom,
    ring_s5316885176909347197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.ring_axioms
thf(fact_842_r__neg2,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
          = Y ) ) ) ).

% r_neg2
thf(fact_843_r__neg1,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
          = Y ) ) ) ).

% r_neg1
thf(fact_844_local_Ominus__add,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
          = ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).

% local.minus_add
thf(fact_845_a__transpose__inv,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( ( add_a_b @ r @ X @ Y )
        = Z )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
              = Y ) ) ) ) ) ).

% a_transpose_inv
thf(fact_846_add_Oinv__solve__right_H,axiom,
    ! [A: a,B2: a,C: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ B2 @ ( a_inv_a_b @ r @ C ) )
              = A )
            = ( B2
              = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).

% add.inv_solve_right'
thf(fact_847_add_Oinv__solve__right,axiom,
    ! [A: a,B2: a,C: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( A
              = ( add_a_b @ r @ B2 @ ( a_inv_a_b @ r @ C ) ) )
            = ( B2
              = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).

% add.inv_solve_right
thf(fact_848_add_Oinv__solve__left_H,axiom,
    ! [A: a,B2: a,C: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B2 ) @ C )
              = A )
            = ( C
              = ( add_a_b @ r @ B2 @ A ) ) ) ) ) ) ).

% add.inv_solve_left'
thf(fact_849_add_Oinv__solve__left,axiom,
    ! [A: a,B2: a,C: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( A
              = ( add_a_b @ r @ ( a_inv_a_b @ r @ B2 ) @ C ) )
            = ( C
              = ( add_a_b @ r @ B2 @ A ) ) ) ) ) ) ).

% add.inv_solve_left
thf(fact_850_add_Oinv__mult__group,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
          = ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).

% add.inv_mult_group
thf(fact_851_a__inv__inj,axiom,
    inj_on_a_a @ ( a_inv_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% a_inv_inj
thf(fact_852_quot__domain__imp__primeideal,axiom,
    ! [P3: set_a] :
      ( ( ideal_a_b @ P3 @ r )
     => ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ P3 ) )
       => ( primeideal_a_b @ P3 @ r ) ) ) ).

% quot_domain_imp_primeideal
thf(fact_853_quot__domain__iff__primeideal,axiom,
    ! [P3: set_a] :
      ( ( ideal_a_b @ P3 @ r )
     => ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ P3 ) )
        = ( primeideal_a_b @ P3 @ r ) ) ) ).

% quot_domain_iff_primeideal
thf(fact_854_l__neg,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
        = ( zero_a_b @ r ) ) ) ).

% l_neg
thf(fact_855_minus__equality,axiom,
    ! [Y: a,X: a] :
      ( ( ( add_a_b @ r @ Y @ X )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( a_inv_a_b @ r @ X )
            = Y ) ) ) ) ).

% minus_equality
thf(fact_856_r__neg,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
        = ( zero_a_b @ r ) ) ) ).

% r_neg
thf(fact_857_sum__zero__eq__neg,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( add_a_b @ r @ X @ Y )
            = ( zero_a_b @ r ) )
         => ( X
            = ( a_inv_a_b @ r @ Y ) ) ) ) ) ).

% sum_zero_eq_neg
thf(fact_858_zeroprimeideal,axiom,
    primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).

% zeroprimeideal
thf(fact_859_comp__apply,axiom,
    ( comp_l364663056984283087et_int
    = ( ^ [F2: list_a > a,G2: list_set_int > list_a,X2: list_set_int] : ( F2 @ ( G2 @ X2 ) ) ) ) ).

% comp_apply
thf(fact_860_comp__apply,axiom,
    ( comp_l1769873867988408929et_int
    = ( ^ [F2: list_set_int > set_list_set_int,G2: set_int > list_set_int,X2: set_int] : ( F2 @ ( G2 @ X2 ) ) ) ) ).

% comp_apply
thf(fact_861_comp__apply,axiom,
    ( comp_l4546862774662178811st_a_a
    = ( ^ [F2: list_a > set_list_a,G2: a > list_a,X2: a] : ( F2 @ ( G2 @ X2 ) ) ) ) ).

% comp_apply
thf(fact_862_comp__apply,axiom,
    ( comp_l2129226576378413585et_int
    = ( ^ [F2: list_list_set_int > set_li2838760105183549752et_int,G2: list_set_int > list_list_set_int,X2: list_set_int] : ( F2 @ ( G2 @ X2 ) ) ) ) ).

% comp_apply
thf(fact_863_quotient__eq__iff__same__a__r__cos,axiom,
    ! [I2: set_a,A: a,B2: a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ ( a_minus_a_b @ r @ A @ B2 ) @ I2 )
            = ( ( a_r_coset_a_b @ r @ I2 @ A )
              = ( a_r_coset_a_b @ r @ I2 @ B2 ) ) ) ) ) ) ).

% quotient_eq_iff_same_a_r_cos
thf(fact_864_zf_Omaximalideal__prime,axiom,
    ! [I2: set_set_int] :
      ( ( maxima6262477034536100350t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( primei350866878734230858t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.maximalideal_prime
thf(fact_865_zf_Oup__a__inv__closed,axiom,
    ! [P2: nat > set_int] :
      ( ( member_nat_set_int @ P2 @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( member_nat_set_int
        @ ^ [I4: nat] : ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( P2 @ I4 ) )
        @ ( up_set1168727741560211120t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.up_a_inv_closed
thf(fact_866_pzf_Oring__axioms,axiom,
    ring_l5834914389771393517t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.ring_axioms
thf(fact_867_zf_Or__neg2,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ Y ) )
          = Y ) ) ) ).

% zf.r_neg2
thf(fact_868_zf_Or__neg1,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) )
          = Y ) ) ) ).

% zf.r_neg1
thf(fact_869_zf_Ominus__add,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) )
          = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) ) ) ) ) ).

% zf.minus_add
thf(fact_870_zf_Oadd_Oinv__solve__right_H,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C ) )
              = A )
            = ( B2
              = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C ) ) ) ) ) ) ).

% zf.add.inv_solve_right'
thf(fact_871_zf_Oadd_Oinv__solve__right,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( A
              = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C ) ) )
            = ( B2
              = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C ) ) ) ) ) ) ).

% zf.add.inv_solve_right
thf(fact_872_zf_Oadd_Oinv__solve__left_H,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 ) @ C )
              = A )
            = ( C
              = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 @ A ) ) ) ) ) ) ).

% zf.add.inv_solve_left'
thf(fact_873_zf_Oadd_Oinv__solve__left,axiom,
    ! [A: set_int,B2: set_int,C: set_int] :
      ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( A
              = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 ) @ C ) )
            = ( C
              = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 @ A ) ) ) ) ) ) ).

% zf.add.inv_solve_left
thf(fact_874_zf_Oadd_Oinv__mult__group,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) )
          = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) ) ) ) ) ).

% zf.add.inv_mult_group
thf(fact_875_zf_Oa__transpose__inv,axiom,
    ! [X: set_int,Y: set_int,Z: set_int] :
      ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
        = Z )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
           => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ Z )
              = Y ) ) ) ) ) ).

% zf.a_transpose_inv
thf(fact_876_zf_Oadd_Oinv__inj,axiom,
    inj_on6435365835345961783et_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ).

% zf.add.inv_inj
thf(fact_877_a__coset__add__inv1,axiom,
    ! [M: set_a,X: a,Y: a] :
      ( ( ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) )
        = M )
     => ( ( member_a @ X @ ( 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 @ X )
              = ( a_r_coset_a_b @ r @ M @ Y ) ) ) ) ) ) ).

% a_coset_add_inv1
thf(fact_878_a__coset__add__inv2,axiom,
    ! [M: set_a,X: a,Y: a] :
      ( ( ( a_r_coset_a_b @ r @ M @ X )
        = ( a_r_coset_a_b @ r @ M @ Y ) )
     => ( ( member_a @ X @ ( 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 @ X @ ( a_inv_a_b @ r @ Y ) ) )
              = M ) ) ) ) ) ).

% a_coset_add_inv2
thf(fact_879_zf_Ominus__eq,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
      = ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) ) ) ).

% zf.minus_eq
thf(fact_880_zf_Oquot__domain__imp__primeideal,axiom,
    ! [P3: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ P3 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( domain2593739924244766771t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P3 ) )
       => ( primei350866878734230858t_unit @ P3 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.quot_domain_imp_primeideal
thf(fact_881_zf_Oquot__domain__iff__primeideal,axiom,
    ! [P3: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ P3 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( domain2593739924244766771t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P3 ) )
        = ( primei350866878734230858t_unit @ P3 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.quot_domain_iff_primeideal
thf(fact_882_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_883_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_884_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_885_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_886_zf_Osum__zero__eq__neg,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
            = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( X
            = ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) ) ) ) ) ).

% zf.sum_zero_eq_neg
thf(fact_887_zf_Or__neg,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) )
        = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.r_neg
thf(fact_888_zf_Ominus__equality,axiom,
    ! [Y: set_int,X: set_int] :
      ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X )
        = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X )
            = Y ) ) ) ) ).

% zf.minus_equality
thf(fact_889_zf_Ol__neg,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ X )
        = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.l_neg
thf(fact_890_zf_Ozeroprimeideal__domainI,axiom,
    ( ( primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
   => ( domain6183376680155302761t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.zeroprimeideal_domainI
thf(fact_891_zf_Odomain__eq__zeroprimeideal,axiom,
    ( ( domain6183376680155302761t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
    = ( primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.domain_eq_zeroprimeideal
thf(fact_892_zf_Ozeroprimeideal,axiom,
    primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.zeroprimeideal
thf(fact_893_add_Oone__in__subset,axiom,
    ! [H: set_a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( H != bot_bot_set_a )
       => ( ! [X3: a] :
              ( ( member_a @ X3 @ H )
             => ( member_a @ ( a_inv_a_b @ r @ X3 ) @ H ) )
         => ( ! [X3: a] :
                ( ( member_a @ X3 @ H )
               => ! [Xa: a] :
                    ( ( member_a @ Xa @ H )
                   => ( member_a @ ( add_a_b @ r @ X3 @ Xa ) @ H ) ) )
           => ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).

% add.one_in_subset
thf(fact_894_ring__iso__imp__img__ring,axiom,
    ! [H2: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ r @ S ) )
     => ( ring_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_ring
thf(fact_895_ring__iso__imp__img__ring,axiom,
    ! [H2: a > set_int,S: partia4934656038542163276t_unit] :
      ( ( member_a_set_int @ H2 @ ( ring_i4118987378922720216t_unit @ r @ S ) )
     => ( ring_s5316885176909347197t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_ring
thf(fact_896_ring__iso__imp__img__ring,axiom,
    ! [H2: a > list_set_int,S: partia6652711048488803900t_unit] :
      ( ( member6392905276882098698et_int @ H2 @ ( ring_i8932118395132445384t_unit @ r @ S ) )
     => ( ring_l5834914389771393517t_unit
        @ ( zero_u3668689415746537772t_unit
          @ ^ [Uu: list_set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_ring
thf(fact_897_ring__iso__imp__img__ring,axiom,
    ! [H2: a > list_list_set_int,S: partia7144473111789202860t_unit] :
      ( ( member2665369038410323738et_int @ H2 @ ( ring_i3530801300774344760t_unit @ r @ S ) )
     => ( ring_l6429277945453189341t_unit
        @ ( zero_u4006987073238113436t_unit
          @ ^ [Uu: list_list_set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_ring
thf(fact_898_ring__iso__imp__img__ring,axiom,
    ! [H2: a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ r @ S ) )
     => ( ring_l6212528067271185461t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_ring
thf(fact_899_pzf_Odomain__axioms,axiom,
    domain1631227219952682457t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.domain_axioms
thf(fact_900_ring__iso__imp__img__domain,axiom,
    ! [H2: a > set_list_set_int,S: partia3352279114339913222t_unit] :
      ( ( member8440910143787153472et_int @ H2 @ ( ring_i3909575963551679762t_unit @ r @ S ) )
     => ( domain6977320662053078051t_unit
        @ ( zero_u393940103915928438t_unit
          @ ^ [Uu: set_list_set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_domain
thf(fact_901_ring__iso__imp__img__domain,axiom,
    ! [H2: a > set_set_int,S: partia3601206958761670294t_unit] :
      ( ( member_a_set_set_int @ H2 @ ( ring_i2195039277548872866t_unit @ r @ S ) )
     => ( domain2593739924244766771t_unit
        @ ( zero_u3927532417453612294t_unit
          @ ^ [Uu: set_set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_domain
thf(fact_902_ring__iso__imp__img__domain,axiom,
    ! [H2: a > set_int,S: partia4934656038542163276t_unit] :
      ( ( member_a_set_int @ H2 @ ( ring_i4118987378922720216t_unit @ r @ S ) )
     => ( domain6183376680155302761t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_domain
thf(fact_903_ring__iso__imp__img__domain,axiom,
    ! [H2: a > set_a,S: partia6043505979758434576t_unit] :
      ( ( member_a_set_a @ H2 @ ( ring_i7849008455817099456t_unit @ r @ S ) )
     => ( domain4236798911309298543t_unit
        @ ( zero_u8960205505688764764t_unit
          @ ^ [Uu: set_a] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_domain
thf(fact_904_ring__iso__imp__img__domain,axiom,
    ! [H2: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ r @ S ) )
     => ( domain_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_domain
thf(fact_905_ring__iso__imp__img__field,axiom,
    ! [H2: a > set_list_set_int,S: partia3352279114339913222t_unit] :
      ( ( member8440910143787153472et_int @ H2 @ ( ring_i3909575963551679762t_unit @ r @ S ) )
     => ( field_6001781496977528197t_unit
        @ ( zero_u393940103915928438t_unit
          @ ^ [Uu: set_list_set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_field
thf(fact_906_ring__iso__imp__img__field,axiom,
    ! [H2: a > set_set_int,S: partia3601206958761670294t_unit] :
      ( ( member_a_set_set_int @ H2 @ ( ring_i2195039277548872866t_unit @ r @ S ) )
     => ( field_4969826516285655701t_unit
        @ ( zero_u3927532417453612294t_unit
          @ ^ [Uu: set_set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_field
thf(fact_907_ring__iso__imp__img__field,axiom,
    ! [H2: a > set_int,S: partia4934656038542163276t_unit] :
      ( ( member_a_set_int @ H2 @ ( ring_i4118987378922720216t_unit @ r @ S ) )
     => ( field_5943785737635511755t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_field
thf(fact_908_ring__iso__imp__img__field,axiom,
    ! [H2: a > set_a,S: partia6043505979758434576t_unit] :
      ( ( member_a_set_a @ H2 @ ( ring_i7849008455817099456t_unit @ r @ S ) )
     => ( field_6045675692312731021t_unit
        @ ( zero_u8960205505688764764t_unit
          @ ^ [Uu: set_a] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_field
thf(fact_909_ring__iso__imp__img__field,axiom,
    ! [H2: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ r @ S ) )
     => ( field_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_a_b @ r ) )
          @ S ) ) ) ).

% ring_iso_imp_img_field
thf(fact_910_zf_Oa__coset__add__inv2,axiom,
    ! [M: set_set_int,X: set_int,Y: set_int] :
      ( ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ X )
        = ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ Y ) )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
           => ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) ) )
              = M ) ) ) ) ) ).

% zf.a_coset_add_inv2
thf(fact_911_zf_Oa__coset__add__inv1,axiom,
    ! [M: set_set_int,X: set_int,Y: set_int] :
      ( ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) ) )
        = M )
     => ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
           => ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ X )
              = ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ Y ) ) ) ) ) ) ).

% zf.a_coset_add_inv1
thf(fact_912_zf_Ozeromaximalideal__fieldI,axiom,
    ( ( maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
   => ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.zeromaximalideal_fieldI
thf(fact_913_zf_Ozeromaximalideal__eq__field,axiom,
    ( ( maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
    = ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.zeromaximalideal_eq_field
thf(fact_914_zf_Oa__card__cosets__equal,axiom,
    ! [C: set_set_int,H: set_set_int] :
      ( ( member_set_set_int @ C @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H ) )
     => ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( finite_card_set_int @ C )
            = ( finite_card_set_int @ H ) ) ) ) ) ).

% zf.a_card_cosets_equal
thf(fact_915_primeideal__iff__prime,axiom,
    ! [P2: a] :
      ( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
     => ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r )
        = ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).

% primeideal_iff_prime
thf(fact_916_zf_Oring__iso__imp__img__ring,axiom,
    ! [H2: set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_ring
thf(fact_917_zf_Oring__iso__imp__img__ring,axiom,
    ! [H2: set_int > set_int,S: partia4934656038542163276t_unit] :
      ( ( member5205197933313416826et_int @ H2 @ ( ring_i7857697894474155344t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_s5316885176909347197t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_ring
thf(fact_918_zf_Oring__iso__imp__img__ring,axiom,
    ! [H2: set_int > list_set_int,S: partia6652711048488803900t_unit] :
      ( ( member7759640485799516682et_int @ H2 @ ( ring_i7208507585547788352t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_l5834914389771393517t_unit
        @ ( zero_u3668689415746537772t_unit
          @ ^ [Uu: list_set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_ring
thf(fact_919_zf_Oring__iso__imp__img__ring,axiom,
    ! [H2: set_int > list_a,S: partia2670972154091845814t_unit] :
      ( ( member6804298160596253954list_a @ H2 @ ( ring_i8159631783994945698t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_l6212528067271185461t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_ring
thf(fact_920_zf_Oring__iso__imp__img__ring,axiom,
    ! [H2: set_int > list_list_set_int,S: partia7144473111789202860t_unit] :
      ( ( member1521596788047370010et_int @ H2 @ ( ring_i5388733263752337840t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( ring_l6429277945453189341t_unit
        @ ( zero_u4006987073238113436t_unit
          @ ^ [Uu: list_list_set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_ring
thf(fact_921_zf_Oring__iso__imp__img__domain,axiom,
    ! [H2: set_int > set_list_set_int,S: partia3352279114339913222t_unit] :
      ( ( member7361235547688761408et_int @ H2 @ ( ring_i8625367162156237450t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( domain6977320662053078051t_unit
        @ ( zero_u393940103915928438t_unit
          @ ^ [Uu: set_list_set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_domain
thf(fact_922_zf_Oring__iso__imp__img__domain,axiom,
    ! [H2: set_int > set_set_int,S: partia3601206958761670294t_unit] :
      ( ( member1123395007370249776et_int @ H2 @ ( ring_i8032482649323388442t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( domain2593739924244766771t_unit
        @ ( zero_u3927532417453612294t_unit
          @ ^ [Uu: set_set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_domain
thf(fact_923_zf_Oring__iso__imp__img__domain,axiom,
    ! [H2: set_int > set_int,S: partia4934656038542163276t_unit] :
      ( ( member5205197933313416826et_int @ H2 @ ( ring_i7857697894474155344t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( domain6183376680155302761t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_domain
thf(fact_924_zf_Oring__iso__imp__img__domain,axiom,
    ! [H2: set_int > set_a,S: partia6043505979758434576t_unit] :
      ( ( member_set_int_set_a @ H2 @ ( ring_i5186029982270319944t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( domain4236798911309298543t_unit
        @ ( zero_u8960205505688764764t_unit
          @ ^ [Uu: set_a] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_domain
thf(fact_925_zf_Oring__iso__imp__img__domain,axiom,
    ! [H2: set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( domain_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_domain
thf(fact_926_zf_Oadd_Oone__in__subset,axiom,
    ! [H: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( H != bot_bot_set_set_int )
       => ( ! [X3: set_int] :
              ( ( member_set_int @ X3 @ H )
             => ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X3 ) @ H ) )
         => ( ! [X3: set_int] :
                ( ( member_set_int @ X3 @ H )
               => ! [Xa: set_int] :
                    ( ( member_set_int @ Xa @ H )
                   => ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X3 @ Xa ) @ H ) ) )
           => ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ H ) ) ) ) ) ).

