%------------------------------------------------------------------------------
% 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    : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00128_004629__17201250_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1767 ( 152 unt; 481 typ;   0 def)
%            Number of atoms       : 5302 (1602 equ;   0 cnn)
%            Maximal formula atoms :   14 (   4 avg)
%            Number of connectives : 27047 ( 171   ~;  38   |; 128   &;23184   @)
%                                         (   0 <=>;3526  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (  10 avg)
%            Number of types       :   58 (  57 usr)
%            Number of type conns  : 3249 (3249   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  427 ( 424 usr;  18 con; 0-4 aty)
%            Number of variables   : 4654 ( 723   ^;3849   !;  82   ?;4654   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 15:37:08.733
%------------------------------------------------------------------------------
% Could-be-implicit typings (57)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
    partia5333488208502193986t_unit: $tType ).

thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
    partia4556295656693239580t_unit: $tType ).

thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
    partia2956882679547061052t_unit: $tType ).

thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
    partia7496981018696276118t_unit: $tType ).

thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_Itf__a_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J_J,type,
    partia2670972154091845814t_unit: $tType ).

thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_Itf__a_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_Itf__a_J_Mt__Product____Type__Ounit_J_J_J,type,
    partia6043505979758434576t_unit: $tType ).

thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
    partia2175431115845679010xt_a_b: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    produc7709606177366032167list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_na3751069889010993891list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    set_se1917860372504128155list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_a_7189831355048152981list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_na7345207451400031092list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_nat_a_a_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_Mtf__a_J_J,type,
    set_nat_list_a_a_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    set_li6773872926390105121list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_li3422455791611400469list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_se5067313844698916539list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    set_li1071299071675007611list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_a_list_a_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_a_a_a_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_Mtf__a_J_J,type,
    set_a_list_a_a_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    set_nat_list_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    set_nat_set_list_a: $tType ).

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

thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_nat_a_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
    set_nat_list_a_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    set_list_list_list_a: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    set_list_set_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_Mtf__a_J_J,type,
    set_nat_a_a_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
    set_a_list_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_list_a_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
    set_list_list_a_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
    set_a_set_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
    set_set_list_a_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    set_a_a_list_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
    set_a_list_a_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_Mtf__a_J_J,type,
    set_a_a_a_a: $tType ).

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
    set_nat_list_a: $tType ).

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

thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
    set_nat_a_a: $tType ).

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

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

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

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

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

thf(ty_n_t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
    set_a_a_a: $tType ).

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

thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
    set_nat_a: $tType ).

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

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

thf(ty_n_t__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__List__Olist_Itf__a_J,type,
    list_a: $tType ).

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

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

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

% Explicit typings (424)
thf(sy_c_AbelCoset_Oa__kernel_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    a_kern7116238624728830086it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > ( list_a > a ) > set_list_a ).

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

thf(sy_c_AbelCoset_Oabelian__group__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    abelia8217020544048703197it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > ( list_a > a ) > $o ).

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

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

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

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
    partia2464479390973590831t_unit: partia2956882679547061052t_unit > set_list_list_a ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J,type,
    partia7265347635606999311t_unit: partia4556295656693239580t_unit > set_list_set_list_a ).

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

thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
    partia141011252114345353t_unit: partia7496981018696276118t_unit > set_set_list_a ).

thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
    partia707051561876973205xt_a_b: partia2175431115845679010xt_a_b > set_a ).

thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001t__List__Olist_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
    partia9041243232023819264t_unit: ( set_list_a > set_list_a ) > partia2670972154091845814t_unit > partia2670972154091845814t_unit ).

thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
    partia8674076737563717228xt_a_b: ( set_a > set_a ) > partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b ).

thf(sy_c_Coset_Oorder_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
    order_3240872107759947550t_unit: partia2670972154091845814t_unit > nat ).

thf(sy_c_Coset_Oorder_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
    order_a_ring_ext_a_b: partia2175431115845679010xt_a_b > nat ).

thf(sy_c_Divisibility_Oessentially__equal_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
    essent703981920984620806t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a > $o ).

thf(sy_c_Divisibility_Ofactors_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
    factor7181967632740204193t_unit: partia2670972154091845814t_unit > list_list_a > list_a > $o ).

thf(sy_c_Divisibility_Omonoid__cancel_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
    monoid4303264861975686087t_unit: partia2670972154091845814t_unit > $o ).

thf(sy_c_Divisibility_Oprime_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
    prime_2011924034616061926t_unit: partia2670972154091845814t_unit > list_a > $o ).

thf(sy_c_Divisibility_Oprime_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
    prime_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > $o ).

thf(sy_c_Embedded__Algebras_Oring_OSpan_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    embedd4402942584324845940t_unit: partia2670972154091845814t_unit > set_list_a > list_list_a > set_list_a ).

thf(sy_c_Embedded__Algebras_Oring_OSpan_001tf__a_001tf__b,type,
    embedded_Span_a_b: partia2175431115845679010xt_a_b > set_a > list_a > set_a ).

thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    embedd1345800358437254783t_unit: partia2670972154091845814t_unit > set_list_a > set_list_a > $o ).

thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001tf__a_001tf__b,type,
    embedd8708762675212832759on_a_b: partia2175431115845679010xt_a_b > set_a > set_a > $o ).

thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    embedd5150658419831591667t_unit: partia2670972154091845814t_unit > set_list_a > list_a > set_list_a > set_list_a ).

thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001tf__a_001tf__b,type,
    embedd971793762689825387on_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a > set_a ).

thf(sy_c_Embedded__Algebras_Osubalgebra_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    embedd1768981623711841426t_unit: set_list_a > set_list_a > partia2670972154091845814t_unit > $o ).

thf(sy_c_Embedded__Algebras_Osubalgebra_001tf__a_001tf__b,type,
    embedd9027525575939734154ra_a_b: set_a > set_a > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_001t__List__Olist_Itf__a_J,type,
    finpro3417560807142560175list_a: partia2956882679547061052t_unit > ( list_a > list_list_a ) > set_list_a > list_list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_001t__Nat__Onat,type,
    finpro4561275463894985573it_nat: partia2956882679547061052t_unit > ( nat > list_list_a ) > set_nat > list_list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_001tf__a,type,
    finpro5596966875920909993unit_a: partia2956882679547061052t_unit > ( a > list_list_a ) > set_a > list_list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    finpro3153191432819564877st_a_a: partia2670972154091845814t_unit > ( ( ( nat > list_a ) > a ) > list_a ) > set_nat_list_a_a > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
    finpro5068724286038259027at_a_a: partia2670972154091845814t_unit > ( ( ( nat > a ) > a ) > list_a ) > set_nat_a_a > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    finpro5783859944493153077st_a_a: partia2670972154091845814t_unit > ( ( ( a > list_a ) > a ) > list_a ) > set_a_list_a_a > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
    finpro3871538611420980027_a_a_a: partia2670972154091845814t_unit > ( ( ( a > a ) > a ) > list_a ) > set_a_a_a > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
    finpro5225711790780397718list_a: partia2670972154091845814t_unit > ( ( nat > list_a ) > list_a ) > set_nat_list_a > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001_062_It__Nat__Onat_Mtf__a_J,type,
    finpro2700224059436637840_nat_a: partia2670972154091845814t_unit > ( ( nat > a ) > list_a ) > set_nat_a > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    finpro1324004282804121728list_a: partia2670972154091845814t_unit > ( ( a > list_a ) > list_a ) > set_a_list_a > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001_062_Itf__a_Mtf__a_J,type,
    finpro4522412760713991034it_a_a: partia2670972154091845814t_unit > ( ( a > a ) > list_a ) > set_a_a > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001t__List__Olist_Itf__a_J,type,
    finpro738134188688310831list_a: partia2670972154091845814t_unit > ( list_a > list_a ) > set_list_a > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001t__Nat__Onat,type,
    finpro1918104735009086181it_nat: partia2670972154091845814t_unit > ( nat > list_a ) > set_nat > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001tf__a,type,
    finpro4329226410377213737unit_a: partia2670972154091845814t_unit > ( a > list_a ) > set_a > list_a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    finpro3076322037277673151st_a_a: partia2175431115845679010xt_a_b > ( ( ( nat > list_a ) > a ) > a ) > set_nat_list_a_a > a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
    finpro7303668365752794053at_a_a: partia2175431115845679010xt_a_b > ( ( ( nat > a ) > a ) > a ) > set_nat_a_a > a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    finpro3356682085724622403st_a_a: partia2175431115845679010xt_a_b > ( ( ( a > list_a ) > a ) > a ) > set_a_list_a_a > a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
    finpro8565617760274797897_a_a_a: partia2175431115845679010xt_a_b > ( ( ( a > a ) > a ) > a ) > set_a_a_a > a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
    finpro4838020199848830884list_a: partia2175431115845679010xt_a_b > ( ( nat > list_a ) > a ) > set_nat_list_a > a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_It__Nat__Onat_Mtf__a_J,type,
    finpro5839458686994656414_nat_a: partia2175431115845679010xt_a_b > ( ( nat > a ) > a ) > set_nat_a > a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    finpro2305829206037604594list_a: partia2175431115845679010xt_a_b > ( ( a > list_a ) > a ) > set_a_list_a > a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_Itf__a_Mtf__a_J,type,
    finpro3012607322079259884_b_a_a: partia2175431115845679010xt_a_b > ( ( a > a ) > a ) > set_a_a > a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001t__List__Olist_Itf__a_J,type,
    finpro6052973074229812797list_a: partia2175431115845679010xt_a_b > ( list_a > a ) > set_list_a > a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    finpro1280035270526425175_b_nat: partia2175431115845679010xt_a_b > ( nat > a ) > set_nat > a ).

thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001tf__a,type,
    finpro205304725090349623_a_b_a: partia2175431115845679010xt_a_b > ( a > a ) > set_a > a ).

thf(sy_c_Finite__Set_Ofinite_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
    finite7239108116303828181at_a_a: set_nat_a_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    finite681744870773793075st_a_a: set_a_list_a_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
    finite_finite_a_a_a: set_a_a_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
    finite7630042315537210004list_a: set_nat_list_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mtf__a_J,type,
    finite_finite_nat_a: set_nat_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    finite8564763978580267522list_a: set_a_list_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001_062_Itf__a_Mtf__a_J,type,
    finite_finite_a_a: set_a_a > $o ).

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

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

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

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

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

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
    finite_finite_set_a: set_set_a > $o ).

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

thf(sy_c_Fun_Ocomp_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J_001tf__a,type,
    comp_l6744607643581155169st_a_a: ( list_list_a > list_a ) > ( a > list_list_a ) > a > list_a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a_001t__Nat__Onat,type,
    comp_l4331022603867890727_a_nat: ( list_list_a > a ) > ( nat > list_list_a ) > nat > a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a_001tf__a,type,
    comp_list_list_a_a_a: ( list_list_a > a ) > ( a > list_list_a ) > a > a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J_001tf__a,type,
    comp_list_a_list_a_a: ( list_a > list_a ) > ( a > list_a ) > a > list_a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001tf__a_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
    comp_l7081298308343408974list_a: ( list_a > a ) > ( ( nat > list_a ) > list_a ) > ( nat > list_a ) > a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
    comp_list_a_a_nat_a: ( list_a > a ) > ( ( nat > a ) > list_a ) > ( nat > a ) > a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001tf__a_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    comp_l1888214947559275976list_a: ( list_a > a ) > ( ( a > list_a ) > list_a ) > ( a > list_a ) > a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001tf__a_001_062_Itf__a_Mtf__a_J,type,
    comp_list_a_a_a_a: ( list_a > a ) > ( ( a > a ) > list_a ) > ( a > a ) > a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001tf__a_001t__List__Olist_Itf__a_J,type,
    comp_list_a_a_list_a: ( list_a > a ) > ( list_a > list_a ) > list_a > a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001tf__a_001t__Nat__Onat,type,
    comp_list_a_a_nat: ( list_a > a ) > ( nat > list_a ) > nat > a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001tf__a_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    comp_l2163758027525145287list_a: ( list_a > a ) > ( set_list_a > list_a ) > set_list_a > a ).

thf(sy_c_Fun_Ocomp_001t__List__Olist_Itf__a_J_001tf__a_001tf__a,type,
    comp_list_a_a_a: ( list_a > a ) > ( a > list_a ) > a > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    comp_a5642107814311143777list_a: ( a > list_a ) > ( list_list_a > a ) > list_list_a > list_a ).

thf(sy_c_Fun_Ocomp_001tf__a_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    comp_a_list_a_list_a: ( a > list_a ) > ( list_a > a ) > list_a > list_a ).

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_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
    comp_a_a_nat_a: ( a > a ) > ( ( nat > a ) > a ) > ( nat > a ) > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001_062_Itf__a_Mtf__a_J,type,
    comp_a_a_a_a: ( a > a ) > ( ( a > a ) > a ) > ( a > a ) > 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__Nat__Onat,type,
    comp_a_a_nat: ( a > a ) > ( nat > a ) > nat > a ).

thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    comp_a_a_set_list_a: ( a > a ) > ( set_list_a > a ) > set_list_a > 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_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
    inj_on_list_a_nat: ( list_a > nat ) > set_list_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001tf__a,type,
    inj_on_list_a_a: ( list_a > a ) > set_list_a > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
    inj_on_nat_list_a: ( nat > list_a ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001tf__a,type,
    inj_on_nat_a: ( nat > a ) > set_nat > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
    inj_on_a_nat_a: ( a > nat > a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001_062_Itf__a_Mtf__a_J,type,
    inj_on_a_a_a: ( a > a > a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__List__Olist_Itf__a_J,type,
    inj_on_a_list_a: ( a > list_a ) > set_a > $o ).

thf(sy_c_Fun_Oinj__on_001tf__a_001t__Nat__Onat,type,
    inj_on_a_nat: ( a > nat ) > set_a > $o ).

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

thf(sy_c_FuncSet_OPi_001_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_001t__List__Olist_Itf__a_J,type,
    pi_nat837482917391966783list_a: set_nat_list_a_a > ( ( ( nat > list_a ) > a ) > set_list_a ) > set_na3751069889010993891list_a ).

thf(sy_c_FuncSet_OPi_001_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_001tf__a,type,
    pi_nat_list_a_a_a: set_nat_list_a_a > ( ( ( nat > list_a ) > a ) > set_a ) > set_nat_list_a_a_a ).

thf(sy_c_FuncSet_OPi_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_001t__List__Olist_Itf__a_J,type,
    pi_nat_a_a_list_a: set_nat_a_a > ( ( ( nat > a ) > a ) > set_list_a ) > set_nat_a_a_list_a ).

thf(sy_c_FuncSet_OPi_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_001tf__a,type,
    pi_nat_a_a_a: set_nat_a_a > ( ( ( nat > a ) > a ) > set_a ) > set_nat_a_a_a ).

thf(sy_c_FuncSet_OPi_001_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_001t__List__Olist_Itf__a_J,type,
    pi_a_list_a_a_list_a: set_a_list_a_a > ( ( ( a > list_a ) > a ) > set_list_a ) > set_a_7189831355048152981list_a ).

thf(sy_c_FuncSet_OPi_001_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_001tf__a,type,
    pi_a_list_a_a_a: set_a_list_a_a > ( ( ( a > list_a ) > a ) > set_a ) > set_a_list_a_a_a ).

thf(sy_c_FuncSet_OPi_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_001t__List__Olist_Itf__a_J,type,
    pi_a_a_a_list_a: set_a_a_a > ( ( ( a > a ) > a ) > set_list_a ) > set_a_a_a_list_a ).

thf(sy_c_FuncSet_OPi_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_001tf__a,type,
    pi_a_a_a_a: set_a_a_a > ( ( ( a > a ) > a ) > set_a ) > set_a_a_a_a ).

thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
    pi_nat_list_a_list_a: set_nat_list_a > ( ( nat > list_a ) > set_list_a ) > set_na7345207451400031092list_a ).

thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_001tf__a,type,
    pi_nat_list_a_a: set_nat_list_a > ( ( nat > list_a ) > set_a ) > set_nat_list_a_a ).

thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mtf__a_J_001t__List__Olist_Itf__a_J,type,
    pi_nat_a_list_a: set_nat_a > ( ( nat > a ) > set_list_a ) > set_nat_a_list_a ).

thf(sy_c_FuncSet_OPi_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
    pi_nat_a_a: set_nat_a > ( ( nat > a ) > set_a ) > set_nat_a_a ).

thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
    pi_a_list_a_list_a: set_a_list_a > ( ( a > list_a ) > set_list_a ) > set_a_list_a_list_a ).

thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J_001tf__a,type,
    pi_a_list_a_a: set_a_list_a > ( ( a > list_a ) > set_a ) > set_a_list_a_a ).

thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001t__List__Olist_Itf__a_J,type,
    pi_a_a_list_a: set_a_a > ( ( a > a ) > set_list_a ) > set_a_a_list_a ).

thf(sy_c_FuncSet_OPi_001_062_Itf__a_Mtf__a_J_001tf__a,type,
    pi_a_a_a: set_a_a > ( ( a > a ) > set_a ) > set_a_a_a ).

thf(sy_c_FuncSet_OPi_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    pi_list_a_list_a: set_list_a > ( list_a > set_list_a ) > set_list_a_list_a ).

thf(sy_c_FuncSet_OPi_001t__List__Olist_Itf__a_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    pi_list_a_set_list_a: set_list_a > ( list_a > set_set_list_a ) > set_li1071299071675007611list_a ).

thf(sy_c_FuncSet_OPi_001t__List__Olist_Itf__a_J_001tf__a,type,
    pi_list_a_a: set_list_a > ( list_a > set_a ) > set_list_a_a ).

thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    pi_nat_list_list_a: set_nat > ( nat > set_list_list_a ) > set_nat_list_list_a ).

thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
    pi_nat_list_a: set_nat > ( nat > set_list_a ) > set_nat_list_a ).

thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    pi_nat_set_list_a: set_nat > ( nat > set_set_list_a ) > set_nat_set_list_a ).

thf(sy_c_FuncSet_OPi_001t__Nat__Onat_001tf__a,type,
    pi_nat_a: set_nat > ( nat > set_a ) > set_nat_a ).

thf(sy_c_FuncSet_OPi_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    pi_a_list_list_a: set_a > ( a > set_list_list_a ) > set_a_list_list_a ).

thf(sy_c_FuncSet_OPi_001tf__a_001t__List__Olist_Itf__a_J,type,
    pi_a_list_a: set_a > ( a > set_list_a ) > set_a_list_a ).

thf(sy_c_FuncSet_OPi_001tf__a_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    pi_a_set_list_a: set_a > ( a > set_set_list_a ) > set_a_set_list_a ).

thf(sy_c_FuncSet_OPi_001tf__a_001tf__a,type,
    pi_a_a: set_a > ( a > set_a ) > set_a_a ).

thf(sy_c_Group_Omonoid_Omult_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J,type,
    mult_l3065349954589089105t_unit: partia5333488208502193986t_unit > list_list_list_a > list_list_list_a > list_list_list_a ).

thf(sy_c_Group_Omonoid_Omult_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
    mult_l4853965630390486993t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).

thf(sy_c_Group_Omonoid_Omult_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J,type,
    mult_l5330480240434472913t_unit: partia4556295656693239580t_unit > list_set_list_a > list_set_list_a > list_set_list_a ).

thf(sy_c_Group_Omonoid_Omult_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
    mult_l7073676228092353617t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).

thf(sy_c_Group_Omonoid_Omult_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
    mult_s7802724872828879953t_unit: partia7496981018696276118t_unit > set_list_a > set_list_a > set_list_a ).

thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
    mult_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a > a ).

thf(sy_c_Group_Omonoid_Oone_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
    one_li8234411390022467901t_unit: partia2956882679547061052t_unit > list_list_a ).

thf(sy_c_Group_Omonoid_Oone_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
    one_li8328186300101108157t_unit: partia2670972154091845814t_unit > list_a ).

thf(sy_c_Group_Omonoid_Oone_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
    one_se1127990129394575805t_unit: partia7496981018696276118t_unit > set_list_a ).

thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
    one_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
    minus_1482667089342205261at_a_a: set_nat_a_a > set_nat_a_a > set_nat_a_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
    minus_4703795076483250129st_a_a: set_a_list_a_a > set_a_list_a_a > set_a_list_a_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
    minus_4881627095085115735_a_a_a: set_a_a_a > set_a_a_a > set_a_a_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
    minus_4169782841487898290list_a: set_nat_list_a > set_nat_list_a > set_nat_list_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
    minus_490503922182417452_nat_a: set_nat_a > set_nat_a > set_nat_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
    minus_3253130117680460026list_a: set_a_list_a > set_a_list_a > set_a_list_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
    minus_minus_set_a_a: set_a_a > set_a_a > set_a_a ).

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__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
    minus_minus_set_a: set_a > set_a > set_a ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Ideal_Ocgenideal_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
    cgenid9131348535277946915t_unit: partia2670972154091845814t_unit > list_a > set_list_a ).

thf(sy_c_Ideal_Ocgenideal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
    cgenid547466209912283029xt_a_b: partia2175431115845679010xt_a_b > a > set_a ).

thf(sy_c_Ideal_Ogenideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    genide3243992037924705879t_unit: partia2670972154091845814t_unit > set_list_a > set_list_a ).

thf(sy_c_Ideal_Omaximalideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    maxima6585700282301356660t_unit: set_list_a > partia2670972154091845814t_unit > $o ).

thf(sy_c_Ideal_Omaximalideal_001tf__a_001tf__b,type,
    maximalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_Ideal_Oprimeideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    primei6309817859076077608t_unit: set_list_a > partia2670972154091845814t_unit > $o ).

thf(sy_c_Ideal_Oprimeideal_001tf__a_001tf__b,type,
    primeideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_Ideal_Oprincipalideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    princi8786919440553033881t_unit: set_list_a > partia2670972154091845814t_unit > $o ).

thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
    principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_If_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    if_list_list_a: $o > list_list_a > list_list_a > list_list_a ).

thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
    if_list_a: $o > list_a > list_a > list_a ).

thf(sy_c_If_001tf__a,type,
    if_a: $o > a > a > a ).

thf(sy_c_Lagrange__Interpolation_Oring_Olagrange__basis__polynomial__aux_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    lagran3534788790333317459t_unit: partia2670972154091845814t_unit > set_list_a > list_list_a ).

thf(sy_c_Lagrange__Interpolation_Oring_Olagrange__basis__polynomial__aux_001tf__a_001tf__b,type,
    lagran9092808442999052491ux_a_b: partia2175431115845679010xt_a_b > set_a > list_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
    inf_inf_set_nat_a_a: set_nat_a_a > set_nat_a_a > set_nat_a_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
    inf_in8941282555790403672st_a_a: set_a_list_a_a > set_a_list_a_a > set_a_list_a_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
    inf_inf_set_a_a_a: set_a_a_a > set_a_a_a > set_a_a_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
    inf_in6652419485960844601list_a: set_nat_list_a > set_nat_list_a > set_nat_list_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
    inf_inf_set_nat_a: set_nat_a > set_nat_a > set_nat_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
    inf_inf_set_a_list_a: set_a_list_a > set_a_list_a > set_a_list_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
    inf_inf_set_a_a: set_a_a > set_a_a > set_a_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    inf_inf_set_list_a: set_list_a > set_list_a > set_list_a ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
    inf_inf_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
    inf_inf_set_a: set_a > set_a > set_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
    sup_su5649930751583389983list_a: set_nat_list_a > set_nat_list_a > set_nat_list_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
    sup_sup_set_nat_a: set_nat_a > set_nat_a > set_nat_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
    sup_sup_set_a_list_a: set_a_list_a > set_a_list_a > set_a_list_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
    sup_sup_set_a_a: set_a_a > set_a_a > set_a_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    sup_sup_set_list_a: set_list_a > set_list_a > set_list_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
    sup_sup_set_a: set_a > set_a > set_a ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
    cons_list_a: list_a > list_list_a > list_list_a ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
    nil_list_a: list_list_a ).

thf(sy_c_List_Olist_ONil_001tf__a,type,
    nil_a: list_a ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001tf__a,type,
    map_list_a_a: ( list_a > a ) > list_list_a > list_a ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_Itf__a_J,type,
    map_a_list_a: ( a > list_a ) > list_a > list_list_a ).

thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
    map_a_a: ( a > a ) > list_a > list_a ).

thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
    set_list_a2: list_list_a > set_list_a ).

thf(sy_c_List_Olist_Oset_001tf__a,type,
    set_a2: list_a > set_a ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    size_s349497388124573686list_a: list_list_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
    size_size_list_a: list_a > nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
    bot_bo3806784159821827511list_a: set_nat_list_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
    bot_bot_set_nat_a: set_nat_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
    bot_bot_set_a_list_a: set_a_list_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
    bot_bot_set_a_a: set_a_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    bot_bot_set_list_a: set_list_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
    bot_bo3186585308812441520list_a: set_set_list_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    bot_bot_set_set_a: set_set_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
    bot_bot_set_a: set_a ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    ord_less_set_list_a: set_list_a > set_list_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_J,type,
    ord_le3509452538356653652at_a_a: set_nat_a_a > set_nat_a_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
    ord_le2847972559472028170st_a_a: set_a_list_a_a > set_a_list_a_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_J,type,
    ord_le7181591058469194768_a_a_a: set_a_a_a > set_a_a_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
    ord_le2145805922479659755list_a: set_nat_list_a > set_nat_list_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
    ord_le871467723717165285_nat_a: set_nat_a > set_nat_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
    ord_le50412136050534657list_a: set_a_list_a > set_a_list_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
    ord_less_eq_set_a_a: set_a_a > set_a_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
    ord_less_eq_set_a: set_a > set_a > $o ).

thf(sy_c_Polynomial__Divisibility_Oring_Oalg__mult_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    polyno1672195411705137432t_unit: partia2956882679547061052t_unit > list_list_list_a > list_list_a > nat ).

thf(sy_c_Polynomial__Divisibility_Oring_Oalg__mult_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    polyno4259638811958763678t_unit: partia2670972154091845814t_unit > list_list_a > list_a > nat ).

thf(sy_c_Polynomial__Divisibility_Oring_Oalg__mult_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    polyno1088517687229135038t_unit: partia7496981018696276118t_unit > list_set_list_a > set_list_a > nat ).

thf(sy_c_Polynomial__Divisibility_Oring_Oalg__mult_001tf__a_001tf__b,type,
    polyno4422430861927485590lt_a_b: partia2175431115845679010xt_a_b > list_a > a > nat ).

thf(sy_c_Polynomial__Divisibility_Oring_Oexp__base_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    polyno3522816881121920896t_unit: partia2670972154091845814t_unit > list_a > nat > list_list_a ).

thf(sy_c_Polynomial__Divisibility_Oring_Oexp__base_001tf__a_001tf__b,type,
    polyno2922411391617481336se_a_b: partia2175431115845679010xt_a_b > a > nat > list_a ).

thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    polyno5142720416380192742t_unit: partia2956882679547061052t_unit > list_list_list_a > list_list_a > $o ).

thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    polyno6951661231331188332t_unit: partia2670972154091845814t_unit > list_list_a > list_a > $o ).

thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    polyno4320237611291262604t_unit: partia7496981018696276118t_unit > list_set_list_a > set_list_a > $o ).

thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001tf__a_001tf__b,type,
    polyno4133073214067823460ot_a_b: partia2175431115845679010xt_a_b > list_a > a > $o ).

thf(sy_c_Polynomial__Divisibility_Oring_Olong__divides_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    polyno6947042923167803568t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a > produc7709606177366032167list_a > $o ).

thf(sy_c_Polynomial__Divisibility_Oring_Olong__divides_001tf__a_001tf__b,type,
    polyno2806191415236617128es_a_b: partia2175431115845679010xt_a_b > list_a > list_a > produc9164743771328383783list_a > $o ).

thf(sy_c_Polynomial__Divisibility_Oring_Opdiv_001tf__a_001tf__b,type,
    polynomial_pdiv_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).

thf(sy_c_Polynomial__Divisibility_Oring_Opmod_001tf__a_001tf__b,type,
    polynomial_pmod_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).

thf(sy_c_Polynomial__Divisibility_Oring_Osplitted_001tf__a_001tf__b,type,
    polyno8329700637149614481ed_a_b: partia2175431115845679010xt_a_b > list_a > $o ).

thf(sy_c_Polynomial__Divisibility_Orupture_001tf__a_001tf__b,type,
    polyno5459750281392823787re_a_b: partia2175431115845679010xt_a_b > set_a > list_a > partia7496981018696276118t_unit ).

thf(sy_c_Polynomials_Oring_Oconst__term_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    const_6738166269504826821t_unit: partia2670972154091845814t_unit > list_list_a > list_a ).

thf(sy_c_Polynomials_Oring_Oconst__term_001tf__a_001tf__b,type,
    const_term_a_b: partia2175431115845679010xt_a_b > list_a > a ).

thf(sy_c_Polynomials_Oring_Oeval_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    eval_l1088911609197519410t_unit: partia2956882679547061052t_unit > list_list_list_a > list_list_a > list_list_a ).

thf(sy_c_Polynomials_Oring_Oeval_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    eval_l34571156754992824t_unit: partia2670972154091845814t_unit > list_list_a > list_a > list_a ).

thf(sy_c_Polynomials_Oring_Oeval_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    eval_s5133945360527818456t_unit: partia7496981018696276118t_unit > list_set_list_a > set_list_a > set_list_a ).

thf(sy_c_Polynomials_Oring_Oeval_001tf__a_001tf__b,type,
    eval_a_b: partia2175431115845679010xt_a_b > list_a > a > a ).

thf(sy_c_Polynomials_Oring_Opoly__of__const_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    poly_o8716471131768098070t_unit: partia2670972154091845814t_unit > list_a > list_list_a ).

thf(sy_c_Polynomials_Oring_Opoly__of__const_001tf__a_001tf__b,type,
    poly_of_const_a_b: partia2175431115845679010xt_a_b > a > list_a ).

thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    univ_p2250591967980070728t_unit: partia2956882679547061052t_unit > set_list_list_a > partia5333488208502193986t_unit ).

thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    univ_p7953238456130426574t_unit: partia2670972154091845814t_unit > set_list_a > partia2956882679547061052t_unit ).

thf(sy_c_Polynomials_Ouniv__poly_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    univ_p863672496597069550t_unit: partia7496981018696276118t_unit > set_set_list_a > partia4556295656693239580t_unit ).

thf(sy_c_Polynomials_Ouniv__poly_001tf__a_001tf__b,type,
    univ_poly_a_b: partia2175431115845679010xt_a_b > set_a > partia2670972154091845814t_unit ).

thf(sy_c_Polynomials_Ovar_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    var_li8453953174693405341t_unit: partia2670972154091845814t_unit > list_list_a ).

thf(sy_c_Polynomials_Ovar_001tf__a_001tf__b,type,
    var_a_b: partia2175431115845679010xt_a_b > list_a ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    produc8696003437204565271list_a: list_list_a > list_list_a > produc7709606177366032167list_a ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).

thf(sy_c_QuotRing_OFactRing_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    factRi3329376332477095402t_unit: partia2670972154091845814t_unit > set_list_a > partia7496981018696276118t_unit ).

thf(sy_c_QuotRing_OFactRing_001tf__a_001tf__b,type,
    factRing_a_b: partia2175431115845679010xt_a_b > set_a > partia6043505979758434576t_unit ).

thf(sy_c_QuotRing_Ois__ring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    is_rin2993610189962786360t_unit: partia2670972154091845814t_unit > partia7496981018696276118t_unit > $o ).

thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    is_rin4843644836746533432t_unit: partia7496981018696276118t_unit > partia2670972154091845814t_unit > $o ).

thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    is_rin5597148638330396976it_a_b: partia7496981018696276118t_unit > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    is_rin6001486760346555702it_a_b: partia6043505979758434576t_unit > partia2175431115845679010xt_a_b > $o ).

thf(sy_c_QuotRing_Ois__ring__iso_001tf__a_001tf__b_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
    is_rin9099215527551818550t_unit: partia2175431115845679010xt_a_b > partia6043505979758434576t_unit > $o ).

thf(sy_c_QuotRing_Oring__iso_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    ring_i8122894263081988538it_a_b: partia7496981018696276118t_unit > partia2175431115845679010xt_a_b > set_set_list_a_a ).

thf(sy_c_Ring_Oa__inv_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    a_inv_8944721093294617173t_unit: partia2670972154091845814t_unit > list_a > list_a ).

thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
    a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).

thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
    a_minu4820293213911669576t_unit: partia5333488208502193986t_unit > list_list_list_a > list_list_list_a > list_list_list_a ).

thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    a_minu2241224857956505934t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).

thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
    a_minu6874796375791416686t_unit: partia4556295656693239580t_unit > list_set_list_a > list_set_list_a > list_set_list_a ).

thf(sy_c_Ring_Oa__minus_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    a_minu3984020753470702548t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).

thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
    a_minus_a_b: partia2175431115845679010xt_a_b > a > a > a ).

thf(sy_c_Ring_Oabelian__monoid_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    abelia3641329199688042803t_unit: partia2956882679547061052t_unit > $o ).

thf(sy_c_Ring_Oabelian__monoid_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    abelia226231641709521465t_unit: partia2670972154091845814t_unit > $o ).

thf(sy_c_Ring_Oabelian__monoid_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    abelia3322010900105369177t_unit: partia7496981018696276118t_unit > $o ).

thf(sy_c_Ring_Oabelian__monoid_001tf__a_001tf__b,type,
    abelian_monoid_a_b: partia2175431115845679010xt_a_b > $o ).

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

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

thf(sy_c_Ring_Odomain_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    domain1617769409708967785t_unit: partia7496981018696276118t_unit > $o ).

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

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

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

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

thf(sy_c_Ring_Ofinsum_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J,type,
    finsum159916373282764644list_a: partia2956882679547061052t_unit > ( list_a > list_list_a ) > set_list_a > list_list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__Nat__Onat,type,
    finsum3990971441743328240it_nat: partia2956882679547061052t_unit > ( nat > list_list_a ) > set_nat > list_list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a,type,
    finsum463596448938265310unit_a: partia2956882679547061052t_unit > ( a > list_list_a ) > set_a > list_list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    finsum2390613145322704222st_a_a: partia2670972154091845814t_unit > ( ( ( nat > list_a ) > a ) > list_a ) > set_nat_list_a_a > list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
    finsum5886066780512034276at_a_a: partia2670972154091845814t_unit > ( ( ( nat > a ) > a ) > list_a ) > set_nat_a_a > list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    finsum416311231956857188st_a_a: partia2670972154091845814t_unit > ( ( ( a > list_a ) > a ) > list_a ) > set_a_list_a_a > list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
    finsum5639492236432915690_a_a_a: partia2670972154091845814t_unit > ( ( ( a > a ) > a ) > list_a ) > set_a_a_a > list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
    finsum4426778018909949125list_a: partia2670972154091845814t_unit > ( ( nat > list_a ) > list_a ) > set_nat_list_a > list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001_062_It__Nat__Onat_Mtf__a_J,type,
    finsum7881878320310621759_nat_a: partia2670972154091845814t_unit > ( ( nat > a ) > list_a ) > set_nat_a > list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    finsum3406705939639355921list_a: partia2670972154091845814t_unit > ( ( a > list_a ) > list_a ) > set_a_list_a > list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001_062_Itf__a_Mtf__a_J,type,
    finsum4909627262586317195it_a_a: partia2670972154091845814t_unit > ( ( a > a ) > list_a ) > set_a_a > list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J,type,
    finsum8721804980556663006list_a: partia2670972154091845814t_unit > ( list_a > list_a ) > set_list_a > list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Nat__Onat,type,
    finsum3497512462216549110it_nat: partia2670972154091845814t_unit > ( nat > list_a ) > set_nat > list_a ).

thf(sy_c_Ring_Ofinsum_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a,type,
    finsum7322697649718157656unit_a: partia2670972154091845814t_unit > ( a > list_a ) > set_a > list_a ).

thf(sy_c_Ring_Ofinsum_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J,type,
    finsum3353672621075604670list_a: partia7496981018696276118t_unit > ( list_a > set_list_a ) > set_list_a > set_list_a ).

thf(sy_c_Ring_Ofinsum_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__Nat__Onat,type,
    finsum1644613019690897174it_nat: partia7496981018696276118t_unit > ( nat > set_list_a ) > set_nat > set_list_a ).

thf(sy_c_Ring_Ofinsum_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a,type,
    finsum8105788327582199096unit_a: partia7496981018696276118t_unit > ( a > set_list_a ) > set_a > set_list_a ).

thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
    finsum1341700292807219277list_a: partia2175431115845679010xt_a_b > ( ( nat > list_a ) > a ) > set_nat_list_a > a ).

thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001_062_It__Nat__Onat_Mtf__a_J,type,
    finsum_a_b_nat_a: partia2175431115845679010xt_a_b > ( ( nat > a ) > a ) > set_nat_a > a ).

thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    finsum_a_b_a_list_a: partia2175431115845679010xt_a_b > ( ( a > list_a ) > a ) > set_a_list_a > a ).

thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001_062_Itf__a_Mtf__a_J,type,
    finsum_a_b_a_a: partia2175431115845679010xt_a_b > ( ( a > a ) > a ) > set_a_a > a ).

thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001t__List__Olist_Itf__a_J,type,
    finsum_a_b_list_a: partia2175431115845679010xt_a_b > ( list_a > a ) > set_list_a > a ).

thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001t__Nat__Onat,type,
    finsum_a_b_nat: partia2175431115845679010xt_a_b > ( nat > a ) > set_nat > a ).

thf(sy_c_Ring_Ofinsum_001tf__a_001tf__b_001tf__a,type,
    finsum_a_b_a: partia2175431115845679010xt_a_b > ( a > a ) > set_a > a ).

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

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

thf(sy_c_Ring_Oring_Oadd_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    add_se2486902527185523630t_unit: partia7496981018696276118t_unit > set_list_a > set_list_a > set_list_a ).

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

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

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

thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    zero_s2910681146719230829t_unit: partia7496981018696276118t_unit > set_list_a ).

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

thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_h5031276006722532742t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > set_li3422455791611400469list_a ).

thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    ring_h8078271382950527358it_a_b: partia2956882679547061052t_unit > partia2175431115845679010xt_a_b > set_list_list_a_a ).

thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h8002040739877300486t_unit: partia2670972154091845814t_unit > partia2956882679547061052t_unit > set_li6773872926390105121list_a ).

thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_h7399960747407462284t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > set_list_a_list_a ).

thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h6188449271506562988t_unit: partia2670972154091845814t_unit > partia7496981018696276118t_unit > set_li1071299071675007611list_a ).

thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    ring_h2895973938487309444it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > set_list_a_a ).

thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_h8038483918290310060t_unit: partia7496981018696276118t_unit > partia2670972154091845814t_unit > set_se5067313844698916539list_a ).

thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h6076331213207892940t_unit: partia7496981018696276118t_unit > partia7496981018696276118t_unit > set_se1917860372504128155list_a ).

thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    ring_h8906680420194085028it_a_b: partia7496981018696276118t_unit > partia2175431115845679010xt_a_b > set_set_list_a_a ).

thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h6858658657455840382t_unit: partia2175431115845679010xt_a_b > partia2956882679547061052t_unit > set_a_list_list_a ).

thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_h405018892823518980t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > set_a_list_a ).

thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h6109298854714515236t_unit: partia2175431115845679010xt_a_b > partia7496981018696276118t_unit > set_a_set_list_a ).

thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001tf__a_001tf__b,type,
    ring_hom_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).

thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h5732965726171697305t_unit: partia5333488208502193986t_unit > partia2956882679547061052t_unit > ( list_list_list_a > list_list_a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_h453377649743177125t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > ( list_list_a > list_a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    ring_h3216284694433613853it_a_b: partia2956882679547061052t_unit > partia2175431115845679010xt_a_b > ( list_list_a > a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h4012356781940393445t_unit: partia4556295656693239580t_unit > partia7496981018696276118t_unit > ( list_set_list_a > set_list_a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h3424142382897944869t_unit: partia2670972154091845814t_unit > partia2956882679547061052t_unit > ( list_a > list_list_a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_h8282015026914974507t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > ( list_a > list_a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h5296475915237130059t_unit: partia2670972154091845814t_unit > partia7496981018696276118t_unit > ( list_a > set_list_a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    ring_h1547129875642963619it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > ( list_a > a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
    ring_h1101743994381864643it_a_b: partia7496981018696276118t_unit > partia2175431115845679010xt_a_b > ( set_list_a > a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001tf__a_001tf__b_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h1996671968938926877t_unit: partia2175431115845679010xt_a_b > partia2956882679547061052t_unit > ( a > list_list_a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_h8279546866833948963t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > ( a > list_a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001tf__a_001tf__b_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    ring_h7527734465757070659t_unit: partia2175431115845679010xt_a_b > partia7496981018696276118t_unit > ( a > set_list_a ) > $o ).

thf(sy_c_Ring_Oring__hom__cring_001tf__a_001tf__b_001tf__a_001tf__b,type,
    ring_h661254511236296859_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > ( a > a ) > $o ).

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

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

thf(sy_c_Ring_Osemiring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    semiri4000464634269493571t_unit: partia7496981018696276118t_unit > $o ).

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

thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    ring_e7478897652244013592t_unit: partia2670972154091845814t_unit > ( list_a > nat ) > $o ).

thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001tf__a_001tf__b,type,
    ring_e8745995371659049232in_a_b: partia2175431115845679010xt_a_b > ( a > nat ) > $o ).

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

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

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

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

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

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

thf(sy_c_Set_OCollect_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
    collect_nat_a_a: ( ( ( nat > a ) > a ) > $o ) > set_nat_a_a ).

thf(sy_c_Set_OCollect_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
    collect_a_a_a: ( ( ( a > a ) > a ) > $o ) > set_a_a_a ).

thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
    collect_nat_list_a: ( ( nat > list_a ) > $o ) > set_nat_list_a ).

thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
    collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).

thf(sy_c_Set_OCollect_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    collect_a_list_a: ( ( a > list_a ) > $o ) > set_a_list_a ).

thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__a_J,type,
    collect_a_a: ( ( a > a ) > $o ) > set_a_a ).

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

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

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
    collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
    collect_set_a: ( set_a > $o ) > set_set_a ).

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

thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    image_8260866953997875467list_a: ( list_a > list_list_a ) > set_list_a > set_list_list_a ).

thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).

thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
    image_list_a_a: ( list_a > a ) > set_list_a > set_a ).

thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001tf__a,type,
    image_set_list_a_a: ( set_list_a > a ) > set_set_list_a > set_a ).

thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
    image_a_list_a: ( a > list_a ) > set_a > set_list_a ).

thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
    image_a_a: ( a > a ) > set_a > set_a ).

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

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

thf(sy_c_Set_Othe__elem_001tf__a,type,
    the_elem_a: set_a > a ).

thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
    subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).

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

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

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

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

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

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

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

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

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

thf(sy_c_UnivPoly_Obound_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    bound_list_list_a: list_list_a > nat > ( nat > list_list_a ) > $o ).

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

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

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

thf(sy_c_UnivPoly_Oup_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
    up_lis8963924889346801084t_unit: partia2956882679547061052t_unit > set_nat_list_list_a ).

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

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

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

thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member4189978660109718316list_a: ( ( ( nat > list_a ) > a ) > list_a ) > set_na3751069889010993891list_a > $o ).

thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_Mtf__a_J,type,
    member7453349880713133094_a_a_a: ( ( ( nat > list_a ) > a ) > a ) > set_nat_list_a_a_a > $o ).

thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member1630437648146986790list_a: ( ( ( nat > a ) > a ) > list_a ) > set_nat_a_a_list_a > $o ).

thf(sy_c_member_001_062_I_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J_Mtf__a_J,type,
    member_nat_a_a_a: ( ( ( nat > a ) > a ) > a ) > set_nat_a_a_a > $o ).

thf(sy_c_member_001_062_I_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member455139831191412214list_a: ( ( ( a > list_a ) > a ) > list_a ) > set_a_7189831355048152981list_a > $o ).

thf(sy_c_member_001_062_I_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J_Mtf__a_J,type,
    member_a_list_a_a_a: ( ( ( a > list_a ) > a ) > a ) > set_a_list_a_a_a > $o ).

thf(sy_c_member_001_062_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member_a_a_a_list_a: ( ( ( a > a ) > a ) > list_a ) > set_a_a_a_list_a > $o ).

thf(sy_c_member_001_062_I_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J_Mtf__a_J,type,
    member_a_a_a_a: ( ( ( a > a ) > a ) > a ) > set_a_a_a_a > $o ).

thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
    member1174670940520477013list_a: ( ( nat > list_a ) > list_a ) > set_na7345207451400031092list_a > $o ).

thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    member_nat_list_a_a: ( ( nat > list_a ) > a ) > set_nat_list_a_a > $o ).

thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member_nat_a_list_a: ( ( nat > a ) > list_a ) > set_nat_a_list_a > $o ).

thf(sy_c_member_001_062_I_062_It__Nat__Onat_Mtf__a_J_Mtf__a_J,type,
    member_nat_a_a: ( ( nat > a ) > a ) > set_nat_a_a > $o ).

thf(sy_c_member_001_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
    member4106537806236033721list_a: ( ( a > list_a ) > list_a ) > set_a_list_a_list_a > $o ).

thf(sy_c_member_001_062_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    member_a_list_a_a: ( ( a > list_a ) > a ) > set_a_list_a_a > $o ).

thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member_a_a_list_a: ( ( a > a ) > list_a ) > set_a_a_list_a > $o ).

thf(sy_c_member_001_062_I_062_Itf__a_Mtf__a_J_Mtf__a_J,type,
    member_a_a_a: ( ( a > a ) > a ) > set_a_a_a > $o ).

thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
    member7168557129179038582list_a: ( list_list_a > list_a ) > set_li3422455791611400469list_a > $o ).

thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
    member_list_list_a_a: ( list_list_a > a ) > set_list_list_a_a > $o ).

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

thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).

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

thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
    member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).

thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    member8650753269014980122list_a: ( nat > list_list_a ) > set_nat_list_list_a > $o ).

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

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

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

thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
    member5910328476188217884list_a: ( set_list_a > list_a ) > set_se5067313844698916539list_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_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
    member_a_list_list_a: ( a > list_list_a ) > set_a_list_list_a > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_member_001t__Set__Oset_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_S,type,
    s: set_a ).

thf(sy_v_x,type,
    x: a ).

% Relevant facts (1278)
thf(fact_0_assms_I3_J,axiom,
    member_a @ x @ ( partia707051561876973205xt_a_b @ r ) ).

% assms(3)
thf(fact_1__092_060open_062a__minus_AR_Ax_A_092_060in_062_AS_A_092_060rightarrow_062_Acarrier_AR_092_060close_062,axiom,
    ( member_a_a @ ( a_minus_a_b @ r @ x )
    @ ( pi_a_a @ s
      @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) ) ).

% \<open>a_minus R x \<in> S \<rightarrow> carrier R\<close>
thf(fact_2__092_060open_062_092_060And_062a_O_Aa_A_092_060in_062_AS_A_092_060Longrightarrow_062_AX_A_092_060ominus_062_092_060_094bsub_062poly__ring_AR_092_060_094esub_062_Apoly__of__const_Aa_A_092_060in_062_Acarrier_A_Ipoly__ring_AR_J_092_060close_062,axiom,
    ! [A: a] :
      ( ( member_a @ A @ s )
     => ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ A ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% \<open>\<And>a. a \<in> S \<Longrightarrow> X \<ominus>\<^bsub>poly_ring R\<^esub> poly_of_const a \<in> carrier (poly_ring R)\<close>
thf(fact_3_assms_I1_J,axiom,
    finite_finite_a @ s ).

% assms(1)
thf(fact_4__092_060open_062_092_060And_062s_O_As_A_092_060in_062_AS_A_092_060Longrightarrow_062_Alocal_Oeval_A_IX_A_092_060ominus_062_092_060_094bsub_062poly__ring_AR_092_060_094esub_062_Apoly__of__const_As_J_Ax_A_061_Ax_A_092_060ominus_062_As_092_060close_062,axiom,
    ! [S: a] :
      ( ( member_a @ S @ s )
     => ( ( eval_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ S ) ) @ x )
        = ( a_minus_a_b @ r @ x @ S ) ) ) ).

% \<open>\<And>s. s \<in> S \<Longrightarrow> local.eval (X \<ominus>\<^bsub>poly_ring R\<^esub> poly_of_const s) x = x \<ominus> s\<close>
thf(fact_5_domain__axioms,axiom,
    domain_a_b @ r ).

% domain_axioms
thf(fact_6_eval__var,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X )
        = X ) ) ).

% eval_var
thf(fact_7_eval__poly__of__const,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ X ) @ Y )
        = X ) ) ).

% eval_poly_of_const
thf(fact_8_finprod__closed,axiom,
    ! [F: ( a > a ) > a,A2: set_a_a] :
      ( ( member_a_a_a @ F
        @ ( pi_a_a_a @ A2
          @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( member_a @ ( finpro3012607322079259884_b_a_a @ r @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% finprod_closed
thf(fact_9_finprod__closed,axiom,
    ! [F: ( a > list_a ) > a,A2: set_a_list_a] :
      ( ( member_a_list_a_a @ F
        @ ( pi_a_list_a_a @ A2
          @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( member_a @ ( finpro2305829206037604594list_a @ r @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% finprod_closed
thf(fact_10_finprod__closed,axiom,
    ! [F: ( nat > a ) > a,A2: set_nat_a] :
      ( ( member_nat_a_a @ F
        @ ( pi_nat_a_a @ A2
          @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( member_a @ ( finpro5839458686994656414_nat_a @ r @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% finprod_closed
thf(fact_11_finprod__closed,axiom,
    ! [F: ( nat > list_a ) > a,A2: set_nat_list_a] :
      ( ( member_nat_list_a_a @ F
        @ ( pi_nat_list_a_a @ A2
          @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( member_a @ ( finpro4838020199848830884list_a @ r @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% finprod_closed
thf(fact_12_finprod__closed,axiom,
    ! [F: nat > a,A2: set_nat] :
      ( ( member_nat_a @ F
        @ ( pi_nat_a @ A2
          @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( member_a @ ( finpro1280035270526425175_b_nat @ r @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% finprod_closed
thf(fact_13_finprod__closed,axiom,
    ! [F: a > a,A2: set_a] :
      ( ( member_a_a @ F
        @ ( pi_a_a @ A2
          @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( member_a @ ( finpro205304725090349623_a_b_a @ r @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% finprod_closed
thf(fact_14_finprod__cong_H,axiom,
    ! [A2: set_a_a_a,B: set_a_a_a,G: ( ( a > a ) > a ) > a,F: ( ( a > a ) > a ) > a] :
      ( ( A2 = B )
     => ( ( member_a_a_a_a @ G
          @ ( pi_a_a_a_a @ B
            @ ^ [Uu: ( a > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro8565617760274797897_a_a_a @ r @ F @ A2 )
            = ( finpro8565617760274797897_a_a_a @ r @ G @ B ) ) ) ) ) ).

% finprod_cong'
thf(fact_15_finprod__cong_H,axiom,
    ! [A2: set_a_list_a_a,B: set_a_list_a_a,G: ( ( a > list_a ) > a ) > a,F: ( ( a > list_a ) > a ) > a] :
      ( ( A2 = B )
     => ( ( member_a_list_a_a_a @ G
          @ ( pi_a_list_a_a_a @ B
            @ ^ [Uu: ( a > list_a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro3356682085724622403st_a_a @ r @ F @ A2 )
            = ( finpro3356682085724622403st_a_a @ r @ G @ B ) ) ) ) ) ).

% finprod_cong'
thf(fact_16_finprod__cong_H,axiom,
    ! [A2: set_nat_a_a,B: set_nat_a_a,G: ( ( nat > a ) > a ) > a,F: ( ( nat > a ) > a ) > a] :
      ( ( A2 = B )
     => ( ( member_nat_a_a_a @ G
          @ ( pi_nat_a_a_a @ B
            @ ^ [Uu: ( nat > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro7303668365752794053at_a_a @ r @ F @ A2 )
            = ( finpro7303668365752794053at_a_a @ r @ G @ B ) ) ) ) ) ).

% finprod_cong'
thf(fact_17_finprod__cong_H,axiom,
    ! [A2: set_nat_list_a_a,B: set_nat_list_a_a,G: ( ( nat > list_a ) > a ) > a,F: ( ( nat > list_a ) > a ) > a] :
      ( ( A2 = B )
     => ( ( member7453349880713133094_a_a_a @ G
          @ ( pi_nat_list_a_a_a @ B
            @ ^ [Uu: ( nat > list_a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ! [I: ( nat > list_a ) > a] :
              ( ( member_nat_list_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro3076322037277673151st_a_a @ r @ F @ A2 )
            = ( finpro3076322037277673151st_a_a @ r @ G @ B ) ) ) ) ) ).

% finprod_cong'
thf(fact_18_finprod__cong_H,axiom,
    ! [A2: set_nat_list_a,B: set_nat_list_a,G: ( nat > list_a ) > a,F: ( nat > list_a ) > a] :
      ( ( A2 = B )
     => ( ( member_nat_list_a_a @ G
          @ ( pi_nat_list_a_a @ B
            @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro4838020199848830884list_a @ r @ F @ A2 )
            = ( finpro4838020199848830884list_a @ r @ G @ B ) ) ) ) ) ).

% finprod_cong'
thf(fact_19_finprod__cong_H,axiom,
    ! [A2: set_nat_a,B: set_nat_a,G: ( nat > a ) > a,F: ( nat > a ) > a] :
      ( ( A2 = B )
     => ( ( member_nat_a_a @ G
          @ ( pi_nat_a_a @ B
            @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro5839458686994656414_nat_a @ r @ F @ A2 )
            = ( finpro5839458686994656414_nat_a @ r @ G @ B ) ) ) ) ) ).

% finprod_cong'
thf(fact_20_finprod__cong_H,axiom,
    ! [A2: set_a_list_a,B: set_a_list_a,G: ( a > list_a ) > a,F: ( a > list_a ) > a] :
      ( ( A2 = B )
     => ( ( member_a_list_a_a @ G
          @ ( pi_a_list_a_a @ B
            @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro2305829206037604594list_a @ r @ F @ A2 )
            = ( finpro2305829206037604594list_a @ r @ G @ B ) ) ) ) ) ).

% finprod_cong'
thf(fact_21_finprod__cong_H,axiom,
    ! [A2: set_a_a,B: set_a_a,G: ( a > a ) > a,F: ( a > a ) > a] :
      ( ( A2 = B )
     => ( ( member_a_a_a @ G
          @ ( pi_a_a_a @ B
            @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro3012607322079259884_b_a_a @ r @ F @ A2 )
            = ( finpro3012607322079259884_b_a_a @ r @ G @ B ) ) ) ) ) ).

% finprod_cong'
thf(fact_22_finprod__cong_H,axiom,
    ! [A2: set_nat,B: set_nat,G: nat > a,F: nat > a] :
      ( ( A2 = B )
     => ( ( member_nat_a @ G
          @ ( pi_nat_a @ B
            @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro1280035270526425175_b_nat @ r @ F @ A2 )
            = ( finpro1280035270526425175_b_nat @ r @ G @ B ) ) ) ) ) ).

% finprod_cong'
thf(fact_23_finprod__cong_H,axiom,
    ! [A2: set_a,B: set_a,G: a > a,F: a > a] :
      ( ( A2 = B )
     => ( ( member_a_a @ G
          @ ( pi_a_a @ B
            @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ! [I: a] :
              ( ( member_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro205304725090349623_a_b_a @ r @ F @ A2 )
            = ( finpro205304725090349623_a_b_a @ r @ G @ B ) ) ) ) ) ).

% finprod_cong'
thf(fact_24_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
    lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).

% ring.lagrange_basis_polynomial_aux.cong
thf(fact_25_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
    lagran3534788790333317459t_unit = lagran3534788790333317459t_unit ).

% ring.lagrange_basis_polynomial_aux.cong
thf(fact_26_assms_I2_J,axiom,
    ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).

% assms(2)
thf(fact_27_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_28_eval__in__carrier__2,axiom,
    ! [X: list_a,Y: a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( eval_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% eval_in_carrier_2
thf(fact_29_up__minus__closed,axiom,
    ! [P: nat > a,Q: nat > a] :
      ( ( member_nat_a @ P @ ( up_a_b @ r ) )
     => ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
       => ( member_nat_a
          @ ^ [I2: nat] : ( a_minus_a_b @ r @ ( P @ I2 ) @ ( Q @ I2 ) )
          @ ( up_a_b @ r ) ) ) ) ).

% up_minus_closed
thf(fact_30_local_Osemiring__axioms,axiom,
    semiring_a_b @ r ).

% local.semiring_axioms
thf(fact_31_x_Oring_Ohom__closed,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( member_a @ ( eval_a_b @ r @ X @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% x.ring.hom_closed
thf(fact_32_x_Ohom__sub,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( eval_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
          = ( a_minus_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).

% x.hom_sub
thf(fact_33_finprod__one__eqI,axiom,
    ! [A2: set_a_a_a,F: ( ( a > a ) > a ) > a] :
      ( ! [X2: ( a > a ) > a] :
          ( ( member_a_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) )
     => ( ( finpro8565617760274797897_a_a_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_one_eqI
thf(fact_34_finprod__one__eqI,axiom,
    ! [A2: set_a_list_a_a,F: ( ( a > list_a ) > a ) > a] :
      ( ! [X2: ( a > list_a ) > a] :
          ( ( member_a_list_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) )
     => ( ( finpro3356682085724622403st_a_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_one_eqI
thf(fact_35_finprod__one__eqI,axiom,
    ! [A2: set_nat_a_a,F: ( ( nat > a ) > a ) > a] :
      ( ! [X2: ( nat > a ) > a] :
          ( ( member_nat_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) )
     => ( ( finpro7303668365752794053at_a_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_one_eqI
thf(fact_36_finprod__one__eqI,axiom,
    ! [A2: set_nat_list_a_a,F: ( ( nat > list_a ) > a ) > a] :
      ( ! [X2: ( nat > list_a ) > a] :
          ( ( member_nat_list_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) )
     => ( ( finpro3076322037277673151st_a_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_one_eqI
thf(fact_37_finprod__one__eqI,axiom,
    ! [A2: set_a,F: a > a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) )
     => ( ( finpro205304725090349623_a_b_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_one_eqI
thf(fact_38_finprod__one__eqI,axiom,
    ! [A2: set_nat,F: nat > a] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) )
     => ( ( finpro1280035270526425175_b_nat @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_one_eqI
thf(fact_39_finprod__one__eqI,axiom,
    ! [A2: set_a_a,F: ( a > a ) > a] :
      ( ! [X2: a > a] :
          ( ( member_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) )
     => ( ( finpro3012607322079259884_b_a_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_one_eqI
thf(fact_40_finprod__one__eqI,axiom,
    ! [A2: set_a_list_a,F: ( a > list_a ) > a] :
      ( ! [X2: a > list_a] :
          ( ( member_a_list_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) )
     => ( ( finpro2305829206037604594list_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_one_eqI
thf(fact_41_finprod__one__eqI,axiom,
    ! [A2: set_nat_a,F: ( nat > a ) > a] :
      ( ! [X2: nat > a] :
          ( ( member_nat_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) )
     => ( ( finpro5839458686994656414_nat_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_one_eqI
thf(fact_42_finprod__one__eqI,axiom,
    ! [A2: set_nat_list_a,F: ( nat > list_a ) > a] :
      ( ! [X2: nat > list_a] :
          ( ( member_nat_list_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_a_ring_ext_a_b @ r ) ) )
     => ( ( finpro4838020199848830884list_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_one_eqI
thf(fact_43_x_Oring__hom__cring__axioms,axiom,
    ( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
    @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) ).

% x.ring_hom_cring_axioms
thf(fact_44_x_Oring_Ois__abelian__group__hom,axiom,
    ( abelia8217020544048703197it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
    @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) ).

% x.ring.is_abelian_group_hom
thf(fact_45_onepideal,axiom,
    principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% onepideal
thf(fact_46_abelian__monoid__axioms,axiom,
    abelian_monoid_a_b @ r ).