% zf.add.one_in_subset
thf(fact_927_zf_Oring__iso__imp__img__field,axiom,
    ! [H2: set_int > set_list_set_int,S: partia3352279114339913222t_unit] :
      ( ( member7361235547688761408et_int @ H2 @ ( ring_i8625367162156237450t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( field_6001781496977528197t_unit
        @ ( zero_u393940103915928438t_unit
          @ ^ [Uu: set_list_set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_field
thf(fact_928_zf_Oring__iso__imp__img__field,axiom,
    ! [H2: set_int > set_set_int,S: partia3601206958761670294t_unit] :
      ( ( member1123395007370249776et_int @ H2 @ ( ring_i8032482649323388442t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( field_4969826516285655701t_unit
        @ ( zero_u3927532417453612294t_unit
          @ ^ [Uu: set_set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_field
thf(fact_929_zf_Oring__iso__imp__img__field,axiom,
    ! [H2: set_int > set_int,S: partia4934656038542163276t_unit] :
      ( ( member5205197933313416826et_int @ H2 @ ( ring_i7857697894474155344t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( field_5943785737635511755t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_field
thf(fact_930_zf_Oring__iso__imp__img__field,axiom,
    ! [H2: set_int > set_a,S: partia6043505979758434576t_unit] :
      ( ( member_set_int_set_a @ H2 @ ( ring_i5186029982270319944t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( field_6045675692312731021t_unit
        @ ( zero_u8960205505688764764t_unit
          @ ^ [Uu: set_a] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_field
thf(fact_931_zf_Oring__iso__imp__img__field,axiom,
    ! [H2: set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) )
     => ( field_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ S ) ) ) ).

% zf.ring_iso_imp_img_field
thf(fact_932_pzf_Oquot__domain__imp__primeideal,axiom,
    ! [P3: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ P3 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( domain6977320662053078051t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P3 ) )
       => ( primei7054412349171268410t_unit @ P3 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.quot_domain_imp_primeideal
thf(fact_933_pzf_Oquot__domain__iff__primeideal,axiom,
    ! [P3: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ P3 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( domain6977320662053078051t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P3 ) )
        = ( primei7054412349171268410t_unit @ P3 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.quot_domain_iff_primeideal
thf(fact_934_field__iff__prime,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
     => ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
        = ( ring_ring_prime_a_b @ r @ A ) ) ) ).

% field_iff_prime
thf(fact_935_domain__iff__prime,axiom,
    ! [A: a] :
      ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
     => ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
        = ( ring_ring_prime_a_b @ r @ A ) ) ) ).

% domain_iff_prime
thf(fact_936_ring__hom__imp__img__ring,axiom,
    ! [H2: a > list_list_set_int,S: partia7144473111789202860t_unit] :
      ( ( member2665369038410323738et_int @ H2 @ ( ring_h7017680599212693454t_unit @ r @ S ) )
     => ( ring_l6429277945453189341t_unit
        @ ( zero_u4006987073238113436t_unit
          @ ^ [Uu: list_list_set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ ( partia2558897084348286356t_unit
            @ ^ [Uu: set_li2838760105183549752et_int] : ( image_4485297384227577915et_int @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
            @ S ) ) ) ) ).

% ring_hom_imp_img_ring
thf(fact_937_ring__hom__imp__img__ring,axiom,
    ! [H2: a > list_a,S: partia2670972154091845814t_unit] :
      ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ r @ S ) )
     => ( ring_l6212528067271185461t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ r ) )
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
            @ S ) ) ) ) ).

% ring_hom_imp_img_ring
thf(fact_938_ring__hom__imp__img__ring,axiom,
    ! [H2: a > set_list_a,S: partia7496981018696276118t_unit] :
      ( ( member_a_set_list_a @ H2 @ ( ring_h6109298854714515236t_unit @ r @ S ) )
     => ( ring_s8247141995668492373t_unit
        @ ( zero_u3422801854947504854t_unit
          @ ^ [Uu: set_list_a] : ( H2 @ ( zero_a_b @ r ) )
          @ ( partia2314006361284536288t_unit
            @ ^ [Uu: set_set_list_a] : ( image_a_set_list_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
            @ S ) ) ) ) ).

% ring_hom_imp_img_ring
thf(fact_939_ring__hom__imp__img__ring,axiom,
    ! [H2: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ r @ S ) )
     => ( ring_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_a_b @ r ) )
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
            @ S ) ) ) ) ).

% ring_hom_imp_img_ring
thf(fact_940_ring__hom__imp__img__ring,axiom,
    ! [H2: a > list_set_int,S: partia6652711048488803900t_unit] :
      ( ( member6392905276882098698et_int @ H2 @ ( ring_h8172015933180347742t_unit @ r @ S ) )
     => ( ring_l5834914389771393517t_unit
        @ ( zero_u3668689415746537772t_unit
          @ ^ [Uu: list_set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : ( image_a_list_set_int @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
            @ S ) ) ) ) ).

% ring_hom_imp_img_ring
thf(fact_941_ring__hom__imp__img__ring,axiom,
    ! [H2: a > set_int,S: partia4934656038542163276t_unit] :
      ( ( member_a_set_int @ H2 @ ( ring_h8337389090535002478t_unit @ r @ S ) )
     => ( ring_s5316885176909347197t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_a_b @ r ) )
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : ( image_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
            @ S ) ) ) ) ).

% ring_hom_imp_img_ring
thf(fact_942_local_Ominus__minus,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
        = X ) ) ).

% local.minus_minus
thf(fact_943_a__inv__closed,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% a_inv_closed
thf(fact_944_minus__closed,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% minus_closed
thf(fact_945_local_Ominus__zero,axiom,
    ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
    = ( zero_a_b @ r ) ) ).

% local.minus_zero
thf(fact_946_zf_Oprimeideal__iff__prime,axiom,
    ! [P2: set_int] :
      ( ( member_set_int @ P2 @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
     => ( ( primei350866878734230858t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
        = ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) ) ) ).

% zf.primeideal_iff_prime
thf(fact_947_trivialideals__eq__field,axiom,
    ( ( ( partia707051561876973205xt_a_b @ r )
     != ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
   => ( ( ( collect_set_a
          @ ^ [I5: set_a] : ( ideal_a_b @ I5 @ r ) )
        = ( insert_set_a @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia707051561876973205xt_a_b @ r ) @ bot_bot_set_set_a ) ) )
      = ( field_a_b @ r ) ) ) ).

% trivialideals_eq_field
thf(fact_948_trivialideals__fieldI,axiom,
    ( ( ( partia707051561876973205xt_a_b @ r )
     != ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
   => ( ( ( collect_set_a
          @ ^ [I5: set_a] : ( ideal_a_b @ I5 @ r ) )
        = ( insert_set_a @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia707051561876973205xt_a_b @ r ) @ bot_bot_set_set_a ) ) )
     => ( field_a_b @ r ) ) ) ).

% trivialideals_fieldI
thf(fact_949_pzf_Oring__iso__imp__img__ring,axiom,
    ! [H2: list_set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( member6759595599802011116_int_a @ H2 @ ( ring_i4082836582328865034it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( ring_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ S ) ) ) ).

% pzf.ring_iso_imp_img_ring
thf(fact_950_pzf_Oring__iso__imp__img__ring,axiom,
    ! [H2: list_set_int > set_int,S: partia4934656038542163276t_unit] :
      ( ( member4589863265046936970et_int @ H2 @ ( ring_i372858595358256864t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( ring_s5316885176909347197t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ S ) ) ) ).

% pzf.ring_iso_imp_img_ring
thf(fact_951_pzf_Oring__iso__imp__img__ring,axiom,
    ! [H2: list_set_int > list_set_int,S: partia6652711048488803900t_unit] :
      ( ( member5734324632954958106et_int @ H2 @ ( ring_i6638472765406904784t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( ring_l5834914389771393517t_unit
        @ ( zero_u3668689415746537772t_unit
          @ ^ [Uu: list_set_int] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ S ) ) ) ).

% pzf.ring_iso_imp_img_ring
thf(fact_952_pzf_Oring__iso__imp__img__ring,axiom,
    ! [H2: list_set_int > list_a,S: partia2670972154091845814t_unit] :
      ( ( member7247483931175078386list_a @ H2 @ ( ring_i1390216875651519250t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( ring_l6212528067271185461t_unit
        @ ( zero_u1196785550890449590t_unit
          @ ^ [Uu: list_a] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ S ) ) ) ).

% pzf.ring_iso_imp_img_ring
thf(fact_953_pzf_Oring__iso__imp__img__ring,axiom,
    ! [H2: list_set_int > list_list_set_int,S: partia7144473111789202860t_unit] :
      ( ( member5915730702165157418et_int @ H2 @ ( ring_i1213622044540416832t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( ring_l6429277945453189341t_unit
        @ ( zero_u4006987073238113436t_unit
          @ ^ [Uu: list_list_set_int] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ S ) ) ) ).

% pzf.ring_iso_imp_img_ring
thf(fact_954_pzf_Oring__iso__imp__img__domain,axiom,
    ! [H2: list_set_int > set_list_set_int,S: partia3352279114339913222t_unit] :
      ( ( member5554658343060950864et_int @ H2 @ ( ring_i4019963246533157914t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( domain6977320662053078051t_unit
        @ ( zero_u393940103915928438t_unit
          @ ^ [Uu: set_list_set_int] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ S ) ) ) ).

% pzf.ring_iso_imp_img_domain
thf(fact_955_pzf_Oring__iso__imp__img__domain,axiom,
    ! [H2: list_set_int > set_set_int,S: partia3601206958761670294t_unit] :
      ( ( member7904089798749087040et_int @ H2 @ ( ring_i3303342158262216618t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( domain2593739924244766771t_unit
        @ ( zero_u3927532417453612294t_unit
          @ ^ [Uu: set_set_int] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ S ) ) ) ).

% pzf.ring_iso_imp_img_domain
thf(fact_956_pzf_Oring__iso__imp__img__domain,axiom,
    ! [H2: list_set_int > set_int,S: partia4934656038542163276t_unit] :
      ( ( member4589863265046936970et_int @ H2 @ ( ring_i372858595358256864t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( domain6183376680155302761t_unit
        @ ( zero_u8982883731493082172t_unit
          @ ^ [Uu: set_int] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ S ) ) ) ).

% pzf.ring_iso_imp_img_domain
thf(fact_957_pzf_Oring__iso__imp__img__domain,axiom,
    ! [H2: list_set_int > set_a,S: partia6043505979758434576t_unit] :
      ( ( member2022743676220373836_set_a @ H2 @ ( ring_i6194641345466501048t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( domain4236798911309298543t_unit
        @ ( zero_u8960205505688764764t_unit
          @ ^ [Uu: set_a] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ S ) ) ) ).

% pzf.ring_iso_imp_img_domain
thf(fact_958_pzf_Oring__iso__imp__img__domain,axiom,
    ! [H2: list_set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( member6759595599802011116_int_a @ H2 @ ( ring_i4082836582328865034it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) )
     => ( domain_a_b
        @ ( zero_update_a_b
          @ ^ [Uu: a] : ( H2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ S ) ) ) ).

% pzf.ring_iso_imp_img_domain
thf(fact_959_pzf_Ozeroprimeideal__domainI,axiom,
    ( ( primei7054412349171268410t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
   => ( domain1631227219952682457t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.zeroprimeideal_domainI
thf(fact_960_pzf_Odomain__eq__zeroprimeideal,axiom,
    ( ( domain1631227219952682457t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
    = ( primei7054412349171268410t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.domain_eq_zeroprimeideal
thf(fact_961_zf_Ofield__iff__prime,axiom,
    ! [A: set_int] :
      ( ( member_set_int @ A @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
     => ( ( field_4969826516285655701t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) ) )
        = ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) ) ) ).

% zf.field_iff_prime
thf(fact_962_zf_Odomain__iff__prime,axiom,
    ! [A: set_int] :
      ( ( member_set_int @ A @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
     => ( ( domain2593739924244766771t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) ) )
        = ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) ) ) ).

% zf.domain_iff_prime
thf(fact_963_zf_Otrivialideals__fieldI,axiom,
    ( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     != ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) )
   => ( ( ( collect_set_set_int
          @ ^ [I5: set_set_int] : ( ideal_7262958097527394045t_unit @ I5 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        = ( insert_set_set_int @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( insert_set_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bo2384636101374064866et_int ) ) )
     => ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.trivialideals_fieldI
thf(fact_964_zf_Otrivialideals__eq__field,axiom,
    ( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     != ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) )
   => ( ( ( collect_set_set_int
          @ ^ [I5: set_set_int] : ( ideal_7262958097527394045t_unit @ I5 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        = ( insert_set_set_int @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( insert_set_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bo2384636101374064866et_int ) ) )
      = ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.trivialideals_eq_field
thf(fact_965_add_Oinv__eq__1__iff,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ( a_inv_a_b @ r @ X )
          = ( zero_a_b @ r ) )
        = ( X
          = ( zero_a_b @ r ) ) ) ) ).

% add.inv_eq_1_iff
thf(fact_966_r__right__minus__eq,axiom,
    ! [A: a,B2: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( a_minus_a_b @ r @ A @ B2 )
            = ( zero_a_b @ r ) )
          = ( A = B2 ) ) ) ) ).

% r_right_minus_eq
thf(fact_967_pzf_Ofield__iff__prime,axiom,
    ! [A: list_set_int] :
      ( ( member_list_set_int @ A @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
     => ( ( field_6001781496977528197t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) ) )
        = ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) ) ) ).

% pzf.field_iff_prime
thf(fact_968_pzf_Odomain__iff__prime,axiom,
    ! [A: list_set_int] :
      ( ( member_list_set_int @ A @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
     => ( ( domain6977320662053078051t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) ) )
        = ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) ) ) ).

% pzf.domain_iff_prime
thf(fact_969_zf_Ominus__minus,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) )
        = X ) ) ).

% zf.minus_minus
thf(fact_970_zf_Oadd_Oinv__closed,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.add.inv_closed
thf(fact_971_zf_Ominus__zero,axiom,
    ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
    = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.minus_zero
thf(fact_972_zf_Oadd_Oinv__eq__1__iff,axiom,
    ! [X: set_int] :
      ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X )
          = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        = ( X
          = ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.add.inv_eq_1_iff
thf(fact_973_h_Ohom__a__inv,axiom,
    ! [X: list_set_int] :
      ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) @ x )
        = ( a_inv_a_b @ r @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X ) @ x ) ) ) ) ).

% h.hom_a_inv
thf(fact_974_ideal_Oquotient__is__ring,axiom,
    ! [I2: set_set_list_set_int,R2: partia3352279114339913222t_unit] :
      ( ( ideal_7892184242768911799t_unit @ I2 @ R2 )
     => ( ring_s2484293242784062849t_unit @ ( factRi8810600641151685442t_unit @ R2 @ I2 ) ) ) ).

% ideal.quotient_is_ring
thf(fact_975_ideal_Oquotient__is__ring,axiom,
    ! [I2: set_set_set_int,R2: partia3601206958761670294t_unit] :
      ( ( ideal_1992573625310827463t_unit @ I2 @ R2 )
     => ( ring_s3163219825908230673t_unit @ ( factRi2699865449404500690t_unit @ R2 @ I2 ) ) ) ).

% ideal.quotient_is_ring
thf(fact_976_ideal_Oquotient__is__ring,axiom,
    ! [I2: set_set_a,R2: partia6043505979758434576t_unit] :
      ( ( ideal_4463284918206690523t_unit @ I2 @ R2 )
     => ( ring_s1894871991113730299t_unit @ ( factRi9077435707053964944t_unit @ R2 @ I2 ) ) ) ).

% ideal.quotient_is_ring
thf(fact_977_ideal_Oquotient__is__ring,axiom,
    ! [I2: set_list_set_int,R2: partia6652711048488803900t_unit] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ R2 )
     => ( ring_s3147478420139132983t_unit @ ( factRi8995599497114439928t_unit @ R2 @ I2 ) ) ) ).

% ideal.quotient_is_ring
thf(fact_978_ideal_Oquotient__is__ring,axiom,
    ! [I2: set_a,R2: partia2175431115845679010xt_a_b] :
      ( ( ideal_a_b @ I2 @ R2 )
     => ( ring_s5549108798478166619t_unit @ ( factRing_a_b @ R2 @ I2 ) ) ) ).

% ideal.quotient_is_ring
thf(fact_979_ideal_Oquotient__is__ring,axiom,
    ! [I2: set_set_int,R2: partia4934656038542163276t_unit] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ R2 )
     => ( ring_s5321399439445376583t_unit @ ( factRi3149420076008518152t_unit @ R2 @ I2 ) ) ) ).

% ideal.quotient_is_ring
thf(fact_980_comp__def,axiom,
    ( comp_l364663056984283087et_int
    = ( ^ [F2: list_a > a,G2: list_set_int > list_a,X2: list_set_int] : ( F2 @ ( G2 @ X2 ) ) ) ) ).

% comp_def
thf(fact_981_comp__def,axiom,
    ( comp_l1769873867988408929et_int
    = ( ^ [F2: list_set_int > set_list_set_int,G2: set_int > list_set_int,X2: set_int] : ( F2 @ ( G2 @ X2 ) ) ) ) ).

% comp_def
thf(fact_982_comp__def,axiom,
    ( comp_l4546862774662178811st_a_a
    = ( ^ [F2: list_a > set_list_a,G2: a > list_a,X2: a] : ( F2 @ ( G2 @ X2 ) ) ) ) ).