% abelian_monoid_axioms
thf(fact_47_poly__of__const__in__carrier,axiom,
    ! [S: a] :
      ( ( member_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_list_a @ ( poly_of_const_a_b @ r @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% poly_of_const_in_carrier
thf(fact_48_finprod__singleton__swap,axiom,
    ! [I3: a,A2: set_a,F: a > a] :
      ( ( member_a @ I3 @ A2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( member_a_a @ F
            @ ( pi_a_a @ A2
              @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro205304725090349623_a_b_a @ r
              @ ^ [J: a] : ( if_a @ ( J = I3 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton_swap
thf(fact_49_finprod__singleton__swap,axiom,
    ! [I3: nat,A2: set_nat,F: nat > a] :
      ( ( member_nat @ I3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( member_nat_a @ F
            @ ( pi_nat_a @ A2
              @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro1280035270526425175_b_nat @ r
              @ ^ [J: nat] : ( if_a @ ( J = I3 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton_swap
thf(fact_50_finprod__singleton__swap,axiom,
    ! [I3: list_a,A2: set_list_a,F: list_a > a] :
      ( ( member_list_a @ I3 @ A2 )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( member_list_a_a @ F
            @ ( pi_list_a_a @ A2
              @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro6052973074229812797list_a @ r
              @ ^ [J: list_a] : ( if_a @ ( J = I3 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton_swap
thf(fact_51_finprod__singleton__swap,axiom,
    ! [I3: a > a,A2: set_a_a,F: ( a > a ) > a] :
      ( ( member_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a @ A2 )
       => ( ( member_a_a_a @ F
            @ ( pi_a_a_a @ A2
              @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro3012607322079259884_b_a_a @ r
              @ ^ [J: a > a] : ( if_a @ ( J = I3 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton_swap
thf(fact_52_finprod__singleton__swap,axiom,
    ! [I3: nat > a,A2: set_nat_a,F: ( nat > a ) > a] :
      ( ( member_nat_a @ I3 @ A2 )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ( member_nat_a_a @ F
            @ ( pi_nat_a_a @ A2
              @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro5839458686994656414_nat_a @ r
              @ ^ [J: nat > a] : ( if_a @ ( J = I3 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton_swap
thf(fact_53_finprod__singleton__swap,axiom,
    ! [I3: a > list_a,A2: set_a_list_a,F: ( a > list_a ) > a] :
      ( ( member_a_list_a @ I3 @ A2 )
     => ( ( finite8564763978580267522list_a @ A2 )
       => ( ( member_a_list_a_a @ F
            @ ( pi_a_list_a_a @ A2
              @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro2305829206037604594list_a @ r
              @ ^ [J: a > list_a] : ( if_a @ ( J = I3 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton_swap
thf(fact_54_finprod__singleton__swap,axiom,
    ! [I3: nat > list_a,A2: set_nat_list_a,F: ( nat > list_a ) > a] :
      ( ( member_nat_list_a @ I3 @ A2 )
     => ( ( finite7630042315537210004list_a @ A2 )
       => ( ( member_nat_list_a_a @ F
            @ ( pi_nat_list_a_a @ A2
              @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro4838020199848830884list_a @ r
              @ ^ [J: nat > list_a] : ( if_a @ ( J = I3 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton_swap
thf(fact_55_finprod__singleton__swap,axiom,
    ! [I3: ( a > a ) > a,A2: set_a_a_a,F: ( ( a > a ) > a ) > a] :
      ( ( member_a_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a_a @ A2 )
       => ( ( member_a_a_a_a @ F
            @ ( pi_a_a_a_a @ A2
              @ ^ [Uu: ( a > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro8565617760274797897_a_a_a @ r
              @ ^ [J: ( a > a ) > a] : ( if_a @ ( J = I3 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton_swap
thf(fact_56_finprod__singleton__swap,axiom,
    ! [I3: ( nat > a ) > a,A2: set_nat_a_a,F: ( ( nat > a ) > a ) > a] :
      ( ( member_nat_a_a @ I3 @ A2 )
     => ( ( finite7239108116303828181at_a_a @ A2 )
       => ( ( member_nat_a_a_a @ F
            @ ( pi_nat_a_a_a @ A2
              @ ^ [Uu: ( nat > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro7303668365752794053at_a_a @ r
              @ ^ [J: ( nat > a ) > a] : ( if_a @ ( J = I3 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton_swap
thf(fact_57_finprod__singleton__swap,axiom,
    ! [I3: ( a > list_a ) > a,A2: set_a_list_a_a,F: ( ( a > list_a ) > a ) > a] :
      ( ( member_a_list_a_a @ I3 @ A2 )
     => ( ( finite681744870773793075st_a_a @ A2 )
       => ( ( member_a_list_a_a_a @ F
            @ ( pi_a_list_a_a_a @ A2
              @ ^ [Uu: ( a > list_a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro3356682085724622403st_a_a @ r
              @ ^ [J: ( a > list_a ) > a] : ( if_a @ ( J = I3 ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton_swap
thf(fact_58_finprod__singleton,axiom,
    ! [I3: a,A2: set_a,F: a > a] :
      ( ( member_a @ I3 @ A2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( member_a_a @ F
            @ ( pi_a_a @ A2
              @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro205304725090349623_a_b_a @ r
              @ ^ [J: a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton
thf(fact_59_finprod__singleton,axiom,
    ! [I3: nat,A2: set_nat,F: nat > a] :
      ( ( member_nat @ I3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( member_nat_a @ F
            @ ( pi_nat_a @ A2
              @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro1280035270526425175_b_nat @ r
              @ ^ [J: nat] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton
thf(fact_60_finprod__singleton,axiom,
    ! [I3: list_a,A2: set_list_a,F: list_a > a] :
      ( ( member_list_a @ I3 @ A2 )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( member_list_a_a @ F
            @ ( pi_list_a_a @ A2
              @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro6052973074229812797list_a @ r
              @ ^ [J: list_a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton
thf(fact_61_finprod__singleton,axiom,
    ! [I3: a > a,A2: set_a_a,F: ( a > a ) > a] :
      ( ( member_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a @ A2 )
       => ( ( member_a_a_a @ F
            @ ( pi_a_a_a @ A2
              @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro3012607322079259884_b_a_a @ r
              @ ^ [J: a > a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton
thf(fact_62_finprod__singleton,axiom,
    ! [I3: nat > a,A2: set_nat_a,F: ( nat > a ) > a] :
      ( ( member_nat_a @ I3 @ A2 )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ( member_nat_a_a @ F
            @ ( pi_nat_a_a @ A2
              @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro5839458686994656414_nat_a @ r
              @ ^ [J: nat > a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton
thf(fact_63_finprod__singleton,axiom,
    ! [I3: a > list_a,A2: set_a_list_a,F: ( a > list_a ) > a] :
      ( ( member_a_list_a @ I3 @ A2 )
     => ( ( finite8564763978580267522list_a @ A2 )
       => ( ( member_a_list_a_a @ F
            @ ( pi_a_list_a_a @ A2
              @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro2305829206037604594list_a @ r
              @ ^ [J: a > list_a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton
thf(fact_64_finprod__singleton,axiom,
    ! [I3: nat > list_a,A2: set_nat_list_a,F: ( nat > list_a ) > a] :
      ( ( member_nat_list_a @ I3 @ A2 )
     => ( ( finite7630042315537210004list_a @ A2 )
       => ( ( member_nat_list_a_a @ F
            @ ( pi_nat_list_a_a @ A2
              @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro4838020199848830884list_a @ r
              @ ^ [J: nat > list_a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton
thf(fact_65_finprod__singleton,axiom,
    ! [I3: ( a > a ) > a,A2: set_a_a_a,F: ( ( a > a ) > a ) > a] :
      ( ( member_a_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a_a @ A2 )
       => ( ( member_a_a_a_a @ F
            @ ( pi_a_a_a_a @ A2
              @ ^ [Uu: ( a > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro8565617760274797897_a_a_a @ r
              @ ^ [J: ( a > a ) > a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton
thf(fact_66_finprod__singleton,axiom,
    ! [I3: ( nat > a ) > a,A2: set_nat_a_a,F: ( ( nat > a ) > a ) > a] :
      ( ( member_nat_a_a @ I3 @ A2 )
     => ( ( finite7239108116303828181at_a_a @ A2 )
       => ( ( member_nat_a_a_a @ F
            @ ( pi_nat_a_a_a @ A2
              @ ^ [Uu: ( nat > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro7303668365752794053at_a_a @ r
              @ ^ [J: ( nat > a ) > a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton
thf(fact_67_finprod__singleton,axiom,
    ! [I3: ( a > list_a ) > a,A2: set_a_list_a_a,F: ( ( a > list_a ) > a ) > a] :
      ( ( member_a_list_a_a @ I3 @ A2 )
     => ( ( finite681744870773793075st_a_a @ A2 )
       => ( ( member_a_list_a_a_a @ F
            @ ( pi_a_list_a_a_a @ A2
              @ ^ [Uu: ( a > list_a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( finpro3356682085724622403st_a_a @ r
              @ ^ [J: ( a > list_a ) > a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( one_a_ring_ext_a_b @ r ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% finprod_singleton
thf(fact_68_one__closed,axiom,
    member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% one_closed
thf(fact_69_finprod__one,axiom,
    ! [A2: set_a] :
      ( ( finpro205304725090349623_a_b_a @ r
        @ ^ [I2: a] : ( one_a_ring_ext_a_b @ r )
        @ A2 )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_one
thf(fact_70_finprod__one,axiom,
    ! [A2: set_nat] :
      ( ( finpro1280035270526425175_b_nat @ r
        @ ^ [I2: nat] : ( one_a_ring_ext_a_b @ r )
        @ A2 )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_one
thf(fact_71_finprod__one,axiom,
    ! [A2: set_a_a] :
      ( ( finpro3012607322079259884_b_a_a @ r
        @ ^ [I2: a > a] : ( one_a_ring_ext_a_b @ r )
        @ A2 )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_one
thf(fact_72_finprod__one,axiom,
    ! [A2: set_a_list_a] :
      ( ( finpro2305829206037604594list_a @ r
        @ ^ [I2: a > list_a] : ( one_a_ring_ext_a_b @ r )
        @ A2 )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_one
thf(fact_73_finprod__one,axiom,
    ! [A2: set_nat_a] :
      ( ( finpro5839458686994656414_nat_a @ r
        @ ^ [I2: nat > a] : ( one_a_ring_ext_a_b @ r )
        @ A2 )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_one
thf(fact_74_finprod__one,axiom,
    ! [A2: set_nat_list_a] :
      ( ( finpro4838020199848830884list_a @ r
        @ ^ [I2: nat > list_a] : ( one_a_ring_ext_a_b @ r )
        @ A2 )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_one
thf(fact_75_finprod__infinite,axiom,
    ! [A2: set_list_a,F: list_a > a] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finpro6052973074229812797list_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_infinite
thf(fact_76_finprod__infinite,axiom,
    ! [A2: set_a,F: a > a] :
      ( ~ ( finite_finite_a @ A2 )
     => ( ( finpro205304725090349623_a_b_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_infinite
thf(fact_77_finprod__infinite,axiom,
    ! [A2: set_nat,F: nat > a] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finpro1280035270526425175_b_nat @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_infinite
thf(fact_78_finprod__infinite,axiom,
    ! [A2: set_a_a,F: ( a > a ) > a] :
      ( ~ ( finite_finite_a_a @ A2 )
     => ( ( finpro3012607322079259884_b_a_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_infinite
thf(fact_79_finprod__infinite,axiom,
    ! [A2: set_a_list_a,F: ( a > list_a ) > a] :
      ( ~ ( finite8564763978580267522list_a @ A2 )
     => ( ( finpro2305829206037604594list_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_infinite
thf(fact_80_finprod__infinite,axiom,
    ! [A2: set_nat_a,F: ( nat > a ) > a] :
      ( ~ ( finite_finite_nat_a @ A2 )
     => ( ( finpro5839458686994656414_nat_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_infinite
thf(fact_81_finprod__infinite,axiom,
    ! [A2: set_nat_list_a,F: ( nat > list_a ) > a] :
      ( ~ ( finite7630042315537210004list_a @ A2 )
     => ( ( finpro4838020199848830884list_a @ r @ F @ A2 )
        = ( one_a_ring_ext_a_b @ r ) ) ) ).

% finprod_infinite
thf(fact_82_x_Ominus__closed,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.minus_closed
thf(fact_83_x_Oonepideal,axiom,
    princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).

% x.onepideal
thf(fact_84_finite__number__of__roots,axiom,
    ! [P: list_a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ r @ P ) ) ) ) ).

% finite_number_of_roots
thf(fact_85_lagrange__basis__polynomial__aux__def,axiom,
    ! [S2: set_a] :
      ( ( lagran9092808442999052491ux_a_b @ r @ S2 )
      = ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
        @ ^ [S3: a] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ S3 ) )
        @ S2 ) ) ).

% lagrange_basis_polynomial_aux_def
thf(fact_86_finprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat,A2: set_nat,G: nat > a,H: nat > a] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_a @ G
                @ ( pi_nat_a @ B
                  @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro1280035270526425175_b_nat @ r @ G @ B )
                = ( finpro1280035270526425175_b_nat @ r @ H @ A2 ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_right
thf(fact_87_finprod__mono__neutral__cong__right,axiom,
    ! [B: set_a,A2: set_a,G: a > a,H: a > a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( ! [I: a] :
              ( ( member_a @ I @ ( minus_minus_set_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a @ G
                @ ( pi_a_a @ B
                  @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro205304725090349623_a_b_a @ r @ G @ B )
                = ( finpro205304725090349623_a_b_a @ r @ H @ A2 ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_right
thf(fact_88_finprod__mono__neutral__cong__right,axiom,
    ! [B: set_list_a,A2: set_list_a,G: list_a > a,H: list_a > a] :
      ( ( finite_finite_list_a @ B )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B )
       => ( ! [I: list_a] :
              ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_list_a_a @ G
                @ ( pi_list_a_a @ B
                  @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro6052973074229812797list_a @ r @ G @ B )
                = ( finpro6052973074229812797list_a @ r @ H @ A2 ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_right
thf(fact_89_finprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_a,A2: set_a_a,G: ( a > a ) > a,H: ( a > a ) > a] :
      ( ( finite_finite_a_a @ B )
     => ( ( ord_less_eq_set_a_a @ A2 @ B )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ ( minus_minus_set_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: a > a] :
                ( ( member_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_a @ G
                @ ( pi_a_a_a @ B
                  @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro3012607322079259884_b_a_a @ r @ G @ B )
                = ( finpro3012607322079259884_b_a_a @ r @ H @ A2 ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_right
thf(fact_90_finprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat_a,A2: set_nat_a,G: ( nat > a ) > a,H: ( nat > a ) > a] :
      ( ( finite_finite_nat_a @ B )
     => ( ( ord_le871467723717165285_nat_a @ A2 @ B )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: nat > a] :
                ( ( member_nat_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_a_a @ G
                @ ( pi_nat_a_a @ B
                  @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro5839458686994656414_nat_a @ r @ G @ B )
                = ( finpro5839458686994656414_nat_a @ r @ H @ A2 ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_right
thf(fact_91_finprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_list_a,A2: set_a_list_a,G: ( a > list_a ) > a,H: ( a > list_a ) > a] :
      ( ( finite8564763978580267522list_a @ B )
     => ( ( ord_le50412136050534657list_a @ A2 @ B )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: a > list_a] :
                ( ( member_a_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_list_a_a @ G
                @ ( pi_a_list_a_a @ B
                  @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro2305829206037604594list_a @ r @ G @ B )
                = ( finpro2305829206037604594list_a @ r @ H @ A2 ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_right
thf(fact_92_finprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat_list_a,A2: set_nat_list_a,G: ( nat > list_a ) > a,H: ( nat > list_a ) > a] :
      ( ( finite7630042315537210004list_a @ B )
     => ( ( ord_le2145805922479659755list_a @ A2 @ B )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: nat > list_a] :
                ( ( member_nat_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_list_a_a @ G
                @ ( pi_nat_list_a_a @ B
                  @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro4838020199848830884list_a @ r @ G @ B )
                = ( finpro4838020199848830884list_a @ r @ H @ A2 ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_right
thf(fact_93_finprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_a_a,A2: set_a_a_a,G: ( ( a > a ) > a ) > a,H: ( ( a > a ) > a ) > a] :
      ( ( finite_finite_a_a_a @ B )
     => ( ( ord_le7181591058469194768_a_a_a @ A2 @ B )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: ( a > a ) > a] :
                ( ( member_a_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_a_a @ G
                @ ( pi_a_a_a_a @ B
                  @ ^ [Uu: ( a > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro8565617760274797897_a_a_a @ r @ G @ B )
                = ( finpro8565617760274797897_a_a_a @ r @ H @ A2 ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_right
thf(fact_94_finprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat_a_a,A2: set_nat_a_a,G: ( ( nat > a ) > a ) > a,H: ( ( nat > a ) > a ) > a] :
      ( ( finite7239108116303828181at_a_a @ B )
     => ( ( ord_le3509452538356653652at_a_a @ A2 @ B )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: ( nat > a ) > a] :
                ( ( member_nat_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_a_a_a @ G
                @ ( pi_nat_a_a_a @ B
                  @ ^ [Uu: ( nat > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro7303668365752794053at_a_a @ r @ G @ B )
                = ( finpro7303668365752794053at_a_a @ r @ H @ A2 ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_right
thf(fact_95_finprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_list_a_a,A2: set_a_list_a_a,G: ( ( a > list_a ) > a ) > a,H: ( ( a > list_a ) > a ) > a] :
      ( ( finite681744870773793075st_a_a @ B )
     => ( ( ord_le2847972559472028170st_a_a @ A2 @ B )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: ( a > list_a ) > a] :
                ( ( member_a_list_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_list_a_a_a @ G
                @ ( pi_a_list_a_a_a @ B
                  @ ^ [Uu: ( a > list_a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro3356682085724622403st_a_a @ r @ G @ B )
                = ( finpro3356682085724622403st_a_a @ r @ H @ A2 ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_right
thf(fact_96_finprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat,A2: set_nat,H: nat > a,G: nat > a] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_a @ H
                @ ( pi_nat_a @ B
                  @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro1280035270526425175_b_nat @ r @ G @ A2 )
                = ( finpro1280035270526425175_b_nat @ r @ H @ B ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_left
thf(fact_97_finprod__mono__neutral__cong__left,axiom,
    ! [B: set_a,A2: set_a,H: a > a,G: a > a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( ! [I: a] :
              ( ( member_a @ I @ ( minus_minus_set_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a @ H
                @ ( pi_a_a @ B
                  @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro205304725090349623_a_b_a @ r @ G @ A2 )
                = ( finpro205304725090349623_a_b_a @ r @ H @ B ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_left
thf(fact_98_finprod__mono__neutral__cong__left,axiom,
    ! [B: set_list_a,A2: set_list_a,H: list_a > a,G: list_a > a] :
      ( ( finite_finite_list_a @ B )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B )
       => ( ! [I: list_a] :
              ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_list_a_a @ H
                @ ( pi_list_a_a @ B
                  @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro6052973074229812797list_a @ r @ G @ A2 )
                = ( finpro6052973074229812797list_a @ r @ H @ B ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_left
thf(fact_99_finprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_a,A2: set_a_a,H: ( a > a ) > a,G: ( a > a ) > a] :
      ( ( finite_finite_a_a @ B )
     => ( ( ord_less_eq_set_a_a @ A2 @ B )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ ( minus_minus_set_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: a > a] :
                ( ( member_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_a @ H
                @ ( pi_a_a_a @ B
                  @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro3012607322079259884_b_a_a @ r @ G @ A2 )
                = ( finpro3012607322079259884_b_a_a @ r @ H @ B ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_left
thf(fact_100_finprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat_a,A2: set_nat_a,H: ( nat > a ) > a,G: ( nat > a ) > a] :
      ( ( finite_finite_nat_a @ B )
     => ( ( ord_le871467723717165285_nat_a @ A2 @ B )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: nat > a] :
                ( ( member_nat_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_a_a @ H
                @ ( pi_nat_a_a @ B
                  @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro5839458686994656414_nat_a @ r @ G @ A2 )
                = ( finpro5839458686994656414_nat_a @ r @ H @ B ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_left
thf(fact_101_finprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_list_a,A2: set_a_list_a,H: ( a > list_a ) > a,G: ( a > list_a ) > a] :
      ( ( finite8564763978580267522list_a @ B )
     => ( ( ord_le50412136050534657list_a @ A2 @ B )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: a > list_a] :
                ( ( member_a_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_list_a_a @ H
                @ ( pi_a_list_a_a @ B
                  @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro2305829206037604594list_a @ r @ G @ A2 )
                = ( finpro2305829206037604594list_a @ r @ H @ B ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_left
thf(fact_102_finprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat_list_a,A2: set_nat_list_a,H: ( nat > list_a ) > a,G: ( nat > list_a ) > a] :
      ( ( finite7630042315537210004list_a @ B )
     => ( ( ord_le2145805922479659755list_a @ A2 @ B )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: nat > list_a] :
                ( ( member_nat_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_list_a_a @ H
                @ ( pi_nat_list_a_a @ B
                  @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro4838020199848830884list_a @ r @ G @ A2 )
                = ( finpro4838020199848830884list_a @ r @ H @ B ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_left
thf(fact_103_finprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_a_a,A2: set_a_a_a,H: ( ( a > a ) > a ) > a,G: ( ( a > a ) > a ) > a] :
      ( ( finite_finite_a_a_a @ B )
     => ( ( ord_le7181591058469194768_a_a_a @ A2 @ B )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: ( a > a ) > a] :
                ( ( member_a_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_a_a @ H
                @ ( pi_a_a_a_a @ B
                  @ ^ [Uu: ( a > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro8565617760274797897_a_a_a @ r @ G @ A2 )
                = ( finpro8565617760274797897_a_a_a @ r @ H @ B ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_left
thf(fact_104_finprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat_a_a,A2: set_nat_a_a,H: ( ( nat > a ) > a ) > a,G: ( ( nat > a ) > a ) > a] :
      ( ( finite7239108116303828181at_a_a @ B )
     => ( ( ord_le3509452538356653652at_a_a @ A2 @ B )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: ( nat > a ) > a] :
                ( ( member_nat_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_a_a_a @ H
                @ ( pi_nat_a_a_a @ B
                  @ ^ [Uu: ( nat > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro7303668365752794053at_a_a @ r @ G @ A2 )
                = ( finpro7303668365752794053at_a_a @ r @ H @ B ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_left
thf(fact_105_finprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_list_a_a,A2: set_a_list_a_a,H: ( ( a > list_a ) > a ) > a,G: ( ( a > list_a ) > a ) > a] :
      ( ( finite681744870773793075st_a_a @ B )
     => ( ( ord_le2847972559472028170st_a_a @ A2 @ B )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [X2: ( a > list_a ) > a] :
                ( ( member_a_list_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_list_a_a_a @ H
                @ ( pi_a_list_a_a_a @ B
                  @ ^ [Uu: ( a > list_a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro3356682085724622403st_a_a @ r @ G @ A2 )
                = ( finpro3356682085724622403st_a_a @ r @ H @ B ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong_left
thf(fact_106_finite__Collect__subsets,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B2: set_nat] : ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_107_finite__Collect__subsets,axiom,
    ! [A2: set_a] :
      ( ( finite_finite_a @ A2 )
     => ( finite_finite_set_a
        @ ( collect_set_a
          @ ^ [B2: set_a] : ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_108_finite__Collect__subsets,axiom,
    ! [A2: set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( finite5282473924520328461list_a
        @ ( collect_set_list_a
          @ ^ [B2: set_list_a] : ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_109_x_Oup__minus__closed,axiom,
    ! [P: nat > list_a,Q: nat > list_a] :
      ( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_nat_list_a @ Q @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member_nat_list_a
          @ ^ [I2: nat] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( P @ I2 ) @ ( Q @ I2 ) )
          @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.up_minus_closed
thf(fact_110_x_Ocarrier__is__subcring,axiom,
    subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).

% x.carrier_is_subcring
thf(fact_111_x_Oring_Ozero__closed,axiom,
    member_a @ ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ).

% x.ring.zero_closed
thf(fact_112_x_Osemiring__axioms,axiom,
    semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).

% x.semiring_axioms
thf(fact_113_finite__Collect__disjI,axiom,
    ! [P3: list_list_a > $o,Q2: list_list_a > $o] :
      ( ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [X3: list_list_a] :
              ( ( P3 @ X3 )
              | ( Q2 @ X3 ) ) ) )
      = ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P3 ) )
        & ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_114_finite__Collect__disjI,axiom,
    ! [P3: a > $o,Q2: a > $o] :
      ( ( finite_finite_a
        @ ( collect_a
          @ ^ [X3: a] :
              ( ( P3 @ X3 )
              | ( Q2 @ X3 ) ) ) )
      = ( ( finite_finite_a @ ( collect_a @ P3 ) )
        & ( finite_finite_a @ ( collect_a @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_115_finite__Collect__disjI,axiom,
    ! [P3: nat > $o,Q2: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( P3 @ X3 )
              | ( Q2 @ X3 ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
        & ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_116_finite__Collect__disjI,axiom,
    ! [P3: list_a > $o,Q2: list_a > $o] :
      ( ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X3: list_a] :
              ( ( P3 @ X3 )
              | ( Q2 @ X3 ) ) ) )
      = ( ( finite_finite_list_a @ ( collect_list_a @ P3 ) )
        & ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) ) ) ).

% finite_Collect_disjI
thf(fact_117_finite__Collect__conjI,axiom,
    ! [P3: list_list_a > $o,Q2: list_list_a > $o] :
      ( ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P3 ) )
        | ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q2 ) ) )
     => ( finite1660835950917165235list_a
        @ ( collect_list_list_a
          @ ^ [X3: list_list_a] :
              ( ( P3 @ X3 )
              & ( Q2 @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_118_finite__Collect__conjI,axiom,
    ! [P3: a > $o,Q2: a > $o] :
      ( ( ( finite_finite_a @ ( collect_a @ P3 ) )
        | ( finite_finite_a @ ( collect_a @ Q2 ) ) )
     => ( finite_finite_a
        @ ( collect_a
          @ ^ [X3: a] :
              ( ( P3 @ X3 )
              & ( Q2 @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_119_finite__Collect__conjI,axiom,
    ! [P3: nat > $o,Q2: nat > $o] :
      ( ( ( finite_finite_nat @ ( collect_nat @ P3 ) )
        | ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X3: nat] :
              ( ( P3 @ X3 )
              & ( Q2 @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_120_finite__Collect__conjI,axiom,
    ! [P3: list_a > $o,Q2: list_a > $o] :
      ( ( ( finite_finite_list_a @ ( collect_list_a @ P3 ) )
        | ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) )
     => ( finite_finite_list_a
        @ ( collect_list_a
          @ ^ [X3: list_a] :
              ( ( P3 @ X3 )
              & ( Q2 @ X3 ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_121_finite__Diff2,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) )
        = ( finite_finite_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_122_finite__Diff2,axiom,
    ! [B: set_list_a,A2: set_list_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B ) )
        = ( finite_finite_list_a @ A2 ) ) ) ).

% finite_Diff2
thf(fact_123_finite__Diff2,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) )
        = ( finite_finite_a @ A2 ) ) ) ).

% finite_Diff2
thf(fact_124_mem__Collect__eq,axiom,
    ! [A: a,P3: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
    ! [A: nat,P3: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
    ! [A: list_a,P3: list_a > $o] :
      ( ( member_list_a @ A @ ( collect_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_127_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_128_mem__Collect__eq,axiom,
    ! [A: a > a,P3: ( a > a ) > $o] :
      ( ( member_a_a @ A @ ( collect_a_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
    ! [A: list_list_a,P3: list_list_a > $o] :
      ( ( member_list_list_a @ A @ ( collect_list_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_130_mem__Collect__eq,axiom,
    ! [A: nat > list_a,P3: ( nat > list_a ) > $o] :
      ( ( member_nat_list_a @ A @ ( collect_nat_list_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_131_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_132_mem__Collect__eq,axiom,
    ! [A: ( a > a ) > a,P3: ( ( a > a ) > a ) > $o] :
      ( ( member_a_a_a @ A @ ( collect_a_a_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_133_mem__Collect__eq,axiom,
    ! [A: ( nat > a ) > a,P3: ( ( nat > a ) > a ) > $o] :
      ( ( member_nat_a_a @ A @ ( collect_nat_a_a @ P3 ) )
      = ( P3 @ A ) ) ).

% mem_Collect_eq
thf(fact_134_Collect__mem__eq,axiom,
    ! [A2: set_a] :
      ( ( collect_a
        @ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_136_Collect__mem__eq,axiom,
    ! [A2: set_list_a] :
      ( ( collect_list_a
        @ ^ [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
    ! [A2: set_nat_a] :
      ( ( collect_nat_a
        @ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
    ! [A2: set_a_a] :
      ( ( collect_a_a
        @ ^ [X3: a > a] : ( member_a_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
    ! [A2: set_list_list_a] :
      ( ( collect_list_list_a
        @ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_140_Collect__mem__eq,axiom,
    ! [A2: set_nat_list_a] :
      ( ( collect_nat_list_a
        @ ^ [X3: nat > list_a] : ( member_nat_list_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_141_Collect__mem__eq,axiom,
    ! [A2: set_a_list_a] :
      ( ( collect_a_list_a
        @ ^ [X3: a > list_a] : ( member_a_list_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_142_Collect__mem__eq,axiom,
    ! [A2: set_a_a_a] :
      ( ( collect_a_a_a
        @ ^ [X3: ( a > a ) > a] : ( member_a_a_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_143_Collect__mem__eq,axiom,
    ! [A2: set_nat_a_a] :
      ( ( collect_nat_a_a
        @ ^ [X3: ( nat > a ) > a] : ( member_nat_a_a @ X3 @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_144_Collect__cong,axiom,
    ! [P3: a > $o,Q2: a > $o] :
      ( ! [X2: a] :
          ( ( P3 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collect_a @ P3 )
        = ( collect_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_145_Collect__cong,axiom,
    ! [P3: nat > $o,Q2: nat > $o] :
      ( ! [X2: nat] :
          ( ( P3 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collect_nat @ P3 )
        = ( collect_nat @ Q2 ) ) ) ).

% Collect_cong
thf(fact_146_Collect__cong,axiom,
    ! [P3: list_a > $o,Q2: list_a > $o] :
      ( ! [X2: list_a] :
          ( ( P3 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collect_list_a @ P3 )
        = ( collect_list_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_147_Collect__cong,axiom,
    ! [P3: list_list_a > $o,Q2: list_list_a > $o] :
      ( ! [X2: list_list_a] :
          ( ( P3 @ X2 )
          = ( Q2 @ X2 ) )
     => ( ( collect_list_list_a @ P3 )
        = ( collect_list_list_a @ Q2 ) ) ) ).

% Collect_cong
thf(fact_148_finite__Diff,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B ) ) ) ).

% finite_Diff
thf(fact_149_finite__Diff,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B ) ) ) ).

% finite_Diff
thf(fact_150_finite__Diff,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( finite_finite_a @ A2 )
     => ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B ) ) ) ).

% finite_Diff
thf(fact_151_x_Ozero__closed,axiom,
    member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% x.zero_closed
thf(fact_152_x_Or__right__minus__eq,axiom,
    ! [A: list_a,B3: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B3 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
          = ( A = B3 ) ) ) ) ).

% x.r_right_minus_eq
thf(fact_153_Diff__infinite__finite,axiom,
    ! [T: set_nat,S2: set_nat] :
      ( ( finite_finite_nat @ T )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T ) ) ) ) ).

% Diff_infinite_finite
thf(fact_154_Diff__infinite__finite,axiom,
    ! [T: set_list_a,S2: set_list_a] :
      ( ( finite_finite_list_a @ T )
     => ( ~ ( finite_finite_list_a @ S2 )
       => ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S2 @ T ) ) ) ) ).

% Diff_infinite_finite
thf(fact_155_Diff__infinite__finite,axiom,
    ! [T: set_a,S2: set_a] :
      ( ( finite_finite_a @ T )
     => ( ~ ( finite_finite_a @ S2 )
       => ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ T ) ) ) ) ).

% Diff_infinite_finite
thf(fact_156_domain_Ofinprod__zero__iff,axiom,
    ! [R: partia2175431115845679010xt_a_b,A2: set_a,F: a > a] :
      ( ( domain_a_b @ R )
     => ( ( finite_finite_a @ A2 )
       => ( ! [A3: a] :
              ( ( member_a @ A3 @ A2 )
             => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
         => ( ( ( finpro205304725090349623_a_b_a @ R @ F @ A2 )
              = ( zero_a_b @ R ) )
            = ( ? [X3: a] :
                  ( ( member_a @ X3 @ A2 )
                  & ( ( F @ X3 )
                    = ( zero_a_b @ R ) ) ) ) ) ) ) ) ).

% domain.finprod_zero_iff
thf(fact_157_domain_Ofinprod__zero__iff,axiom,
    ! [R: partia2175431115845679010xt_a_b,A2: set_nat,F: nat > a] :
      ( ( domain_a_b @ R )
     => ( ( finite_finite_nat @ A2 )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A2 )
             => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
         => ( ( ( finpro1280035270526425175_b_nat @ R @ F @ A2 )
              = ( zero_a_b @ R ) )
            = ( ? [X3: nat] :
                  ( ( member_nat @ X3 @ A2 )
                  & ( ( F @ X3 )
                    = ( zero_a_b @ R ) ) ) ) ) ) ) ) ).

% domain.finprod_zero_iff
thf(fact_158_domain_Ofinprod__zero__iff,axiom,
    ! [R: partia2175431115845679010xt_a_b,A2: set_list_a,F: list_a > a] :
      ( ( domain_a_b @ R )
     => ( ( finite_finite_list_a @ A2 )
       => ( ! [A3: list_a] :
              ( ( member_list_a @ A3 @ A2 )
             => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
         => ( ( ( finpro6052973074229812797list_a @ R @ F @ A2 )
              = ( zero_a_b @ R ) )
            = ( ? [X3: list_a] :
                  ( ( member_list_a @ X3 @ A2 )
                  & ( ( F @ X3 )
                    = ( zero_a_b @ R ) ) ) ) ) ) ) ) ).

% domain.finprod_zero_iff
thf(fact_159_domain_Ofinprod__zero__iff,axiom,
    ! [R: partia2670972154091845814t_unit,A2: set_nat,F: nat > list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( finite_finite_nat @ A2 )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A2 )
             => ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ R ) ) )
         => ( ( ( finpro1918104735009086181it_nat @ R @ F @ A2 )
              = ( zero_l4142658623432671053t_unit @ R ) )
            = ( ? [X3: nat] :
                  ( ( member_nat @ X3 @ A2 )
                  & ( ( F @ X3 )
                    = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).

% domain.finprod_zero_iff
thf(fact_160_domain_Ofinprod__zero__iff,axiom,
    ! [R: partia2670972154091845814t_unit,A2: set_a,F: a > list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( finite_finite_a @ A2 )
       => ( ! [A3: a] :
              ( ( member_a @ A3 @ A2 )
             => ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ R ) ) )
         => ( ( ( finpro4329226410377213737unit_a @ R @ F @ A2 )
              = ( zero_l4142658623432671053t_unit @ R ) )
            = ( ? [X3: a] :
                  ( ( member_a @ X3 @ A2 )
                  & ( ( F @ X3 )
                    = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).

% domain.finprod_zero_iff
thf(fact_161_domain_Ofinprod__zero__iff,axiom,
    ! [R: partia2175431115845679010xt_a_b,A2: set_a_a,F: ( a > a ) > a] :
      ( ( domain_a_b @ R )
     => ( ( finite_finite_a_a @ A2 )
       => ( ! [A3: a > a] :
              ( ( member_a_a @ A3 @ A2 )
             => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
         => ( ( ( finpro3012607322079259884_b_a_a @ R @ F @ A2 )
              = ( zero_a_b @ R ) )
            = ( ? [X3: a > a] :
                  ( ( member_a_a @ X3 @ A2 )
                  & ( ( F @ X3 )
                    = ( zero_a_b @ R ) ) ) ) ) ) ) ) ).

% domain.finprod_zero_iff
thf(fact_162_domain_Ofinprod__zero__iff,axiom,
    ! [R: partia2175431115845679010xt_a_b,A2: set_nat_a,F: ( nat > a ) > a] :
      ( ( domain_a_b @ R )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ! [A3: nat > a] :
              ( ( member_nat_a @ A3 @ A2 )
             => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ R ) ) )
         => ( ( ( finpro5839458686994656414_nat_a @ R @ F @ A2 )
              = ( zero_a_b @ R ) )
            = ( ? [X3: nat > a] :
                  ( ( member_nat_a @ X3 @ A2 )
                  & ( ( F @ X3 )
                    = ( zero_a_b @ R ) ) ) ) ) ) ) ) ).

% domain.finprod_zero_iff
thf(fact_163_domain_Ofinprod__zero__iff,axiom,
    ! [R: partia2670972154091845814t_unit,A2: set_list_a,F: list_a > list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( finite_finite_list_a @ A2 )
       => ( ! [A3: list_a] :
              ( ( member_list_a @ A3 @ A2 )
             => ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ R ) ) )
         => ( ( ( finpro738134188688310831list_a @ R @ F @ A2 )
              = ( zero_l4142658623432671053t_unit @ R ) )
            = ( ? [X3: list_a] :
                  ( ( member_list_a @ X3 @ A2 )
                  & ( ( F @ X3 )
                    = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ) ).

% domain.finprod_zero_iff
thf(fact_164_domain_Ofinprod__zero__iff,axiom,
    ! [R: partia2956882679547061052t_unit,A2: set_a,F: a > list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( finite_finite_a @ A2 )
       => ( ! [A3: a] :
              ( ( member_a @ A3 @ A2 )
             => ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ R ) ) )
         => ( ( ( finpro5596966875920909993unit_a @ R @ F @ A2 )
              = ( zero_l347298301471573063t_unit @ R ) )
            = ( ? [X3: a] :
                  ( ( member_a @ X3 @ A2 )
                  & ( ( F @ X3 )
                    = ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ) ).

% domain.finprod_zero_iff
thf(fact_165_domain_Ofinprod__zero__iff,axiom,
    ! [R: partia2956882679547061052t_unit,A2: set_nat,F: nat > list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( finite_finite_nat @ A2 )
       => ( ! [A3: nat] :
              ( ( member_nat @ A3 @ A2 )
             => ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ R ) ) )
         => ( ( ( finpro4561275463894985573it_nat @ R @ F @ A2 )
              = ( zero_l347298301471573063t_unit @ R ) )
            = ( ? [X3: nat] :
                  ( ( member_nat @ X3 @ A2 )
                  & ( ( F @ X3 )
                    = ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ) ).

% domain.finprod_zero_iff
thf(fact_166_pigeonhole__infinite__rel,axiom,
    ! [A2: set_a,B: set_a,R: a > a > $o] :
      ( ~ ( finite_finite_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ A2 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: a] :
              ( ( member_a @ X2 @ B )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A4: a] :
                        ( ( member_a @ A4 @ A2 )
                        & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_167_pigeonhole__infinite__rel,axiom,
    ! [A2: set_a,B: set_nat,R: a > nat > $o] :
      ( ~ ( finite_finite_a @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A4: a] :
                        ( ( member_a @ A4 @ A2 )
                        & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_168_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_a,R: nat > a > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: a] :
              ( ( member_a @ X2 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_169_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_170_pigeonhole__infinite__rel,axiom,
    ! [A2: set_a,B: set_list_a,R: a > list_a > $o] :
      ( ~ ( finite_finite_a @ A2 )
     => ( ( finite_finite_list_a @ B )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ A2 )
             => ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: list_a] :
              ( ( member_list_a @ X2 @ B )
              & ~ ( finite_finite_a
                  @ ( collect_a
                    @ ^ [A4: a] :
                        ( ( member_a @ A4 @ A2 )
                        & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_171_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B: set_list_a,R: nat > list_a > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_list_a @ B )
       => ( ! [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
             => ? [Xa: list_a] :
                  ( ( member_list_a @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: list_a] :
              ( ( member_list_a @ X2 @ B )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A4: nat] :
                        ( ( member_nat @ A4 @ A2 )
                        & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_172_pigeonhole__infinite__rel,axiom,
    ! [A2: set_list_a,B: set_a,R: list_a > a > $o] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ! [X2: list_a] :
              ( ( member_list_a @ X2 @ A2 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: a] :
              ( ( member_a @ X2 @ B )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A4: list_a] :
                        ( ( member_list_a @ A4 @ A2 )
                        & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_173_pigeonhole__infinite__rel,axiom,
    ! [A2: set_list_a,B: set_nat,R: list_a > nat > $o] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ! [X2: list_a] :
              ( ( member_list_a @ X2 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: nat] :
              ( ( member_nat @ X2 @ B )
              & ~ ( finite_finite_list_a
                  @ ( collect_list_a
                    @ ^ [A4: list_a] :
                        ( ( member_list_a @ A4 @ A2 )
                        & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_174_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat_a,B: set_a,R: ( nat > a ) > a > $o] :
      ( ~ ( finite_finite_nat_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ! [X2: nat > a] :
              ( ( member_nat_a @ X2 @ A2 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: a] :
              ( ( member_a @ X2 @ B )
              & ~ ( finite_finite_nat_a
                  @ ( collect_nat_a
                    @ ^ [A4: nat > a] :
                        ( ( member_nat_a @ A4 @ A2 )
                        & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_175_pigeonhole__infinite__rel,axiom,
    ! [A2: set_a_a,B: set_a,R: ( a > a ) > a > $o] :
      ( ~ ( finite_finite_a_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ! [X2: a > a] :
              ( ( member_a_a @ X2 @ A2 )
             => ? [Xa: a] :
                  ( ( member_a @ Xa @ B )
                  & ( R @ X2 @ Xa ) ) )
         => ? [X2: a] :
              ( ( member_a @ X2 @ B )
              & ~ ( finite_finite_a_a
                  @ ( collect_a_a
                    @ ^ [A4: a > a] :
                        ( ( member_a_a @ A4 @ A2 )
                        & ( R @ A4 @ X2 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_176_not__finite__existsD,axiom,
    ! [P3: list_list_a > $o] :
      ( ~ ( finite1660835950917165235list_a @ ( collect_list_list_a @ P3 ) )
     => ? [X_1: list_list_a] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_177_not__finite__existsD,axiom,
    ! [P3: a > $o] :
      ( ~ ( finite_finite_a @ ( collect_a @ P3 ) )
     => ? [X_1: a] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_178_not__finite__existsD,axiom,
    ! [P3: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P3 ) )
     => ? [X_1: nat] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_179_not__finite__existsD,axiom,
    ! [P3: list_a > $o] :
      ( ~ ( finite_finite_list_a @ ( collect_list_a @ P3 ) )
     => ? [X_1: list_a] : ( P3 @ X_1 ) ) ).

% not_finite_existsD
thf(fact_180_finite__has__minimal2,axiom,
    ! [A2: set_set_a,A: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( member_set_a @ A @ A2 )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A2 )
            & ( ord_less_eq_set_a @ X2 @ A )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A2 )
               => ( ( ord_less_eq_set_a @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_181_finite__has__minimal2,axiom,
    ! [A2: set_set_list_a,A: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( member_set_list_a @ A @ A2 )
       => ? [X2: set_list_a] :
            ( ( member_set_list_a @ X2 @ A2 )
            & ( ord_le8861187494160871172list_a @ X2 @ A )
            & ! [Xa: set_list_a] :
                ( ( member_set_list_a @ Xa @ A2 )
               => ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_182_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A2 )
            & ( ord_less_eq_nat @ X2 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_183_finite__has__maximal2,axiom,
    ! [A2: set_set_a,A: set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( member_set_a @ A @ A2 )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A2 )
            & ( ord_less_eq_set_a @ A @ X2 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A2 )
               => ( ( ord_less_eq_set_a @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_184_finite__has__maximal2,axiom,
    ! [A2: set_set_list_a,A: set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( member_set_list_a @ A @ A2 )
       => ? [X2: set_list_a] :
            ( ( member_set_list_a @ X2 @ A2 )
            & ( ord_le8861187494160871172list_a @ A @ X2 )
            & ! [Xa: set_list_a] :
                ( ( member_set_list_a @ Xa @ A2 )
               => ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_185_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A2 )
            & ( ord_less_eq_nat @ A @ X2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_186_rev__finite__subset,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_187_rev__finite__subset,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( finite_finite_a @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_188_rev__finite__subset,axiom,
    ! [B: set_list_a,A2: set_list_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B )
       => ( finite_finite_list_a @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_189_infinite__super,axiom,
    ! [S2: set_nat,T: set_nat] :
      ( ( ord_less_eq_set_nat @ S2 @ T )
     => ( ~ ( finite_finite_nat @ S2 )
       => ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_super
thf(fact_190_infinite__super,axiom,
    ! [S2: set_a,T: set_a] :
      ( ( ord_less_eq_set_a @ S2 @ T )
     => ( ~ ( finite_finite_a @ S2 )
       => ~ ( finite_finite_a @ T ) ) ) ).

% infinite_super
thf(fact_191_infinite__super,axiom,
    ! [S2: set_list_a,T: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ S2 @ T )
     => ( ~ ( finite_finite_list_a @ S2 )
       => ~ ( finite_finite_list_a @ T ) ) ) ).

% infinite_super
thf(fact_192_finite__subset,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B )
     => ( ( finite_finite_nat @ B )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_193_finite__subset,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ A2 @ B )
     => ( ( finite_finite_a @ B )
       => ( finite_finite_a @ A2 ) ) ) ).

% finite_subset
thf(fact_194_finite__subset,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ B )
     => ( ( finite_finite_list_a @ B )
       => ( finite_finite_list_a @ A2 ) ) ) ).

% finite_subset
thf(fact_195_x_Obound__upD,axiom,
    ! [F: nat > list_a] :
      ( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ? [N: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F ) ) ).

% x.bound_upD
thf(fact_196_domain_Ofinite__number__of__roots,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( finite1660835950917165235list_a @ ( collect_list_list_a @ ( polyno5142720416380192742t_unit @ R @ P ) ) ) ) ) ).

% domain.finite_number_of_roots
thf(fact_197_domain_Ofinite__number__of__roots,axiom,
    ! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
       => ( finite5282473924520328461list_a @ ( collect_set_list_a @ ( polyno4320237611291262604t_unit @ R @ P ) ) ) ) ) ).

% domain.finite_number_of_roots
thf(fact_198_domain_Ofinite__number__of__roots,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ R @ P ) ) ) ) ) ).

% domain.finite_number_of_roots
thf(fact_199_domain_Ofinite__number__of__roots,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( finite_finite_list_a @ ( collect_list_a @ ( polyno6951661231331188332t_unit @ R @ P ) ) ) ) ) ).

% domain.finite_number_of_roots
thf(fact_200_x_Oa__lcos__mult__one,axiom,
    ! [M: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M )
        = M ) ) ).

% x.a_lcos_mult_one
thf(fact_201_finprod__mono__neutral__cong,axiom,
    ! [B: set_nat,A2: set_nat,H: nat > a,G: nat > a] :
      ( ( finite_finite_nat @ B )
     => ( ( finite_finite_nat @ A2 )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [I: nat] :
                ( ( member_nat @ I @ ( minus_minus_set_nat @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_a_ring_ext_a_b @ r ) ) )
           => ( ! [X2: nat] :
                  ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_nat_a @ G
                  @ ( pi_nat_a @ A2
                    @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( member_nat_a @ H
                    @ ( pi_nat_a @ B
                      @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
                 => ( ( finpro1280035270526425175_b_nat @ r @ G @ A2 )
                    = ( finpro1280035270526425175_b_nat @ r @ H @ B ) ) ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong
thf(fact_202_finprod__mono__neutral__cong,axiom,
    ! [B: set_a,A2: set_a,H: a > a,G: a > a] :
      ( ( finite_finite_a @ B )
     => ( ( finite_finite_a @ A2 )
       => ( ! [I: a] :
              ( ( member_a @ I @ ( minus_minus_set_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [I: a] :
                ( ( member_a @ I @ ( minus_minus_set_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_a_ring_ext_a_b @ r ) ) )
           => ( ! [X2: a] :
                  ( ( member_a @ X2 @ ( inf_inf_set_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_a @ G
                  @ ( pi_a_a @ A2
                    @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( member_a_a @ H
                    @ ( pi_a_a @ B
                      @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
                 => ( ( finpro205304725090349623_a_b_a @ r @ G @ A2 )
                    = ( finpro205304725090349623_a_b_a @ r @ H @ B ) ) ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong
thf(fact_203_finprod__mono__neutral__cong,axiom,
    ! [B: set_list_a,A2: set_list_a,H: list_a > a,G: list_a > a] :
      ( ( finite_finite_list_a @ B )
     => ( ( finite_finite_list_a @ A2 )
       => ( ! [I: list_a] :
              ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [I: list_a] :
                ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_a_ring_ext_a_b @ r ) ) )
           => ( ! [X2: list_a] :
                  ( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_list_a_a @ G
                  @ ( pi_list_a_a @ A2
                    @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( member_list_a_a @ H
                    @ ( pi_list_a_a @ B
                      @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
                 => ( ( finpro6052973074229812797list_a @ r @ G @ A2 )
                    = ( finpro6052973074229812797list_a @ r @ H @ B ) ) ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong
thf(fact_204_finprod__mono__neutral__cong,axiom,
    ! [B: set_a_a,A2: set_a_a,H: ( a > a ) > a,G: ( a > a ) > a] :
      ( ( finite_finite_a_a @ B )
     => ( ( finite_finite_a_a @ A2 )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ ( minus_minus_set_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [I: a > a] :
                ( ( member_a_a @ I @ ( minus_minus_set_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_a_ring_ext_a_b @ r ) ) )
           => ( ! [X2: a > a] :
                  ( ( member_a_a @ X2 @ ( inf_inf_set_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_a_a @ G
                  @ ( pi_a_a_a @ A2
                    @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( member_a_a_a @ H
                    @ ( pi_a_a_a @ B
                      @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
                 => ( ( finpro3012607322079259884_b_a_a @ r @ G @ A2 )
                    = ( finpro3012607322079259884_b_a_a @ r @ H @ B ) ) ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong
thf(fact_205_finprod__mono__neutral__cong,axiom,
    ! [B: set_nat_a,A2: set_nat_a,H: ( nat > a ) > a,G: ( nat > a ) > a] :
      ( ( finite_finite_nat_a @ B )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [I: nat > a] :
                ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_a_ring_ext_a_b @ r ) ) )
           => ( ! [X2: nat > a] :
                  ( ( member_nat_a @ X2 @ ( inf_inf_set_nat_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_nat_a_a @ G
                  @ ( pi_nat_a_a @ A2
                    @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( member_nat_a_a @ H
                    @ ( pi_nat_a_a @ B
                      @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
                 => ( ( finpro5839458686994656414_nat_a @ r @ G @ A2 )
                    = ( finpro5839458686994656414_nat_a @ r @ H @ B ) ) ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong
thf(fact_206_finprod__mono__neutral__cong,axiom,
    ! [B: set_a_list_a,A2: set_a_list_a,H: ( a > list_a ) > a,G: ( a > list_a ) > a] :
      ( ( finite8564763978580267522list_a @ B )
     => ( ( finite8564763978580267522list_a @ A2 )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [I: a > list_a] :
                ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_a_ring_ext_a_b @ r ) ) )
           => ( ! [X2: a > list_a] :
                  ( ( member_a_list_a @ X2 @ ( inf_inf_set_a_list_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_list_a_a @ G
                  @ ( pi_a_list_a_a @ A2
                    @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( member_a_list_a_a @ H
                    @ ( pi_a_list_a_a @ B
                      @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
                 => ( ( finpro2305829206037604594list_a @ r @ G @ A2 )
                    = ( finpro2305829206037604594list_a @ r @ H @ B ) ) ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong
thf(fact_207_finprod__mono__neutral__cong,axiom,
    ! [B: set_nat_list_a,A2: set_nat_list_a,H: ( nat > list_a ) > a,G: ( nat > list_a ) > a] :
      ( ( finite7630042315537210004list_a @ B )
     => ( ( finite7630042315537210004list_a @ A2 )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [I: nat > list_a] :
                ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_a_ring_ext_a_b @ r ) ) )
           => ( ! [X2: nat > list_a] :
                  ( ( member_nat_list_a @ X2 @ ( inf_in6652419485960844601list_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_nat_list_a_a @ G
                  @ ( pi_nat_list_a_a @ A2
                    @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( member_nat_list_a_a @ H
                    @ ( pi_nat_list_a_a @ B
                      @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
                 => ( ( finpro4838020199848830884list_a @ r @ G @ A2 )
                    = ( finpro4838020199848830884list_a @ r @ H @ B ) ) ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong
thf(fact_208_finprod__mono__neutral__cong,axiom,
    ! [B: set_a_a_a,A2: set_a_a_a,H: ( ( a > a ) > a ) > a,G: ( ( a > a ) > a ) > a] :
      ( ( finite_finite_a_a_a @ B )
     => ( ( finite_finite_a_a_a @ A2 )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [I: ( a > a ) > a] :
                ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_a_ring_ext_a_b @ r ) ) )
           => ( ! [X2: ( a > a ) > a] :
                  ( ( member_a_a_a @ X2 @ ( inf_inf_set_a_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_a_a_a @ G
                  @ ( pi_a_a_a_a @ A2
                    @ ^ [Uu: ( a > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( member_a_a_a_a @ H
                    @ ( pi_a_a_a_a @ B
                      @ ^ [Uu: ( a > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
                 => ( ( finpro8565617760274797897_a_a_a @ r @ G @ A2 )
                    = ( finpro8565617760274797897_a_a_a @ r @ H @ B ) ) ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong
thf(fact_209_finprod__mono__neutral__cong,axiom,
    ! [B: set_nat_a_a,A2: set_nat_a_a,H: ( ( nat > a ) > a ) > a,G: ( ( nat > a ) > a ) > a] :
      ( ( finite7239108116303828181at_a_a @ B )
     => ( ( finite7239108116303828181at_a_a @ A2 )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [I: ( nat > a ) > a] :
                ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_a_ring_ext_a_b @ r ) ) )
           => ( ! [X2: ( nat > a ) > a] :
                  ( ( member_nat_a_a @ X2 @ ( inf_inf_set_nat_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_nat_a_a_a @ G
                  @ ( pi_nat_a_a_a @ A2
                    @ ^ [Uu: ( nat > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( member_nat_a_a_a @ H
                    @ ( pi_nat_a_a_a @ B
                      @ ^ [Uu: ( nat > a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
                 => ( ( finpro7303668365752794053at_a_a @ r @ G @ A2 )
                    = ( finpro7303668365752794053at_a_a @ r @ H @ B ) ) ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong
thf(fact_210_finprod__mono__neutral__cong,axiom,
    ! [B: set_a_list_a_a,A2: set_a_list_a_a,H: ( ( a > list_a ) > a ) > a,G: ( ( a > list_a ) > a ) > a] :
      ( ( finite681744870773793075st_a_a @ B )
     => ( ( finite681744870773793075st_a_a @ A2 )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_a_ring_ext_a_b @ r ) ) )
         => ( ! [I: ( a > list_a ) > a] :
                ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_a_ring_ext_a_b @ r ) ) )
           => ( ! [X2: ( a > list_a ) > a] :
                  ( ( member_a_list_a_a @ X2 @ ( inf_in8941282555790403672st_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_list_a_a_a @ G
                  @ ( pi_a_list_a_a_a @ A2
                    @ ^ [Uu: ( a > list_a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( member_a_list_a_a_a @ H
                    @ ( pi_a_list_a_a_a @ B
                      @ ^ [Uu: ( a > list_a ) > a] : ( partia707051561876973205xt_a_b @ r ) ) )
                 => ( ( finpro3356682085724622403st_a_a @ r @ G @ A2 )
                    = ( finpro3356682085724622403st_a_a @ r @ H @ B ) ) ) ) ) ) ) ) ) ).

% finprod_mono_neutral_cong
thf(fact_211_x_Oa__l__coset__subset__G,axiom,
    ! [H2: set_list_a,X: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ H2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.a_l_coset_subset_G
thf(fact_212_x_Ocarrier__is__subalgebra,axiom,
    ! [K: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( embedd1768981623711841426t_unit @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.carrier_is_subalgebra
thf(fact_213_x_Osubalgebra__in__carrier,axiom,
    ! [K: set_list_a,V: set_list_a] :
      ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.subalgebra_in_carrier
thf(fact_214_x_Ogenideal__self,axiom,
    ! [S2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ S2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ord_le8861187494160871172list_a @ S2 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) ) ) ).

% x.genideal_self
thf(fact_215_x_Osubset__Idl__subset,axiom,
    ! [I4: set_list_a,H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_le8861187494160871172list_a @ H2 @ I4 )
       => ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 ) ) ) ) ).

% x.subset_Idl_subset
thf(fact_216_eval__cring__hom,axiom,
    ! [K: set_a,A: a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ r @ K ) @ r
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ A ) ) ) ) ).

% eval_cring_hom
thf(fact_217_x_Oup__smult__closed,axiom,
    ! [A: list_a,P: nat > list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member_nat_list_a
          @ ^ [I2: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( P @ I2 ) )
          @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.up_smult_closed
thf(fact_218_zero__not__one,axiom,
    ( ( zero_a_b @ r )
   != ( one_a_ring_ext_a_b @ r ) ) ).

% zero_not_one
thf(fact_219_subring__inter,axiom,
    ! [I4: set_a,J2: set_a] :
      ( ( subring_a_b @ I4 @ r )
     => ( ( subring_a_b @ J2 @ r )
       => ( subring_a_b @ ( inf_inf_set_a @ I4 @ J2 ) @ r ) ) ) ).

% subring_inter
thf(fact_220_carrier__is__subring,axiom,
    subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).

% carrier_is_subring
thf(fact_221_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_222_finprod__zero__iff,axiom,
    ! [A2: set_a,F: a > a] :
      ( ( finite_finite_a @ A2 )
     => ( ! [A3: a] :
            ( ( member_a @ A3 @ A2 )
           => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ( finpro205304725090349623_a_b_a @ r @ F @ A2 )
            = ( zero_a_b @ r ) )
          = ( ? [X3: a] :
                ( ( member_a @ X3 @ A2 )
                & ( ( F @ X3 )
                  = ( zero_a_b @ r ) ) ) ) ) ) ) ).

% finprod_zero_iff
thf(fact_223_finprod__zero__iff,axiom,
    ! [A2: set_nat,F: nat > a] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A2 )
           => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ( finpro1280035270526425175_b_nat @ r @ F @ A2 )
            = ( zero_a_b @ r ) )
          = ( ? [X3: nat] :
                ( ( member_nat @ X3 @ A2 )
                & ( ( F @ X3 )
                  = ( zero_a_b @ r ) ) ) ) ) ) ) ).

% finprod_zero_iff
thf(fact_224_finprod__zero__iff,axiom,
    ! [A2: set_list_a,F: list_a > a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ! [A3: list_a] :
            ( ( member_list_a @ A3 @ A2 )
           => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ( finpro6052973074229812797list_a @ r @ F @ A2 )
            = ( zero_a_b @ r ) )
          = ( ? [X3: list_a] :
                ( ( member_list_a @ X3 @ A2 )
                & ( ( F @ X3 )
                  = ( zero_a_b @ r ) ) ) ) ) ) ) ).

% finprod_zero_iff
thf(fact_225_finprod__zero__iff,axiom,
    ! [A2: set_a_a,F: ( a > a ) > a] :
      ( ( finite_finite_a_a @ A2 )
     => ( ! [A3: a > a] :
            ( ( member_a_a @ A3 @ A2 )
           => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ( finpro3012607322079259884_b_a_a @ r @ F @ A2 )
            = ( zero_a_b @ r ) )
          = ( ? [X3: a > a] :
                ( ( member_a_a @ X3 @ A2 )
                & ( ( F @ X3 )
                  = ( zero_a_b @ r ) ) ) ) ) ) ) ).

% finprod_zero_iff
thf(fact_226_finprod__zero__iff,axiom,
    ! [A2: set_nat_a,F: ( nat > a ) > a] :
      ( ( finite_finite_nat_a @ A2 )
     => ( ! [A3: nat > a] :
            ( ( member_nat_a @ A3 @ A2 )
           => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ( finpro5839458686994656414_nat_a @ r @ F @ A2 )
            = ( zero_a_b @ r ) )
          = ( ? [X3: nat > a] :
                ( ( member_nat_a @ X3 @ A2 )
                & ( ( F @ X3 )
                  = ( zero_a_b @ r ) ) ) ) ) ) ) ).

% finprod_zero_iff
thf(fact_227_finprod__zero__iff,axiom,
    ! [A2: set_a_list_a,F: ( a > list_a ) > a] :
      ( ( finite8564763978580267522list_a @ A2 )
     => ( ! [A3: a > list_a] :
            ( ( member_a_list_a @ A3 @ A2 )
           => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ( finpro2305829206037604594list_a @ r @ F @ A2 )
            = ( zero_a_b @ r ) )
          = ( ? [X3: a > list_a] :
                ( ( member_a_list_a @ X3 @ A2 )
                & ( ( F @ X3 )
                  = ( zero_a_b @ r ) ) ) ) ) ) ) ).

% finprod_zero_iff
thf(fact_228_finprod__zero__iff,axiom,
    ! [A2: set_nat_list_a,F: ( nat > list_a ) > a] :
      ( ( finite7630042315537210004list_a @ A2 )
     => ( ! [A3: nat > list_a] :
            ( ( member_nat_list_a @ A3 @ A2 )
           => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ( finpro4838020199848830884list_a @ r @ F @ A2 )
            = ( zero_a_b @ r ) )
          = ( ? [X3: nat > list_a] :
                ( ( member_nat_list_a @ X3 @ A2 )
                & ( ( F @ X3 )
                  = ( zero_a_b @ r ) ) ) ) ) ) ) ).

% finprod_zero_iff
thf(fact_229_finprod__zero__iff,axiom,
    ! [A2: set_a_a_a,F: ( ( a > a ) > a ) > a] :
      ( ( finite_finite_a_a_a @ A2 )
     => ( ! [A3: ( a > a ) > a] :
            ( ( member_a_a_a @ A3 @ A2 )
           => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ( finpro8565617760274797897_a_a_a @ r @ F @ A2 )
            = ( zero_a_b @ r ) )
          = ( ? [X3: ( a > a ) > a] :
                ( ( member_a_a_a @ X3 @ A2 )
                & ( ( F @ X3 )
                  = ( zero_a_b @ r ) ) ) ) ) ) ) ).

% finprod_zero_iff
thf(fact_230_finprod__zero__iff,axiom,
    ! [A2: set_nat_a_a,F: ( ( nat > a ) > a ) > a] :
      ( ( finite7239108116303828181at_a_a @ A2 )
     => ( ! [A3: ( nat > a ) > a] :
            ( ( member_nat_a_a @ A3 @ A2 )
           => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ( finpro7303668365752794053at_a_a @ r @ F @ A2 )
            = ( zero_a_b @ r ) )
          = ( ? [X3: ( nat > a ) > a] :
                ( ( member_nat_a_a @ X3 @ A2 )
                & ( ( F @ X3 )
                  = ( zero_a_b @ r ) ) ) ) ) ) ) ).

% finprod_zero_iff
thf(fact_231_finprod__zero__iff,axiom,
    ! [A2: set_a_list_a_a,F: ( ( a > list_a ) > a ) > a] :
      ( ( finite681744870773793075st_a_a @ A2 )
     => ( ! [A3: ( a > list_a ) > a] :
            ( ( member_a_list_a_a @ A3 @ A2 )
           => ( member_a @ ( F @ A3 ) @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ( finpro3356682085724622403st_a_a @ r @ F @ A2 )
            = ( zero_a_b @ r ) )
          = ( ? [X3: ( a > list_a ) > a] :
                ( ( member_a_list_a_a @ X3 @ A2 )
                & ( ( F @ X3 )
                  = ( zero_a_b @ r ) ) ) ) ) ) ) ).

% finprod_zero_iff
thf(fact_232_x_Osubalgebra__inter,axiom,
    ! [K: set_list_a,V: set_list_a,V2: set_list_a] :
      ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( embedd1768981623711841426t_unit @ K @ V2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( embedd1768981623711841426t_unit @ K @ ( inf_inf_set_list_a @ V @ V2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.subalgebra_inter
thf(fact_233_x_Osubcring__inter,axiom,
    ! [I4: set_list_a,J2: set_list_a] :
      ( ( subcri7763218559781929323t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( subcri7763218559781929323t_unit @ J2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( subcri7763218559781929323t_unit @ ( inf_inf_set_list_a @ I4 @ J2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.subcring_inter
thf(fact_234_x_Om__lcomm,axiom,
    ! [X: list_a,Y: list_a,Z: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
            = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).

% x.m_lcomm
thf(fact_235_x_Om__comm,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
          = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).

% x.m_comm
thf(fact_236_x_Om__assoc,axiom,
    ! [X: list_a,Y: list_a,Z: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
            = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).

% x.m_assoc
thf(fact_237_var__closed_I1_J,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).

% var_closed(1)
thf(fact_238_is__root__poly__mult__imp__is__root,axiom,
    ! [P: list_a,Q: list_a,X: a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X )
         => ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
            | ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).

% is_root_poly_mult_imp_is_root
thf(fact_239_univ__poly__a__minus__consistent,axiom,
    ! [K: set_a,Q: list_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
          = ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ).

% univ_poly_a_minus_consistent
thf(fact_240_finite__Int,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( ( finite_finite_nat @ F2 )
        | ( finite_finite_nat @ G2 ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).

% finite_Int
thf(fact_241_finite__Int,axiom,
    ! [F2: set_a,G2: set_a] :
      ( ( ( finite_finite_a @ F2 )
        | ( finite_finite_a @ G2 ) )
     => ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G2 ) ) ) ).

% finite_Int
thf(fact_242_finite__Int,axiom,
    ! [F2: set_list_a,G2: set_list_a] :
      ( ( ( finite_finite_list_a @ F2 )
        | ( finite_finite_list_a @ G2 ) )
     => ( finite_finite_list_a @ ( inf_inf_set_list_a @ F2 @ G2 ) ) ) ).

% finite_Int
thf(fact_243_zero__closed,axiom,
    member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% zero_closed
thf(fact_244_funcset__Int__left,axiom,
    ! [F: nat > list_a,A2: set_nat,C: set_list_a,B: set_nat] :
      ( ( member_nat_list_a @ F
        @ ( pi_nat_list_a @ A2
          @ ^ [Uu: nat] : C ) )
     => ( ( member_nat_list_a @ F
          @ ( pi_nat_list_a @ B
            @ ^ [Uu: nat] : C ) )
       => ( member_nat_list_a @ F
          @ ( pi_nat_list_a @ ( inf_inf_set_nat @ A2 @ B )
            @ ^ [Uu: nat] : C ) ) ) ) ).