% comp_def
thf(fact_983_comp__def,axiom,
    ( comp_l2129226576378413585et_int
    = ( ^ [F2: list_list_set_int > set_li2838760105183549752et_int,G2: list_set_int > list_list_set_int,X2: list_set_int] : ( F2 @ ( G2 @ X2 ) ) ) ) ).

% comp_def
thf(fact_984_comp__assoc,axiom,
    ! [F: list_a > set_list_a,G: a > list_a,H2: a > a] :
      ( ( comp_a_set_list_a_a @ ( comp_l4546862774662178811st_a_a @ F @ G ) @ H2 )
      = ( comp_l4546862774662178811st_a_a @ F @ ( comp_a_list_a_a @ G @ H2 ) ) ) ).

% comp_assoc
thf(fact_985_comp__assoc,axiom,
    ! [F: a > a,G: list_a > a,H2: list_set_int > list_a] :
      ( ( comp_l364663056984283087et_int @ ( comp_a_a_list_a @ F @ G ) @ H2 )
      = ( comp_a6958603637840890197et_int @ F @ ( comp_l364663056984283087et_int @ G @ H2 ) ) ) ).

% comp_assoc
thf(fact_986_comp__assoc,axiom,
    ! [F: list_a > a,G: list_a > list_a,H2: list_set_int > list_a] :
      ( ( comp_l364663056984283087et_int @ ( comp_list_a_a_list_a @ F @ G ) @ H2 )
      = ( comp_l364663056984283087et_int @ F @ ( comp_l5162460535484027477et_int @ G @ H2 ) ) ) ).

% comp_assoc
thf(fact_987_comp__assoc,axiom,
    ! [F: list_a > set_list_a,G: list_a > list_a,H2: a > list_a] :
      ( ( comp_l4546862774662178811st_a_a @ ( comp_l5379884480430907905list_a @ F @ G ) @ H2 )
      = ( comp_l4546862774662178811st_a_a @ F @ ( comp_list_a_list_a_a @ G @ H2 ) ) ) ).

% comp_assoc
thf(fact_988_comp__assoc,axiom,
    ! [F: list_a > a,G: list_set_int > list_a,H2: list_set_int > list_set_int] :
      ( ( comp_l1642282121434624219et_int @ ( comp_l364663056984283087et_int @ F @ G ) @ H2 )
      = ( comp_l364663056984283087et_int @ F @ ( comp_l7485871278972073313et_int @ G @ H2 ) ) ) ).

% comp_assoc
thf(fact_989_comp__assoc,axiom,
    ! [F: set_list_a > set_list_a,G: list_a > set_list_a,H2: a > list_a] :
      ( ( comp_l4546862774662178811st_a_a @ ( comp_s1152226898037786081list_a @ F @ G ) @ H2 )
      = ( comp_s7504242949508807643st_a_a @ F @ ( comp_l4546862774662178811st_a_a @ G @ H2 ) ) ) ).

% comp_assoc
thf(fact_990_comp__assoc,axiom,
    ! [F: list_set_int > set_list_set_int,G: set_int > list_set_int,H2: set_int > set_int] :
      ( ( comp_s4919011123983036881et_int @ ( comp_l1769873867988408929et_int @ F @ G ) @ H2 )
      = ( comp_l1769873867988408929et_int @ F @ ( comp_s883782994277854363et_int @ G @ H2 ) ) ) ).

% comp_assoc
thf(fact_991_comp__assoc,axiom,
    ! [F: list_set_int > set_list_set_int,G: list_set_int > list_set_int,H2: set_int > list_set_int] :
      ( ( comp_l1769873867988408929et_int @ ( comp_l7560170755363187953et_int @ F @ G ) @ H2 )
      = ( comp_l1769873867988408929et_int @ F @ ( comp_l9103437134707987755et_int @ G @ H2 ) ) ) ).

% comp_assoc
thf(fact_992_comp__assoc,axiom,
    ! [F: set_list_set_int > set_list_set_int,G: list_set_int > set_list_set_int,H2: set_int > list_set_int] :
      ( ( comp_l1769873867988408929et_int @ ( comp_s3671312503824928807et_int @ F @ G ) @ H2 )
      = ( comp_s2717484327125700759et_int @ F @ ( comp_l1769873867988408929et_int @ G @ H2 ) ) ) ).

% comp_assoc
thf(fact_993_comp__assoc,axiom,
    ! [F: list_list_set_int > set_li2838760105183549752et_int,G: list_set_int > list_list_set_int,H2: list_set_int > list_set_int] :
      ( ( comp_l8904473517367846913et_int @ ( comp_l2129226576378413585et_int @ F @ G ) @ H2 )
      = ( comp_l2129226576378413585et_int @ F @ ( comp_l7131751364485660491et_int @ G @ H2 ) ) ) ).

% comp_assoc
thf(fact_994_comp__eq__dest,axiom,
    ! [A: list_a > a,B2: list_set_int > list_a,C: list_a > a,D: list_set_int > list_a,V2: list_set_int] :
      ( ( ( comp_l364663056984283087et_int @ A @ B2 )
        = ( comp_l364663056984283087et_int @ C @ D ) )
     => ( ( A @ ( B2 @ V2 ) )
        = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_dest
thf(fact_995_comp__eq__dest,axiom,
    ! [A: list_set_int > set_list_set_int,B2: set_int > list_set_int,C: list_set_int > set_list_set_int,D: set_int > list_set_int,V2: set_int] :
      ( ( ( comp_l1769873867988408929et_int @ A @ B2 )
        = ( comp_l1769873867988408929et_int @ C @ D ) )
     => ( ( A @ ( B2 @ V2 ) )
        = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_dest
thf(fact_996_comp__eq__dest,axiom,
    ! [A: list_a > set_list_a,B2: a > list_a,C: list_a > set_list_a,D: a > list_a,V2: a] :
      ( ( ( comp_l4546862774662178811st_a_a @ A @ B2 )
        = ( comp_l4546862774662178811st_a_a @ C @ D ) )
     => ( ( A @ ( B2 @ V2 ) )
        = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_dest
thf(fact_997_comp__eq__dest,axiom,
    ! [A: list_list_set_int > set_li2838760105183549752et_int,B2: list_set_int > list_list_set_int,C: list_list_set_int > set_li2838760105183549752et_int,D: list_set_int > list_list_set_int,V2: list_set_int] :
      ( ( ( comp_l2129226576378413585et_int @ A @ B2 )
        = ( comp_l2129226576378413585et_int @ C @ D ) )
     => ( ( A @ ( B2 @ V2 ) )
        = ( C @ ( D @ V2 ) ) ) ) ).

% comp_eq_dest
thf(fact_998_comp__eq__elim,axiom,
    ! [A: list_a > a,B2: list_set_int > list_a,C: list_a > a,D: list_set_int > list_a] :
      ( ( ( comp_l364663056984283087et_int @ A @ B2 )
        = ( comp_l364663056984283087et_int @ C @ D ) )
     => ! [V3: list_set_int] :
          ( ( A @ ( B2 @ V3 ) )
          = ( C @ ( D @ V3 ) ) ) ) ).

% comp_eq_elim
thf(fact_999_comp__eq__elim,axiom,
    ! [A: list_set_int > set_list_set_int,B2: set_int > list_set_int,C: list_set_int > set_list_set_int,D: set_int > list_set_int] :
      ( ( ( comp_l1769873867988408929et_int @ A @ B2 )
        = ( comp_l1769873867988408929et_int @ C @ D ) )
     => ! [V3: set_int] :
          ( ( A @ ( B2 @ V3 ) )
          = ( C @ ( D @ V3 ) ) ) ) ).

% comp_eq_elim
thf(fact_1000_comp__eq__elim,axiom,
    ! [A: list_a > set_list_a,B2: a > list_a,C: list_a > set_list_a,D: a > list_a] :
      ( ( ( comp_l4546862774662178811st_a_a @ A @ B2 )
        = ( comp_l4546862774662178811st_a_a @ C @ D ) )
     => ! [V3: a] :
          ( ( A @ ( B2 @ V3 ) )
          = ( C @ ( D @ V3 ) ) ) ) ).

% comp_eq_elim
thf(fact_1001_comp__eq__elim,axiom,
    ! [A: list_list_set_int > set_li2838760105183549752et_int,B2: list_set_int > list_list_set_int,C: list_list_set_int > set_li2838760105183549752et_int,D: list_set_int > list_list_set_int] :
      ( ( ( comp_l2129226576378413585et_int @ A @ B2 )
        = ( comp_l2129226576378413585et_int @ C @ D ) )
     => ! [V3: list_set_int] :
          ( ( A @ ( B2 @ V3 ) )
          = ( C @ ( D @ V3 ) ) ) ) ).

% comp_eq_elim
thf(fact_1002_comp__eq__dest__lhs,axiom,
    ! [A: list_a > a,B2: list_set_int > list_a,C: list_set_int > a,V2: list_set_int] :
      ( ( ( comp_l364663056984283087et_int @ A @ B2 )
        = C )
     => ( ( A @ ( B2 @ V2 ) )
        = ( C @ V2 ) ) ) ).

% comp_eq_dest_lhs
thf(fact_1003_comp__eq__dest__lhs,axiom,
    ! [A: list_set_int > set_list_set_int,B2: set_int > list_set_int,C: set_int > set_list_set_int,V2: set_int] :
      ( ( ( comp_l1769873867988408929et_int @ A @ B2 )
        = C )
     => ( ( A @ ( B2 @ V2 ) )
        = ( C @ V2 ) ) ) ).

% comp_eq_dest_lhs
thf(fact_1004_comp__eq__dest__lhs,axiom,
    ! [A: list_a > set_list_a,B2: a > list_a,C: a > set_list_a,V2: a] :
      ( ( ( comp_l4546862774662178811st_a_a @ A @ B2 )
        = C )
     => ( ( A @ ( B2 @ V2 ) )
        = ( C @ V2 ) ) ) ).

% comp_eq_dest_lhs
thf(fact_1005_comp__eq__dest__lhs,axiom,
    ! [A: list_list_set_int > set_li2838760105183549752et_int,B2: list_set_int > list_list_set_int,C: list_set_int > set_li2838760105183549752et_int,V2: list_set_int] :
      ( ( ( comp_l2129226576378413585et_int @ A @ B2 )
        = C )
     => ( ( A @ ( B2 @ V2 ) )
        = ( C @ V2 ) ) ) ).

% comp_eq_dest_lhs
thf(fact_1006_ring_Oring__iso__imp__img__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R2 )
     => ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R2 @ S ) )
       => ( ring_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring.ring_iso_imp_img_ring
thf(fact_1007_ring_Oring__iso__imp__img__ring,axiom,
    ! [R2: partia2670972154091845814t_unit,H2: list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R2 @ S ) )
       => ( ring_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring.ring_iso_imp_img_ring
thf(fact_1008_ring_Oring__iso__imp__img__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > set_int,S: partia4934656038542163276t_unit] :
      ( ( ring_a_b @ R2 )
     => ( ( member_a_set_int @ H2 @ ( ring_i4118987378922720216t_unit @ R2 @ S ) )
       => ( ring_s5316885176909347197t_unit
          @ ( zero_u8982883731493082172t_unit
            @ ^ [Uu: set_int] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring.ring_iso_imp_img_ring
thf(fact_1009_ring_Oring__iso__imp__img__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > list_a,S: partia2670972154091845814t_unit] :
      ( ( ring_a_b @ R2 )
     => ( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R2 @ S ) )
       => ( ring_l6212528067271185461t_unit
          @ ( zero_u1196785550890449590t_unit
            @ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring.ring_iso_imp_img_ring
thf(fact_1010_ring_Oring__iso__imp__img__ring,axiom,
    ! [R2: partia4934656038542163276t_unit,H2: set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R2 @ S ) )
       => ( ring_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring.ring_iso_imp_img_ring
thf(fact_1011_ring_Oring__iso__imp__img__ring,axiom,
    ! [R2: partia2670972154091845814t_unit,H2: list_a > set_int,S: partia4934656038542163276t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( member5309957867533224448et_int @ H2 @ ( ring_i5545097000213460560t_unit @ R2 @ S ) )
       => ( ring_s5316885176909347197t_unit
          @ ( zero_u8982883731493082172t_unit
            @ ^ [Uu: set_int] : ( H2 @ ( zero_l4142658623432671053t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring.ring_iso_imp_img_ring
thf(fact_1012_ring_Oring__iso__imp__img__ring,axiom,
    ! [R2: partia2670972154091845814t_unit,H2: list_a > list_a,S: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R2 @ S ) )
       => ( ring_l6212528067271185461t_unit
          @ ( zero_u1196785550890449590t_unit
            @ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring.ring_iso_imp_img_ring
thf(fact_1013_ring_Oring__iso__imp__img__ring,axiom,
    ! [R2: partia6652711048488803900t_unit,H2: list_set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( ring_l5834914389771393517t_unit @ R2 )
     => ( ( member6759595599802011116_int_a @ H2 @ ( ring_i4082836582328865034it_a_b @ R2 @ S ) )
       => ( ring_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_l5318887016005761237t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring.ring_iso_imp_img_ring
thf(fact_1014_ring_Oring__iso__imp__img__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > list_set_int,S: partia6652711048488803900t_unit] :
      ( ( ring_a_b @ R2 )
     => ( ( member6392905276882098698et_int @ H2 @ ( ring_i8932118395132445384t_unit @ R2 @ S ) )
       => ( ring_l5834914389771393517t_unit
          @ ( zero_u3668689415746537772t_unit
            @ ^ [Uu: list_set_int] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% ring.ring_iso_imp_img_ring
thf(fact_1015_ring_Oring__iso__imp__img__ring,axiom,
    ! [R2: partia4934656038542163276t_unit,H2: set_int > set_int,S: partia4934656038542163276t_unit] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( member5205197933313416826et_int @ H2 @ ( ring_i7857697894474155344t_unit @ R2 @ S ) )
       => ( ring_s5316885176909347197t_unit
          @ ( zero_u8982883731493082172t_unit
            @ ^ [Uu: set_int] : ( H2 @ ( zero_s6269048424454532197t_unit @ R2 ) )
            @ S ) ) ) ) ).

% ring.ring_iso_imp_img_ring
thf(fact_1016_domain_Oring__iso__imp__img__domain,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( domain_a_b @ R2 )
     => ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R2 @ S ) )
       => ( domain_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% domain.ring_iso_imp_img_domain
thf(fact_1017_domain_Oring__iso__imp__img__domain,axiom,
    ! [R2: partia6043505979758434576t_unit,H2: set_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( domain4236798911309298543t_unit @ R2 )
     => ( ( member_set_a_a @ H2 @ ( ring_i4751279688611836608it_a_b @ R2 @ S ) )
       => ( domain_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_s2174465271003423091t_unit @ R2 ) )
            @ S ) ) ) ) ).

% domain.ring_iso_imp_img_domain
thf(fact_1018_domain_Oring__iso__imp__img__domain,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > set_int,S: partia4934656038542163276t_unit] :
      ( ( domain_a_b @ R2 )
     => ( ( member_a_set_int @ H2 @ ( ring_i4118987378922720216t_unit @ R2 @ S ) )
       => ( domain6183376680155302761t_unit
          @ ( zero_u8982883731493082172t_unit
            @ ^ [Uu: set_int] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% domain.ring_iso_imp_img_domain
thf(fact_1019_domain_Oring__iso__imp__img__domain,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > set_a,S: partia6043505979758434576t_unit] :
      ( ( domain_a_b @ R2 )
     => ( ( member_a_set_a @ H2 @ ( ring_i7849008455817099456t_unit @ R2 @ S ) )
       => ( domain4236798911309298543t_unit
          @ ( zero_u8960205505688764764t_unit
            @ ^ [Uu: set_a] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% domain.ring_iso_imp_img_domain
thf(fact_1020_domain_Oring__iso__imp__img__domain,axiom,
    ! [R2: partia4934656038542163276t_unit,H2: set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( domain6183376680155302761t_unit @ R2 )
     => ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R2 @ S ) )
       => ( domain_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R2 ) )
            @ S ) ) ) ) ).

% domain.ring_iso_imp_img_domain
thf(fact_1021_domain_Oring__iso__imp__img__domain,axiom,
    ! [R2: partia3601206958761670294t_unit,H2: set_set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( domain2593739924244766771t_unit @ R2 )
     => ( ( member_set_set_int_a @ H2 @ ( ring_i5461826592427486820it_a_b @ R2 @ S ) )
       => ( domain_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_s2946895028292161839t_unit @ R2 ) )
            @ S ) ) ) ) ).

% domain.ring_iso_imp_img_domain
thf(fact_1022_domain_Oring__iso__imp__img__domain,axiom,
    ! [R2: partia6043505979758434576t_unit,H2: set_a > set_int,S: partia4934656038542163276t_unit] :
      ( ( domain4236798911309298543t_unit @ R2 )
     => ( ( member_set_a_set_int @ H2 @ ( ring_i1340567420592345514t_unit @ R2 @ S ) )
       => ( domain6183376680155302761t_unit
          @ ( zero_u8982883731493082172t_unit
            @ ^ [Uu: set_int] : ( H2 @ ( zero_s2174465271003423091t_unit @ R2 ) )
            @ S ) ) ) ) ).

% domain.ring_iso_imp_img_domain
thf(fact_1023_domain_Oring__iso__imp__img__domain,axiom,
    ! [R2: partia6043505979758434576t_unit,H2: set_a > set_a,S: partia6043505979758434576t_unit] :
      ( ( domain4236798911309298543t_unit @ R2 )
     => ( ( member_set_a_set_a @ H2 @ ( ring_i7821126338072249198t_unit @ R2 @ S ) )
       => ( domain4236798911309298543t_unit
          @ ( zero_u8960205505688764764t_unit
            @ ^ [Uu: set_a] : ( H2 @ ( zero_s2174465271003423091t_unit @ R2 ) )
            @ S ) ) ) ) ).

% domain.ring_iso_imp_img_domain
thf(fact_1024_domain_Oring__iso__imp__img__domain,axiom,
    ! [R2: partia6652711048488803900t_unit,H2: list_set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( domain1631227219952682457t_unit @ R2 )
     => ( ( member6759595599802011116_int_a @ H2 @ ( ring_i4082836582328865034it_a_b @ R2 @ S ) )
       => ( domain_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_l5318887016005761237t_unit @ R2 ) )
            @ S ) ) ) ) ).

% domain.ring_iso_imp_img_domain
thf(fact_1025_domain_Oring__iso__imp__img__domain,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > set_set_int,S: partia3601206958761670294t_unit] :
      ( ( domain_a_b @ R2 )
     => ( ( member_a_set_set_int @ H2 @ ( ring_i2195039277548872866t_unit @ R2 @ S ) )
       => ( domain2593739924244766771t_unit
          @ ( zero_u3927532417453612294t_unit
            @ ^ [Uu: set_set_int] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% domain.ring_iso_imp_img_domain
thf(fact_1026_field_Oring__iso__imp__img__field,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( field_a_b @ R2 )
     => ( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R2 @ S ) )
       => ( field_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% field.ring_iso_imp_img_field
thf(fact_1027_field_Oring__iso__imp__img__field,axiom,
    ! [R2: partia6043505979758434576t_unit,H2: set_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( field_6045675692312731021t_unit @ R2 )
     => ( ( member_set_a_a @ H2 @ ( ring_i4751279688611836608it_a_b @ R2 @ S ) )
       => ( field_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_s2174465271003423091t_unit @ R2 ) )
            @ S ) ) ) ) ).

% field.ring_iso_imp_img_field
thf(fact_1028_field_Oring__iso__imp__img__field,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > set_int,S: partia4934656038542163276t_unit] :
      ( ( field_a_b @ R2 )
     => ( ( member_a_set_int @ H2 @ ( ring_i4118987378922720216t_unit @ R2 @ S ) )
       => ( field_5943785737635511755t_unit
          @ ( zero_u8982883731493082172t_unit
            @ ^ [Uu: set_int] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% field.ring_iso_imp_img_field
thf(fact_1029_field_Oring__iso__imp__img__field,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > set_a,S: partia6043505979758434576t_unit] :
      ( ( field_a_b @ R2 )
     => ( ( member_a_set_a @ H2 @ ( ring_i7849008455817099456t_unit @ R2 @ S ) )
       => ( field_6045675692312731021t_unit
          @ ( zero_u8960205505688764764t_unit
            @ ^ [Uu: set_a] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% field.ring_iso_imp_img_field
thf(fact_1030_field_Oring__iso__imp__img__field,axiom,
    ! [R2: partia4934656038542163276t_unit,H2: set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( field_5943785737635511755t_unit @ R2 )
     => ( ( member_set_int_a @ H2 @ ( ring_i2213127016525301402it_a_b @ R2 @ S ) )
       => ( field_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R2 ) )
            @ S ) ) ) ) ).

% field.ring_iso_imp_img_field
thf(fact_1031_field_Oring__iso__imp__img__field,axiom,
    ! [R2: partia3601206958761670294t_unit,H2: set_set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( field_4969826516285655701t_unit @ R2 )
     => ( ( member_set_set_int_a @ H2 @ ( ring_i5461826592427486820it_a_b @ R2 @ S ) )
       => ( field_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_s2946895028292161839t_unit @ R2 ) )
            @ S ) ) ) ) ).

% field.ring_iso_imp_img_field
thf(fact_1032_field_Oring__iso__imp__img__field,axiom,
    ! [R2: partia6043505979758434576t_unit,H2: set_a > set_int,S: partia4934656038542163276t_unit] :
      ( ( field_6045675692312731021t_unit @ R2 )
     => ( ( member_set_a_set_int @ H2 @ ( ring_i1340567420592345514t_unit @ R2 @ S ) )
       => ( field_5943785737635511755t_unit
          @ ( zero_u8982883731493082172t_unit
            @ ^ [Uu: set_int] : ( H2 @ ( zero_s2174465271003423091t_unit @ R2 ) )
            @ S ) ) ) ) ).

% field.ring_iso_imp_img_field
thf(fact_1033_field_Oring__iso__imp__img__field,axiom,
    ! [R2: partia6043505979758434576t_unit,H2: set_a > set_a,S: partia6043505979758434576t_unit] :
      ( ( field_6045675692312731021t_unit @ R2 )
     => ( ( member_set_a_set_a @ H2 @ ( ring_i7821126338072249198t_unit @ R2 @ S ) )
       => ( field_6045675692312731021t_unit
          @ ( zero_u8960205505688764764t_unit
            @ ^ [Uu: set_a] : ( H2 @ ( zero_s2174465271003423091t_unit @ R2 ) )
            @ S ) ) ) ) ).

% field.ring_iso_imp_img_field
thf(fact_1034_field_Oring__iso__imp__img__field,axiom,
    ! [R2: partia6652711048488803900t_unit,H2: list_set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( field_7411436548976123707t_unit @ R2 )
     => ( ( member6759595599802011116_int_a @ H2 @ ( ring_i4082836582328865034it_a_b @ R2 @ S ) )
       => ( field_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_l5318887016005761237t_unit @ R2 ) )
            @ S ) ) ) ) ).

% field.ring_iso_imp_img_field
thf(fact_1035_field_Oring__iso__imp__img__field,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > set_set_int,S: partia3601206958761670294t_unit] :
      ( ( field_a_b @ R2 )
     => ( ( member_a_set_set_int @ H2 @ ( ring_i2195039277548872866t_unit @ R2 @ S ) )
       => ( field_4969826516285655701t_unit
          @ ( zero_u3927532417453612294t_unit
            @ ^ [Uu: set_set_int] : ( H2 @ ( zero_a_b @ R2 ) )
            @ S ) ) ) ) ).

% field.ring_iso_imp_img_field
thf(fact_1036_inj__on__id2,axiom,
    ! [A2: set_a] :
      ( inj_on_a_a
      @ ^ [X2: a] : X2
      @ A2 ) ).

% inj_on_id2
thf(fact_1037_inj__on__id2,axiom,
    ! [A2: set_set_int] :
      ( inj_on6435365835345961783et_int
      @ ^ [X2: set_int] : X2
      @ A2 ) ).

% inj_on_id2
thf(fact_1038_inj__on__id2,axiom,
    ! [A2: set_list_set_int] :
      ( inj_on3170377866336917463et_int
      @ ^ [X2: list_set_int] : X2
      @ A2 ) ).

% inj_on_id2
thf(fact_1039_image__eq__imp__comp,axiom,
    ! [F: a > a,A2: set_a,G: a > a,B: set_a,H2: a > a] :
      ( ( ( image_a_a @ F @ A2 )
        = ( image_a_a @ G @ B ) )
     => ( ( image_a_a @ ( comp_a_a_a @ H2 @ F ) @ A2 )
        = ( image_a_a @ ( comp_a_a_a @ H2 @ G ) @ B ) ) ) ).

% image_eq_imp_comp
thf(fact_1040_image__eq__imp__comp,axiom,
    ! [F: a > set_a,A2: set_a,G: a > set_a,B: set_a,H2: set_a > a] :
      ( ( ( image_a_set_a @ F @ A2 )
        = ( image_a_set_a @ G @ B ) )
     => ( ( image_a_a @ ( comp_set_a_a_a @ H2 @ F ) @ A2 )
        = ( image_a_a @ ( comp_set_a_a_a @ H2 @ G ) @ B ) ) ) ).

% image_eq_imp_comp
thf(fact_1041_image__eq__imp__comp,axiom,
    ! [F: a > a,A2: set_a,G: a > a,B: set_a,H2: a > set_a] :
      ( ( ( image_a_a @ F @ A2 )
        = ( image_a_a @ G @ B ) )
     => ( ( image_a_set_a @ ( comp_a_set_a_a @ H2 @ F ) @ A2 )
        = ( image_a_set_a @ ( comp_a_set_a_a @ H2 @ G ) @ B ) ) ) ).

% image_eq_imp_comp
thf(fact_1042_image__eq__imp__comp,axiom,
    ! [F: a > set_int,A2: set_a,G: set_int > set_int,B: set_set_int,H2: set_int > a] :
      ( ( ( image_a_set_int @ F @ A2 )
        = ( image_524474410958335435et_int @ G @ B ) )
     => ( ( image_a_a @ ( comp_set_int_a_a @ H2 @ F ) @ A2 )
        = ( image_set_int_a @ ( comp_s6482183999417960411et_int @ H2 @ G ) @ B ) ) ) ).

% image_eq_imp_comp
thf(fact_1043_image__eq__imp__comp,axiom,
    ! [F: set_int > a,A2: set_set_int,G: a > a,B: set_a,H2: a > set_int] :
      ( ( ( image_set_int_a @ F @ A2 )
        = ( image_a_a @ G @ B ) )
     => ( ( image_524474410958335435et_int @ ( comp_a7082422283589945889et_int @ H2 @ F ) @ A2 )
        = ( image_a_set_int @ ( comp_a_set_int_a @ H2 @ G ) @ B ) ) ) ).

% image_eq_imp_comp
thf(fact_1044_image__eq__imp__comp,axiom,
    ! [F: set_int > set_int,A2: set_set_int,G: a > set_int,B: set_a,H2: set_int > a] :
      ( ( ( image_524474410958335435et_int @ F @ A2 )
        = ( image_a_set_int @ G @ B ) )
     => ( ( image_set_int_a @ ( comp_s6482183999417960411et_int @ H2 @ F ) @ A2 )
        = ( image_a_a @ ( comp_set_int_a_a @ H2 @ G ) @ B ) ) ) ).

% image_eq_imp_comp
thf(fact_1045_image__eq__imp__comp,axiom,
    ! [F: a > set_a,A2: set_a,G: a > set_a,B: set_a,H2: set_a > set_a] :
      ( ( ( image_a_set_a @ F @ A2 )
        = ( image_a_set_a @ G @ B ) )
     => ( ( image_a_set_a @ ( comp_set_a_set_a_a @ H2 @ F ) @ A2 )
        = ( image_a_set_a @ ( comp_set_a_set_a_a @ H2 @ G ) @ B ) ) ) ).

% image_eq_imp_comp
thf(fact_1046_image__eq__imp__comp,axiom,
    ! [F: a > a,A2: set_a,G: set_int > a,B: set_set_int,H2: a > set_int] :
      ( ( ( image_a_a @ F @ A2 )
        = ( image_set_int_a @ G @ B ) )
     => ( ( image_a_set_int @ ( comp_a_set_int_a @ H2 @ F ) @ A2 )
        = ( image_524474410958335435et_int @ ( comp_a7082422283589945889et_int @ H2 @ G ) @ B ) ) ) ).

% image_eq_imp_comp
thf(fact_1047_image__eq__imp__comp,axiom,
    ! [F: a > list_a,A2: set_a,G: a > list_a,B: set_a,H2: list_a > set_list_a] :
      ( ( ( image_a_list_a @ F @ A2 )
        = ( image_a_list_a @ G @ B ) )
     => ( ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ H2 @ F ) @ A2 )
        = ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ H2 @ G ) @ B ) ) ) ).

% image_eq_imp_comp
thf(fact_1048_image__eq__imp__comp,axiom,
    ! [F: list_set_int > list_a,A2: set_list_set_int,G: a > list_a,B: set_a,H2: list_a > a] :
      ( ( ( image_1311554247746015051list_a @ F @ A2 )
        = ( image_a_list_a @ G @ B ) )
     => ( ( image_list_set_int_a @ ( comp_l364663056984283087et_int @ H2 @ F ) @ A2 )
        = ( image_a_a @ ( comp_list_a_a_a @ H2 @ G ) @ B ) ) ) ).

% image_eq_imp_comp
thf(fact_1049_image__comp,axiom,
    ! [F: a > a,G: a > a,R5: set_a] :
      ( ( image_a_a @ F @ ( image_a_a @ G @ R5 ) )
      = ( image_a_a @ ( comp_a_a_a @ F @ G ) @ R5 ) ) ).

% image_comp
thf(fact_1050_image__comp,axiom,
    ! [F: set_a > a,G: a > set_a,R5: set_a] :
      ( ( image_set_a_a @ F @ ( image_a_set_a @ G @ R5 ) )
      = ( image_a_a @ ( comp_set_a_a_a @ F @ G ) @ R5 ) ) ).

% image_comp
thf(fact_1051_image__comp,axiom,
    ! [F: a > set_a,G: a > a,R5: set_a] :
      ( ( image_a_set_a @ F @ ( image_a_a @ G @ R5 ) )
      = ( image_a_set_a @ ( comp_a_set_a_a @ F @ G ) @ R5 ) ) ).

% image_comp
thf(fact_1052_image__comp,axiom,
    ! [F: set_a > set_a,G: a > set_a,R5: set_a] :
      ( ( image_set_a_set_a @ F @ ( image_a_set_a @ G @ R5 ) )
      = ( image_a_set_a @ ( comp_set_a_set_a_a @ F @ G ) @ R5 ) ) ).

% image_comp
thf(fact_1053_image__comp,axiom,
    ! [F: list_a > a,G: list_set_int > list_a,R5: set_list_set_int] :
      ( ( image_list_a_a @ F @ ( image_1311554247746015051list_a @ G @ R5 ) )
      = ( image_list_set_int_a @ ( comp_l364663056984283087et_int @ F @ G ) @ R5 ) ) ).

% image_comp
thf(fact_1054_image__comp,axiom,
    ! [F: list_a > set_list_a,G: a > list_a,R5: set_a] :
      ( ( image_5464838071766335845list_a @ F @ ( image_a_list_a @ G @ R5 ) )
      = ( image_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ F @ G ) @ R5 ) ) ).

% image_comp
thf(fact_1055_image__comp,axiom,
    ! [F: set_list_set_int > a,G: a > set_list_set_int,R5: set_a] :
      ( ( image_5898968216206821007_int_a @ F @ ( image_1101092336288225889et_int @ G @ R5 ) )
      = ( image_a_a @ ( comp_s10382787549540875nt_a_a @ F @ G ) @ R5 ) ) ).

% image_comp
thf(fact_1056_image__comp,axiom,
    ! [F: set_int > set_int,G: set_int > set_int,R5: set_set_int] :
      ( ( image_524474410958335435et_int @ F @ ( image_524474410958335435et_int @ G @ R5 ) )
      = ( image_524474410958335435et_int @ ( comp_s3539599085587913099et_int @ F @ G ) @ R5 ) ) ).

% image_comp
thf(fact_1057_image__comp,axiom,
    ! [F: a > a,G: set_list_set_int > a,R5: set_set_list_set_int] :
      ( ( image_a_a @ F @ ( image_5898968216206821007_int_a @ G @ R5 ) )
      = ( image_5898968216206821007_int_a @ ( comp_a3704501331711576715et_int @ F @ G ) @ R5 ) ) ).

% image_comp
thf(fact_1058_image__comp,axiom,
    ! [F: set_set_int > set_int,G: set_int > set_set_int,R5: set_set_int] :
      ( ( image_3513010637850279041et_int @ F @ ( image_1010086626112315521et_int @ G @ R5 ) )
      = ( image_524474410958335435et_int @ ( comp_s3581714896423519297et_int @ F @ G ) @ R5 ) ) ).

% image_comp
thf(fact_1059_subset__inj__on,axiom,
    ! [F: list_set_int > a,B: set_list_set_int,A2: set_list_set_int] :
      ( ( inj_on3061584747197171161_int_a @ F @ B )
     => ( ( ord_le4527193859438875528et_int @ A2 @ B )
       => ( inj_on3061584747197171161_int_a @ F @ A2 ) ) ) ).

% subset_inj_on
thf(fact_1060_subset__inj__on,axiom,
    ! [F: list_set_int > list_set_int,B: set_list_set_int,A2: set_list_set_int] :
      ( ( inj_on3170377866336917463et_int @ F @ B )
     => ( ( ord_le4527193859438875528et_int @ A2 @ B )
       => ( inj_on3170377866336917463et_int @ F @ A2 ) ) ) ).

% subset_inj_on
thf(fact_1061_subset__inj__on,axiom,
    ! [F: set_int > set_int,B: set_set_int,A2: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ B )
     => ( ( ord_le4403425263959731960et_int @ A2 @ B )
       => ( inj_on6435365835345961783et_int @ F @ A2 ) ) ) ).

% subset_inj_on
thf(fact_1062_subset__inj__on,axiom,
    ! [F: a > a,B: set_a,A2: set_a] :
      ( ( inj_on_a_a @ F @ B )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( inj_on_a_a @ F @ A2 ) ) ) ).

% subset_inj_on
thf(fact_1063_inj__on__subset,axiom,
    ! [F: list_set_int > a,A2: set_list_set_int,B: set_list_set_int] :
      ( ( inj_on3061584747197171161_int_a @ F @ A2 )
     => ( ( ord_le4527193859438875528et_int @ B @ A2 )
       => ( inj_on3061584747197171161_int_a @ F @ B ) ) ) ).

% inj_on_subset
thf(fact_1064_inj__on__subset,axiom,
    ! [F: list_set_int > list_set_int,A2: set_list_set_int,B: set_list_set_int] :
      ( ( inj_on3170377866336917463et_int @ F @ A2 )
     => ( ( ord_le4527193859438875528et_int @ B @ A2 )
       => ( inj_on3170377866336917463et_int @ F @ B ) ) ) ).

% inj_on_subset
thf(fact_1065_inj__on__subset,axiom,
    ! [F: set_int > set_int,A2: set_set_int,B: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ A2 )
     => ( ( ord_le4403425263959731960et_int @ B @ A2 )
       => ( inj_on6435365835345961783et_int @ F @ B ) ) ) ).

% inj_on_subset
thf(fact_1066_inj__on__subset,axiom,
    ! [F: a > a,A2: set_a,B: set_a] :
      ( ( inj_on_a_a @ F @ A2 )
     => ( ( ord_less_eq_set_a @ B @ A2 )
       => ( inj_on_a_a @ F @ B ) ) ) ).

% inj_on_subset
thf(fact_1067_inj__on__image__iff,axiom,
    ! [A2: set_a,G: a > a,F: a > a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A2 )
         => ! [Xa: a] :
              ( ( member_a @ Xa @ A2 )
             => ( ( ( G @ ( F @ X3 ) )
                  = ( G @ ( F @ Xa ) ) )
                = ( ( G @ X3 )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on_a_a @ F @ A2 )
       => ( ( inj_on_a_a @ G @ ( image_a_a @ F @ A2 ) )
          = ( inj_on_a_a @ G @ A2 ) ) ) ) ).

% inj_on_image_iff
thf(fact_1068_inj__on__image__iff,axiom,
    ! [A2: set_set_int,G: set_int > set_int,F: set_int > set_int] :
      ( ! [X3: set_int] :
          ( ( member_set_int @ X3 @ A2 )
         => ! [Xa: set_int] :
              ( ( member_set_int @ Xa @ A2 )
             => ( ( ( G @ ( F @ X3 ) )
                  = ( G @ ( F @ Xa ) ) )
                = ( ( G @ X3 )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on6435365835345961783et_int @ F @ A2 )
       => ( ( inj_on6435365835345961783et_int @ G @ ( image_524474410958335435et_int @ F @ A2 ) )
          = ( inj_on6435365835345961783et_int @ G @ A2 ) ) ) ) ).