% funcset_Int_left
thf(fact_245_funcset__Int__left,axiom,
    ! [F: nat > a,A2: set_nat,C: set_a,B: set_nat] :
      ( ( member_nat_a @ F
        @ ( pi_nat_a @ A2
          @ ^ [Uu: nat] : C ) )
     => ( ( member_nat_a @ F
          @ ( pi_nat_a @ B
            @ ^ [Uu: nat] : C ) )
       => ( member_nat_a @ F
          @ ( pi_nat_a @ ( inf_inf_set_nat @ A2 @ B )
            @ ^ [Uu: nat] : C ) ) ) ) ).

% funcset_Int_left
thf(fact_246_funcset__Int__left,axiom,
    ! [F: ( a > a ) > a,A2: set_a_a,C: set_a,B: set_a_a] :
      ( ( member_a_a_a @ F
        @ ( pi_a_a_a @ A2
          @ ^ [Uu: a > a] : C ) )
     => ( ( member_a_a_a @ F
          @ ( pi_a_a_a @ B
            @ ^ [Uu: a > a] : C ) )
       => ( member_a_a_a @ F
          @ ( pi_a_a_a @ ( inf_inf_set_a_a @ A2 @ B )
            @ ^ [Uu: a > a] : C ) ) ) ) ).

% funcset_Int_left
thf(fact_247_funcset__Int__left,axiom,
    ! [F: ( a > list_a ) > a,A2: set_a_list_a,C: set_a,B: set_a_list_a] :
      ( ( member_a_list_a_a @ F
        @ ( pi_a_list_a_a @ A2
          @ ^ [Uu: a > list_a] : C ) )
     => ( ( member_a_list_a_a @ F
          @ ( pi_a_list_a_a @ B
            @ ^ [Uu: a > list_a] : C ) )
       => ( member_a_list_a_a @ F
          @ ( pi_a_list_a_a @ ( inf_inf_set_a_list_a @ A2 @ B )
            @ ^ [Uu: a > list_a] : C ) ) ) ) ).

% funcset_Int_left
thf(fact_248_funcset__Int__left,axiom,
    ! [F: ( nat > a ) > a,A2: set_nat_a,C: set_a,B: set_nat_a] :
      ( ( member_nat_a_a @ F
        @ ( pi_nat_a_a @ A2
          @ ^ [Uu: nat > a] : C ) )
     => ( ( member_nat_a_a @ F
          @ ( pi_nat_a_a @ B
            @ ^ [Uu: nat > a] : C ) )
       => ( member_nat_a_a @ F
          @ ( pi_nat_a_a @ ( inf_inf_set_nat_a @ A2 @ B )
            @ ^ [Uu: nat > a] : C ) ) ) ) ).

% funcset_Int_left
thf(fact_249_funcset__Int__left,axiom,
    ! [F: ( nat > list_a ) > a,A2: set_nat_list_a,C: set_a,B: set_nat_list_a] :
      ( ( member_nat_list_a_a @ F
        @ ( pi_nat_list_a_a @ A2
          @ ^ [Uu: nat > list_a] : C ) )
     => ( ( member_nat_list_a_a @ F
          @ ( pi_nat_list_a_a @ B
            @ ^ [Uu: nat > list_a] : C ) )
       => ( member_nat_list_a_a @ F
          @ ( pi_nat_list_a_a @ ( inf_in6652419485960844601list_a @ A2 @ B )
            @ ^ [Uu: nat > list_a] : C ) ) ) ) ).

% funcset_Int_left
thf(fact_250_funcset__Int__left,axiom,
    ! [F: a > list_a,A2: set_a,C: set_list_a,B: set_a] :
      ( ( member_a_list_a @ F
        @ ( pi_a_list_a @ A2
          @ ^ [Uu: a] : C ) )
     => ( ( member_a_list_a @ F
          @ ( pi_a_list_a @ B
            @ ^ [Uu: a] : C ) )
       => ( member_a_list_a @ F
          @ ( pi_a_list_a @ ( inf_inf_set_a @ A2 @ B )
            @ ^ [Uu: a] : C ) ) ) ) ).

% funcset_Int_left
thf(fact_251_funcset__Int__left,axiom,
    ! [F: a > a,A2: set_a,C: set_a,B: set_a] :
      ( ( member_a_a @ F
        @ ( pi_a_a @ A2
          @ ^ [Uu: a] : C ) )
     => ( ( member_a_a @ F
          @ ( pi_a_a @ B
            @ ^ [Uu: a] : C ) )
       => ( member_a_a @ F
          @ ( pi_a_a @ ( inf_inf_set_a @ A2 @ B )
            @ ^ [Uu: a] : C ) ) ) ) ).

% funcset_Int_left
thf(fact_252_r__right__minus__eq,axiom,
    ! [A: a,B3: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( a_minus_a_b @ r @ A @ B3 )
            = ( zero_a_b @ r ) )
          = ( A = B3 ) ) ) ) ).

% r_right_minus_eq
thf(fact_253_x_Om__closed,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.m_closed
thf(fact_254_carrier__polynomial__shell,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% carrier_polynomial_shell
thf(fact_255_x_Or__null,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.r_null
thf(fact_256_x_Ol__null,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.l_null
thf(fact_257_x_Oring_Ohom__zero,axiom,
    ( ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
    = ( zero_a_b @ r ) ) ).

% x.ring.hom_zero
thf(fact_258_domain_Ois__root__poly__mult__imp__is__root,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,X: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
         => ( ( polyno5142720416380192742t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q ) @ X )
           => ( ( polyno5142720416380192742t_unit @ R @ P @ X )
              | ( polyno5142720416380192742t_unit @ R @ Q @ X ) ) ) ) ) ) ).

% domain.is_root_poly_mult_imp_is_root
thf(fact_259_domain_Ois__root__poly__mult__imp__is__root,axiom,
    ! [R: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a,X: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
       => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
         => ( ( polyno4320237611291262604t_unit @ R @ ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ P @ Q ) @ X )
           => ( ( polyno4320237611291262604t_unit @ R @ P @ X )
              | ( polyno4320237611291262604t_unit @ R @ Q @ X ) ) ) ) ) ) ).

% domain.is_root_poly_mult_imp_is_root
thf(fact_260_domain_Ois__root__poly__mult__imp__is__root,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X: a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
         => ( ( polyno4133073214067823460ot_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q ) @ X )
           => ( ( polyno4133073214067823460ot_a_b @ R @ P @ X )
              | ( polyno4133073214067823460ot_a_b @ R @ Q @ X ) ) ) ) ) ) ).

% domain.is_root_poly_mult_imp_is_root
thf(fact_261_domain_Ois__root__poly__mult__imp__is__root,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
         => ( ( polyno6951661231331188332t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q ) @ X )
           => ( ( polyno6951661231331188332t_unit @ R @ P @ X )
              | ( polyno6951661231331188332t_unit @ R @ Q @ X ) ) ) ) ) ) ).

% domain.is_root_poly_mult_imp_is_root
thf(fact_262_ring_Ois__root_Ocong,axiom,
    polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).

% ring.is_root.cong
thf(fact_263_ring_Ois__root_Ocong,axiom,
    polyno6951661231331188332t_unit = polyno6951661231331188332t_unit ).

% ring.is_root.cong
thf(fact_264_mem__upI,axiom,
    ! [F: nat > a,R: partia2175431115845679010xt_a_b] :
      ( ! [N: nat] : ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R ) )
     => ( ? [N2: nat] : ( bound_a @ ( zero_a_b @ R ) @ N2 @ F )
       => ( member_nat_a @ F @ ( up_a_b @ R ) ) ) ) ).

% mem_upI
thf(fact_265_mem__upI,axiom,
    ! [F: nat > list_a,R: partia2670972154091845814t_unit] :
      ( ! [N: nat] : ( member_list_a @ ( F @ N ) @ ( partia5361259788508890537t_unit @ R ) )
     => ( ? [N2: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ N2 @ F )
       => ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ).

% mem_upI
thf(fact_266_mem__upI,axiom,
    ! [F: nat > list_list_a,R: partia2956882679547061052t_unit] :
      ( ! [N: nat] : ( member_list_list_a @ ( F @ N ) @ ( partia2464479390973590831t_unit @ R ) )
     => ( ? [N2: nat] : ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ N2 @ F )
       => ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) ) ) ) ).

% mem_upI
thf(fact_267_mem__upI,axiom,
    ! [F: nat > set_list_a,R: partia7496981018696276118t_unit] :
      ( ! [N: nat] : ( member_set_list_a @ ( F @ N ) @ ( partia141011252114345353t_unit @ R ) )
     => ( ? [N2: nat] : ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ N2 @ F )
       => ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R ) ) ) ) ).

% mem_upI
thf(fact_268_ring__primeE_I1_J,axiom,
    ! [P: a] :
      ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_ring_prime_a_b @ r @ P )
       => ( P
         != ( zero_a_b @ r ) ) ) ) ).

% ring_primeE(1)
thf(fact_269_bound__upD,axiom,
    ! [F: nat > a] :
      ( ( member_nat_a @ F @ ( up_a_b @ r ) )
     => ? [N: nat] : ( bound_a @ ( zero_a_b @ r ) @ N @ F ) ) ).

% bound_upD
thf(fact_270_x_Omonoid__cancelI,axiom,
    ( ! [A3: list_a,B4: list_a,C2: list_a] :
        ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A3 )
          = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B4 ) )
       => ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
             => ( A3 = B4 ) ) ) ) )
   => ( ! [A3: list_a,B4: list_a,C2: list_a] :
          ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ C2 )
            = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B4 @ C2 ) )
         => ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
             => ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
               => ( A3 = B4 ) ) ) ) )
     => ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.monoid_cancelI
thf(fact_271_ring__irreducibleE_I1_J,axiom,
    ! [R2: a] :
      ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_r999134135267193926le_a_b @ r @ R2 )
       => ( R2
         != ( zero_a_b @ r ) ) ) ) ).

% ring_irreducibleE(1)
thf(fact_272_domain_Oeval__cring__hom,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,A: a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ R @ K ) @ R
            @ ^ [P2: list_a] : ( eval_a_b @ R @ P2 @ A ) ) ) ) ) ).

% domain.eval_cring_hom
thf(fact_273_domain_Oeval__cring__hom,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
         => ( ring_h453377649743177125t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ R
            @ ^ [P2: list_list_a] : ( eval_l34571156754992824t_unit @ R @ P2 @ A ) ) ) ) ) ).

% domain.eval_cring_hom
thf(fact_274_domain_Oeval__cring__hom,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,A: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
         => ( ring_h5732965726171697305t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ R
            @ ^ [P2: list_list_list_a] : ( eval_l1088911609197519410t_unit @ R @ P2 @ A ) ) ) ) ) ).

% domain.eval_cring_hom
thf(fact_275_domain_Oeval__cring__hom,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,A: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
         => ( ring_h4012356781940393445t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ R
            @ ^ [P2: list_set_list_a] : ( eval_s5133945360527818456t_unit @ R @ P2 @ A ) ) ) ) ) ).

% domain.eval_cring_hom
thf(fact_276_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a,P: list_list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( subrin3541368690557094692t_unit @ K @ R )
       => ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
         => ( ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P @ Q )
            = ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_277_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia7496981018696276118t_unit,K: set_set_list_a,Q: list_set_list_a,P: list_set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( subrin5643252653130547402t_unit @ K @ R )
       => ( ( member5524387281408368019list_a @ Q @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
         => ( ( a_minu6874796375791416686t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P @ Q )
            = ( a_minu6874796375791416686t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_278_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a,P: list_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
         => ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q )
            = ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_279_domain_Ouniv__poly__a__minus__consistent,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a,P: list_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
         => ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q )
            = ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).

% domain.univ_poly_a_minus_consistent
thf(fact_280_x_Opoly__of__const__in__carrier,axiom,
    ! [S: list_a] :
      ( ( member_list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( member_list_list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).

% x.poly_of_const_in_carrier
thf(fact_281_x_OboundD__carrier,axiom,
    ! [N3: nat,F: nat > list_a,M2: nat] :
      ( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N3 @ F )
     => ( ( ord_less_nat @ N3 @ M2 )
       => ( member_list_a @ ( F @ M2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.boundD_carrier
thf(fact_282_eval__is__hom,axiom,
    ! [K: set_a,A: a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ A )
          @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ K ) @ r ) ) ) ) ).

% eval_is_hom
thf(fact_283_x_Ocarrier__is__subring,axiom,
    subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).

% x.carrier_is_subring
thf(fact_284_x_Osubring__inter,axiom,
    ! [I4: set_list_a,J2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( subrin6918843898125473962t_unit @ J2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( subrin6918843898125473962t_unit @ ( inf_inf_set_list_a @ I4 @ J2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.subring_inter
thf(fact_285_x_OsubcringI_H,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.subcringI'
thf(fact_286_x_Oeval__poly__of__const,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y )
        = X ) ) ).

% x.eval_poly_of_const
thf(fact_287_x_OsubcringI,axiom,
    ! [H2: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ! [H1: list_a,H22: list_a] :
            ( ( member_list_a @ H1 @ H2 )
           => ( ( member_list_a @ H22 @ H2 )
             => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 )
                = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H22 @ H1 ) ) ) )
       => ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.subcringI
thf(fact_288_x_Oring__hom__restrict,axiom,
    ! [F: list_a > a,S2: partia2175431115845679010xt_a_b,G: list_a > a] :
      ( ( member_list_a_a @ F @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) )
     => ( ! [R3: list_a] :
            ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( F @ R3 )
              = ( G @ R3 ) ) )
       => ( member_list_a_a @ G @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) ) ) ) ).

% x.ring_hom_restrict
thf(fact_289_x_Ofinprod__closed,axiom,
    ! [F: nat > list_a,A2: set_nat] :
      ( ( member_nat_list_a @ F
        @ ( pi_nat_list_a @ A2
          @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( member_list_a @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_closed
thf(fact_290_x_Ofinprod__closed,axiom,
    ! [F: a > list_a,A2: set_a] :
      ( ( member_a_list_a @ F
        @ ( pi_a_list_a @ A2
          @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( member_list_a @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_closed
thf(fact_291_x_Ofinprod__cong_H,axiom,
    ! [A2: set_nat_list_a,B: set_nat_list_a,G: ( nat > list_a ) > list_a,F: ( nat > list_a ) > list_a] :
      ( ( A2 = B )
     => ( ( member1174670940520477013list_a @ G
          @ ( pi_nat_list_a_list_a @ B
            @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finprod_cong'
thf(fact_292_x_Ofinprod__cong_H,axiom,
    ! [A2: set_nat_a,B: set_nat_a,G: ( nat > a ) > list_a,F: ( nat > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member_nat_a_list_a @ G
          @ ( pi_nat_a_list_a @ B
            @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finprod_cong'
thf(fact_293_x_Ofinprod__cong_H,axiom,
    ! [A2: set_a_list_a,B: set_a_list_a,G: ( a > list_a ) > list_a,F: ( a > list_a ) > list_a] :
      ( ( A2 = B )
     => ( ( member4106537806236033721list_a @ G
          @ ( pi_a_list_a_list_a @ B
            @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finprod_cong'
thf(fact_294_x_Ofinprod__cong_H,axiom,
    ! [A2: set_a_a,B: set_a_a,G: ( a > a ) > list_a,F: ( a > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member_a_a_list_a @ G
          @ ( pi_a_a_list_a @ B
            @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finprod_cong'
thf(fact_295_x_Ofinprod__cong_H,axiom,
    ! [A2: set_a_a_a,B: set_a_a_a,G: ( ( a > a ) > a ) > list_a,F: ( ( a > a ) > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member_a_a_a_list_a @ G
          @ ( pi_a_a_a_list_a @ B
            @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finprod_cong'
thf(fact_296_x_Ofinprod__cong_H,axiom,
    ! [A2: set_a_list_a_a,B: set_a_list_a_a,G: ( ( a > list_a ) > a ) > list_a,F: ( ( a > list_a ) > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member455139831191412214list_a @ G
          @ ( pi_a_list_a_a_list_a @ B
            @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finprod_cong'
thf(fact_297_x_Ofinprod__cong_H,axiom,
    ! [A2: set_nat_a_a,B: set_nat_a_a,G: ( ( nat > a ) > a ) > list_a,F: ( ( nat > a ) > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member1630437648146986790list_a @ G
          @ ( pi_nat_a_a_list_a @ B
            @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finprod_cong'
thf(fact_298_x_Ofinprod__cong_H,axiom,
    ! [A2: set_nat_list_a_a,B: set_nat_list_a_a,G: ( ( nat > list_a ) > a ) > list_a,F: ( ( nat > list_a ) > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member4189978660109718316list_a @ G
          @ ( pi_nat837482917391966783list_a @ B
            @ ^ [Uu: ( nat > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: ( nat > list_a ) > a] :
              ( ( member_nat_list_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro3153191432819564877st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finpro3153191432819564877st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finprod_cong'
thf(fact_299_x_Ofinprod__cong_H,axiom,
    ! [A2: set_nat,B: set_nat,G: nat > list_a,F: nat > list_a] :
      ( ( A2 = B )
     => ( ( member_nat_list_a @ G
          @ ( pi_nat_list_a @ B
            @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finprod_cong'
thf(fact_300_x_Ofinprod__cong_H,axiom,
    ! [A2: set_a,B: set_a,G: a > list_a,F: a > list_a] :
      ( ( A2 = B )
     => ( ( member_a_list_a @ G
          @ ( pi_a_list_a @ B
            @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: a] :
              ( ( member_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finprod_cong'
thf(fact_301_boundD__carrier,axiom,
    ! [N3: nat,F: nat > a,M2: nat] :
      ( ( bound_a @ ( zero_a_b @ r ) @ N3 @ F )
     => ( ( ord_less_nat @ N3 @ M2 )
       => ( member_a @ ( F @ M2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% boundD_carrier
thf(fact_302_finite__Collect__less__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_nat @ N4 @ K2 ) ) ) ).

% finite_Collect_less_nat
thf(fact_303_x_Oeval__in__carrier__2,axiom,
    ! [X: list_list_a,Y: list_a] :
      ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.eval_in_carrier_2
thf(fact_304_bound_Ointro,axiom,
    ! [N3: nat,F: nat > list_a,Z: list_a] :
      ( ! [M3: nat] :
          ( ( ord_less_nat @ N3 @ M3 )
         => ( ( F @ M3 )
            = Z ) )
     => ( bound_list_a @ Z @ N3 @ F ) ) ).

% bound.intro
thf(fact_305_bound_Ointro,axiom,
    ! [N3: nat,F: nat > a,Z: a] :
      ( ! [M3: nat] :
          ( ( ord_less_nat @ N3 @ M3 )
         => ( ( F @ M3 )
            = Z ) )
     => ( bound_a @ Z @ N3 @ F ) ) ).

% bound.intro
thf(fact_306_x_Ocarrier__polynomial__shell,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
       => ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).

% x.carrier_polynomial_shell
thf(fact_307_x_Ofinprod__multf,axiom,
    ! [F: nat > list_a,A2: set_nat,G: nat > list_a] :
      ( ( member_nat_list_a @ F
        @ ( pi_nat_list_a @ A2
          @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( member_nat_list_a @ G
          @ ( pi_nat_list_a @ A2
            @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
            @ ^ [X3: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ X3 ) @ ( G @ X3 ) )
            @ A2 )
          = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) ) ) ) ) ).

% x.finprod_multf
thf(fact_308_x_Ofinprod__multf,axiom,
    ! [F: a > list_a,A2: set_a,G: a > list_a] :
      ( ( member_a_list_a @ F
        @ ( pi_a_list_a @ A2
          @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( member_a_list_a @ G
          @ ( pi_a_list_a @ A2
            @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
            @ ^ [X3: a] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ X3 ) @ ( G @ X3 ) )
            @ A2 )
          = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) ) ) ) ) ).

% x.finprod_multf
thf(fact_309_x_Oring_Ohomh,axiom,
    ( member_list_a_a
    @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
    @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r ) ) ).

% x.ring.homh
thf(fact_310_bound__def,axiom,
    ( bound_list_a
    = ( ^ [Z2: list_a,N4: nat,F3: nat > list_a] :
        ! [M4: nat] :
          ( ( ord_less_nat @ N4 @ M4 )
         => ( ( F3 @ M4 )
            = Z2 ) ) ) ) ).

% bound_def
thf(fact_311_bound__def,axiom,
    ( bound_a
    = ( ^ [Z2: a,N4: nat,F3: nat > a] :
        ! [M4: nat] :
          ( ( ord_less_nat @ N4 @ M4 )
         => ( ( F3 @ M4 )
            = Z2 ) ) ) ) ).

% bound_def
thf(fact_312_bound_Obound,axiom,
    ! [Z: list_a,N3: nat,F: nat > list_a,M2: nat] :
      ( ( bound_list_a @ Z @ N3 @ F )
     => ( ( ord_less_nat @ N3 @ M2 )
       => ( ( F @ M2 )
          = Z ) ) ) ).

% bound.bound
thf(fact_313_bound_Obound,axiom,
    ! [Z: a,N3: nat,F: nat > a,M2: nat] :
      ( ( bound_a @ Z @ N3 @ F )
     => ( ( ord_less_nat @ N3 @ M2 )
       => ( ( F @ M2 )
          = Z ) ) ) ).

% bound.bound
thf(fact_314_abelian__monoid_OboundD__carrier,axiom,
    ! [G2: partia2175431115845679010xt_a_b,N3: nat,F: nat > a,M2: nat] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( bound_a @ ( zero_a_b @ G2 ) @ N3 @ F )
       => ( ( ord_less_nat @ N3 @ M2 )
         => ( member_a @ ( F @ M2 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ) ).

% abelian_monoid.boundD_carrier
thf(fact_315_abelian__monoid_OboundD__carrier,axiom,
    ! [G2: partia2670972154091845814t_unit,N3: nat,F: nat > list_a,M2: nat] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ G2 ) @ N3 @ F )
       => ( ( ord_less_nat @ N3 @ M2 )
         => ( member_list_a @ ( F @ M2 ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) ) ) ).

% abelian_monoid.boundD_carrier
thf(fact_316_abelian__monoid_OboundD__carrier,axiom,
    ! [G2: partia2956882679547061052t_unit,N3: nat,F: nat > list_list_a,M2: nat] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ G2 ) @ N3 @ F )
       => ( ( ord_less_nat @ N3 @ M2 )
         => ( member_list_list_a @ ( F @ M2 ) @ ( partia2464479390973590831t_unit @ G2 ) ) ) ) ) ).

% abelian_monoid.boundD_carrier
thf(fact_317_abelian__monoid_OboundD__carrier,axiom,
    ! [G2: partia7496981018696276118t_unit,N3: nat,F: nat > set_list_a,M2: nat] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ G2 ) @ N3 @ F )
       => ( ( ord_less_nat @ N3 @ M2 )
         => ( member_set_list_a @ ( F @ M2 ) @ ( partia141011252114345353t_unit @ G2 ) ) ) ) ) ).

% abelian_monoid.boundD_carrier
thf(fact_318_ring_Oeval_Ocong,axiom,
    eval_a_b = eval_a_b ).

% ring.eval.cong
thf(fact_319_ring_Oeval_Ocong,axiom,
    eval_l34571156754992824t_unit = eval_l34571156754992824t_unit ).

% ring.eval.cong
thf(fact_320_ring_Opoly__of__const_Ocong,axiom,
    poly_of_const_a_b = poly_of_const_a_b ).

% ring.poly_of_const.cong
thf(fact_321_ring_Opoly__of__const_Ocong,axiom,
    poly_o8716471131768098070t_unit = poly_o8716471131768098070t_unit ).

% ring.poly_of_const.cong
thf(fact_322_bound__below,axiom,
    ! [Z: list_a,M2: nat,F: nat > list_a,N3: nat] :
      ( ( bound_list_a @ Z @ M2 @ F )
     => ( ( ( F @ N3 )
         != Z )
       => ( ord_less_eq_nat @ N3 @ M2 ) ) ) ).

% bound_below
thf(fact_323_bound__below,axiom,
    ! [Z: a,M2: nat,F: nat > a,N3: nat] :
      ( ( bound_a @ Z @ M2 @ F )
     => ( ( ( F @ N3 )
         != Z )
       => ( ord_less_eq_nat @ N3 @ M2 ) ) ) ).

% bound_below
thf(fact_324_mem__upD,axiom,
    ! [F: nat > a,R: partia2175431115845679010xt_a_b,N3: nat] :
      ( ( member_nat_a @ F @ ( up_a_b @ R ) )
     => ( member_a @ ( F @ N3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% mem_upD
thf(fact_325_mem__upD,axiom,
    ! [F: nat > list_a,R: partia2670972154091845814t_unit,N3: nat] :
      ( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) )
     => ( member_list_a @ ( F @ N3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% mem_upD
thf(fact_326_mem__upD,axiom,
    ! [F: nat > list_list_a,R: partia2956882679547061052t_unit,N3: nat] :
      ( ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) )
     => ( member_list_list_a @ ( F @ N3 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% mem_upD
thf(fact_327_mem__upD,axiom,
    ! [F: nat > set_list_a,R: partia7496981018696276118t_unit,N3: nat] :
      ( ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R ) )
     => ( member_set_list_a @ ( F @ N3 ) @ ( partia141011252114345353t_unit @ R ) ) ) ).

% mem_upD
thf(fact_328_domain_Ouniv__poly__is__domain,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.univ_poly_is_domain
thf(fact_329_domain_Ouniv__poly__is__domain,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_domain
thf(fact_330_domain_Ovar__closed_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( member_list_a @ ( var_a_b @ R ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ).

% domain.var_closed(1)
thf(fact_331_domain_Ovar__closed_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( member_list_list_a @ ( var_li8453953174693405341t_unit @ R ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ).

% domain.var_closed(1)
thf(fact_332_x_Olagrange__basis__polynomial__aux__def,axiom,
    ! [S2: set_list_a] :
      ( ( lagran3534788790333317459t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 )
      = ( finpro3417560807142560175list_a @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        @ ^ [S3: list_a] : ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S3 ) )
        @ S2 ) ) ).

% x.lagrange_basis_polynomial_aux_def
thf(fact_333_x_Oeval__var,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
        = X ) ) ).

% x.eval_var
thf(fact_334_ring__primeI,axiom,
    ! [P: a] :
      ( ( P
       != ( zero_a_b @ r ) )
     => ( ( prime_a_ring_ext_a_b @ r @ P )
       => ( ring_ring_prime_a_b @ r @ P ) ) ) ).

% ring_primeI
thf(fact_335_ring__primeE_I3_J,axiom,
    ! [P: a] :
      ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ring_ring_prime_a_b @ r @ P )
       => ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).

% ring_primeE(3)
thf(fact_336_x_Ofinprod__Un__Int,axiom,
    ! [A2: set_nat,B: set_nat,G: nat > list_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( member_nat_list_a @ G
            @ ( pi_nat_list_a @ A2
              @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( member_nat_list_a @ G
              @ ( pi_nat_list_a @ B
                @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_nat @ A2 @ B ) ) @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( inf_inf_set_nat @ A2 @ B ) ) )
              = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ).

% x.finprod_Un_Int
thf(fact_337_x_Ofinprod__Un__Int,axiom,
    ! [A2: set_a,B: set_a,G: a > list_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ( member_a_list_a @ G
            @ ( pi_a_list_a @ A2
              @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( member_a_list_a @ G
              @ ( pi_a_list_a @ B
                @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_a @ A2 @ B ) ) @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( inf_inf_set_a @ A2 @ B ) ) )
              = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ).

% x.finprod_Un_Int
thf(fact_338_x_Ofinprod__Un__Int,axiom,
    ! [A2: set_list_a,B: set_list_a,G: list_a > list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_list_a @ B )
       => ( ( member_list_a_list_a @ G
            @ ( pi_list_a_list_a @ A2
              @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( member_list_a_list_a @ G
              @ ( pi_list_a_list_a @ B
                @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_list_a @ A2 @ B ) ) @ ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( inf_inf_set_list_a @ A2 @ B ) ) )
              = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ).