% inj_on_image_iff
thf(fact_1069_inj__on__image__iff,axiom,
    ! [A2: set_list_set_int,G: list_set_int > a,F: list_set_int > list_set_int] :
      ( ! [X3: list_set_int] :
          ( ( member_list_set_int @ X3 @ A2 )
         => ! [Xa: list_set_int] :
              ( ( member_list_set_int @ Xa @ A2 )
             => ( ( ( G @ ( F @ X3 ) )
                  = ( G @ ( F @ Xa ) ) )
                = ( ( G @ X3 )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on3170377866336917463et_int @ F @ A2 )
       => ( ( inj_on3061584747197171161_int_a @ G @ ( image_586265162069257835et_int @ F @ A2 ) )
          = ( inj_on3061584747197171161_int_a @ G @ A2 ) ) ) ) ).

% inj_on_image_iff
thf(fact_1070_inj__on__image__iff,axiom,
    ! [A2: set_list_set_int,G: list_set_int > list_set_int,F: list_set_int > list_set_int] :
      ( ! [X3: list_set_int] :
          ( ( member_list_set_int @ X3 @ A2 )
         => ! [Xa: list_set_int] :
              ( ( member_list_set_int @ Xa @ A2 )
             => ( ( ( G @ ( F @ X3 ) )
                  = ( G @ ( F @ Xa ) ) )
                = ( ( G @ X3 )
                  = ( G @ Xa ) ) ) ) )
     => ( ( inj_on3170377866336917463et_int @ F @ A2 )
       => ( ( inj_on3170377866336917463et_int @ G @ ( image_586265162069257835et_int @ F @ A2 ) )
          = ( inj_on3170377866336917463et_int @ G @ A2 ) ) ) ) ).

% inj_on_image_iff
thf(fact_1071_inj__on__imageI2,axiom,
    ! [F3: list_set_int > set_list_set_int,F: set_int > list_set_int,A2: set_set_int] :
      ( ( inj_on6725311424131697661et_int @ ( comp_l1769873867988408929et_int @ F3 @ F ) @ A2 )
     => ( inj_on2182408081467232967et_int @ F @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1072_inj__on__imageI2,axiom,
    ! [F3: list_a > set_list_a,F: a > list_a,A2: set_a] :
      ( ( inj_on_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ F3 @ F ) @ A2 )
     => ( inj_on_a_list_a @ F @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1073_inj__on__imageI2,axiom,
    ! [F3: list_list_set_int > set_li2838760105183549752et_int,F: list_set_int > list_list_set_int,A2: set_list_set_int] :
      ( ( inj_on144625271008030877et_int @ ( comp_l2129226576378413585et_int @ F3 @ F ) @ A2 )
     => ( inj_on5847296377779290343et_int @ F @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1074_inj__on__imageI2,axiom,
    ! [F3: list_a > a,F: list_set_int > list_a,A2: set_list_set_int] :
      ( ( inj_on3061584747197171161_int_a @ ( comp_l364663056984283087et_int @ F3 @ F ) @ A2 )
     => ( inj_on1856771975478933727list_a @ F @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1075_inj__on__imageI2,axiom,
    ! [F3: a > a,F: list_set_int > a,A2: set_list_set_int] :
      ( ( inj_on3061584747197171161_int_a @ ( comp_a6958603637840890197et_int @ F3 @ F ) @ A2 )
     => ( inj_on3061584747197171161_int_a @ F @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1076_inj__on__imageI2,axiom,
    ! [F3: list_set_int > a,F: list_set_int > list_set_int,A2: set_list_set_int] :
      ( ( inj_on3061584747197171161_int_a @ ( comp_l1642282121434624219et_int @ F3 @ F ) @ A2 )
     => ( inj_on3170377866336917463et_int @ F @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1077_inj__on__imageI2,axiom,
    ! [F3: a > a,F: a > a,A2: set_a] :
      ( ( inj_on_a_a @ ( comp_a_a_a @ F3 @ F ) @ A2 )
     => ( inj_on_a_a @ F @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1078_inj__on__imageI2,axiom,
    ! [F3: set_int > set_int,F: set_int > set_int,A2: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ ( comp_s3539599085587913099et_int @ F3 @ F ) @ A2 )
     => ( inj_on6435365835345961783et_int @ F @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1079_inj__on__imageI2,axiom,
    ! [F3: a > list_set_int,F: list_set_int > a,A2: set_list_set_int] :
      ( ( inj_on3170377866336917463et_int @ ( comp_a4633020022310225089et_int @ F3 @ F ) @ A2 )
     => ( inj_on3061584747197171161_int_a @ F @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1080_inj__on__imageI2,axiom,
    ! [F3: list_set_int > list_set_int,F: list_set_int > list_set_int,A2: set_list_set_int] :
      ( ( inj_on3170377866336917463et_int @ ( comp_l1653122045001841339et_int @ F3 @ F ) @ A2 )
     => ( inj_on3170377866336917463et_int @ F @ A2 ) ) ).

% inj_on_imageI2
thf(fact_1081_ring_Oring__hom__imp__img__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
      ( ( ring_a_b @ R2 )
     => ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R2 @ S ) )
       => ( ring_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_a_b @ R2 ) )
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : ( image_a_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
              @ S ) ) ) ) ) ).

% ring.ring_hom_imp_img_ring
thf(fact_1082_ring_Oring__hom__imp__img__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > list_a,S: partia2670972154091845814t_unit] :
      ( ( ring_a_b @ R2 )
     => ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R2 @ S ) )
       => ( ring_l6212528067271185461t_unit
          @ ( zero_u1196785550890449590t_unit
            @ ^ [Uu: list_a] : ( H2 @ ( zero_a_b @ R2 ) )
            @ ( partia9041243232023819264t_unit
              @ ^ [Uu: set_list_a] : ( image_a_list_a @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
              @ S ) ) ) ) ) ).

% ring.ring_hom_imp_img_ring
thf(fact_1083_ring_Oring__hom__imp__img__ring,axiom,
    ! [R2: partia2670972154091845814t_unit,H2: list_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R2 @ S ) )
       => ( ring_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R2 ) )
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : ( image_list_a_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
              @ S ) ) ) ) ) ).

% ring.ring_hom_imp_img_ring
thf(fact_1084_ring_Oring__hom__imp__img__ring,axiom,
    ! [R2: partia4934656038542163276t_unit,H2: set_int > a,S: partia2175431115845679010xt_a_b] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( member_set_int_a @ H2 @ ( ring_h6431528728137583664it_a_b @ R2 @ S ) )
       => ( ring_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R2 ) )
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : ( image_set_int_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
              @ S ) ) ) ) ) ).

% ring.ring_hom_imp_img_ring
thf(fact_1085_ring_Oring__hom__imp__img__ring,axiom,
    ! [R2: partia6043505979758434576t_unit,H2: set_a > a,S: partia2175431115845679010xt_a_b] :
      ( ( ring_s5549108798478166619t_unit @ R2 )
     => ( ( member_set_a_a @ H2 @ ( ring_h4811522740288071338it_a_b @ R2 @ S ) )
       => ( ring_a_b
          @ ( zero_update_a_b
            @ ^ [Uu: a] : ( H2 @ ( zero_s2174465271003423091t_unit @ R2 ) )
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : ( image_set_a_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
              @ S ) ) ) ) ) ).

% ring.ring_hom_imp_img_ring
thf(fact_1086_ring_Oring__hom__imp__img__ring,axiom,
    ! [R2: partia2175431115845679010xt_a_b,H2: a > set_int,S: partia4934656038542163276t_unit] :
      ( ( ring_a_b @ R2 )
     => ( ( member_a_set_int @ H2 @ ( ring_h8337389090535002478t_unit @ R2 @ S ) )
       => ( ring_s5316885176909347197t_unit
          @ ( zero_u8982883731493082172t_unit
            @ ^ [Uu: set_int] : ( H2 @ ( zero_a_b @ R2 ) )
            @ ( partia2914870419229397556t_unit
              @ ^ [Uu: set_set_int] : ( image_a_set_int @ H2 @ ( partia707051561876973205xt_a_b @ R2 ) )
              @ S ) ) ) ) ) ).

% ring.ring_hom_imp_img_ring
thf(fact_1087_ring_Oring__hom__imp__img__ring,axiom,
    ! [R2: partia2670972154091845814t_unit,H2: list_a > list_a,S: partia2670972154091845814t_unit] :
      ( ( ring_l6212528067271185461t_unit @ R2 )
     => ( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R2 @ S ) )
       => ( ring_l6212528067271185461t_unit
          @ ( zero_u1196785550890449590t_unit
            @ ^ [Uu: list_a] : ( H2 @ ( zero_l4142658623432671053t_unit @ R2 ) )
            @ ( partia9041243232023819264t_unit
              @ ^ [Uu: set_list_a] : ( image_list_a_list_a @ H2 @ ( partia5361259788508890537t_unit @ R2 ) )
              @ S ) ) ) ) ) ).

% ring.ring_hom_imp_img_ring
thf(fact_1088_ring_Oring__hom__imp__img__ring,axiom,
    ! [R2: partia4934656038542163276t_unit,H2: set_int > list_a,S: partia2670972154091845814t_unit] :
      ( ( ring_s5316885176909347197t_unit @ R2 )
     => ( ( member6804298160596253954list_a @ H2 @ ( ring_h8006428831458247992t_unit @ R2 @ S ) )
       => ( ring_l6212528067271185461t_unit
          @ ( zero_u1196785550890449590t_unit
            @ ^ [Uu: list_a] : ( H2 @ ( zero_s6269048424454532197t_unit @ R2 ) )
            @ ( partia9041243232023819264t_unit
              @ ^ [Uu: set_list_a] : ( image_set_int_list_a @ H2 @ ( partia966996272515721803t_unit @ R2 ) )
              @ S ) ) ) ) ) ).

% ring.ring_hom_imp_img_ring
thf(fact_1089_ring_Oring__hom__imp__img__ring,axiom,
    ! [R2: partia6043505979758434576t_unit,H2: set_a > list_a,S: partia2670972154091845814t_unit] :
      ( ( ring_s5549108798478166619t_unit @ R2 )
     => ( ( member_set_a_list_a @ H2 @ ( ring_h6465547548369483954t_unit @ R2 @ S ) )
       => ( ring_l6212528067271185461t_unit
          @ ( zero_u1196785550890449590t_unit
            @ ^ [Uu: list_a] : ( H2 @ ( zero_s2174465271003423091t_unit @ R2 ) )
            @ ( partia9041243232023819264t_unit
              @ ^ [Uu: set_list_a] : ( image_set_a_list_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
              @ S ) ) ) ) ) ).

% ring.ring_hom_imp_img_ring
thf(fact_1090_ring_Oring__hom__imp__img__ring,axiom,
    ! [R2: partia6043505979758434576t_unit,H2: set_a > set_a,S: partia6043505979758434576t_unit] :
      ( ( ring_s5549108798478166619t_unit @ R2 )
     => ( ( member_set_a_set_a @ H2 @ ( ring_h4754363464466234712t_unit @ R2 @ S ) )
       => ( ring_s5549108798478166619t_unit
          @ ( zero_u8960205505688764764t_unit
            @ ^ [Uu: set_a] : ( H2 @ ( zero_s2174465271003423091t_unit @ R2 ) )
            @ ( partia7874839658741621210t_unit
              @ ^ [Uu: set_set_a] : ( image_set_a_set_a @ H2 @ ( partia5907974310037520643t_unit @ R2 ) )
              @ S ) ) ) ) ) ).

% ring.ring_hom_imp_img_ring
thf(fact_1091_principal__domain_Ofield__iff__prime,axiom,
    ! [R2: partia2670972154091845814t_unit,A: list_a] :
      ( ( ring_p8098905331641078952t_unit @ R2 )
     => ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R2 ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ bot_bot_set_list_a ) ) )
       => ( ( field_26233345952514695t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( cgenid9131348535277946915t_unit @ R2 @ A ) ) )
          = ( ring_r6430282645014804837t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.field_iff_prime
thf(fact_1092_principal__domain_Ofield__iff__prime,axiom,
    ! [R2: partia7144473111789202860t_unit,A: list_list_set_int] :
      ( ( ring_p1105522777314784042t_unit @ R2 )
     => ( ( member7440485371949894425et_int @ A @ ( minus_678714269029007121et_int @ ( partia3237421854401958059t_unit @ R2 ) @ ( insert5982949434426573746et_int @ ( zero_l4263514192383796165t_unit @ R2 ) @ bot_bo8087486759604424652et_int ) ) )
       => ( ( field_7531927466519412469t_unit @ ( factRi1079867023605223016t_unit @ R2 @ ( cgenid7864102925118505423t_unit @ R2 @ A ) ) )
          = ( ring_r7068619596278734765t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.field_iff_prime
thf(fact_1093_principal__domain_Ofield__iff__prime,axiom,
    ! [R2: partia3352279114339913222t_unit,A: set_list_set_int] :
      ( ( ring_p4083217851854698756t_unit @ R2 )
     => ( ( member4487585027056014399et_int @ A @ ( minus_8524384805321567159et_int @ ( partia7954853462395303941t_unit @ R2 ) @ ( insert4179525362955487576et_int @ ( zero_s8971087499367319327t_unit @ R2 ) @ bot_bo4503415735588907506et_int ) ) )
       => ( ( field_2520730373575520975t_unit @ ( factRi8810600641151685442t_unit @ R2 @ ( cgenid3413961061359118467t_unit @ R2 @ A ) ) )
          = ( ring_r1442117713053202183t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.field_iff_prime
thf(fact_1094_principal__domain_Ofield__iff__prime,axiom,
    ! [R2: partia3601206958761670294t_unit,A: set_set_int] :
      ( ( ring_p4005655891433666964t_unit @ R2 )
     => ( ( member_set_set_int @ A @ ( minus_6857623457997529383et_int @ ( partia1981568256555877781t_unit @ R2 ) @ ( insert_set_set_int @ ( zero_s2946895028292161839t_unit @ R2 ) @ bot_bo2384636101374064866et_int ) ) )
       => ( ( field_1116283816648818271t_unit @ ( factRi2699865449404500690t_unit @ R2 @ ( cgenid8920438795327392163t_unit @ R2 @ A ) ) )
          = ( ring_r8238242118552278295t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.field_iff_prime
thf(fact_1095_principal__domain_Ofield__iff__prime,axiom,
    ! [R2: partia6043505979758434576t_unit,A: set_a] :
      ( ( ring_p2862007038493914190t_unit @ R2 )
     => ( ( member_set_a @ A @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
       => ( ( field_6196704264051520557t_unit @ ( factRi9077435707053964944t_unit @ R2 @ ( cgenid6682780793756002467t_unit @ R2 @ A ) ) )
          = ( ring_r6795642478576035723t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.field_iff_prime
thf(fact_1096_principal__domain_Ofield__iff__prime,axiom,
    ! [R2: partia4934656038542163276t_unit,A: set_int] :
      ( ( ring_p2413783099608688842t_unit @ R2 )
     => ( ( member_set_int @ A @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ R2 ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
       => ( ( field_4969826516285655701t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( cgenid8502489213727343375t_unit @ R2 @ A ) ) )
          = ( ring_r2768735206093983821t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.field_iff_prime
thf(fact_1097_principal__domain_Ofield__iff__prime,axiom,
    ! [R2: partia2175431115845679010xt_a_b,A: a] :
      ( ( ring_p8803135361686045600in_a_b @ R2 )
     => ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
       => ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ R2 @ ( cgenid547466209912283029xt_a_b @ R2 @ A ) ) )
          = ( ring_ring_prime_a_b @ R2 @ A ) ) ) ) ).

% principal_domain.field_iff_prime
thf(fact_1098_principal__domain_Ofield__iff__prime,axiom,
    ! [R2: partia6652711048488803900t_unit,A: list_set_int] :
      ( ( ring_p1500425057661402810t_unit @ R2 )
     => ( ( member_list_set_int @ A @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ R2 ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ R2 ) @ bot_bo7344713902834881468et_int ) ) )
       => ( ( field_6001781496977528197t_unit @ ( factRi8995599497114439928t_unit @ R2 @ ( cgenid1447163913751362543t_unit @ R2 @ A ) ) )
          = ( ring_r7101923692879464637t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.field_iff_prime
thf(fact_1099_principal__domain_Odomain__iff__prime,axiom,
    ! [R2: partia2670972154091845814t_unit,A: list_a] :
      ( ( ring_p8098905331641078952t_unit @ R2 )
     => ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R2 ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R2 ) @ bot_bot_set_list_a ) ) )
       => ( ( domain1617769409708967785t_unit @ ( factRi3329376332477095402t_unit @ R2 @ ( cgenid9131348535277946915t_unit @ R2 @ A ) ) )
          = ( ring_r6430282645014804837t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.domain_iff_prime
thf(fact_1100_principal__domain_Odomain__iff__prime,axiom,
    ! [R2: partia7144473111789202860t_unit,A: list_list_set_int] :
      ( ( ring_p1105522777314784042t_unit @ R2 )
     => ( ( member7440485371949894425et_int @ A @ ( minus_678714269029007121et_int @ ( partia3237421854401958059t_unit @ R2 ) @ ( insert5982949434426573746et_int @ ( zero_l4263514192383796165t_unit @ R2 ) @ bot_bo8087486759604424652et_int ) ) )
       => ( ( domain5452931957204584083t_unit @ ( factRi1079867023605223016t_unit @ R2 @ ( cgenid7864102925118505423t_unit @ R2 @ A ) ) )
          = ( ring_r7068619596278734765t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.domain_iff_prime
thf(fact_1101_principal__domain_Odomain__iff__prime,axiom,
    ! [R2: partia3352279114339913222t_unit,A: set_list_set_int] :
      ( ( ring_p4083217851854698756t_unit @ R2 )
     => ( ( member4487585027056014399et_int @ A @ ( minus_8524384805321567159et_int @ ( partia7954853462395303941t_unit @ R2 ) @ ( insert4179525362955487576et_int @ ( zero_s8971087499367319327t_unit @ R2 ) @ bot_bo4503415735588907506et_int ) ) )
       => ( ( domain6191935110319139181t_unit @ ( factRi8810600641151685442t_unit @ R2 @ ( cgenid3413961061359118467t_unit @ R2 @ A ) ) )
          = ( ring_r1442117713053202183t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.domain_iff_prime
thf(fact_1102_principal__domain_Odomain__iff__prime,axiom,
    ! [R2: partia3601206958761670294t_unit,A: set_set_int] :
      ( ( ring_p4005655891433666964t_unit @ R2 )
     => ( ( member_set_set_int @ A @ ( minus_6857623457997529383et_int @ ( partia1981568256555877781t_unit @ R2 ) @ ( insert_set_set_int @ ( zero_s2946895028292161839t_unit @ R2 ) @ bot_bo2384636101374064866et_int ) ) )
       => ( ( domain336178477442980861t_unit @ ( factRi2699865449404500690t_unit @ R2 @ ( cgenid8920438795327392163t_unit @ R2 @ A ) ) )
          = ( ring_r8238242118552278295t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.domain_iff_prime
thf(fact_1103_principal__domain_Odomain__iff__prime,axiom,
    ! [R2: partia6043505979758434576t_unit,A: set_a] :
      ( ( ring_p2862007038493914190t_unit @ R2 )
     => ( ( member_set_a @ A @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R2 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R2 ) @ bot_bot_set_set_a ) ) )
       => ( ( domain5937373053834922511t_unit @ ( factRi9077435707053964944t_unit @ R2 @ ( cgenid6682780793756002467t_unit @ R2 @ A ) ) )
          = ( ring_r6795642478576035723t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.domain_iff_prime
thf(fact_1104_principal__domain_Odomain__iff__prime,axiom,
    ! [R2: partia4934656038542163276t_unit,A: set_int] :
      ( ( ring_p2413783099608688842t_unit @ R2 )
     => ( ( member_set_int @ A @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ R2 ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ R2 ) @ bot_bot_set_set_int ) ) )
       => ( ( domain2593739924244766771t_unit @ ( factRi3149420076008518152t_unit @ R2 @ ( cgenid8502489213727343375t_unit @ R2 @ A ) ) )
          = ( ring_r2768735206093983821t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.domain_iff_prime
thf(fact_1105_principal__domain_Odomain__iff__prime,axiom,
    ! [R2: partia2175431115845679010xt_a_b,A: a] :
      ( ( ring_p8803135361686045600in_a_b @ R2 )
     => ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R2 ) @ ( insert_a @ ( zero_a_b @ R2 ) @ bot_bot_set_a ) ) )
       => ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ R2 @ ( cgenid547466209912283029xt_a_b @ R2 @ A ) ) )
          = ( ring_ring_prime_a_b @ R2 @ A ) ) ) ) ).

% principal_domain.domain_iff_prime
thf(fact_1106_principal__domain_Odomain__iff__prime,axiom,
    ! [R2: partia6652711048488803900t_unit,A: list_set_int] :
      ( ( ring_p1500425057661402810t_unit @ R2 )
     => ( ( member_list_set_int @ A @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ R2 ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ R2 ) @ bot_bo7344713902834881468et_int ) ) )
       => ( ( domain6977320662053078051t_unit @ ( factRi8995599497114439928t_unit @ R2 @ ( cgenid1447163913751362543t_unit @ R2 @ A ) ) )
          = ( ring_r7101923692879464637t_unit @ R2 @ A ) ) ) ) ).

% principal_domain.domain_iff_prime
thf(fact_1107_inj__on__image__eq__iff,axiom,
    ! [F: set_list_set_int > a,C2: set_set_list_set_int,A2: set_set_list_set_int,B: set_set_list_set_int] :
      ( ( inj_on5590897182042520099_int_a @ F @ C2 )
     => ( ( ord_le8104031629355235774et_int @ A2 @ C2 )
       => ( ( ord_le8104031629355235774et_int @ B @ C2 )
         => ( ( ( image_5898968216206821007_int_a @ F @ A2 )
              = ( image_5898968216206821007_int_a @ F @ B ) )
            = ( A2 = B ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1108_inj__on__image__eq__iff,axiom,
    ! [F: list_set_int > a,C2: set_list_set_int,A2: set_list_set_int,B: set_list_set_int] :
      ( ( inj_on3061584747197171161_int_a @ F @ C2 )
     => ( ( ord_le4527193859438875528et_int @ A2 @ C2 )
       => ( ( ord_le4527193859438875528et_int @ B @ C2 )
         => ( ( ( image_list_set_int_a @ F @ A2 )
              = ( image_list_set_int_a @ F @ B ) )
            = ( A2 = B ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1109_inj__on__image__eq__iff,axiom,
    ! [F: list_set_int > list_set_int,C2: set_list_set_int,A2: set_list_set_int,B: set_list_set_int] :
      ( ( inj_on3170377866336917463et_int @ F @ C2 )
     => ( ( ord_le4527193859438875528et_int @ A2 @ C2 )
       => ( ( ord_le4527193859438875528et_int @ B @ C2 )
         => ( ( ( image_586265162069257835et_int @ F @ A2 )
              = ( image_586265162069257835et_int @ F @ B ) )
            = ( A2 = B ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1110_inj__on__image__eq__iff,axiom,
    ! [F: set_int > set_set_int,C2: set_set_int,A2: set_set_int,B: set_set_int] :
      ( ( inj_on6285404204842837485et_int @ F @ C2 )
     => ( ( ord_le4403425263959731960et_int @ A2 @ C2 )
       => ( ( ord_le4403425263959731960et_int @ B @ C2 )
         => ( ( ( image_1010086626112315521et_int @ F @ A2 )
              = ( image_1010086626112315521et_int @ F @ B ) )
            = ( A2 = B ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1111_inj__on__image__eq__iff,axiom,
    ! [F: set_int > set_int,C2: set_set_int,A2: set_set_int,B: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ C2 )
     => ( ( ord_le4403425263959731960et_int @ A2 @ C2 )
       => ( ( ord_le4403425263959731960et_int @ B @ C2 )
         => ( ( ( image_524474410958335435et_int @ F @ A2 )
              = ( image_524474410958335435et_int @ F @ B ) )
            = ( A2 = B ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1112_inj__on__image__eq__iff,axiom,
    ! [F: a > set_a,C2: set_a,A2: set_a,B: set_a] :
      ( ( inj_on_a_set_a @ F @ C2 )
     => ( ( ord_less_eq_set_a @ A2 @ C2 )
       => ( ( ord_less_eq_set_a @ B @ C2 )
         => ( ( ( image_a_set_a @ F @ A2 )
              = ( image_a_set_a @ F @ B ) )
            = ( A2 = B ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1113_inj__on__image__eq__iff,axiom,
    ! [F: a > a,C2: set_a,A2: set_a,B: set_a] :
      ( ( inj_on_a_a @ F @ C2 )
     => ( ( ord_less_eq_set_a @ A2 @ C2 )
       => ( ( ord_less_eq_set_a @ B @ C2 )
         => ( ( ( image_a_a @ F @ A2 )
              = ( image_a_a @ F @ B ) )
            = ( A2 = B ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1114_inj__on__image__mem__iff,axiom,
    ! [F: a > a,B: set_a,A: a,A2: set_a] :
      ( ( inj_on_a_a @ F @ B )
     => ( ( member_a @ A @ B )
       => ( ( ord_less_eq_set_a @ A2 @ B )
         => ( ( member_a @ ( F @ A ) @ ( image_a_a @ F @ A2 ) )
            = ( member_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1115_inj__on__image__mem__iff,axiom,
    ! [F: set_int > a,B: set_set_int,A: set_int,A2: set_set_int] :
      ( ( inj_on_set_int_a @ F @ B )
     => ( ( member_set_int @ A @ B )
       => ( ( ord_le4403425263959731960et_int @ A2 @ B )
         => ( ( member_a @ ( F @ A ) @ ( image_set_int_a @ F @ A2 ) )
            = ( member_set_int @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1116_inj__on__image__mem__iff,axiom,
    ! [F: set_a > a,B: set_set_a,A: set_a,A2: set_set_a] :
      ( ( inj_on_set_a_a @ F @ B )
     => ( ( member_set_a @ A @ B )
       => ( ( ord_le3724670747650509150_set_a @ A2 @ B )
         => ( ( member_a @ ( F @ A ) @ ( image_set_a_a @ F @ A2 ) )
            = ( member_set_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1117_inj__on__image__mem__iff,axiom,
    ! [F: a > set_a,B: set_a,A: a,A2: set_a] :
      ( ( inj_on_a_set_a @ F @ B )
     => ( ( member_a @ A @ B )
       => ( ( ord_less_eq_set_a @ A2 @ B )
         => ( ( member_set_a @ ( F @ A ) @ ( image_a_set_a @ F @ A2 ) )
            = ( member_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1118_inj__on__image__mem__iff,axiom,
    ! [F: list_set_int > a,B: set_list_set_int,A: list_set_int,A2: set_list_set_int] :
      ( ( inj_on3061584747197171161_int_a @ F @ B )
     => ( ( member_list_set_int @ A @ B )
       => ( ( ord_le4527193859438875528et_int @ A2 @ B )
         => ( ( member_a @ ( F @ A ) @ ( image_list_set_int_a @ F @ A2 ) )
            = ( member_list_set_int @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1119_inj__on__image__mem__iff,axiom,
    ! [F: ( nat > a ) > a,B: set_nat_a,A: nat > a,A2: set_nat_a] :
      ( ( inj_on_nat_a_a @ F @ B )
     => ( ( member_nat_a @ A @ B )
       => ( ( ord_le871467723717165285_nat_a @ A2 @ B )
         => ( ( member_a @ ( F @ A ) @ ( image_nat_a_a @ F @ A2 ) )
            = ( member_nat_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1120_inj__on__image__mem__iff,axiom,
    ! [F: set_set_int > a,B: set_set_set_int,A: set_set_int,A2: set_set_set_int] :
      ( ( inj_on_set_set_int_a @ F @ B )
     => ( ( member_set_set_int @ A @ B )
       => ( ( ord_le4317611570275147438et_int @ A2 @ B )
         => ( ( member_a @ ( F @ A ) @ ( image_set_set_int_a @ F @ A2 ) )
            = ( member_set_set_int @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1121_inj__on__image__mem__iff,axiom,
    ! [F: set_int > set_int,B: set_set_int,A: set_int,A2: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ B )
     => ( ( member_set_int @ A @ B )
       => ( ( ord_le4403425263959731960et_int @ A2 @ B )
         => ( ( member_set_int @ ( F @ A ) @ ( image_524474410958335435et_int @ F @ A2 ) )
            = ( member_set_int @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1122_inj__on__image__mem__iff,axiom,
    ! [F: a > nat > a,B: set_a,A: a,A2: set_a] :
      ( ( inj_on_a_nat_a @ F @ B )
     => ( ( member_a @ A @ B )
       => ( ( ord_less_eq_set_a @ A2 @ B )
         => ( ( member_nat_a @ ( F @ A ) @ ( image_a_nat_a @ F @ A2 ) )
            = ( member_a @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1123_inj__on__image__mem__iff,axiom,
    ! [F: set_list_set_int > a,B: set_set_list_set_int,A: set_list_set_int,A2: set_set_list_set_int] :
      ( ( inj_on5590897182042520099_int_a @ F @ B )
     => ( ( member4487585027056014399et_int @ A @ B )
       => ( ( ord_le8104031629355235774et_int @ A2 @ B )
         => ( ( member_a @ ( F @ A ) @ ( image_5898968216206821007_int_a @ F @ A2 ) )
            = ( member4487585027056014399et_int @ A @ A2 ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1124_inj__img__insertE,axiom,
    ! [F: a > a,A2: set_a,X: a,B: set_a] :
      ( ( inj_on_a_a @ F @ A2 )
     => ( ~ ( member_a @ X @ B )
       => ( ( ( insert_a @ X @ B )
            = ( image_a_a @ F @ A2 ) )
         => ~ ! [X6: a,A6: set_a] :
                ( ~ ( member_a @ X6 @ A6 )
               => ( ( A2
                    = ( insert_a @ X6 @ A6 ) )
                 => ( ( X
                      = ( F @ X6 ) )
                   => ( B
                     != ( image_a_a @ F @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1125_inj__img__insertE,axiom,
    ! [F: a > set_int,A2: set_a,X: set_int,B: set_set_int] :
      ( ( inj_on_a_set_int @ F @ A2 )
     => ( ~ ( member_set_int @ X @ B )
       => ( ( ( insert_set_int @ X @ B )
            = ( image_a_set_int @ F @ A2 ) )
         => ~ ! [X6: a,A6: set_a] :
                ( ~ ( member_a @ X6 @ A6 )
               => ( ( A2
                    = ( insert_a @ X6 @ A6 ) )
                 => ( ( X
                      = ( F @ X6 ) )
                   => ( B
                     != ( image_a_set_int @ F @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1126_inj__img__insertE,axiom,
    ! [F: a > set_a,A2: set_a,X: set_a,B: set_set_a] :
      ( ( inj_on_a_set_a @ F @ A2 )
     => ( ~ ( member_set_a @ X @ B )
       => ( ( ( insert_set_a @ X @ B )
            = ( image_a_set_a @ F @ A2 ) )
         => ~ ! [X6: a,A6: set_a] :
                ( ~ ( member_a @ X6 @ A6 )
               => ( ( A2
                    = ( insert_a @ X6 @ A6 ) )
                 => ( ( X
                      = ( F @ X6 ) )
                   => ( B
                     != ( image_a_set_a @ F @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1127_inj__img__insertE,axiom,
    ! [F: set_int > a,A2: set_set_int,X: a,B: set_a] :
      ( ( inj_on_set_int_a @ F @ A2 )
     => ( ~ ( member_a @ X @ B )
       => ( ( ( insert_a @ X @ B )
            = ( image_set_int_a @ F @ A2 ) )
         => ~ ! [X6: set_int,A6: set_set_int] :
                ( ~ ( member_set_int @ X6 @ A6 )
               => ( ( A2
                    = ( insert_set_int @ X6 @ A6 ) )
                 => ( ( X
                      = ( F @ X6 ) )
                   => ( B
                     != ( image_set_int_a @ F @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1128_inj__img__insertE,axiom,
    ! [F: set_a > a,A2: set_set_a,X: a,B: set_a] :
      ( ( inj_on_set_a_a @ F @ A2 )
     => ( ~ ( member_a @ X @ B )
       => ( ( ( insert_a @ X @ B )
            = ( image_set_a_a @ F @ A2 ) )
         => ~ ! [X6: set_a,A6: set_set_a] :
                ( ~ ( member_set_a @ X6 @ A6 )
               => ( ( A2
                    = ( insert_set_a @ X6 @ A6 ) )
                 => ( ( X
                      = ( F @ X6 ) )
                   => ( B
                     != ( image_set_a_a @ F @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1129_inj__img__insertE,axiom,
    ! [F: set_int > set_int,A2: set_set_int,X: set_int,B: set_set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ A2 )
     => ( ~ ( member_set_int @ X @ B )
       => ( ( ( insert_set_int @ X @ B )
            = ( image_524474410958335435et_int @ F @ A2 ) )
         => ~ ! [X6: set_int,A6: set_set_int] :
                ( ~ ( member_set_int @ X6 @ A6 )
               => ( ( A2
                    = ( insert_set_int @ X6 @ A6 ) )
                 => ( ( X
                      = ( F @ X6 ) )
                   => ( B
                     != ( image_524474410958335435et_int @ F @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1130_inj__img__insertE,axiom,
    ! [F: set_a > set_int,A2: set_set_a,X: set_int,B: set_set_int] :
      ( ( inj_on_set_a_set_int @ F @ A2 )
     => ( ~ ( member_set_int @ X @ B )
       => ( ( ( insert_set_int @ X @ B )
            = ( image_set_a_set_int @ F @ A2 ) )
         => ~ ! [X6: set_a,A6: set_set_a] :
                ( ~ ( member_set_a @ X6 @ A6 )
               => ( ( A2
                    = ( insert_set_a @ X6 @ A6 ) )
                 => ( ( X
                      = ( F @ X6 ) )
                   => ( B
                     != ( image_set_a_set_int @ F @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1131_inj__img__insertE,axiom,
    ! [F: set_int > set_a,A2: set_set_int,X: set_a,B: set_set_a] :
      ( ( inj_on_set_int_set_a @ F @ A2 )
     => ( ~ ( member_set_a @ X @ B )
       => ( ( ( insert_set_a @ X @ B )
            = ( image_set_int_set_a @ F @ A2 ) )
         => ~ ! [X6: set_int,A6: set_set_int] :
                ( ~ ( member_set_int @ X6 @ A6 )
               => ( ( A2
                    = ( insert_set_int @ X6 @ A6 ) )
                 => ( ( X
                      = ( F @ X6 ) )
                   => ( B
                     != ( image_set_int_set_a @ F @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1132_inj__img__insertE,axiom,
    ! [F: set_a > set_a,A2: set_set_a,X: set_a,B: set_set_a] :
      ( ( inj_on_set_a_set_a @ F @ A2 )
     => ( ~ ( member_set_a @ X @ B )
       => ( ( ( insert_set_a @ X @ B )
            = ( image_set_a_set_a @ F @ A2 ) )
         => ~ ! [X6: set_a,A6: set_set_a] :
                ( ~ ( member_set_a @ X6 @ A6 )
               => ( ( A2
                    = ( insert_set_a @ X6 @ A6 ) )
                 => ( ( X
                      = ( F @ X6 ) )
                   => ( B
                     != ( image_set_a_set_a @ F @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1133_inj__img__insertE,axiom,
    ! [F: a > set_set_int,A2: set_a,X: set_set_int,B: set_set_set_int] :
      ( ( inj_on_a_set_set_int @ F @ A2 )
     => ( ~ ( member_set_set_int @ X @ B )
       => ( ( ( insert_set_set_int @ X @ B )
            = ( image_a_set_set_int @ F @ A2 ) )
         => ~ ! [X6: a,A6: set_a] :
                ( ~ ( member_a @ X6 @ A6 )
               => ( ( A2
                    = ( insert_a @ X6 @ A6 ) )
                 => ( ( X
                      = ( F @ X6 ) )
                   => ( B
                     != ( image_a_set_set_int @ F @ A6 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1134_comp__inj__on,axiom,
    ! [F: a > list_set_int,A2: set_a,G: list_set_int > a] :
      ( ( inj_on7524766755751376063et_int @ F @ A2 )
     => ( ( inj_on3061584747197171161_int_a @ G @ ( image_a_list_set_int @ F @ A2 ) )
       => ( inj_on_a_a @ ( comp_l2879352502086878165nt_a_a @ G @ F ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1135_comp__inj__on,axiom,
    ! [F: set_list_set_int > a,A2: set_set_list_set_int,G: a > a] :
      ( ( inj_on5590897182042520099_int_a @ F @ A2 )
     => ( ( inj_on_a_a @ G @ ( image_5898968216206821007_int_a @ F @ A2 ) )
       => ( inj_on5590897182042520099_int_a @ ( comp_a3704501331711576715et_int @ G @ F ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1136_comp__inj__on,axiom,
    ! [F: list_set_int > a,A2: set_list_set_int,G: a > list_set_int] :
      ( ( inj_on3061584747197171161_int_a @ F @ A2 )
     => ( ( inj_on7524766755751376063et_int @ G @ ( image_list_set_int_a @ F @ A2 ) )
       => ( inj_on3170377866336917463et_int @ ( comp_a4633020022310225089et_int @ G @ F ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1137_comp__inj__on,axiom,
    ! [F: list_set_int > a,A2: set_list_set_int,G: a > a] :
      ( ( inj_on3061584747197171161_int_a @ F @ A2 )
     => ( ( inj_on_a_a @ G @ ( image_list_set_int_a @ F @ A2 ) )
       => ( inj_on3061584747197171161_int_a @ ( comp_a6958603637840890197et_int @ G @ F ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1138_comp__inj__on,axiom,
    ! [F: a > a,A2: set_a,G: a > a] :
      ( ( inj_on_a_a @ F @ A2 )
     => ( ( inj_on_a_a @ G @ ( image_a_a @ F @ A2 ) )
       => ( inj_on_a_a @ ( comp_a_a_a @ G @ F ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1139_comp__inj__on,axiom,
    ! [F: set_int > set_int,A2: set_set_int,G: set_int > set_int] :
      ( ( inj_on6435365835345961783et_int @ F @ A2 )
     => ( ( inj_on6435365835345961783et_int @ G @ ( image_524474410958335435et_int @ F @ A2 ) )
       => ( inj_on6435365835345961783et_int @ ( comp_s3539599085587913099et_int @ G @ F ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1140_comp__inj__on,axiom,
    ! [F: list_set_int > list_set_int,A2: set_list_set_int,G: list_set_int > a] :
      ( ( inj_on3170377866336917463et_int @ F @ A2 )
     => ( ( inj_on3061584747197171161_int_a @ G @ ( image_586265162069257835et_int @ F @ A2 ) )
       => ( inj_on3061584747197171161_int_a @ ( comp_l1642282121434624219et_int @ G @ F ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1141_comp__inj__on,axiom,
    ! [F: list_set_int > list_set_int,A2: set_list_set_int,G: list_set_int > list_set_int] :
      ( ( inj_on3170377866336917463et_int @ F @ A2 )
     => ( ( inj_on3170377866336917463et_int @ G @ ( image_586265162069257835et_int @ F @ A2 ) )
       => ( inj_on3170377866336917463et_int @ ( comp_l1653122045001841339et_int @ G @ F ) @ A2 ) ) ) ).