% x.finprod_Un_Int
thf(fact_339_x_Oring_Oinj__on__domain,axiom,
    ( ( inj_on_list_a_a
      @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
      @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
   => ( ( domain_a_b @ r )
     => ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.ring.inj_on_domain
thf(fact_340_x_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_nat,A2: set_nat,H: nat > list_a,G: nat > list_a] :
      ( ( finite_finite_nat @ B )
     => ( ( finite_finite_nat @ A2 )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: nat] :
                ( ( member_nat @ I @ ( minus_minus_set_nat @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: nat] :
                  ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_nat_list_a @ G
                  @ ( pi_nat_list_a @ A2
                    @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_nat_list_a @ H
                    @ ( pi_nat_list_a @ B
                      @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong
thf(fact_341_x_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_a,A2: set_a,H: a > list_a,G: a > list_a] :
      ( ( finite_finite_a @ B )
     => ( ( finite_finite_a @ A2 )
       => ( ! [I: a] :
              ( ( member_a @ I @ ( minus_minus_set_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: a] :
                ( ( member_a @ I @ ( minus_minus_set_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: a] :
                  ( ( member_a @ X2 @ ( inf_inf_set_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_list_a @ G
                  @ ( pi_a_list_a @ A2
                    @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_a_list_a @ H
                    @ ( pi_a_list_a @ B
                      @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong
thf(fact_342_x_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_list_a,A2: set_list_a,H: list_a > list_a,G: list_a > list_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( finite_finite_list_a @ A2 )
       => ( ! [I: list_a] :
              ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: list_a] :
                ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: list_a] :
                  ( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_list_a_list_a @ G
                  @ ( pi_list_a_list_a @ A2
                    @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_list_a_list_a @ H
                    @ ( pi_list_a_list_a @ B
                      @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong
thf(fact_343_x_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_nat_a,A2: set_nat_a,H: ( nat > a ) > list_a,G: ( nat > a ) > list_a] :
      ( ( finite_finite_nat_a @ B )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: nat > a] :
                ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: nat > a] :
                  ( ( member_nat_a @ X2 @ ( inf_inf_set_nat_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_nat_a_list_a @ G
                  @ ( pi_nat_a_list_a @ A2
                    @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_nat_a_list_a @ H
                    @ ( pi_nat_a_list_a @ B
                      @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong
thf(fact_344_x_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_a_a,A2: set_a_a,H: ( a > a ) > list_a,G: ( a > a ) > list_a] :
      ( ( finite_finite_a_a @ B )
     => ( ( finite_finite_a_a @ A2 )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ ( minus_minus_set_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: a > a] :
                ( ( member_a_a @ I @ ( minus_minus_set_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: a > a] :
                  ( ( member_a_a @ X2 @ ( inf_inf_set_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_a_list_a @ G
                  @ ( pi_a_a_list_a @ A2
                    @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_a_a_list_a @ H
                    @ ( pi_a_a_list_a @ B
                      @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong
thf(fact_345_x_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_nat_list_a,A2: set_nat_list_a,H: ( nat > list_a ) > list_a,G: ( nat > list_a ) > list_a] :
      ( ( finite7630042315537210004list_a @ B )
     => ( ( finite7630042315537210004list_a @ A2 )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: nat > list_a] :
                ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: nat > list_a] :
                  ( ( member_nat_list_a @ X2 @ ( inf_in6652419485960844601list_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member1174670940520477013list_a @ G
                  @ ( pi_nat_list_a_list_a @ A2
                    @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member1174670940520477013list_a @ H
                    @ ( pi_nat_list_a_list_a @ B
                      @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong
thf(fact_346_x_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_a_list_a,A2: set_a_list_a,H: ( a > list_a ) > list_a,G: ( a > list_a ) > list_a] :
      ( ( finite8564763978580267522list_a @ B )
     => ( ( finite8564763978580267522list_a @ A2 )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: a > list_a] :
                ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: a > list_a] :
                  ( ( member_a_list_a @ X2 @ ( inf_inf_set_a_list_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member4106537806236033721list_a @ G
                  @ ( pi_a_list_a_list_a @ A2
                    @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member4106537806236033721list_a @ H
                    @ ( pi_a_list_a_list_a @ B
                      @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong
thf(fact_347_x_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_a_a_a,A2: set_a_a_a,H: ( ( a > a ) > a ) > list_a,G: ( ( a > a ) > a ) > list_a] :
      ( ( finite_finite_a_a_a @ B )
     => ( ( finite_finite_a_a_a @ A2 )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: ( a > a ) > a] :
                ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: ( a > a ) > a] :
                  ( ( member_a_a_a @ X2 @ ( inf_inf_set_a_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_a_a_list_a @ G
                  @ ( pi_a_a_a_list_a @ A2
                    @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_a_a_a_list_a @ H
                    @ ( pi_a_a_a_list_a @ B
                      @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong
thf(fact_348_x_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_nat_a_a,A2: set_nat_a_a,H: ( ( nat > a ) > a ) > list_a,G: ( ( nat > a ) > a ) > list_a] :
      ( ( finite7239108116303828181at_a_a @ B )
     => ( ( finite7239108116303828181at_a_a @ A2 )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: ( nat > a ) > a] :
                ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: ( nat > a ) > a] :
                  ( ( member_nat_a_a @ X2 @ ( inf_inf_set_nat_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member1630437648146986790list_a @ G
                  @ ( pi_nat_a_a_list_a @ A2
                    @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member1630437648146986790list_a @ H
                    @ ( pi_nat_a_a_list_a @ B
                      @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong
thf(fact_349_x_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_a_list_a_a,A2: set_a_list_a_a,H: ( ( a > list_a ) > a ) > list_a,G: ( ( a > list_a ) > a ) > list_a] :
      ( ( finite681744870773793075st_a_a @ B )
     => ( ( finite681744870773793075st_a_a @ A2 )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: ( a > list_a ) > a] :
                ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: ( a > list_a ) > a] :
                  ( ( member_a_list_a_a @ X2 @ ( inf_in8941282555790403672st_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member455139831191412214list_a @ G
                  @ ( pi_a_list_a_a_list_a @ A2
                    @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member455139831191412214list_a @ H
                    @ ( pi_a_list_a_a_list_a @ B
                      @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong
thf(fact_350_x_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat,A2: set_nat,G: nat > list_a,H: nat > list_a] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_list_a @ G
                @ ( pi_nat_list_a @ B
                  @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_right
thf(fact_351_x_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_a,A2: set_a,G: a > list_a,H: a > list_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( ! [I: a] :
              ( ( member_a @ I @ ( minus_minus_set_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_list_a @ G
                @ ( pi_a_list_a @ B
                  @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_right
thf(fact_352_x_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_list_a,A2: set_list_a,G: list_a > list_a,H: list_a > list_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B )
       => ( ! [I: list_a] :
              ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_list_a_list_a @ G
                @ ( pi_list_a_list_a @ B
                  @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_right
thf(fact_353_x_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat_a,A2: set_nat_a,G: ( nat > a ) > list_a,H: ( nat > a ) > list_a] :
      ( ( finite_finite_nat_a @ B )
     => ( ( ord_le871467723717165285_nat_a @ A2 @ B )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat > a] :
                ( ( member_nat_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_a_list_a @ G
                @ ( pi_nat_a_list_a @ B
                  @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_right
thf(fact_354_x_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_a,A2: set_a_a,G: ( a > a ) > list_a,H: ( a > a ) > list_a] :
      ( ( finite_finite_a_a @ B )
     => ( ( ord_less_eq_set_a_a @ A2 @ B )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ ( minus_minus_set_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a > a] :
                ( ( member_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_list_a @ G
                @ ( pi_a_a_list_a @ B
                  @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_right
thf(fact_355_x_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat_list_a,A2: set_nat_list_a,G: ( nat > list_a ) > list_a,H: ( nat > list_a ) > list_a] :
      ( ( finite7630042315537210004list_a @ B )
     => ( ( ord_le2145805922479659755list_a @ A2 @ B )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat > list_a] :
                ( ( member_nat_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member1174670940520477013list_a @ G
                @ ( pi_nat_list_a_list_a @ B
                  @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_right
thf(fact_356_x_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_list_a,A2: set_a_list_a,G: ( a > list_a ) > list_a,H: ( a > list_a ) > list_a] :
      ( ( finite8564763978580267522list_a @ B )
     => ( ( ord_le50412136050534657list_a @ A2 @ B )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a > list_a] :
                ( ( member_a_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member4106537806236033721list_a @ G
                @ ( pi_a_list_a_list_a @ B
                  @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_right
thf(fact_357_x_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_a_a,A2: set_a_a_a,G: ( ( a > a ) > a ) > list_a,H: ( ( a > a ) > a ) > list_a] :
      ( ( finite_finite_a_a_a @ B )
     => ( ( ord_le7181591058469194768_a_a_a @ A2 @ B )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( a > a ) > a] :
                ( ( member_a_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_a_list_a @ G
                @ ( pi_a_a_a_list_a @ B
                  @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_right
thf(fact_358_x_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat_a_a,A2: set_nat_a_a,G: ( ( nat > a ) > a ) > list_a,H: ( ( nat > a ) > a ) > list_a] :
      ( ( finite7239108116303828181at_a_a @ B )
     => ( ( ord_le3509452538356653652at_a_a @ A2 @ B )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( nat > a ) > a] :
                ( ( member_nat_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member1630437648146986790list_a @ G
                @ ( pi_nat_a_a_list_a @ B
                  @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_right
thf(fact_359_x_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_list_a_a,A2: set_a_list_a_a,G: ( ( a > list_a ) > a ) > list_a,H: ( ( a > list_a ) > a ) > list_a] :
      ( ( finite681744870773793075st_a_a @ B )
     => ( ( ord_le2847972559472028170st_a_a @ A2 @ B )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( a > list_a ) > a] :
                ( ( member_a_list_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member455139831191412214list_a @ G
                @ ( pi_a_list_a_a_list_a @ B
                  @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_right
thf(fact_360_x_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat,A2: set_nat,H: nat > list_a,G: nat > list_a] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_list_a @ H
                @ ( pi_nat_list_a @ B
                  @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_left
thf(fact_361_x_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_a,A2: set_a,H: a > list_a,G: a > list_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( ! [I: a] :
              ( ( member_a @ I @ ( minus_minus_set_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_list_a @ H
                @ ( pi_a_list_a @ B
                  @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_left
thf(fact_362_x_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_list_a,A2: set_list_a,H: list_a > list_a,G: list_a > list_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B )
       => ( ! [I: list_a] :
              ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_list_a_list_a @ H
                @ ( pi_list_a_list_a @ B
                  @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_left
thf(fact_363_x_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat_a,A2: set_nat_a,H: ( nat > a ) > list_a,G: ( nat > a ) > list_a] :
      ( ( finite_finite_nat_a @ B )
     => ( ( ord_le871467723717165285_nat_a @ A2 @ B )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat > a] :
                ( ( member_nat_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_a_list_a @ H
                @ ( pi_nat_a_list_a @ B
                  @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_left
thf(fact_364_x_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_a,A2: set_a_a,H: ( a > a ) > list_a,G: ( a > a ) > list_a] :
      ( ( finite_finite_a_a @ B )
     => ( ( ord_less_eq_set_a_a @ A2 @ B )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ ( minus_minus_set_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a > a] :
                ( ( member_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_list_a @ H
                @ ( pi_a_a_list_a @ B
                  @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_left
thf(fact_365_x_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat_list_a,A2: set_nat_list_a,H: ( nat > list_a ) > list_a,G: ( nat > list_a ) > list_a] :
      ( ( finite7630042315537210004list_a @ B )
     => ( ( ord_le2145805922479659755list_a @ A2 @ B )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat > list_a] :
                ( ( member_nat_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member1174670940520477013list_a @ H
                @ ( pi_nat_list_a_list_a @ B
                  @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_left
thf(fact_366_x_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_list_a,A2: set_a_list_a,H: ( a > list_a ) > list_a,G: ( a > list_a ) > list_a] :
      ( ( finite8564763978580267522list_a @ B )
     => ( ( ord_le50412136050534657list_a @ A2 @ B )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a > list_a] :
                ( ( member_a_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member4106537806236033721list_a @ H
                @ ( pi_a_list_a_list_a @ B
                  @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_left
thf(fact_367_x_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_a_a,A2: set_a_a_a,H: ( ( a > a ) > a ) > list_a,G: ( ( a > a ) > a ) > list_a] :
      ( ( finite_finite_a_a_a @ B )
     => ( ( ord_le7181591058469194768_a_a_a @ A2 @ B )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( a > a ) > a] :
                ( ( member_a_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_a_list_a @ H
                @ ( pi_a_a_a_list_a @ B
                  @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_left
thf(fact_368_x_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat_a_a,A2: set_nat_a_a,H: ( ( nat > a ) > a ) > list_a,G: ( ( nat > a ) > a ) > list_a] :
      ( ( finite7239108116303828181at_a_a @ B )
     => ( ( ord_le3509452538356653652at_a_a @ A2 @ B )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( nat > a ) > a] :
                ( ( member_nat_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member1630437648146986790list_a @ H
                @ ( pi_nat_a_a_list_a @ B
                  @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_left
thf(fact_369_x_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_list_a_a,A2: set_a_list_a_a,H: ( ( a > list_a ) > a ) > list_a,G: ( ( a > list_a ) > a ) > list_a] :
      ( ( finite681744870773793075st_a_a @ B )
     => ( ( ord_le2847972559472028170st_a_a @ A2 @ B )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( a > list_a ) > a] :
                ( ( member_a_list_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member455139831191412214list_a @ H
                @ ( pi_a_list_a_a_list_a @ B
                  @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.finprod_mono_neutral_cong_left
thf(fact_370_x_Oeval__is__hom,axiom,
    ! [K: set_list_a,A: list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member7168557129179038582list_a
          @ ^ [P2: list_list_a] : ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ A )
          @ ( ring_h5031276006722532742t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.eval_is_hom
thf(fact_371_x_Oabelian__monoid__axioms,axiom,
    abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).

% x.abelian_monoid_axioms
thf(fact_372_univ__poly__is__abelian__monoid,axiom,
    ! [K: set_a] :
      ( ( subring_a_b @ K @ r )
     => ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_is_abelian_monoid
thf(fact_373_zero__is__prime_I1_J,axiom,
    prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).

% zero_is_prime(1)
thf(fact_374_x_Ofinprod__one__eqI,axiom,
    ! [A2: set_nat_list_a,F: ( nat > list_a ) > list_a] :
      ( ! [X2: nat > list_a] :
          ( ( member_nat_list_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_one_eqI
thf(fact_375_x_Ofinprod__one__eqI,axiom,
    ! [A2: set_nat_a,F: ( nat > a ) > list_a] :
      ( ! [X2: nat > a] :
          ( ( member_nat_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_one_eqI
thf(fact_376_x_Ofinprod__one__eqI,axiom,
    ! [A2: set_a_list_a,F: ( a > list_a ) > list_a] :
      ( ! [X2: a > list_a] :
          ( ( member_a_list_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_one_eqI
thf(fact_377_x_Ofinprod__one__eqI,axiom,
    ! [A2: set_a_a,F: ( a > a ) > list_a] :
      ( ! [X2: a > a] :
          ( ( member_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_one_eqI
thf(fact_378_x_Ofinprod__one__eqI,axiom,
    ! [A2: set_nat,F: nat > list_a] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_one_eqI
thf(fact_379_x_Ofinprod__one__eqI,axiom,
    ! [A2: set_a_a_a,F: ( ( a > a ) > a ) > list_a] :
      ( ! [X2: ( a > a ) > a] :
          ( ( member_a_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_one_eqI
thf(fact_380_x_Ofinprod__one__eqI,axiom,
    ! [A2: set_a_list_a_a,F: ( ( a > list_a ) > a ) > list_a] :
      ( ! [X2: ( a > list_a ) > a] :
          ( ( member_a_list_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_one_eqI
thf(fact_381_x_Ofinprod__one__eqI,axiom,
    ! [A2: set_nat_a_a,F: ( ( nat > a ) > a ) > list_a] :
      ( ! [X2: ( nat > a ) > a] :
          ( ( member_nat_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_one_eqI
thf(fact_382_x_Ofinprod__one__eqI,axiom,
    ! [A2: set_nat_list_a_a,F: ( ( nat > list_a ) > a ) > list_a] :
      ( ! [X2: ( nat > list_a ) > a] :
          ( ( member_nat_list_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finpro3153191432819564877st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_one_eqI
thf(fact_383_x_Ofinprod__one__eqI,axiom,
    ! [A2: set_a,F: a > list_a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_one_eqI
thf(fact_384_x_Oone__unique,axiom,
    ! [U: list_a] :
      ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ! [X2: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
              = X2 ) )
       => ( U
          = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.one_unique
thf(fact_385_x_Oinv__unique,axiom,
    ! [Y: list_a,X: list_a,Y2: list_a] :
      ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y2 )
          = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
             => ( Y = Y2 ) ) ) ) ) ) ).

% x.inv_unique
thf(fact_386_finite__Un,axiom,
    ! [F2: set_a,G2: set_a] :
      ( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) )
      = ( ( finite_finite_a @ F2 )
        & ( finite_finite_a @ G2 ) ) ) ).

% finite_Un
thf(fact_387_finite__Un,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
      = ( ( finite_finite_nat @ F2 )
        & ( finite_finite_nat @ G2 ) ) ) ).

% finite_Un
thf(fact_388_finite__Un,axiom,
    ! [F2: set_list_a,G2: set_list_a] :
      ( ( finite_finite_list_a @ ( sup_sup_set_list_a @ F2 @ G2 ) )
      = ( ( finite_finite_list_a @ F2 )
        & ( finite_finite_list_a @ G2 ) ) ) ).

% finite_Un
thf(fact_389_x_Ofinprod__singleton,axiom,
    ! [I3: nat,A2: set_nat,F: nat > list_a] :
      ( ( member_nat @ I3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( member_nat_list_a @ F
            @ ( pi_nat_list_a @ A2
              @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton
thf(fact_390_x_Ofinprod__singleton,axiom,
    ! [I3: a,A2: set_a,F: a > list_a] :
      ( ( member_a @ I3 @ A2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( member_a_list_a @ F
            @ ( pi_a_list_a @ A2
              @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton
thf(fact_391_x_Ofinprod__singleton,axiom,
    ! [I3: list_a,A2: set_list_a,F: list_a > list_a] :
      ( ( member_list_a @ I3 @ A2 )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( member_list_a_list_a @ F
            @ ( pi_list_a_list_a @ A2
              @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: list_a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton
thf(fact_392_x_Ofinprod__singleton,axiom,
    ! [I3: nat > a,A2: set_nat_a,F: ( nat > a ) > list_a] :
      ( ( member_nat_a @ I3 @ A2 )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ( member_nat_a_list_a @ F
            @ ( pi_nat_a_list_a @ A2
              @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat > a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton
thf(fact_393_x_Ofinprod__singleton,axiom,
    ! [I3: a > a,A2: set_a_a,F: ( a > a ) > list_a] :
      ( ( member_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a @ A2 )
       => ( ( member_a_a_list_a @ F
            @ ( pi_a_a_list_a @ A2
              @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a > a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton
thf(fact_394_x_Ofinprod__singleton,axiom,
    ! [I3: nat > list_a,A2: set_nat_list_a,F: ( nat > list_a ) > list_a] :
      ( ( member_nat_list_a @ I3 @ A2 )
     => ( ( finite7630042315537210004list_a @ A2 )
       => ( ( member1174670940520477013list_a @ F
            @ ( pi_nat_list_a_list_a @ A2
              @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat > list_a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton
thf(fact_395_x_Ofinprod__singleton,axiom,
    ! [I3: a > list_a,A2: set_a_list_a,F: ( a > list_a ) > list_a] :
      ( ( member_a_list_a @ I3 @ A2 )
     => ( ( finite8564763978580267522list_a @ A2 )
       => ( ( member4106537806236033721list_a @ F
            @ ( pi_a_list_a_list_a @ A2
              @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a > list_a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton
thf(fact_396_x_Ofinprod__singleton,axiom,
    ! [I3: ( a > a ) > a,A2: set_a_a_a,F: ( ( a > a ) > a ) > list_a] :
      ( ( member_a_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a_a @ A2 )
       => ( ( member_a_a_a_list_a @ F
            @ ( pi_a_a_a_list_a @ A2
              @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( a > a ) > a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton
thf(fact_397_x_Ofinprod__singleton,axiom,
    ! [I3: ( nat > a ) > a,A2: set_nat_a_a,F: ( ( nat > a ) > a ) > list_a] :
      ( ( member_nat_a_a @ I3 @ A2 )
     => ( ( finite7239108116303828181at_a_a @ A2 )
       => ( ( member1630437648146986790list_a @ F
            @ ( pi_nat_a_a_list_a @ A2
              @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( nat > a ) > a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton
thf(fact_398_x_Ofinprod__singleton,axiom,
    ! [I3: ( a > list_a ) > a,A2: set_a_list_a_a,F: ( ( a > list_a ) > a ) > list_a] :
      ( ( member_a_list_a_a @ I3 @ A2 )
     => ( ( finite681744870773793075st_a_a @ A2 )
       => ( ( member455139831191412214list_a @ F
            @ ( pi_a_list_a_a_list_a @ A2
              @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( a > list_a ) > a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton
thf(fact_399_x_Ofinprod__singleton__swap,axiom,
    ! [I3: nat,A2: set_nat,F: nat > list_a] :
      ( ( member_nat @ I3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( member_nat_list_a @ F
            @ ( pi_nat_list_a @ A2
              @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton_swap
thf(fact_400_x_Ofinprod__singleton__swap,axiom,
    ! [I3: a,A2: set_a,F: a > list_a] :
      ( ( member_a @ I3 @ A2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( member_a_list_a @ F
            @ ( pi_a_list_a @ A2
              @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton_swap
thf(fact_401_x_Ofinprod__singleton__swap,axiom,
    ! [I3: list_a,A2: set_list_a,F: list_a > list_a] :
      ( ( member_list_a @ I3 @ A2 )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( member_list_a_list_a @ F
            @ ( pi_list_a_list_a @ A2
              @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: list_a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton_swap
thf(fact_402_x_Ofinprod__singleton__swap,axiom,
    ! [I3: nat > a,A2: set_nat_a,F: ( nat > a ) > list_a] :
      ( ( member_nat_a @ I3 @ A2 )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ( member_nat_a_list_a @ F
            @ ( pi_nat_a_list_a @ A2
              @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat > a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton_swap
thf(fact_403_x_Ofinprod__singleton__swap,axiom,
    ! [I3: a > a,A2: set_a_a,F: ( a > a ) > list_a] :
      ( ( member_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a @ A2 )
       => ( ( member_a_a_list_a @ F
            @ ( pi_a_a_list_a @ A2
              @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a > a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton_swap
thf(fact_404_x_Ofinprod__singleton__swap,axiom,
    ! [I3: nat > list_a,A2: set_nat_list_a,F: ( nat > list_a ) > list_a] :
      ( ( member_nat_list_a @ I3 @ A2 )
     => ( ( finite7630042315537210004list_a @ A2 )
       => ( ( member1174670940520477013list_a @ F
            @ ( pi_nat_list_a_list_a @ A2
              @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat > list_a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton_swap
thf(fact_405_x_Ofinprod__singleton__swap,axiom,
    ! [I3: a > list_a,A2: set_a_list_a,F: ( a > list_a ) > list_a] :
      ( ( member_a_list_a @ I3 @ A2 )
     => ( ( finite8564763978580267522list_a @ A2 )
       => ( ( member4106537806236033721list_a @ F
            @ ( pi_a_list_a_list_a @ A2
              @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a > list_a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton_swap
thf(fact_406_x_Ofinprod__singleton__swap,axiom,
    ! [I3: ( a > a ) > a,A2: set_a_a_a,F: ( ( a > a ) > a ) > list_a] :
      ( ( member_a_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a_a @ A2 )
       => ( ( member_a_a_a_list_a @ F
            @ ( pi_a_a_a_list_a @ A2
              @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro3871538611420980027_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( a > a ) > a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton_swap
thf(fact_407_x_Ofinprod__singleton__swap,axiom,
    ! [I3: ( nat > a ) > a,A2: set_nat_a_a,F: ( ( nat > a ) > a ) > list_a] :
      ( ( member_nat_a_a @ I3 @ A2 )
     => ( ( finite7239108116303828181at_a_a @ A2 )
       => ( ( member1630437648146986790list_a @ F
            @ ( pi_nat_a_a_list_a @ A2
              @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro5068724286038259027at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( nat > a ) > a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton_swap
thf(fact_408_x_Ofinprod__singleton__swap,axiom,
    ! [I3: ( a > list_a ) > a,A2: set_a_list_a_a,F: ( ( a > list_a ) > a ) > list_a] :
      ( ( member_a_list_a_a @ I3 @ A2 )
     => ( ( finite681744870773793075st_a_a @ A2 )
       => ( ( member455139831191412214list_a @ F
            @ ( pi_a_list_a_a_list_a @ A2
              @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finpro5783859944493153077st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( a > list_a ) > a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finprod_singleton_swap
thf(fact_409_funcset__Un__left,axiom,
    ! [F: nat > list_a,A2: set_nat,B: set_nat,C: set_list_a] :
      ( ( member_nat_list_a @ F
        @ ( pi_nat_list_a @ ( sup_sup_set_nat @ A2 @ B )
          @ ^ [Uu: nat] : C ) )
      = ( ( member_nat_list_a @ F
          @ ( pi_nat_list_a @ A2
            @ ^ [Uu: nat] : C ) )
        & ( member_nat_list_a @ F
          @ ( pi_nat_list_a @ B
            @ ^ [Uu: nat] : C ) ) ) ) ).

% funcset_Un_left
thf(fact_410_funcset__Un__left,axiom,
    ! [F: nat > a,A2: set_nat,B: set_nat,C: set_a] :
      ( ( member_nat_a @ F
        @ ( pi_nat_a @ ( sup_sup_set_nat @ A2 @ B )
          @ ^ [Uu: nat] : C ) )
      = ( ( member_nat_a @ F
          @ ( pi_nat_a @ A2
            @ ^ [Uu: nat] : C ) )
        & ( member_nat_a @ F
          @ ( pi_nat_a @ B
            @ ^ [Uu: nat] : C ) ) ) ) ).

% funcset_Un_left
thf(fact_411_funcset__Un__left,axiom,
    ! [F: a > list_a,A2: set_a,B: set_a,C: set_list_a] :
      ( ( member_a_list_a @ F
        @ ( pi_a_list_a @ ( sup_sup_set_a @ A2 @ B )
          @ ^ [Uu: a] : C ) )
      = ( ( member_a_list_a @ F
          @ ( pi_a_list_a @ A2
            @ ^ [Uu: a] : C ) )
        & ( member_a_list_a @ F
          @ ( pi_a_list_a @ B
            @ ^ [Uu: a] : C ) ) ) ) ).

% funcset_Un_left
thf(fact_412_funcset__Un__left,axiom,
    ! [F: a > a,A2: set_a,B: set_a,C: set_a] :
      ( ( member_a_a @ F
        @ ( pi_a_a @ ( sup_sup_set_a @ A2 @ B )
          @ ^ [Uu: a] : C ) )
      = ( ( member_a_a @ F
          @ ( pi_a_a @ A2
            @ ^ [Uu: a] : C ) )
        & ( member_a_a @ F
          @ ( pi_a_a @ B
            @ ^ [Uu: a] : C ) ) ) ) ).

% funcset_Un_left
thf(fact_413_funcset__Un__left,axiom,
    ! [F: ( a > a ) > a,A2: set_a_a,B: set_a_a,C: set_a] :
      ( ( member_a_a_a @ F
        @ ( pi_a_a_a @ ( sup_sup_set_a_a @ A2 @ B )
          @ ^ [Uu: a > a] : C ) )
      = ( ( member_a_a_a @ F
          @ ( pi_a_a_a @ A2
            @ ^ [Uu: a > a] : C ) )
        & ( member_a_a_a @ F
          @ ( pi_a_a_a @ B
            @ ^ [Uu: a > a] : C ) ) ) ) ).

% funcset_Un_left
thf(fact_414_funcset__Un__left,axiom,
    ! [F: ( a > list_a ) > a,A2: set_a_list_a,B: set_a_list_a,C: set_a] :
      ( ( member_a_list_a_a @ F
        @ ( pi_a_list_a_a @ ( sup_sup_set_a_list_a @ A2 @ B )
          @ ^ [Uu: a > list_a] : C ) )
      = ( ( member_a_list_a_a @ F
          @ ( pi_a_list_a_a @ A2
            @ ^ [Uu: a > list_a] : C ) )
        & ( member_a_list_a_a @ F
          @ ( pi_a_list_a_a @ B
            @ ^ [Uu: a > list_a] : C ) ) ) ) ).

% funcset_Un_left
thf(fact_415_funcset__Un__left,axiom,
    ! [F: ( nat > a ) > a,A2: set_nat_a,B: set_nat_a,C: set_a] :
      ( ( member_nat_a_a @ F
        @ ( pi_nat_a_a @ ( sup_sup_set_nat_a @ A2 @ B )
          @ ^ [Uu: nat > a] : C ) )
      = ( ( member_nat_a_a @ F
          @ ( pi_nat_a_a @ A2
            @ ^ [Uu: nat > a] : C ) )
        & ( member_nat_a_a @ F
          @ ( pi_nat_a_a @ B
            @ ^ [Uu: nat > a] : C ) ) ) ) ).

% funcset_Un_left
thf(fact_416_funcset__Un__left,axiom,
    ! [F: ( nat > list_a ) > a,A2: set_nat_list_a,B: set_nat_list_a,C: set_a] :
      ( ( member_nat_list_a_a @ F
        @ ( pi_nat_list_a_a @ ( sup_su5649930751583389983list_a @ A2 @ B )
          @ ^ [Uu: nat > list_a] : C ) )
      = ( ( member_nat_list_a_a @ F
          @ ( pi_nat_list_a_a @ A2
            @ ^ [Uu: nat > list_a] : C ) )
        & ( member_nat_list_a_a @ F
          @ ( pi_nat_list_a_a @ B
            @ ^ [Uu: nat > list_a] : C ) ) ) ) ).

% funcset_Un_left
thf(fact_417_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_418_x_Oone__closed,axiom,
    member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% x.one_closed
thf(fact_419_x_Ofinprod__one,axiom,
    ! [A2: set_a] :
      ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
        @ ^ [I2: a] : ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
        @ A2 )
      = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.finprod_one
thf(fact_420_x_Or__one,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        = X ) ) ).

% x.r_one
thf(fact_421_x_Ol__one,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
        = X ) ) ).

% x.l_one
thf(fact_422_x_Ofinprod__infinite,axiom,
    ! [A2: set_nat,F: nat > list_a] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_infinite
thf(fact_423_x_Ofinprod__infinite,axiom,
    ! [A2: set_list_a,F: list_a > list_a] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_infinite
thf(fact_424_x_Ofinprod__infinite,axiom,
    ! [A2: set_a,F: a > list_a] :
      ( ~ ( finite_finite_a @ A2 )
     => ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finprod_infinite
thf(fact_425_x_Oring_Ohom__one,axiom,
    ( ( eval_a_b @ r @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
    = ( one_a_ring_ext_a_b @ r ) ) ).

% x.ring.hom_one
thf(fact_426_finite__UnI,axiom,
    ! [F2: set_a,G2: set_a] :
      ( ( finite_finite_a @ F2 )
     => ( ( finite_finite_a @ G2 )
       => ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_427_finite__UnI,axiom,
    ! [F2: set_nat,G2: set_nat] :
      ( ( finite_finite_nat @ F2 )
     => ( ( finite_finite_nat @ G2 )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_428_finite__UnI,axiom,
    ! [F2: set_list_a,G2: set_list_a] :
      ( ( finite_finite_list_a @ F2 )
     => ( ( finite_finite_list_a @ G2 )
       => ( finite_finite_list_a @ ( sup_sup_set_list_a @ F2 @ G2 ) ) ) ) ).

% finite_UnI
thf(fact_429_Un__infinite,axiom,
    ! [S2: set_a,T: set_a] :
      ( ~ ( finite_finite_a @ S2 )
     => ~ ( finite_finite_a @ ( sup_sup_set_a @ S2 @ T ) ) ) ).

% Un_infinite
thf(fact_430_Un__infinite,axiom,
    ! [S2: set_nat,T: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T ) ) ) ).

% Un_infinite
thf(fact_431_Un__infinite,axiom,
    ! [S2: set_list_a,T: set_list_a] :
      ( ~ ( finite_finite_list_a @ S2 )
     => ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S2 @ T ) ) ) ).

% Un_infinite
thf(fact_432_infinite__Un,axiom,
    ! [S2: set_a,T: set_a] :
      ( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S2 @ T ) ) )
      = ( ~ ( finite_finite_a @ S2 )
        | ~ ( finite_finite_a @ T ) ) ) ).

% infinite_Un
thf(fact_433_infinite__Un,axiom,
    ! [S2: set_nat,T: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T ) ) )
      = ( ~ ( finite_finite_nat @ S2 )
        | ~ ( finite_finite_nat @ T ) ) ) ).

% infinite_Un
thf(fact_434_infinite__Un,axiom,
    ! [S2: set_list_a,T: set_list_a] :
      ( ( ~ ( finite_finite_list_a @ ( sup_sup_set_list_a @ S2 @ T ) ) )
      = ( ~ ( finite_finite_list_a @ S2 )
        | ~ ( finite_finite_list_a @ T ) ) ) ).

% infinite_Un
thf(fact_435_finite__inverse__image__gen,axiom,
    ! [A2: set_a,F: a > a,D: set_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( inj_on_a_a @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_a @ ( F @ J ) @ A2 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_436_finite__inverse__image__gen,axiom,
    ! [A2: set_a,F: nat > a,D: set_nat] :
      ( ( finite_finite_a @ A2 )
     => ( ( inj_on_nat_a @ F @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_a @ ( F @ J ) @ A2 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_437_finite__inverse__image__gen,axiom,
    ! [A2: set_nat,F: a > nat,D: set_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( inj_on_a_nat @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_nat @ ( F @ J ) @ A2 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_438_finite__inverse__image__gen,axiom,
    ! [A2: set_nat,F: nat > nat,D: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( inj_on_nat_nat @ F @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_nat @ ( F @ J ) @ A2 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_439_finite__inverse__image__gen,axiom,
    ! [A2: set_a,F: list_a > a,D: set_list_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( inj_on_list_a_a @ F @ D )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [J: list_a] :
                ( ( member_list_a @ J @ D )
                & ( member_a @ ( F @ J ) @ A2 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_440_finite__inverse__image__gen,axiom,
    ! [A2: set_nat,F: list_a > nat,D: set_list_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( inj_on_list_a_nat @ F @ D )
       => ( finite_finite_list_a
          @ ( collect_list_a
            @ ^ [J: list_a] :
                ( ( member_list_a @ J @ D )
                & ( member_nat @ ( F @ J ) @ A2 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_441_finite__inverse__image__gen,axiom,
    ! [A2: set_list_a,F: a > list_a,D: set_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( inj_on_a_list_a @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_list_a @ ( F @ J ) @ A2 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_442_finite__inverse__image__gen,axiom,
    ! [A2: set_list_a,F: nat > list_a,D: set_nat] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( inj_on_nat_list_a @ F @ D )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [J: nat] :
                ( ( member_nat @ J @ D )
                & ( member_list_a @ ( F @ J ) @ A2 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_443_finite__inverse__image__gen,axiom,
    ! [A2: set_nat_a,F: a > nat > a,D: set_a] :
      ( ( finite_finite_nat_a @ A2 )
     => ( ( inj_on_a_nat_a @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_nat_a @ ( F @ J ) @ A2 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_444_finite__inverse__image__gen,axiom,
    ! [A2: set_a_a,F: a > a > a,D: set_a] :
      ( ( finite_finite_a_a @ A2 )
     => ( ( inj_on_a_a_a @ F @ D )
       => ( finite_finite_a
          @ ( collect_a
            @ ^ [J: a] :
                ( ( member_a @ J @ D )
                & ( member_a_a @ ( F @ J ) @ A2 ) ) ) ) ) ) ).