% comp_inj_on
thf(fact_1142_pzf_Osubdomain__iff,axiom,
    ! [H: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( subdom6469697055745072292t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
        = ( domain1631227219952682457t_unit
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : H
            @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.subdomain_iff
thf(fact_1143_pzf_Osubdomain__is__domain,axiom,
    ! [H: set_list_set_int] :
      ( ( subdom6469697055745072292t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( domain1631227219952682457t_unit
        @ ( partia488849630590579876t_unit
          @ ^ [Uu: set_list_set_int] : H
          @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.subdomain_is_domain
thf(fact_1144_subdomain__iff,axiom,
    ! [H: set_a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( subdomain_a_b @ H @ r )
        = ( domain_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H
            @ r ) ) ) ) ).

% subdomain_iff
thf(fact_1145_pzf_Oadd_Oderived__set__in__carrier,axiom,
    ! [H: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ord_le4527193859438875528et_int
        @ ( comple3081917594375107323et_int
          @ ( image_8713293071697434529et_int
            @ ^ [H1: list_set_int] :
                ( comple3081917594375107323et_int
                @ ( image_8713293071697434529et_int
                  @ ^ [H22: list_set_int] : ( insert_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H1 @ H22 ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H1 ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H22 ) ) @ bot_bo7344713902834881468et_int )
                  @ H ) )
            @ H ) )
        @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.add.derived_set_in_carrier
thf(fact_1146_h_Oinduced__ring__hom,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ring_h6525298532103594462it_a_b
        @ ( partia488849630590579876t_unit
          @ ^ [Uu: set_list_set_int] : K
          @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
        @ r
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) ).

% h.induced_ring_hom
thf(fact_1147_union__genideal,axiom,
    ! [I2: set_a,J: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ideal_a_b @ J @ r )
       => ( ( genideal_a_b @ r @ ( sup_sup_set_a @ I2 @ J ) )
          = ( set_add_a_b @ r @ I2 @ J ) ) ) ) ).

% union_genideal
thf(fact_1148_subdomain__is__domain,axiom,
    ! [H: set_a] :
      ( ( subdomain_a_b @ H @ r )
     => ( domain_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H
          @ r ) ) ) ).

% subdomain_is_domain
thf(fact_1149_zf_Ounion__genideal,axiom,
    ! [I2: set_set_int,J: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
       => ( ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( sup_sup_set_set_int @ I2 @ J ) )
          = ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ J ) ) ) ) ).

% zf.union_genideal
thf(fact_1150_pzf_Ocarrier__is__subring,axiom,
    subrin291743070294386232t_unit @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.carrier_is_subring
thf(fact_1151_pzf_OsubcringI_H,axiom,
    ! [H: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( subcri4185096033226064695t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.subcringI'
thf(fact_1152_pzf_OsubdomainI_H,axiom,
    ! [H: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( subdom6469697055745072292t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.subdomainI'
thf(fact_1153_pzf_Ochar__consistent,axiom,
    ! [H: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ring_c4550445564046854380t_unit
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : H
            @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        = ( ring_c4550445564046854380t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.char_consistent
thf(fact_1154_pzf_Oline__extension__consistent,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( embedd942231993492570031t_unit
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : K
            @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        = ( embedd942231993492570031t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.line_extension_consistent
thf(fact_1155_pzf_Osubring__is__domain,axiom,
    ! [H: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( domain1631227219952682457t_unit
        @ ( partia488849630590579876t_unit
          @ ^ [Uu: set_list_set_int] : H
          @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.subring_is_domain
thf(fact_1156_pzf_Osubring__is__ring,axiom,
    ! [H: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ring_l5834914389771393517t_unit
        @ ( partia488849630590579876t_unit
          @ ^ [Uu: set_list_set_int] : H
          @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.subring_is_ring
thf(fact_1157_pzf_Oquot__ideal__imp__ring__ideal,axiom,
    ! [I2: set_list_set_int,J: set_set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ideal_7892184242768911799t_unit @ J @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) )
       => ( ideal_3307802554209726317t_unit @ ( comple3081917594375107323et_int @ J ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.quot_ideal_imp_ring_ideal
thf(fact_1158_pzf_Oring__incl__imp__subring,axiom,
    ! [H: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( ring_l5834914389771393517t_unit
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : H
            @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
       => ( subrin291743070294386232t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.ring_incl_imp_subring
thf(fact_1159_pzf_Osubring__iff,axiom,
    ! [H: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( subrin291743070294386232t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
        = ( ring_l5834914389771393517t_unit
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : H
            @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.subring_iff
thf(fact_1160_pzf_Oideal__incl__iff,axiom,
    ! [I2: set_list_set_int,J: set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ideal_3307802554209726317t_unit @ J @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
       => ( ( ord_le4527193859438875528et_int @ I2 @ J )
          = ( J
            = ( comple3081917594375107323et_int @ ( image_8713293071697434529et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) @ J ) ) ) ) ) ) ).

% pzf.ideal_incl_iff
thf(fact_1161_pzf_Ocanonical__proj__vimage__in__carrier,axiom,
    ! [I2: set_list_set_int,J: set_set_list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ord_le8104031629355235774et_int @ J @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) ) )
       => ( ord_le4527193859438875528et_int @ ( comple3081917594375107323et_int @ J ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.canonical_proj_vimage_in_carrier
thf(fact_1162_pzf_Ocanonical__proj__vimage__mem__iff,axiom,
    ! [I2: set_list_set_int,J: set_set_list_set_int,A: list_set_int] :
      ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( ord_le8104031629355235774et_int @ J @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) ) )
       => ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
         => ( ( member_list_set_int @ A @ ( comple3081917594375107323et_int @ J ) )
            = ( member4487585027056014399et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ A ) @ J ) ) ) ) ) ).

% pzf.canonical_proj_vimage_mem_iff
thf(fact_1163_pzf_Oadd_Omono__derived__set,axiom,
    ! [K: set_list_set_int,H: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ K @ H )
     => ( ord_le4527193859438875528et_int
        @ ( comple3081917594375107323et_int
          @ ( image_8713293071697434529et_int
            @ ^ [H1: list_set_int] :
                ( comple3081917594375107323et_int
                @ ( image_8713293071697434529et_int
                  @ ^ [H22: list_set_int] : ( insert_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H1 @ H22 ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H1 ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H22 ) ) @ bot_bo7344713902834881468et_int )
                  @ K ) )
            @ K ) )
        @ ( comple3081917594375107323et_int
          @ ( image_8713293071697434529et_int
            @ ^ [H1: list_set_int] :
                ( comple3081917594375107323et_int
                @ ( image_8713293071697434529et_int
                  @ ^ [H22: list_set_int] : ( insert_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H1 @ H22 ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H1 ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H22 ) ) @ bot_bo7344713902834881468et_int )
                  @ H ) )
            @ H ) ) ) ) ).

% pzf.add.mono_derived_set
thf(fact_1164_quot__ideal__imp__ring__ideal,axiom,
    ! [I2: set_a,J: set_set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ideal_4463284918206690523t_unit @ J @ ( factRing_a_b @ r @ I2 ) )
       => ( ideal_a_b @ ( comple2307003609928055243_set_a @ J ) @ r ) ) ) ).

% quot_ideal_imp_ring_ideal
thf(fact_1165_ideal__incl__iff,axiom,
    ! [I2: set_a,J: set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ideal_a_b @ J @ r )
       => ( ( ord_less_eq_set_a @ I2 @ J )
          = ( J
            = ( comple2307003609928055243_set_a @ ( image_a_set_a @ ( a_r_coset_a_b @ r @ I2 ) @ J ) ) ) ) ) ) ).

% ideal_incl_iff
thf(fact_1166_zf_Oquot__ideal__imp__ring__ideal,axiom,
    ! [I2: set_set_int,J: set_set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( ideal_1992573625310827463t_unit @ J @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) )
       => ( ideal_7262958097527394045t_unit @ ( comple7281953568134767595et_int @ J ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.quot_ideal_imp_ring_ideal
thf(fact_1167_canonical__proj__vimage__in__carrier,axiom,
    ! [I2: set_a,J: set_set_a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
       => ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ J ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% canonical_proj_vimage_in_carrier
thf(fact_1168_canonical__proj__vimage__mem__iff,axiom,
    ! [I2: set_a,J: set_set_a,A: a] :
      ( ( ideal_a_b @ I2 @ r )
     => ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ A @ ( comple2307003609928055243_set_a @ J ) )
            = ( member_set_a @ ( a_r_coset_a_b @ r @ I2 @ A ) @ J ) ) ) ) ) ).

% canonical_proj_vimage_mem_iff
thf(fact_1169_add_Omono__derived__set,axiom,
    ! [K: set_a,H: set_a] :
      ( ( ord_less_eq_set_a @ K @ H )
     => ( ord_less_eq_set_a
        @ ( comple2307003609928055243_set_a
          @ ( image_a_set_a
            @ ^ [H1: a] :
                ( comple2307003609928055243_set_a
                @ ( image_a_set_a
                  @ ^ [H22: a] : ( insert_a @ ( add_a_b @ r @ ( add_a_b @ r @ ( add_a_b @ r @ H1 @ H22 ) @ ( a_inv_a_b @ r @ H1 ) ) @ ( a_inv_a_b @ r @ H22 ) ) @ bot_bot_set_a )
                  @ K ) )
            @ K ) )
        @ ( comple2307003609928055243_set_a
          @ ( image_a_set_a
            @ ^ [H1: a] :
                ( comple2307003609928055243_set_a
                @ ( image_a_set_a
                  @ ^ [H22: a] : ( insert_a @ ( add_a_b @ r @ ( add_a_b @ r @ ( add_a_b @ r @ H1 @ H22 ) @ ( a_inv_a_b @ r @ H1 ) ) @ ( a_inv_a_b @ r @ H22 ) ) @ bot_bot_set_a )
                  @ H ) )
            @ H ) ) ) ) ).

% add.mono_derived_set
thf(fact_1170_zf_Oideal__incl__iff,axiom,
    ! [I2: set_set_int,J: set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
       => ( ( ord_le4403425263959731960et_int @ I2 @ J )
          = ( J
            = ( comple7281953568134767595et_int @ ( image_1010086626112315521et_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) @ J ) ) ) ) ) ) ).

% zf.ideal_incl_iff
thf(fact_1171_add_Oderived__set__in__carrier,axiom,
    ! [H: set_a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ord_less_eq_set_a
        @ ( comple2307003609928055243_set_a
          @ ( image_a_set_a
            @ ^ [H1: a] :
                ( comple2307003609928055243_set_a
                @ ( image_a_set_a
                  @ ^ [H22: a] : ( insert_a @ ( add_a_b @ r @ ( add_a_b @ r @ ( add_a_b @ r @ H1 @ H22 ) @ ( a_inv_a_b @ r @ H1 ) ) @ ( a_inv_a_b @ r @ H22 ) ) @ bot_bot_set_a )
                  @ H ) )
            @ H ) )
        @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% add.derived_set_in_carrier
thf(fact_1172_zf_Ocanonical__proj__vimage__in__carrier,axiom,
    ! [I2: set_set_int,J: set_set_set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( ord_le4317611570275147438et_int @ J @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) ) )
       => ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ J ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.canonical_proj_vimage_in_carrier
thf(fact_1173_zf_Ocanonical__proj__vimage__mem__iff,axiom,
    ! [I2: set_set_int,J: set_set_set_int,A: set_int] :
      ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( ord_le4317611570275147438et_int @ J @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) ) )
       => ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
         => ( ( member_set_int @ A @ ( comple7281953568134767595et_int @ J ) )
            = ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ A ) @ J ) ) ) ) ) ).

% zf.canonical_proj_vimage_mem_iff
thf(fact_1174_zf_Oadd_Omono__derived__set,axiom,
    ! [K: set_set_int,H: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ K @ H )
     => ( ord_le4403425263959731960et_int
        @ ( comple7281953568134767595et_int
          @ ( image_1010086626112315521et_int
            @ ^ [H1: set_int] :
                ( comple7281953568134767595et_int
                @ ( image_1010086626112315521et_int
                  @ ^ [H22: set_int] : ( insert_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H1 @ H22 ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H1 ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H22 ) ) @ bot_bot_set_set_int )
                  @ K ) )
            @ K ) )
        @ ( comple7281953568134767595et_int
          @ ( image_1010086626112315521et_int
            @ ^ [H1: set_int] :
                ( comple7281953568134767595et_int
                @ ( image_1010086626112315521et_int
                  @ ^ [H22: set_int] : ( insert_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H1 @ H22 ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H1 ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H22 ) ) @ bot_bot_set_set_int )
                  @ H ) )
            @ H ) ) ) ) ).

% zf.add.mono_derived_set
thf(fact_1175_zf_Oadd_Oderived__set__in__carrier,axiom,
    ! [H: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ord_le4403425263959731960et_int
        @ ( comple7281953568134767595et_int
          @ ( image_1010086626112315521et_int
            @ ^ [H1: set_int] :
                ( comple7281953568134767595et_int
                @ ( image_1010086626112315521et_int
                  @ ^ [H22: set_int] : ( insert_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H1 @ H22 ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H1 ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H22 ) ) @ bot_bot_set_set_int )
                  @ H ) )
            @ H ) )
        @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.add.derived_set_in_carrier
thf(fact_1176_zf_Osubdomain__iff,axiom,
    ! [H: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( subdom1520866149873910708t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
        = ( domain6183376680155302761t_unit
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : H
            @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.subdomain_iff
thf(fact_1177_h_Oprimeideal__vimage,axiom,
    ! [P3: set_a] :
      ( ( cring_8461301442745498818t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( primeideal_a_b @ P3 @ r )
       => ( primei7054412349171268410t_unit
          @ ( collect_list_set_int
            @ ^ [R: list_set_int] :
                ( ( member_list_set_int @ R @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
                & ( member_a @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ R ) @ x ) @ P3 ) ) )
          @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% h.primeideal_vimage
thf(fact_1178_h_Oimg__is__subring,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( subring_a_b
        @ ( image_list_set_int_a
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
          @ K )
        @ r ) ) ).

% h.img_is_subring
thf(fact_1179_carrier__is__subring,axiom,
    subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% carrier_is_subring
thf(fact_1180_subdomainI_H,axiom,
    ! [H: set_a] :
      ( ( subring_a_b @ H @ r )
     => ( subdomain_a_b @ H @ r ) ) ).

% subdomainI'
thf(fact_1181_char__consistent,axiom,
    ! [H: set_a] :
      ( ( subring_a_b @ H @ r )
     => ( ( ring_char_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H
            @ r ) )
        = ( ring_char_a_b @ r ) ) ) ).

% char_consistent
thf(fact_1182_line__extension__consistent,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( embedd971793762689825387on_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : K
            @ r ) )
        = ( embedd971793762689825387on_a_b @ r ) ) ) ).

% line_extension_consistent
thf(fact_1183_subring__is__domain,axiom,
    ! [H: set_a] :
      ( ( subring_a_b @ H @ r )
     => ( domain_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H
          @ r ) ) ) ).

% subring_is_domain
thf(fact_1184_subring__is__ring,axiom,
    ! [H: set_a] :
      ( ( subring_a_b @ H @ r )
     => ( ring_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H
          @ r ) ) ) ).

% subring_is_ring
thf(fact_1185_pzf_Ois__cring,axiom,
    cring_8461301442745498818t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% pzf.is_cring
thf(fact_1186_subring__iff,axiom,
    ! [H: set_a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( subring_a_b @ H @ r )
        = ( ring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H
            @ r ) ) ) ) ).

% subring_iff
thf(fact_1187_ring__incl__imp__subring,axiom,
    ! [H: set_a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H
            @ r ) )
       => ( subring_a_b @ H @ r ) ) ) ).

% ring_incl_imp_subring
thf(fact_1188_zf_Osubdomain__is__domain,axiom,
    ! [H: set_set_int] :
      ( ( subdom1520866149873910708t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( domain6183376680155302761t_unit
        @ ( partia2914870419229397556t_unit
          @ ^ [Uu: set_set_int] : H
          @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.subdomain_is_domain
thf(fact_1189_pzf_Osubcring__iff,axiom,
    ! [H: set_list_set_int] :
      ( ( ord_le4527193859438875528et_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
     => ( ( subcri4185096033226064695t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
        = ( cring_8461301442745498818t_unit
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : H
            @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.subcring_iff
thf(fact_1190_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_1191_h_Oimg__is__cring,axiom,
    ( ( cring_a_b @ r )
   => ( cring_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] :
            ( image_list_set_int_a
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
            @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        @ r ) ) ) ).

% h.img_is_cring
thf(fact_1192_is__cring,axiom,
    cring_a_b @ r ).

% is_cring
thf(fact_1193_zf_Ois__cring,axiom,
    cring_3079150759069666002t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.is_cring
thf(fact_1194_h_Oinv__ring__hom,axiom,
    ! [K: set_list_set_int] :
      ( ( inj_on3061584747197171161_int_a
        @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
        @ K )
     => ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
       => ( ring_h2151208308052399004t_unit
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] :
                ( image_list_set_int_a
                @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
                @ K )
            @ r )
          @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( hilber7965021363618566206_int_a @ K
            @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) ) ) ) ) ).