% finite_inverse_image_gen
thf(fact_445_domain_Ouniv__poly__is__abelian__monoid,axiom,
    ! [R: partia2175431115845679010xt_a_b,K: set_a] :
      ( ( domain_a_b @ R )
     => ( ( subring_a_b @ K @ R )
       => ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).

% domain.univ_poly_is_abelian_monoid
thf(fact_446_domain_Ouniv__poly__is__abelian__monoid,axiom,
    ! [R: partia2670972154091845814t_unit,K: set_list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( subrin6918843898125473962t_unit @ K @ R )
       => ( abelia3641329199688042803t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).

% domain.univ_poly_is_abelian_monoid
thf(fact_447_x_OsubdomainI,axiom,
    ! [H2: set_list_a] :
      ( ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
         != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ! [H1: list_a,H22: list_a] :
              ( ( member_list_a @ H1 @ H2 )
             => ( ( member_list_a @ H22 @ H2 )
               => ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 )
                    = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
                 => ( ( H1
                      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
                    | ( H22
                      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) )
         => ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.subdomainI
thf(fact_448_x_Ofinprod__Un__disjoint,axiom,
    ! [A2: set_nat,B: set_nat,G: nat > list_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( ( inf_inf_set_nat @ A2 @ B )
            = bot_bot_set_nat )
         => ( ( member_nat_list_a @ G
              @ ( pi_nat_list_a @ A2
                @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( member_nat_list_a @ G
                @ ( pi_nat_list_a @ B
                  @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_nat @ A2 @ B ) )
                = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ) ).

% x.finprod_Un_disjoint
thf(fact_449_x_Ofinprod__Un__disjoint,axiom,
    ! [A2: set_a,B: set_a,G: a > list_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ( ( inf_inf_set_a @ A2 @ B )
            = bot_bot_set_a )
         => ( ( member_a_list_a @ G
              @ ( pi_a_list_a @ A2
                @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( member_a_list_a @ G
                @ ( pi_a_list_a @ B
                  @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_a @ A2 @ B ) )
                = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ) ).

% x.finprod_Un_disjoint
thf(fact_450_x_Ofinprod__Un__disjoint,axiom,
    ! [A2: set_list_a,B: set_list_a,G: list_a > list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_list_a @ B )
       => ( ( ( inf_inf_set_list_a @ A2 @ B )
            = bot_bot_set_list_a )
         => ( ( member_list_a_list_a @ G
              @ ( pi_list_a_list_a @ A2
                @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( member_list_a_list_a @ G
                @ ( pi_list_a_list_a @ B
                  @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_list_a @ A2 @ B ) )
                = ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ) ).

% x.finprod_Un_disjoint
thf(fact_451_x_Oring_Oeval__hom,axiom,
    ! [K: set_list_a,A: list_a,P: list_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
         => ( ( eval_a_b @ r @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ x )
            = ( eval_a_b @ r
              @ ( map_list_a_a
                @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
                @ P )
              @ ( eval_a_b @ r @ A @ x ) ) ) ) ) ) ).

% x.ring.eval_hom
thf(fact_452_x_Oup__one__closed,axiom,
    ( member_nat_list_a
    @ ^ [N4: nat] : ( if_list_a @ ( N4 = zero_zero_nat ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
    @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.up_one_closed
thf(fact_453_x_Ohom__finprod,axiom,
    ! [F: nat > list_a,A2: set_nat] :
      ( ( member_nat_list_a @ F
        @ ( pi_nat_list_a @ A2
          @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( eval_a_b @ r @ ( finpro1918104735009086181it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ x )
        = ( finpro1280035270526425175_b_nat @ r
          @ ( comp_list_a_a_nat
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ F )
          @ A2 ) ) ) ).

% x.hom_finprod
thf(fact_454_x_Ohom__finprod,axiom,
    ! [F: ( a > a ) > list_a,A2: set_a_a] :
      ( ( member_a_a_list_a @ F
        @ ( pi_a_a_list_a @ A2
          @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( eval_a_b @ r @ ( finpro4522412760713991034it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ x )
        = ( finpro3012607322079259884_b_a_a @ r
          @ ( comp_list_a_a_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ F )
          @ A2 ) ) ) ).

% x.hom_finprod
thf(fact_455_x_Ohom__finprod,axiom,
    ! [F: ( a > list_a ) > list_a,A2: set_a_list_a] :
      ( ( member4106537806236033721list_a @ F
        @ ( pi_a_list_a_list_a @ A2
          @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( eval_a_b @ r @ ( finpro1324004282804121728list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ x )
        = ( finpro2305829206037604594list_a @ r
          @ ( comp_l1888214947559275976list_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ F )
          @ A2 ) ) ) ).

% x.hom_finprod
thf(fact_456_x_Ohom__finprod,axiom,
    ! [F: ( nat > a ) > list_a,A2: set_nat_a] :
      ( ( member_nat_a_list_a @ F
        @ ( pi_nat_a_list_a @ A2
          @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( eval_a_b @ r @ ( finpro2700224059436637840_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ x )
        = ( finpro5839458686994656414_nat_a @ r
          @ ( comp_list_a_a_nat_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ F )
          @ A2 ) ) ) ).

% x.hom_finprod
thf(fact_457_x_Ohom__finprod,axiom,
    ! [F: ( nat > list_a ) > list_a,A2: set_nat_list_a] :
      ( ( member1174670940520477013list_a @ F
        @ ( pi_nat_list_a_list_a @ A2
          @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( eval_a_b @ r @ ( finpro5225711790780397718list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ x )
        = ( finpro4838020199848830884list_a @ r
          @ ( comp_l7081298308343408974list_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ F )
          @ A2 ) ) ) ).

% x.hom_finprod
thf(fact_458_x_Ohom__finprod,axiom,
    ! [F: a > list_a,A2: set_a] :
      ( ( member_a_list_a @ F
        @ ( pi_a_list_a @ A2
          @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( eval_a_b @ r @ ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ x )
        = ( finpro205304725090349623_a_b_a @ r
          @ ( comp_list_a_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ F )
          @ A2 ) ) ) ).

% x.hom_finprod
thf(fact_459_x_Oring_Oimg__is__subfield_I1_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ( one_a_ring_ext_a_b @ r )
         != ( zero_a_b @ r ) )
       => ( inj_on_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
          @ K ) ) ) ).

% x.ring.img_is_subfield(1)
thf(fact_460_finprod__Un__Int,axiom,
    ! [A2: set_nat,B: set_nat,G: nat > a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( member_nat_a @ G
            @ ( pi_nat_a @ A2
              @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( member_nat_a @ G
              @ ( pi_nat_a @ B
                @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ G @ ( sup_sup_set_nat @ A2 @ B ) ) @ ( finpro1280035270526425175_b_nat @ r @ G @ ( inf_inf_set_nat @ A2 @ B ) ) )
              = ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ G @ A2 ) @ ( finpro1280035270526425175_b_nat @ r @ G @ B ) ) ) ) ) ) ) ).

% finprod_Un_Int
thf(fact_461_finprod__Un__Int,axiom,
    ! [A2: set_a_a,B: set_a_a,G: ( a > a ) > a] :
      ( ( finite_finite_a_a @ A2 )
     => ( ( finite_finite_a_a @ B )
       => ( ( member_a_a_a @ G
            @ ( pi_a_a_a @ A2
              @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( member_a_a_a @ G
              @ ( pi_a_a_a @ B
                @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( mult_a_ring_ext_a_b @ r @ ( finpro3012607322079259884_b_a_a @ r @ G @ ( sup_sup_set_a_a @ A2 @ B ) ) @ ( finpro3012607322079259884_b_a_a @ r @ G @ ( inf_inf_set_a_a @ A2 @ B ) ) )
              = ( mult_a_ring_ext_a_b @ r @ ( finpro3012607322079259884_b_a_a @ r @ G @ A2 ) @ ( finpro3012607322079259884_b_a_a @ r @ G @ B ) ) ) ) ) ) ) ).

% finprod_Un_Int
thf(fact_462_finprod__Un__Int,axiom,
    ! [A2: set_a_list_a,B: set_a_list_a,G: ( a > list_a ) > a] :
      ( ( finite8564763978580267522list_a @ A2 )
     => ( ( finite8564763978580267522list_a @ B )
       => ( ( member_a_list_a_a @ G
            @ ( pi_a_list_a_a @ A2
              @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( member_a_list_a_a @ G
              @ ( pi_a_list_a_a @ B
                @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( mult_a_ring_ext_a_b @ r @ ( finpro2305829206037604594list_a @ r @ G @ ( sup_sup_set_a_list_a @ A2 @ B ) ) @ ( finpro2305829206037604594list_a @ r @ G @ ( inf_inf_set_a_list_a @ A2 @ B ) ) )
              = ( mult_a_ring_ext_a_b @ r @ ( finpro2305829206037604594list_a @ r @ G @ A2 ) @ ( finpro2305829206037604594list_a @ r @ G @ B ) ) ) ) ) ) ) ).

% finprod_Un_Int
thf(fact_463_finprod__Un__Int,axiom,
    ! [A2: set_nat_a,B: set_nat_a,G: ( nat > a ) > a] :
      ( ( finite_finite_nat_a @ A2 )
     => ( ( finite_finite_nat_a @ B )
       => ( ( member_nat_a_a @ G
            @ ( pi_nat_a_a @ A2
              @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( member_nat_a_a @ G
              @ ( pi_nat_a_a @ B
                @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( mult_a_ring_ext_a_b @ r @ ( finpro5839458686994656414_nat_a @ r @ G @ ( sup_sup_set_nat_a @ A2 @ B ) ) @ ( finpro5839458686994656414_nat_a @ r @ G @ ( inf_inf_set_nat_a @ A2 @ B ) ) )
              = ( mult_a_ring_ext_a_b @ r @ ( finpro5839458686994656414_nat_a @ r @ G @ A2 ) @ ( finpro5839458686994656414_nat_a @ r @ G @ B ) ) ) ) ) ) ) ).

% finprod_Un_Int
thf(fact_464_finprod__Un__Int,axiom,
    ! [A2: set_nat_list_a,B: set_nat_list_a,G: ( nat > list_a ) > a] :
      ( ( finite7630042315537210004list_a @ A2 )
     => ( ( finite7630042315537210004list_a @ B )
       => ( ( member_nat_list_a_a @ G
            @ ( pi_nat_list_a_a @ A2
              @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( member_nat_list_a_a @ G
              @ ( pi_nat_list_a_a @ B
                @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( mult_a_ring_ext_a_b @ r @ ( finpro4838020199848830884list_a @ r @ G @ ( sup_su5649930751583389983list_a @ A2 @ B ) ) @ ( finpro4838020199848830884list_a @ r @ G @ ( inf_in6652419485960844601list_a @ A2 @ B ) ) )
              = ( mult_a_ring_ext_a_b @ r @ ( finpro4838020199848830884list_a @ r @ G @ A2 ) @ ( finpro4838020199848830884list_a @ r @ G @ B ) ) ) ) ) ) ) ).

% finprod_Un_Int
thf(fact_465_finprod__Un__Int,axiom,
    ! [A2: set_a,B: set_a,G: a > a] :
      ( ( finite_finite_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ( member_a_a @ G
            @ ( pi_a_a @ A2
              @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( member_a_a @ G
              @ ( pi_a_a @ B
                @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ ( sup_sup_set_a @ A2 @ B ) ) @ ( finpro205304725090349623_a_b_a @ r @ G @ ( inf_inf_set_a @ A2 @ B ) ) )
              = ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ A2 ) @ ( finpro205304725090349623_a_b_a @ r @ G @ B ) ) ) ) ) ) ) ).

% finprod_Un_Int
thf(fact_466_finprod__Un__Int,axiom,
    ! [A2: set_list_a,B: set_list_a,G: list_a > a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_list_a @ B )
       => ( ( member_list_a_a @ G
            @ ( pi_list_a_a @ A2
              @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( member_list_a_a @ G
              @ ( pi_list_a_a @ B
                @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ ( sup_sup_set_list_a @ A2 @ B ) ) @ ( finpro6052973074229812797list_a @ r @ G @ ( inf_inf_set_list_a @ A2 @ B ) ) )
              = ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ A2 ) @ ( finpro6052973074229812797list_a @ r @ G @ B ) ) ) ) ) ) ) ).

% finprod_Un_Int
thf(fact_467_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_nat,A2: set_nat,H: nat > list_a,G: nat > list_a] :
      ( ( finite_finite_nat @ B )
     => ( ( finite_finite_nat @ A2 )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: nat] :
                ( ( member_nat @ I @ ( minus_minus_set_nat @ A2 @ B ) )
               => ( ( G @ I )
                  = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: nat] :
                  ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_nat_list_a @ G
                  @ ( pi_nat_list_a @ A2
                    @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_nat_list_a @ H
                    @ ( pi_nat_list_a @ B
                      @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong
thf(fact_468_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_a,A2: set_a,H: a > list_a,G: a > list_a] :
      ( ( finite_finite_a @ B )
     => ( ( finite_finite_a @ A2 )
       => ( ! [I: a] :
              ( ( member_a @ I @ ( minus_minus_set_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: a] :
                ( ( member_a @ I @ ( minus_minus_set_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: a] :
                  ( ( member_a @ X2 @ ( inf_inf_set_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_list_a @ G
                  @ ( pi_a_list_a @ A2
                    @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_a_list_a @ H
                    @ ( pi_a_list_a @ B
                      @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong
thf(fact_469_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_list_a,A2: set_list_a,H: list_a > list_a,G: list_a > list_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( finite_finite_list_a @ A2 )
       => ( ! [I: list_a] :
              ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: list_a] :
                ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: list_a] :
                  ( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_list_a_list_a @ G
                  @ ( pi_list_a_list_a @ A2
                    @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_list_a_list_a @ H
                    @ ( pi_list_a_list_a @ B
                      @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong
thf(fact_470_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_nat_a,A2: set_nat_a,H: ( nat > a ) > list_a,G: ( nat > a ) > list_a] :
      ( ( finite_finite_nat_a @ B )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: nat > a] :
                ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: nat > a] :
                  ( ( member_nat_a @ X2 @ ( inf_inf_set_nat_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_nat_a_list_a @ G
                  @ ( pi_nat_a_list_a @ A2
                    @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_nat_a_list_a @ H
                    @ ( pi_nat_a_list_a @ B
                      @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong
thf(fact_471_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_a_a,A2: set_a_a,H: ( a > a ) > list_a,G: ( a > a ) > list_a] :
      ( ( finite_finite_a_a @ B )
     => ( ( finite_finite_a_a @ A2 )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ ( minus_minus_set_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: a > a] :
                ( ( member_a_a @ I @ ( minus_minus_set_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: a > a] :
                  ( ( member_a_a @ X2 @ ( inf_inf_set_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_a_list_a @ G
                  @ ( pi_a_a_list_a @ A2
                    @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_a_a_list_a @ H
                    @ ( pi_a_a_list_a @ B
                      @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong
thf(fact_472_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_nat_list_a,A2: set_nat_list_a,H: ( nat > list_a ) > list_a,G: ( nat > list_a ) > list_a] :
      ( ( finite7630042315537210004list_a @ B )
     => ( ( finite7630042315537210004list_a @ A2 )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: nat > list_a] :
                ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: nat > list_a] :
                  ( ( member_nat_list_a @ X2 @ ( inf_in6652419485960844601list_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member1174670940520477013list_a @ G
                  @ ( pi_nat_list_a_list_a @ A2
                    @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member1174670940520477013list_a @ H
                    @ ( pi_nat_list_a_list_a @ B
                      @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong
thf(fact_473_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_a_list_a,A2: set_a_list_a,H: ( a > list_a ) > list_a,G: ( a > list_a ) > list_a] :
      ( ( finite8564763978580267522list_a @ B )
     => ( ( finite8564763978580267522list_a @ A2 )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: a > list_a] :
                ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: a > list_a] :
                  ( ( member_a_list_a @ X2 @ ( inf_inf_set_a_list_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member4106537806236033721list_a @ G
                  @ ( pi_a_list_a_list_a @ A2
                    @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member4106537806236033721list_a @ H
                    @ ( pi_a_list_a_list_a @ B
                      @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong
thf(fact_474_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_a_a_a,A2: set_a_a_a,H: ( ( a > a ) > a ) > list_a,G: ( ( a > a ) > a ) > list_a] :
      ( ( finite_finite_a_a_a @ B )
     => ( ( finite_finite_a_a_a @ A2 )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: ( a > a ) > a] :
                ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: ( a > a ) > a] :
                  ( ( member_a_a_a @ X2 @ ( inf_inf_set_a_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member_a_a_a_list_a @ G
                  @ ( pi_a_a_a_list_a @ A2
                    @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member_a_a_a_list_a @ H
                    @ ( pi_a_a_a_list_a @ B
                      @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong
thf(fact_475_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_nat_a_a,A2: set_nat_a_a,H: ( ( nat > a ) > a ) > list_a,G: ( ( nat > a ) > a ) > list_a] :
      ( ( finite7239108116303828181at_a_a @ B )
     => ( ( finite7239108116303828181at_a_a @ A2 )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: ( nat > a ) > a] :
                ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: ( nat > a ) > a] :
                  ( ( member_nat_a_a @ X2 @ ( inf_inf_set_nat_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member1630437648146986790list_a @ G
                  @ ( pi_nat_a_a_list_a @ A2
                    @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member1630437648146986790list_a @ H
                    @ ( pi_nat_a_a_list_a @ B
                      @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong
thf(fact_476_x_Oadd_Ofinprod__mono__neutral__cong,axiom,
    ! [B: set_a_list_a_a,A2: set_a_list_a_a,H: ( ( a > list_a ) > a ) > list_a,G: ( ( a > list_a ) > a ) > list_a] :
      ( ( finite681744870773793075st_a_a @ B )
     => ( ( finite681744870773793075st_a_a @ A2 )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [I: ( a > list_a ) > a] :
                ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ A2 @ B ) )
               => ( ( G @ I )
                  = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ! [X2: ( a > list_a ) > a] :
                  ( ( member_a_list_a_a @ X2 @ ( inf_in8941282555790403672st_a_a @ A2 @ B ) )
                 => ( ( G @ X2 )
                    = ( H @ X2 ) ) )
             => ( ( member455139831191412214list_a @ G
                  @ ( pi_a_list_a_a_list_a @ A2
                    @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
               => ( ( member455139831191412214list_a @ H
                    @ ( pi_a_list_a_a_list_a @ B
                      @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
                 => ( ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                    = ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong
thf(fact_477_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat,A2: set_nat,H: nat > list_a,G: nat > list_a] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_list_a @ H
                @ ( pi_nat_list_a @ B
                  @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_left
thf(fact_478_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_a,A2: set_a,H: a > list_a,G: a > list_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( ! [I: a] :
              ( ( member_a @ I @ ( minus_minus_set_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_list_a @ H
                @ ( pi_a_list_a @ B
                  @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_left
thf(fact_479_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_list_a,A2: set_list_a,H: list_a > list_a,G: list_a > list_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B )
       => ( ! [I: list_a] :
              ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_list_a_list_a @ H
                @ ( pi_list_a_list_a @ B
                  @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_left
thf(fact_480_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat_a,A2: set_nat_a,H: ( nat > a ) > list_a,G: ( nat > a ) > list_a] :
      ( ( finite_finite_nat_a @ B )
     => ( ( ord_le871467723717165285_nat_a @ A2 @ B )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat > a] :
                ( ( member_nat_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_a_list_a @ H
                @ ( pi_nat_a_list_a @ B
                  @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_left
thf(fact_481_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_a,A2: set_a_a,H: ( a > a ) > list_a,G: ( a > a ) > list_a] :
      ( ( finite_finite_a_a @ B )
     => ( ( ord_less_eq_set_a_a @ A2 @ B )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ ( minus_minus_set_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a > a] :
                ( ( member_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_list_a @ H
                @ ( pi_a_a_list_a @ B
                  @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_left
thf(fact_482_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat_list_a,A2: set_nat_list_a,H: ( nat > list_a ) > list_a,G: ( nat > list_a ) > list_a] :
      ( ( finite7630042315537210004list_a @ B )
     => ( ( ord_le2145805922479659755list_a @ A2 @ B )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat > list_a] :
                ( ( member_nat_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member1174670940520477013list_a @ H
                @ ( pi_nat_list_a_list_a @ B
                  @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_left
thf(fact_483_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_list_a,A2: set_a_list_a,H: ( a > list_a ) > list_a,G: ( a > list_a ) > list_a] :
      ( ( finite8564763978580267522list_a @ B )
     => ( ( ord_le50412136050534657list_a @ A2 @ B )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a > list_a] :
                ( ( member_a_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member4106537806236033721list_a @ H
                @ ( pi_a_list_a_list_a @ B
                  @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_left
thf(fact_484_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_a_a,A2: set_a_a_a,H: ( ( a > a ) > a ) > list_a,G: ( ( a > a ) > a ) > list_a] :
      ( ( finite_finite_a_a_a @ B )
     => ( ( ord_le7181591058469194768_a_a_a @ A2 @ B )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( a > a ) > a] :
                ( ( member_a_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_a_list_a @ H
                @ ( pi_a_a_a_list_a @ B
                  @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_left
thf(fact_485_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_nat_a_a,A2: set_nat_a_a,H: ( ( nat > a ) > a ) > list_a,G: ( ( nat > a ) > a ) > list_a] :
      ( ( finite7239108116303828181at_a_a @ B )
     => ( ( ord_le3509452538356653652at_a_a @ A2 @ B )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( nat > a ) > a] :
                ( ( member_nat_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member1630437648146986790list_a @ H
                @ ( pi_nat_a_a_list_a @ B
                  @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_left
thf(fact_486_x_Oadd_Ofinprod__mono__neutral__cong__left,axiom,
    ! [B: set_a_list_a_a,A2: set_a_list_a_a,H: ( ( a > list_a ) > a ) > list_a,G: ( ( a > list_a ) > a ) > list_a] :
      ( ( finite681744870773793075st_a_a @ B )
     => ( ( ord_le2847972559472028170st_a_a @ A2 @ B )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ B @ A2 ) )
             => ( ( H @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( a > list_a ) > a] :
                ( ( member_a_list_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member455139831191412214list_a @ H
                @ ( pi_a_list_a_a_list_a @ B
                  @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 )
                = ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ B ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_left
thf(fact_487_carrier__not__empty,axiom,
    ( ( partia707051561876973205xt_a_b @ r )
   != bot_bot_set_a ) ).

% carrier_not_empty
thf(fact_488_m__assoc,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
            = ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).

% m_assoc
thf(fact_489_m__comm,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
          = ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).

% m_comm
thf(fact_490_m__lcomm,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
            = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).

% m_lcomm
thf(fact_491_x_Ocarrier__not__empty,axiom,
    ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
   != bot_bot_set_list_a ) ).

% x.carrier_not_empty
thf(fact_492_local_Ointegral,axiom,
    ! [A: a,B3: a] :
      ( ( ( mult_a_ring_ext_a_b @ r @ A @ B3 )
        = ( zero_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( A
              = ( zero_a_b @ r ) )
            | ( B3
              = ( zero_a_b @ r ) ) ) ) ) ) ).

% local.integral
thf(fact_493_integral__iff,axiom,
    ! [A: a,B3: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( ( mult_a_ring_ext_a_b @ r @ A @ B3 )
            = ( zero_a_b @ r ) )
          = ( ( A
              = ( zero_a_b @ r ) )
            | ( B3
              = ( zero_a_b @ r ) ) ) ) ) ) ).

% integral_iff
thf(fact_494_m__lcancel,axiom,
    ! [A: a,B3: a,C3: a] :
      ( ( A
       != ( zero_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( ( mult_a_ring_ext_a_b @ r @ A @ B3 )
                = ( mult_a_ring_ext_a_b @ r @ A @ C3 ) )
              = ( B3 = C3 ) ) ) ) ) ) ).

% m_lcancel
thf(fact_495_m__rcancel,axiom,
    ! [A: a,B3: a,C3: a] :
      ( ( A
       != ( zero_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( ( mult_a_ring_ext_a_b @ r @ B3 @ A )
                = ( mult_a_ring_ext_a_b @ r @ C3 @ A ) )
              = ( B3 = C3 ) ) ) ) ) ) ).

% m_rcancel
thf(fact_496_x_Osubring__props_I4_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( K != bot_bot_set_list_a ) ) ).

% x.subring_props(4)
thf(fact_497_inv__unique,axiom,
    ! [Y: a,X: a,Y2: a] :
      ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
        = ( one_a_ring_ext_a_b @ r ) )
     => ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
          = ( one_a_ring_ext_a_b @ r ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
             => ( Y = Y2 ) ) ) ) ) ) ).

% inv_unique
thf(fact_498_one__unique,axiom,
    ! [U: a] :
      ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ! [X2: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
              = X2 ) )
       => ( U
          = ( one_a_ring_ext_a_b @ r ) ) ) ) ).

% one_unique
thf(fact_499_up__smult__closed,axiom,
    ! [A: a,P: nat > a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_nat_a @ P @ ( up_a_b @ r ) )
       => ( member_nat_a
          @ ^ [I2: nat] : ( mult_a_ring_ext_a_b @ r @ A @ ( P @ I2 ) )
          @ ( up_a_b @ r ) ) ) ) ).

% up_smult_closed
thf(fact_500_x_Osubring__props_I2_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).

% x.subring_props(2)
thf(fact_501_x_Oadd_Ofinprod__one__eqI,axiom,
    ! [A2: set_nat_list_a,F: ( nat > list_a ) > list_a] :
      ( ! [X2: nat > list_a] :
          ( ( member_nat_list_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.finprod_one_eqI
thf(fact_502_x_Oadd_Ofinprod__one__eqI,axiom,
    ! [A2: set_nat_a,F: ( nat > a ) > list_a] :
      ( ! [X2: nat > a] :
          ( ( member_nat_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.finprod_one_eqI
thf(fact_503_x_Oadd_Ofinprod__one__eqI,axiom,
    ! [A2: set_a_list_a,F: ( a > list_a ) > list_a] :
      ( ! [X2: a > list_a] :
          ( ( member_a_list_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.finprod_one_eqI
thf(fact_504_x_Oadd_Ofinprod__one__eqI,axiom,
    ! [A2: set_a_a,F: ( a > a ) > list_a] :
      ( ! [X2: a > a] :
          ( ( member_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.finprod_one_eqI
thf(fact_505_x_Oadd_Ofinprod__one__eqI,axiom,
    ! [A2: set_a,F: a > list_a] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.finprod_one_eqI
thf(fact_506_x_Oadd_Ofinprod__one__eqI,axiom,
    ! [A2: set_nat,F: nat > list_a] :
      ( ! [X2: nat] :
          ( ( member_nat @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.finprod_one_eqI
thf(fact_507_x_Oadd_Ofinprod__one__eqI,axiom,
    ! [A2: set_a_a_a,F: ( ( a > a ) > a ) > list_a] :
      ( ! [X2: ( a > a ) > a] :
          ( ( member_a_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.finprod_one_eqI
thf(fact_508_x_Oadd_Ofinprod__one__eqI,axiom,
    ! [A2: set_a_list_a_a,F: ( ( a > list_a ) > a ) > list_a] :
      ( ! [X2: ( a > list_a ) > a] :
          ( ( member_a_list_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.finprod_one_eqI
thf(fact_509_x_Oadd_Ofinprod__one__eqI,axiom,
    ! [A2: set_nat_a_a,F: ( ( nat > a ) > a ) > list_a] :
      ( ! [X2: ( nat > a ) > a] :
          ( ( member_nat_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.finprod_one_eqI
thf(fact_510_x_Oadd_Ofinprod__one__eqI,axiom,
    ! [A2: set_nat_list_a_a,F: ( ( nat > list_a ) > a ) > list_a] :
      ( ! [X2: ( nat > list_a ) > a] :
          ( ( member_nat_list_a_a @ X2 @ A2 )
         => ( ( F @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( finsum2390613145322704222st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.finprod_one_eqI
thf(fact_511_x_Osubring__props_I6_J,axiom,
    ! [K: set_list_a,H12: list_a,H23: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_list_a @ H12 @ K )
       => ( ( member_list_a @ H23 @ K )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H23 ) @ K ) ) ) ) ).

% x.subring_props(6)
thf(fact_512_x_Osubring__props_I3_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).

% x.subring_props(3)
thf(fact_513_up__one__closed,axiom,
    ( member_nat_a
    @ ^ [N4: nat] : ( if_a @ ( N4 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ r ) @ ( zero_a_b @ r ) )
    @ ( up_a_b @ r ) ) ).

% up_one_closed
thf(fact_514_x_Osubring__props_I1_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.subring_props(1)
thf(fact_515_x_Ofinsum__closed,axiom,
    ! [F: nat > list_a,A2: set_nat] :
      ( ( member_nat_list_a @ F
        @ ( pi_nat_list_a @ A2
          @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( member_list_a @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finsum_closed
thf(fact_516_x_Ofinsum__closed,axiom,
    ! [F: a > list_a,A2: set_a] :
      ( ( member_a_list_a @ F
        @ ( pi_a_list_a @ A2
          @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( member_list_a @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finsum_closed
thf(fact_517_x_Ofinsum__cong_H,axiom,
    ! [A2: set_nat_list_a,B: set_nat_list_a,G: ( nat > list_a ) > list_a,F: ( nat > list_a ) > list_a] :
      ( ( A2 = B )
     => ( ( member1174670940520477013list_a @ G
          @ ( pi_nat_list_a_list_a @ B
            @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finsum_cong'
thf(fact_518_x_Ofinsum__cong_H,axiom,
    ! [A2: set_nat_a,B: set_nat_a,G: ( nat > a ) > list_a,F: ( nat > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member_nat_a_list_a @ G
          @ ( pi_nat_a_list_a @ B
            @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finsum_cong'
thf(fact_519_x_Ofinsum__cong_H,axiom,
    ! [A2: set_a_list_a,B: set_a_list_a,G: ( a > list_a ) > list_a,F: ( a > list_a ) > list_a] :
      ( ( A2 = B )
     => ( ( member4106537806236033721list_a @ G
          @ ( pi_a_list_a_list_a @ B
            @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finsum_cong'
thf(fact_520_x_Ofinsum__cong_H,axiom,
    ! [A2: set_a_a,B: set_a_a,G: ( a > a ) > list_a,F: ( a > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member_a_a_list_a @ G
          @ ( pi_a_a_list_a @ B
            @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finsum_cong'
thf(fact_521_x_Ofinsum__cong_H,axiom,
    ! [A2: set_a_a_a,B: set_a_a_a,G: ( ( a > a ) > a ) > list_a,F: ( ( a > a ) > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member_a_a_a_list_a @ G
          @ ( pi_a_a_a_list_a @ B
            @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finsum_cong'
thf(fact_522_x_Ofinsum__cong_H,axiom,
    ! [A2: set_a_list_a_a,B: set_a_list_a_a,G: ( ( a > list_a ) > a ) > list_a,F: ( ( a > list_a ) > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member455139831191412214list_a @ G
          @ ( pi_a_list_a_a_list_a @ B
            @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finsum_cong'
thf(fact_523_x_Ofinsum__cong_H,axiom,
    ! [A2: set_nat_a_a,B: set_nat_a_a,G: ( ( nat > a ) > a ) > list_a,F: ( ( nat > a ) > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member1630437648146986790list_a @ G
          @ ( pi_nat_a_a_list_a @ B
            @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finsum_cong'
thf(fact_524_x_Ofinsum__cong_H,axiom,
    ! [A2: set_nat_list_a_a,B: set_nat_list_a_a,G: ( ( nat > list_a ) > a ) > list_a,F: ( ( nat > list_a ) > a ) > list_a] :
      ( ( A2 = B )
     => ( ( member4189978660109718316list_a @ G
          @ ( pi_nat837482917391966783list_a @ B
            @ ^ [Uu: ( nat > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: ( nat > list_a ) > a] :
              ( ( member_nat_list_a_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finsum2390613145322704222st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finsum2390613145322704222st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finsum_cong'
thf(fact_525_x_Ofinsum__cong_H,axiom,
    ! [A2: set_nat,B: set_nat,G: nat > list_a,F: nat > list_a] :
      ( ( A2 = B )
     => ( ( member_nat_list_a @ G
          @ ( pi_nat_list_a @ B
            @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finsum_cong'
thf(fact_526_x_Ofinsum__cong_H,axiom,
    ! [A2: set_a,B: set_a,G: a > list_a,F: a > list_a] :
      ( ( A2 = B )
     => ( ( member_a_list_a @ G
          @ ( pi_a_list_a @ B
            @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [I: a] :
              ( ( member_a @ I @ B )
             => ( ( F @ I )
                = ( G @ I ) ) )
         => ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
            = ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ).