% h.inv_ring_hom
thf(fact_1195_add__additive__subgroups,axiom,
    ! [H: set_a,K: set_a] :
      ( ( additi2834746164131130830up_a_b @ H @ r )
     => ( ( additi2834746164131130830up_a_b @ K @ r )
       => ( additi2834746164131130830up_a_b @ ( set_add_a_b @ r @ H @ K ) @ r ) ) ) ).

% add_additive_subgroups
thf(fact_1196_zf_Ocarrier__is__subring,axiom,
    subrin7689096310803670856t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.carrier_is_subring
thf(fact_1197_zf_Oadd__additive__subgroups,axiom,
    ! [H: set_set_int,K: set_set_int] :
      ( ( additi7073586575563672860t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( additi7073586575563672860t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
       => ( additi7073586575563672860t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ K ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.add_additive_subgroups
thf(fact_1198_zf_OsubdomainI_H,axiom,
    ! [H: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( subdom1520866149873910708t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.subdomainI'
thf(fact_1199_zf_Ouniv__poly__not__field,axiom,
    ! [K: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ~ ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ).

% zf.univ_poly_not_field
thf(fact_1200_zf_Ouniv__poly__is__cring,axiom,
    ! [K: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( cring_8461301442745498818t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ).

% zf.univ_poly_is_cring
thf(fact_1201_zf_Ouniv__poly__is__domain,axiom,
    ! [K: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( domain1631227219952682457t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ).

% zf.univ_poly_is_domain
thf(fact_1202_zf_Ouniv__poly__is__ring,axiom,
    ! [K: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ring_l5834914389771393517t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ).

% zf.univ_poly_is_ring
thf(fact_1203_zf_Ouniv__poly__consistent,axiom,
    ! [K: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( univ_p8160232691235018276t_unit
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : K
            @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        = ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.univ_poly_consistent
thf(fact_1204_zf_Osubring__is__domain,axiom,
    ! [H: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( domain6183376680155302761t_unit
        @ ( partia2914870419229397556t_unit
          @ ^ [Uu: set_set_int] : H
          @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.subring_is_domain
thf(fact_1205_zf_Osubring__is__ring,axiom,
    ! [H: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ring_s5316885176909347197t_unit
        @ ( partia2914870419229397556t_unit
          @ ^ [Uu: set_set_int] : H
          @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.subring_is_ring
thf(fact_1206_zf_Ouniv__poly__a__inv__consistent,axiom,
    ! [K: set_set_int,P2: list_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
       => ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 )
          = ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ) ).

% zf.univ_poly_a_inv_consistent
thf(fact_1207_zf_Ouniv__poly__a__minus__consistent,axiom,
    ! [K: set_set_int,Q2: list_set_int,P2: list_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( member_list_set_int @ Q2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
       => ( ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 @ Q2 )
          = ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q2 ) ) ) ) ).

% zf.univ_poly_a_minus_consistent
thf(fact_1208_zf_Oring__incl__imp__subring,axiom,
    ! [H: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ring_s5316885176909347197t_unit
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : H
            @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
       => ( subrin7689096310803670856t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.ring_incl_imp_subring
thf(fact_1209_zf_Osubring__iff,axiom,
    ! [H: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( subrin7689096310803670856t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
        = ( ring_s5316885176909347197t_unit
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : H
            @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.subring_iff
thf(fact_1210_zf_Ocarrier__polynomial__shell,axiom,
    ! [K: set_set_int,P2: list_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
       => ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% zf.carrier_polynomial_shell
thf(fact_1211_zf_Osubcring__iff,axiom,
    ! [H: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( subcri1024317279029940167t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
        = ( cring_3079150759069666002t_unit
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : H
            @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.subcring_iff
thf(fact_1212_carrier__is__subcring,axiom,
    subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% carrier_is_subcring
thf(fact_1213_subcringI_H,axiom,
    ! [H: set_a] :
      ( ( subring_a_b @ H @ r )
     => ( subcring_a_b @ H @ r ) ) ).

% subcringI'
thf(fact_1214_zf_Ocarrier__is__subcring,axiom,
    subcri1024317279029940167t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).

% zf.carrier_is_subcring
thf(fact_1215_zf_OsubcringI_H,axiom,
    ! [H: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( subcri1024317279029940167t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.subcringI'
thf(fact_1216_finite__Collect__le__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).

% finite_Collect_le_nat
thf(fact_1217_subcring__iff,axiom,
    ! [H: set_a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( subcring_a_b @ H @ r )
        = ( cring_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H
            @ r ) ) ) ) ).

% subcring_iff
thf(fact_1218_zf_Orupture__surj__hom_I1_J,axiom,
    ! [K: set_set_int,P2: list_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
       => ( member5554658343060950864et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) ) @ ( ring_h7954335661761407236t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( polyno6134647158531630271t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K @ P2 ) ) ) ) ) ).

% zf.rupture_surj_hom(1)
thf(fact_1219_zf_Orupture__surj__hom_I2_J,axiom,
    ! [K: set_set_int,P2: list_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
       => ( ring_h4009149326251492422t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( polyno6134647158531630271t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K @ P2 ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) ) ) ) ) ).

% zf.rupture_surj_hom(2)
thf(fact_1220_zf_Orupture__surj__norm__is__hom,axiom,
    ! [K: set_set_int,P2: list_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
       => ( member7361235547688761408et_int @ ( comp_l1769873867988408929et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) ) @ ( poly_o1995067004181805532t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
          @ ( ring_h2330097277019473012t_unit
            @ ( partia2914870419229397556t_unit
              @ ^ [Uu: set_set_int] : K
              @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
            @ ( polyno6134647158531630271t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K @ P2 ) ) ) ) ) ).

% zf.rupture_surj_norm_is_hom
thf(fact_1221_univ__poly__is__cring,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_is_cring
thf(fact_1222_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_1223_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_1224_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_1225_rupture__surj__hom_I1_J,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member4263473470251683292list_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) @ ( ring_h6188449271506562988t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) ) ) ) ) ).

% rupture_surj_hom(1)
thf(fact_1226_rupture__surj__hom_I2_J,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ring_h6719293541701509614t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ) ).

% rupture_surj_hom(2)
thf(fact_1227_univ__poly__a__minus__consistent,axiom,
    ! [K: set_a,Q2: list_a,P2: 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 ) @ P2 @ Q2 )
          = ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q2 ) ) ) ) ).

% univ_poly_a_minus_consistent
thf(fact_1228_univ__poly__a__inv__consistent,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ) ).

% univ_poly_a_inv_consistent
thf(fact_1229_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_1230_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
          @ ^ [P: list_a] : ( eval_a_b @ r @ P @ A )
          @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ K ) @ r ) ) ) ) ).

% eval_is_hom
thf(fact_1231_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
          @ ^ [P: list_a] : ( eval_a_b @ r @ P @ A ) ) ) ) ).

% eval_ring_hom
thf(fact_1232_pzf_Ouniv__poly__is__cring,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( cring_2652521809446997298t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ).

% pzf.univ_poly_is_cring
thf(fact_1233_pzf_Ouniv__poly__not__field,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ~ ( field_4857232450668044075t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ).

% pzf.univ_poly_not_field
thf(fact_1234_pzf_Ouniv__poly__is__domain,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( domain1893962698156629705t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ).

% pzf.univ_poly_is_domain
thf(fact_1235_pzf_Ouniv__poly__is__ring,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ring_l6429277945453189341t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ).

% pzf.univ_poly_is_ring
thf(fact_1236_pzf_Orupture__surj__hom_I1_J,axiom,
    ! [K: set_list_set_int,P2: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( member8798303191263431536et_int @ ( a_r_co2531912773843275078t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( cgenid7864102925118505423t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) ) @ ( ring_h2819912158977254660t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( polyno4866556214859693103t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ P2 ) ) ) ) ) ).

% pzf.rupture_surj_hom(1)
thf(fact_1237_pzf_Orupture__surj__hom_I2_J,axiom,
    ! [K: set_list_set_int,P2: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( ring_h5795583995102309958t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( polyno4866556214859693103t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ P2 ) @ ( a_r_co2531912773843275078t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( cgenid7864102925118505423t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) ) ) ) ) ).

% pzf.rupture_surj_hom(2)
thf(fact_1238_pzf_Ouniv__poly__a__inv__consistent,axiom,
    ! [K: set_list_set_int,P2: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 )
          = ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ P2 ) ) ) ) ).

% pzf.univ_poly_a_inv_consistent
thf(fact_1239_pzf_Ouniv__poly__a__minus__consistent,axiom,
    ! [K: set_list_set_int,Q2: list_list_set_int,P2: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ Q2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( ( a_minu1053138715659470590t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 @ Q2 )
          = ( a_minu1053138715659470590t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ P2 @ Q2 ) ) ) ) ).

% pzf.univ_poly_a_minus_consistent
thf(fact_1240_zf_Opoly__of__const__consistent,axiom,
    ! [K: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ( poly_o1995067004181805532t_unit
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : K
            @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        = ( poly_o1995067004181805532t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.poly_of_const_consistent
thf(fact_1241_pzf_Ouniv__poly__consistent,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( univ_p7181306125638219540t_unit
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : K
            @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        = ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.univ_poly_consistent
thf(fact_1242_zf_Ocanonical__embedding__is__hom,axiom,
    ! [K: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( member7759640485799516682et_int @ ( poly_o1995067004181805532t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
        @ ( ring_h8178984891965966890t_unit
          @ ( partia2914870419229397556t_unit
            @ ^ [Uu: set_set_int] : K
            @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
          @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ) ).

% zf.canonical_embedding_is_hom
thf(fact_1243_zf_Ocanonical__embedding__ring__hom,axiom,
    ! [K: set_set_int] :
      ( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ring_h615793168808539756t_unit
        @ ( partia2914870419229397556t_unit
          @ ^ [Uu: set_set_int] : K
          @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
        @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K )
        @ ( poly_o1995067004181805532t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.canonical_embedding_ring_hom
thf(fact_1244_carrier__polynomial__shell,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% carrier_polynomial_shell
thf(fact_1245_pzf_Ocarrier__polynomial__shell,axiom,
    ! [K: set_list_set_int,P2: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).

% pzf.carrier_polynomial_shell
thf(fact_1246_pzf_Ofinite__number__of__roots,axiom,
    ! [P2: list_list_set_int] :
      ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) )
     => ( finite7825103155802847241et_int @ ( collect_list_set_int @ ( polyno5593083006719613942t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ) ).

% pzf.finite_number_of_roots
thf(fact_1247_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_1248_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_1249_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_1250_finite__number__of__roots,axiom,
    ! [P2: list_a] :
      ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ r @ P2 ) ) ) ) ).

% finite_number_of_roots
thf(fact_1251_pzf_Opoly__of__const__consistent,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( poly_o1782926711940423372t_unit
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : K
            @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
        = ( poly_o1782926711940423372t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.poly_of_const_consistent
thf(fact_1252_pzf_Ocanonical__embedding__ring__hom,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ring_h8400476872904818028t_unit
        @ ( partia488849630590579876t_unit
          @ ^ [Uu: set_list_set_int] : K
          @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
        @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K )
        @ ( poly_o1782926711940423372t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).

% pzf.canonical_embedding_ring_hom
thf(fact_1253_pzf_Ocanonical__embedding__is__hom,axiom,
    ! [K: set_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( member5915730702165157418et_int @ ( poly_o1782926711940423372t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
        @ ( ring_h7978285497051570986t_unit
          @ ( partia488849630590579876t_unit
            @ ^ [Uu: set_list_set_int] : K
            @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
          @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ) ).

% pzf.canonical_embedding_is_hom
thf(fact_1254_rupture__surj__norm__is__hom,axiom,
    ! [K: set_a,P2: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_a_set_list_a @ ( comp_l4546862774662178811st_a_a @ ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) @ ( poly_of_const_a_b @ r ) )
          @ ( ring_h6109298854714515236t_unit
            @ ( partia8674076737563717228xt_a_b
              @ ^ [Uu: set_a] : K
              @ r )
            @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) ) ) ) ) ).

% rupture_surj_norm_is_hom
thf(fact_1255_pzf_Orupture__surj__norm__is__hom,axiom,
    ! [K: set_list_set_int,P2: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( member1423258033164139232et_int @ ( comp_l2129226576378413585et_int @ ( a_r_co2531912773843275078t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( cgenid7864102925118505423t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) ) @ ( poly_o1782926711940423372t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
          @ ( ring_h6922798158891550452t_unit
            @ ( partia488849630590579876t_unit
              @ ^ [Uu: set_list_set_int] : K
              @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
            @ ( polyno4866556214859693103t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ P2 ) ) ) ) ) ).

% pzf.rupture_surj_norm_is_hom
thf(fact_1256_gauss__poly__carr,axiom,
    ! [N: nat] : ( member_list_a @ ( card_I2373409586816755191ly_a_b @ r @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% gauss_poly_carr
thf(fact_1257_pzf_Oconst__term__simprules__shell_I4_J,axiom,
    ! [K: set_list_set_int,P2: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) )
          = ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ) ) ).

% pzf.const_term_simprules_shell(4)
thf(fact_1258_pzf_Oconst__term__simprules__shell_I1_J,axiom,
    ! [K: set_list_set_int,P2: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( member_list_set_int @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) @ K ) ) ) ).

% pzf.const_term_simprules_shell(1)
thf(fact_1259_pzf_Oconst__term__simprules__shell_I3_J,axiom,
    ! [K: set_list_set_int,P2: list_list_set_int,Q2: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( ( member7440485371949894425et_int @ Q2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
         => ( ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 @ Q2 ) )
            = ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Q2 ) ) ) ) ) ) ).

% pzf.const_term_simprules_shell(3)
thf(fact_1260_pzf_Opderiv__inv,axiom,
    ! [K: set_list_set_int,F: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ F @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ F ) )
          = ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ F ) ) ) ) ) ).

% pzf.pderiv_inv
thf(fact_1261_pzf_Orcosets__subset__PowG,axiom,
    ! [H: set_list_set_int] :
      ( ( additi4441901441605882380t_unit @ H @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ord_le8104031629355235774et_int @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H ) @ ( pow_list_set_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).

% pzf.rcosets_subset_PowG
thf(fact_1262_pzf_Opderiv__zero,axiom,
    ! [K: set_list_set_int] :
      ( ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l4263514192383796165t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
      = ( zero_l4263514192383796165t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ).

% pzf.pderiv_zero
thf(fact_1263_pzf_Opderiv__carr,axiom,
    ! [K: set_list_set_int,F: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ F @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( member7440485371949894425et_int @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ F ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ) ) ).

% pzf.pderiv_carr
thf(fact_1264_pzf_Opderiv__add,axiom,
    ! [K: set_list_set_int,F: list_list_set_int,G: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ F @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( ( member7440485371949894425et_int @ G @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
         => ( ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ F @ G ) )
            = ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ F ) @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ G ) ) ) ) ) ) ).

% pzf.pderiv_add
thf(fact_1265_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
          @ ^ [P: list_a] : ( eval_a_b @ r @ P @ A ) ) ) ) ).

% eval_cring_hom
thf(fact_1266_a__card__cosets__equal,axiom,
    ! [C: set_a,H: set_a] :
      ( ( member_set_a @ C @ ( a_RCOSETS_a_b @ r @ H ) )
     => ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( finite_card_a @ C )
            = ( finite_card_a @ H ) ) ) ) ) ).

% a_card_cosets_equal
thf(fact_1267_a__rcosetsI,axiom,
    ! [H: set_a,X: a] :
      ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_set_a @ ( a_r_coset_a_b @ r @ H @ X ) @ ( a_RCOSETS_a_b @ r @ H ) ) ) ) ).

% a_rcosetsI
thf(fact_1268_rcosets__subset__PowG,axiom,
    ! [H: set_a] :
      ( ( additi2834746164131130830up_a_b @ H @ r )
     => ( ord_le3724670747650509150_set_a @ ( a_RCOSETS_a_b @ r @ H ) @ ( pow_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% rcosets_subset_PowG
thf(fact_1269_zf_Orcosets__subset__PowG,axiom,
    ! [H: set_set_int] :
      ( ( additi7073586575563672860t_unit @ H @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ord_le4317611570275147438et_int @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H ) @ ( pow_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% zf.rcosets_subset_PowG
thf(fact_1270_pzf_Opderiv__mult,axiom,
    ! [K: set_list_set_int,F: list_list_set_int,G: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ F @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( ( member7440485371949894425et_int @ G @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
         => ( ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ F @ G ) )
            = ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ F ) @ G ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ F @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ G ) ) ) ) ) ) ) ).

% pzf.pderiv_mult
thf(fact_1271_zf_Oideal__is__subalgebra,axiom,
    ! [K: set_set_int,I2: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
       => ( embedd2743979684206749024t_unit @ K @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.ideal_is_subalgebra
thf(fact_1272_zf_Ocarrier__is__subalgebra,axiom,
    ! [K: set_set_int] :
      ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
     => ( embedd2743979684206749024t_unit @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).

% zf.carrier_is_subalgebra
thf(fact_1273_zf_Osubalgebra__in__carrier,axiom,
    ! [K: set_set_int,V: set_set_int] :
      ( ( embedd2743979684206749024t_unit @ K @ V @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
     => ( ord_le4403425263959731960et_int @ V @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).

% zf.subalgebra_in_carrier
thf(fact_1274_pzf_Ois__root__poly__mult__imp__is__root,axiom,
    ! [P2: list_list_set_int,Q2: list_list_set_int,X: list_set_int] :
      ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) )
     => ( ( member7440485371949894425et_int @ Q2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) )
       => ( ( polyno5593083006719613942t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ P2 @ Q2 ) @ X )
         => ( ( polyno5593083006719613942t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ X )
            | ( polyno5593083006719613942t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Q2 @ X ) ) ) ) ) ).

% pzf.is_root_poly_mult_imp_is_root
thf(fact_1275_pzf_Oconst__term__simprules__shell_I2_J,axiom,
    ! [K: set_list_set_int,P2: list_list_set_int,Q2: list_list_set_int] :
      ( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
     => ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
       => ( ( member7440485371949894425et_int @ Q2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
         => ( ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 @ Q2 ) )
            = ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Q2 ) ) ) ) ) ) ).

% pzf.const_term_simprules_shell(2)
thf(fact_1276_pzf_Ocring__fieldI,axiom,
    ( ( ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
      = ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
   => ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).

% pzf.cring_fieldI
thf(fact_1277_m__lcomm,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
            = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).

% m_lcomm

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( image_list_set_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ y )
    = ( image_list_set_int_a
      @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x )
      @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
        @ ( a_kern885551317661704990it_a_b @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ r
          @ ^ [P: list_set_int] : ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ P ) @ x ) )
        @ y ) ) ) ).

%------------------------------------------------------------------------------