% x.finsum_cong'
thf(fact_527_x_Ofinsum__singleton,axiom,
    ! [I3: a,A2: set_a,F: a > list_a] :
      ( ( member_a @ I3 @ A2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( member_a_list_a @ F
            @ ( pi_a_list_a @ A2
              @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finsum_singleton
thf(fact_528_x_Ofinsum__singleton,axiom,
    ! [I3: nat,A2: set_nat,F: nat > list_a] :
      ( ( member_nat @ I3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( member_nat_list_a @ F
            @ ( pi_nat_list_a @ A2
              @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finsum_singleton
thf(fact_529_x_Ofinsum__singleton,axiom,
    ! [I3: list_a,A2: set_list_a,F: list_a > list_a] :
      ( ( member_list_a @ I3 @ A2 )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( member_list_a_list_a @ F
            @ ( pi_list_a_list_a @ A2
              @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: list_a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finsum_singleton
thf(fact_530_x_Ofinsum__singleton,axiom,
    ! [I3: nat > a,A2: set_nat_a,F: ( nat > a ) > list_a] :
      ( ( member_nat_a @ I3 @ A2 )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ( member_nat_a_list_a @ F
            @ ( pi_nat_a_list_a @ A2
              @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat > a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finsum_singleton
thf(fact_531_x_Ofinsum__singleton,axiom,
    ! [I3: a > a,A2: set_a_a,F: ( a > a ) > list_a] :
      ( ( member_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a @ A2 )
       => ( ( member_a_a_list_a @ F
            @ ( pi_a_a_list_a @ A2
              @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a > a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finsum_singleton
thf(fact_532_x_Ofinsum__singleton,axiom,
    ! [I3: nat > list_a,A2: set_nat_list_a,F: ( nat > list_a ) > list_a] :
      ( ( member_nat_list_a @ I3 @ A2 )
     => ( ( finite7630042315537210004list_a @ A2 )
       => ( ( member1174670940520477013list_a @ F
            @ ( pi_nat_list_a_list_a @ A2
              @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat > list_a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finsum_singleton
thf(fact_533_x_Ofinsum__singleton,axiom,
    ! [I3: a > list_a,A2: set_a_list_a,F: ( a > list_a ) > list_a] :
      ( ( member_a_list_a @ I3 @ A2 )
     => ( ( finite8564763978580267522list_a @ A2 )
       => ( ( member4106537806236033721list_a @ F
            @ ( pi_a_list_a_list_a @ A2
              @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a > list_a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finsum_singleton
thf(fact_534_x_Ofinsum__singleton,axiom,
    ! [I3: ( a > a ) > a,A2: set_a_a_a,F: ( ( a > a ) > a ) > list_a] :
      ( ( member_a_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a_a @ A2 )
       => ( ( member_a_a_a_list_a @ F
            @ ( pi_a_a_a_list_a @ A2
              @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( a > a ) > a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finsum_singleton
thf(fact_535_x_Ofinsum__singleton,axiom,
    ! [I3: ( nat > a ) > a,A2: set_nat_a_a,F: ( ( nat > a ) > a ) > list_a] :
      ( ( member_nat_a_a @ I3 @ A2 )
     => ( ( finite7239108116303828181at_a_a @ A2 )
       => ( ( member1630437648146986790list_a @ F
            @ ( pi_nat_a_a_list_a @ A2
              @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( nat > a ) > a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finsum_singleton
thf(fact_536_x_Ofinsum__singleton,axiom,
    ! [I3: ( a > list_a ) > a,A2: set_a_list_a_a,F: ( ( a > list_a ) > a ) > list_a] :
      ( ( member_a_list_a_a @ I3 @ A2 )
     => ( ( finite681744870773793075st_a_a @ A2 )
       => ( ( member455139831191412214list_a @ F
            @ ( pi_a_list_a_a_list_a @ A2
              @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( a > list_a ) > a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.finsum_singleton
thf(fact_537_x_Oadd_Ofinprod__singleton__swap,axiom,
    ! [I3: a,A2: set_a,F: a > list_a] :
      ( ( member_a @ I3 @ A2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( member_a_list_a @ F
            @ ( pi_a_list_a @ A2
              @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.add.finprod_singleton_swap
thf(fact_538_x_Oadd_Ofinprod__singleton__swap,axiom,
    ! [I3: nat,A2: set_nat,F: nat > list_a] :
      ( ( member_nat @ I3 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( member_nat_list_a @ F
            @ ( pi_nat_list_a @ A2
              @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.add.finprod_singleton_swap
thf(fact_539_x_Oadd_Ofinprod__singleton__swap,axiom,
    ! [I3: list_a,A2: set_list_a,F: list_a > list_a] :
      ( ( member_list_a @ I3 @ A2 )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( member_list_a_list_a @ F
            @ ( pi_list_a_list_a @ A2
              @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: list_a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.add.finprod_singleton_swap
thf(fact_540_x_Oadd_Ofinprod__singleton__swap,axiom,
    ! [I3: nat > a,A2: set_nat_a,F: ( nat > a ) > list_a] :
      ( ( member_nat_a @ I3 @ A2 )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ( member_nat_a_list_a @ F
            @ ( pi_nat_a_list_a @ A2
              @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat > a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.add.finprod_singleton_swap
thf(fact_541_x_Oadd_Ofinprod__singleton__swap,axiom,
    ! [I3: a > a,A2: set_a_a,F: ( a > a ) > list_a] :
      ( ( member_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a @ A2 )
       => ( ( member_a_a_list_a @ F
            @ ( pi_a_a_list_a @ A2
              @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a > a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.add.finprod_singleton_swap
thf(fact_542_x_Oadd_Ofinprod__singleton__swap,axiom,
    ! [I3: nat > list_a,A2: set_nat_list_a,F: ( nat > list_a ) > list_a] :
      ( ( member_nat_list_a @ I3 @ A2 )
     => ( ( finite7630042315537210004list_a @ A2 )
       => ( ( member1174670940520477013list_a @ F
            @ ( pi_nat_list_a_list_a @ A2
              @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: nat > list_a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.add.finprod_singleton_swap
thf(fact_543_x_Oadd_Ofinprod__singleton__swap,axiom,
    ! [I3: a > list_a,A2: set_a_list_a,F: ( a > list_a ) > list_a] :
      ( ( member_a_list_a @ I3 @ A2 )
     => ( ( finite8564763978580267522list_a @ A2 )
       => ( ( member4106537806236033721list_a @ F
            @ ( pi_a_list_a_list_a @ A2
              @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: a > list_a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.add.finprod_singleton_swap
thf(fact_544_x_Oadd_Ofinprod__singleton__swap,axiom,
    ! [I3: ( a > a ) > a,A2: set_a_a_a,F: ( ( a > a ) > a ) > list_a] :
      ( ( member_a_a_a @ I3 @ A2 )
     => ( ( finite_finite_a_a_a @ A2 )
       => ( ( member_a_a_a_list_a @ F
            @ ( pi_a_a_a_list_a @ A2
              @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( a > a ) > a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.add.finprod_singleton_swap
thf(fact_545_x_Oadd_Ofinprod__singleton__swap,axiom,
    ! [I3: ( nat > a ) > a,A2: set_nat_a_a,F: ( ( nat > a ) > a ) > list_a] :
      ( ( member_nat_a_a @ I3 @ A2 )
     => ( ( finite7239108116303828181at_a_a @ A2 )
       => ( ( member1630437648146986790list_a @ F
            @ ( pi_nat_a_a_list_a @ A2
              @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( nat > a ) > a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.add.finprod_singleton_swap
thf(fact_546_x_Oadd_Ofinprod__singleton__swap,axiom,
    ! [I3: ( a > list_a ) > a,A2: set_a_list_a_a,F: ( ( a > list_a ) > a ) > list_a] :
      ( ( member_a_list_a_a @ I3 @ A2 )
     => ( ( finite681744870773793075st_a_a @ A2 )
       => ( ( member455139831191412214list_a @ F
            @ ( pi_a_list_a_a_list_a @ A2
              @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [J: ( a > list_a ) > a] : ( if_list_a @ ( J = I3 ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
              @ A2 )
            = ( F @ I3 ) ) ) ) ) ).

% x.add.finprod_singleton_swap
thf(fact_547_x_Ofinsum__rdistr,axiom,
    ! [A2: set_a,A: list_a,F: a > list_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_a_list_a @ F
            @ ( pi_a_list_a @ A2
              @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) )
            = ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [I2: a] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( F @ I2 ) )
              @ A2 ) ) ) ) ) ).

% x.finsum_rdistr
thf(fact_548_x_Ofinsum__rdistr,axiom,
    ! [A2: set_nat,A: list_a,F: nat > list_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_nat_list_a @ F
            @ ( pi_nat_list_a @ A2
              @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) )
            = ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [I2: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( F @ I2 ) )
              @ A2 ) ) ) ) ) ).

% x.finsum_rdistr
thf(fact_549_x_Ofinsum__rdistr,axiom,
    ! [A2: set_list_a,A: list_a,F: list_a > list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a_list_a @ F
            @ ( pi_list_a_list_a @ A2
              @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) )
            = ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [I2: list_a] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( F @ I2 ) )
              @ A2 ) ) ) ) ) ).

% x.finsum_rdistr
thf(fact_550_x_Ofinsum__ldistr,axiom,
    ! [A2: set_a,A: list_a,F: a > list_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_a_list_a @ F
            @ ( pi_a_list_a @ A2
              @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ A )
            = ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [I2: a] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ I2 ) @ A )
              @ A2 ) ) ) ) ) ).

% x.finsum_ldistr
thf(fact_551_x_Ofinsum__ldistr,axiom,
    ! [A2: set_nat,A: list_a,F: nat > list_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_nat_list_a @ F
            @ ( pi_nat_list_a @ A2
              @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ A )
            = ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [I2: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ I2 ) @ A )
              @ A2 ) ) ) ) ) ).

% x.finsum_ldistr
thf(fact_552_x_Ofinsum__ldistr,axiom,
    ! [A2: set_list_a,A: list_a,F: list_a > list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a_list_a @ F
            @ ( pi_list_a_list_a @ A2
              @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ A )
            = ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ^ [I2: list_a] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ I2 ) @ A )
              @ A2 ) ) ) ) ) ).

% x.finsum_ldistr
thf(fact_553_finprod__Un__disjoint,axiom,
    ! [A2: set_nat,B: set_nat,G: nat > a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( ( inf_inf_set_nat @ A2 @ B )
            = bot_bot_set_nat )
         => ( ( member_nat_a @ G
              @ ( pi_nat_a @ A2
                @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( member_nat_a @ G
                @ ( pi_nat_a @ B
                  @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro1280035270526425175_b_nat @ r @ G @ ( sup_sup_set_nat @ A2 @ B ) )
                = ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ G @ A2 ) @ ( finpro1280035270526425175_b_nat @ r @ G @ B ) ) ) ) ) ) ) ) ).

% finprod_Un_disjoint
thf(fact_554_finprod__Un__disjoint,axiom,
    ! [A2: set_a_a,B: set_a_a,G: ( a > a ) > a] :
      ( ( finite_finite_a_a @ A2 )
     => ( ( finite_finite_a_a @ B )
       => ( ( ( inf_inf_set_a_a @ A2 @ B )
            = bot_bot_set_a_a )
         => ( ( member_a_a_a @ G
              @ ( pi_a_a_a @ A2
                @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( member_a_a_a @ G
                @ ( pi_a_a_a @ B
                  @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro3012607322079259884_b_a_a @ r @ G @ ( sup_sup_set_a_a @ A2 @ B ) )
                = ( mult_a_ring_ext_a_b @ r @ ( finpro3012607322079259884_b_a_a @ r @ G @ A2 ) @ ( finpro3012607322079259884_b_a_a @ r @ G @ B ) ) ) ) ) ) ) ) ).

% finprod_Un_disjoint
thf(fact_555_finprod__Un__disjoint,axiom,
    ! [A2: set_a_list_a,B: set_a_list_a,G: ( a > list_a ) > a] :
      ( ( finite8564763978580267522list_a @ A2 )
     => ( ( finite8564763978580267522list_a @ B )
       => ( ( ( inf_inf_set_a_list_a @ A2 @ B )
            = bot_bot_set_a_list_a )
         => ( ( member_a_list_a_a @ G
              @ ( pi_a_list_a_a @ A2
                @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( member_a_list_a_a @ G
                @ ( pi_a_list_a_a @ B
                  @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro2305829206037604594list_a @ r @ G @ ( sup_sup_set_a_list_a @ A2 @ B ) )
                = ( mult_a_ring_ext_a_b @ r @ ( finpro2305829206037604594list_a @ r @ G @ A2 ) @ ( finpro2305829206037604594list_a @ r @ G @ B ) ) ) ) ) ) ) ) ).

% finprod_Un_disjoint
thf(fact_556_finprod__Un__disjoint,axiom,
    ! [A2: set_nat_a,B: set_nat_a,G: ( nat > a ) > a] :
      ( ( finite_finite_nat_a @ A2 )
     => ( ( finite_finite_nat_a @ B )
       => ( ( ( inf_inf_set_nat_a @ A2 @ B )
            = bot_bot_set_nat_a )
         => ( ( member_nat_a_a @ G
              @ ( pi_nat_a_a @ A2
                @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( member_nat_a_a @ G
                @ ( pi_nat_a_a @ B
                  @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro5839458686994656414_nat_a @ r @ G @ ( sup_sup_set_nat_a @ A2 @ B ) )
                = ( mult_a_ring_ext_a_b @ r @ ( finpro5839458686994656414_nat_a @ r @ G @ A2 ) @ ( finpro5839458686994656414_nat_a @ r @ G @ B ) ) ) ) ) ) ) ) ).

% finprod_Un_disjoint
thf(fact_557_finprod__Un__disjoint,axiom,
    ! [A2: set_nat_list_a,B: set_nat_list_a,G: ( nat > list_a ) > a] :
      ( ( finite7630042315537210004list_a @ A2 )
     => ( ( finite7630042315537210004list_a @ B )
       => ( ( ( inf_in6652419485960844601list_a @ A2 @ B )
            = bot_bo3806784159821827511list_a )
         => ( ( member_nat_list_a_a @ G
              @ ( pi_nat_list_a_a @ A2
                @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( member_nat_list_a_a @ G
                @ ( pi_nat_list_a_a @ B
                  @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro4838020199848830884list_a @ r @ G @ ( sup_su5649930751583389983list_a @ A2 @ B ) )
                = ( mult_a_ring_ext_a_b @ r @ ( finpro4838020199848830884list_a @ r @ G @ A2 ) @ ( finpro4838020199848830884list_a @ r @ G @ B ) ) ) ) ) ) ) ) ).

% finprod_Un_disjoint
thf(fact_558_finprod__Un__disjoint,axiom,
    ! [A2: set_a,B: set_a,G: a > a] :
      ( ( finite_finite_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ( ( inf_inf_set_a @ A2 @ B )
            = bot_bot_set_a )
         => ( ( member_a_a @ G
              @ ( pi_a_a @ A2
                @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( member_a_a @ G
                @ ( pi_a_a @ B
                  @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro205304725090349623_a_b_a @ r @ G @ ( sup_sup_set_a @ A2 @ B ) )
                = ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ G @ A2 ) @ ( finpro205304725090349623_a_b_a @ r @ G @ B ) ) ) ) ) ) ) ) ).

% finprod_Un_disjoint
thf(fact_559_finprod__Un__disjoint,axiom,
    ! [A2: set_list_a,B: set_list_a,G: list_a > a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_list_a @ B )
       => ( ( ( inf_inf_set_list_a @ A2 @ B )
            = bot_bot_set_list_a )
         => ( ( member_list_a_a @ G
              @ ( pi_list_a_a @ A2
                @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ( member_list_a_a @ G
                @ ( pi_list_a_a @ B
                  @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
             => ( ( finpro6052973074229812797list_a @ r @ G @ ( sup_sup_set_list_a @ A2 @ B ) )
                = ( mult_a_ring_ext_a_b @ r @ ( finpro6052973074229812797list_a @ r @ G @ A2 ) @ ( finpro6052973074229812797list_a @ r @ G @ B ) ) ) ) ) ) ) ) ).

% finprod_Un_disjoint
thf(fact_560_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat,A2: set_nat,G: nat > list_a,H: nat > list_a] :
      ( ( finite_finite_nat @ B )
     => ( ( ord_less_eq_set_nat @ A2 @ B )
       => ( ! [I: nat] :
              ( ( member_nat @ I @ ( minus_minus_set_nat @ B @ A2 ) )
             => ( ( G @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_list_a @ G
                @ ( pi_nat_list_a @ B
                  @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_right
thf(fact_561_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_a,A2: set_a,G: a > list_a,H: a > list_a] :
      ( ( finite_finite_a @ B )
     => ( ( ord_less_eq_set_a @ A2 @ B )
       => ( ! [I: a] :
              ( ( member_a @ I @ ( minus_minus_set_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_list_a @ G
                @ ( pi_a_list_a @ B
                  @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_right
thf(fact_562_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_list_a,A2: set_list_a,G: list_a > list_a,H: list_a > list_a] :
      ( ( finite_finite_list_a @ B )
     => ( ( ord_le8861187494160871172list_a @ A2 @ B )
       => ( ! [I: list_a] :
              ( ( member_list_a @ I @ ( minus_646659088055828811list_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_list_a_list_a @ G
                @ ( pi_list_a_list_a @ B
                  @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_right
thf(fact_563_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat_a,A2: set_nat_a,G: ( nat > a ) > list_a,H: ( nat > a ) > list_a] :
      ( ( finite_finite_nat_a @ B )
     => ( ( ord_le871467723717165285_nat_a @ A2 @ B )
       => ( ! [I: nat > a] :
              ( ( member_nat_a @ I @ ( minus_490503922182417452_nat_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat > a] :
                ( ( member_nat_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_nat_a_list_a @ G
                @ ( pi_nat_a_list_a @ B
                  @ ^ [Uu: nat > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finsum7881878320310621759_nat_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_right
thf(fact_564_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_a,A2: set_a_a,G: ( a > a ) > list_a,H: ( a > a ) > list_a] :
      ( ( finite_finite_a_a @ B )
     => ( ( ord_less_eq_set_a_a @ A2 @ B )
       => ( ! [I: a > a] :
              ( ( member_a_a @ I @ ( minus_minus_set_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a > a] :
                ( ( member_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_list_a @ G
                @ ( pi_a_a_list_a @ B
                  @ ^ [Uu: a > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finsum4909627262586317195it_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_right
thf(fact_565_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat_list_a,A2: set_nat_list_a,G: ( nat > list_a ) > list_a,H: ( nat > list_a ) > list_a] :
      ( ( finite7630042315537210004list_a @ B )
     => ( ( ord_le2145805922479659755list_a @ A2 @ B )
       => ( ! [I: nat > list_a] :
              ( ( member_nat_list_a @ I @ ( minus_4169782841487898290list_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: nat > list_a] :
                ( ( member_nat_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member1174670940520477013list_a @ G
                @ ( pi_nat_list_a_list_a @ B
                  @ ^ [Uu: nat > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finsum4426778018909949125list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_right
thf(fact_566_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_list_a,A2: set_a_list_a,G: ( a > list_a ) > list_a,H: ( a > list_a ) > list_a] :
      ( ( finite8564763978580267522list_a @ B )
     => ( ( ord_le50412136050534657list_a @ A2 @ B )
       => ( ! [I: a > list_a] :
              ( ( member_a_list_a @ I @ ( minus_3253130117680460026list_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: a > list_a] :
                ( ( member_a_list_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member4106537806236033721list_a @ G
                @ ( pi_a_list_a_list_a @ B
                  @ ^ [Uu: a > list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finsum3406705939639355921list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_right
thf(fact_567_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_a_a,A2: set_a_a_a,G: ( ( a > a ) > a ) > list_a,H: ( ( a > a ) > a ) > list_a] :
      ( ( finite_finite_a_a_a @ B )
     => ( ( ord_le7181591058469194768_a_a_a @ A2 @ B )
       => ( ! [I: ( a > a ) > a] :
              ( ( member_a_a_a @ I @ ( minus_4881627095085115735_a_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( a > a ) > a] :
                ( ( member_a_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member_a_a_a_list_a @ G
                @ ( pi_a_a_a_list_a @ B
                  @ ^ [Uu: ( a > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finsum5639492236432915690_a_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_right
thf(fact_568_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_nat_a_a,A2: set_nat_a_a,G: ( ( nat > a ) > a ) > list_a,H: ( ( nat > a ) > a ) > list_a] :
      ( ( finite7239108116303828181at_a_a @ B )
     => ( ( ord_le3509452538356653652at_a_a @ A2 @ B )
       => ( ! [I: ( nat > a ) > a] :
              ( ( member_nat_a_a @ I @ ( minus_1482667089342205261at_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( nat > a ) > a] :
                ( ( member_nat_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member1630437648146986790list_a @ G
                @ ( pi_nat_a_a_list_a @ B
                  @ ^ [Uu: ( nat > a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finsum5886066780512034276at_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_right
thf(fact_569_x_Oadd_Ofinprod__mono__neutral__cong__right,axiom,
    ! [B: set_a_list_a_a,A2: set_a_list_a_a,G: ( ( a > list_a ) > a ) > list_a,H: ( ( a > list_a ) > a ) > list_a] :
      ( ( finite681744870773793075st_a_a @ B )
     => ( ( ord_le2847972559472028170st_a_a @ A2 @ B )
       => ( ! [I: ( a > list_a ) > a] :
              ( ( member_a_list_a_a @ I @ ( minus_4703795076483250129st_a_a @ B @ A2 ) )
             => ( ( G @ I )
                = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ! [X2: ( a > list_a ) > a] :
                ( ( member_a_list_a_a @ X2 @ A2 )
               => ( ( G @ X2 )
                  = ( H @ X2 ) ) )
           => ( ( member455139831191412214list_a @ G
                @ ( pi_a_list_a_a_list_a @ B
                  @ ^ [Uu: ( a > list_a ) > a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B )
                = ( finsum416311231956857188st_a_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ A2 ) ) ) ) ) ) ) ).

% x.add.finprod_mono_neutral_cong_right
thf(fact_570_m__closed,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% m_closed
thf(fact_571_l__null,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
        = ( zero_a_b @ r ) ) ) ).

% l_null
thf(fact_572_r__null,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
        = ( zero_a_b @ r ) ) ) ).

% r_null
thf(fact_573_l__one,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
        = X ) ) ).

% l_one
thf(fact_574_r__one,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
        = X ) ) ).

% r_one
thf(fact_575_finprod__empty,axiom,
    ! [F: nat > a] :
      ( ( finpro1280035270526425175_b_nat @ r @ F @ bot_bot_set_nat )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_empty
thf(fact_576_finprod__empty,axiom,
    ! [F: ( a > a ) > a] :
      ( ( finpro3012607322079259884_b_a_a @ r @ F @ bot_bot_set_a_a )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_empty
thf(fact_577_finprod__empty,axiom,
    ! [F: ( a > list_a ) > a] :
      ( ( finpro2305829206037604594list_a @ r @ F @ bot_bot_set_a_list_a )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_empty
thf(fact_578_finprod__empty,axiom,
    ! [F: ( nat > a ) > a] :
      ( ( finpro5839458686994656414_nat_a @ r @ F @ bot_bot_set_nat_a )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_empty
thf(fact_579_finprod__empty,axiom,
    ! [F: ( nat > list_a ) > a] :
      ( ( finpro4838020199848830884list_a @ r @ F @ bot_bo3806784159821827511list_a )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_empty
thf(fact_580_finprod__empty,axiom,
    ! [F: a > a] :
      ( ( finpro205304725090349623_a_b_a @ r @ F @ bot_bot_set_a )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_empty
thf(fact_581_finprod__empty,axiom,
    ! [F: list_a > a] :
      ( ( finpro6052973074229812797list_a @ r @ F @ bot_bot_set_list_a )
      = ( one_a_ring_ext_a_b @ r ) ) ).

% finprod_empty
thf(fact_582_finprod__multf,axiom,
    ! [F: a > a,A2: set_a,G: a > a] :
      ( ( member_a_a @ F
        @ ( pi_a_a @ A2
          @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_a_a @ G
          @ ( pi_a_a @ A2
            @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( finpro205304725090349623_a_b_a @ r
            @ ^ [X3: a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
            @ A2 )
          = ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ F @ A2 ) @ ( finpro205304725090349623_a_b_a @ r @ G @ A2 ) ) ) ) ) ).

% finprod_multf
thf(fact_583_finprod__multf,axiom,
    ! [F: nat > a,A2: set_nat,G: nat > a] :
      ( ( member_nat_a @ F
        @ ( pi_nat_a @ A2
          @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_nat_a @ G
          @ ( pi_nat_a @ A2
            @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( finpro1280035270526425175_b_nat @ r
            @ ^ [X3: nat] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
            @ A2 )
          = ( mult_a_ring_ext_a_b @ r @ ( finpro1280035270526425175_b_nat @ r @ F @ A2 ) @ ( finpro1280035270526425175_b_nat @ r @ G @ A2 ) ) ) ) ) ).

% finprod_multf
thf(fact_584_finprod__multf,axiom,
    ! [F: ( a > a ) > a,A2: set_a_a,G: ( a > a ) > a] :
      ( ( member_a_a_a @ F
        @ ( pi_a_a_a @ A2
          @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_a_a_a @ G
          @ ( pi_a_a_a @ A2
            @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( finpro3012607322079259884_b_a_a @ r
            @ ^ [X3: a > a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
            @ A2 )
          = ( mult_a_ring_ext_a_b @ r @ ( finpro3012607322079259884_b_a_a @ r @ F @ A2 ) @ ( finpro3012607322079259884_b_a_a @ r @ G @ A2 ) ) ) ) ) ).

% finprod_multf
thf(fact_585_finprod__multf,axiom,
    ! [F: ( a > list_a ) > a,A2: set_a_list_a,G: ( a > list_a ) > a] :
      ( ( member_a_list_a_a @ F
        @ ( pi_a_list_a_a @ A2
          @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_a_list_a_a @ G
          @ ( pi_a_list_a_a @ A2
            @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( finpro2305829206037604594list_a @ r
            @ ^ [X3: a > list_a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
            @ A2 )
          = ( mult_a_ring_ext_a_b @ r @ ( finpro2305829206037604594list_a @ r @ F @ A2 ) @ ( finpro2305829206037604594list_a @ r @ G @ A2 ) ) ) ) ) ).

% finprod_multf
thf(fact_586_finprod__multf,axiom,
    ! [F: ( nat > a ) > a,A2: set_nat_a,G: ( nat > a ) > a] :
      ( ( member_nat_a_a @ F
        @ ( pi_nat_a_a @ A2
          @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_nat_a_a @ G
          @ ( pi_nat_a_a @ A2
            @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( finpro5839458686994656414_nat_a @ r
            @ ^ [X3: nat > a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
            @ A2 )
          = ( mult_a_ring_ext_a_b @ r @ ( finpro5839458686994656414_nat_a @ r @ F @ A2 ) @ ( finpro5839458686994656414_nat_a @ r @ G @ A2 ) ) ) ) ) ).

% finprod_multf
thf(fact_587_finprod__multf,axiom,
    ! [F: ( nat > list_a ) > a,A2: set_nat_list_a,G: ( nat > list_a ) > a] :
      ( ( member_nat_list_a_a @ F
        @ ( pi_nat_list_a_a @ A2
          @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_nat_list_a_a @ G
          @ ( pi_nat_list_a_a @ A2
            @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( finpro4838020199848830884list_a @ r
            @ ^ [X3: nat > list_a] : ( mult_a_ring_ext_a_b @ r @ ( F @ X3 ) @ ( G @ X3 ) )
            @ A2 )
          = ( mult_a_ring_ext_a_b @ r @ ( finpro4838020199848830884list_a @ r @ F @ A2 ) @ ( finpro4838020199848830884list_a @ r @ G @ A2 ) ) ) ) ) ).

% finprod_multf
thf(fact_588_x_Ofinsum__empty,axiom,
    ! [F: a > list_a] :
      ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ bot_bot_set_a )
      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.finsum_empty
thf(fact_589_x_Ofinsum__empty,axiom,
    ! [F: list_a > list_a] :
      ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ bot_bot_set_list_a )
      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.finsum_empty
thf(fact_590_x_Ofinsum__infinite,axiom,
    ! [A2: set_a,F: a > list_a] :
      ( ~ ( finite_finite_a @ A2 )
     => ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finsum_infinite
thf(fact_591_x_Ofinsum__infinite,axiom,
    ! [A2: set_nat,F: nat > list_a] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finsum_infinite
thf(fact_592_x_Ofinsum__infinite,axiom,
    ! [A2: set_list_a,F: list_a > list_a] :
      ( ~ ( finite_finite_list_a @ A2 )
     => ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finsum_infinite
thf(fact_593_x_Ofinprod__empty,axiom,
    ! [F: a > list_a] :
      ( ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ bot_bot_set_a )
      = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.finprod_empty
thf(fact_594_x_Ofinprod__empty,axiom,
    ! [F: list_a > list_a] :
      ( ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ bot_bot_set_list_a )
      = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.finprod_empty
thf(fact_595_x_Oring_Ohom__mult,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( eval_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
          = ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).

% x.ring.hom_mult
thf(fact_596_infinite__imp__nonempty,axiom,
    ! [S2: set_nat] :
      ( ~ ( finite_finite_nat @ S2 )
     => ( S2 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_597_infinite__imp__nonempty,axiom,
    ! [S2: set_a] :
      ( ~ ( finite_finite_a @ S2 )
     => ( S2 != bot_bot_set_a ) ) ).

% infinite_imp_nonempty
thf(fact_598_infinite__imp__nonempty,axiom,
    ! [S2: set_list_a] :
      ( ~ ( finite_finite_list_a @ S2 )
     => ( S2 != bot_bot_set_list_a ) ) ).

% infinite_imp_nonempty
thf(fact_599_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_600_finite_OemptyI,axiom,
    finite_finite_a @ bot_bot_set_a ).

% finite.emptyI
thf(fact_601_finite_OemptyI,axiom,
    finite_finite_list_a @ bot_bot_set_list_a ).

% finite.emptyI
thf(fact_602_finite__has__minimal,axiom,
    ! [A2: set_set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A2 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A2 )
               => ( ( ord_less_eq_set_a @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_603_finite__has__minimal,axiom,
    ! [A2: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ? [X2: set_list_a] :
            ( ( member_set_list_a @ X2 @ A2 )
            & ! [Xa: set_list_a] :
                ( ( member_set_list_a @ Xa @ A2 )
               => ( ( ord_le8861187494160871172list_a @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_604_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X2 )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_605_finite__has__maximal,axiom,
    ! [A2: set_set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ? [X2: set_a] :
            ( ( member_set_a @ X2 @ A2 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A2 )
               => ( ( ord_less_eq_set_a @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_606_finite__has__maximal,axiom,
    ! [A2: set_set_list_a] :
      ( ( finite5282473924520328461list_a @ A2 )
     => ( ( A2 != bot_bo3186585308812441520list_a )
       => ? [X2: set_list_a] :
            ( ( member_set_list_a @ X2 @ A2 )
            & ! [Xa: set_list_a] :
                ( ( member_set_list_a @ Xa @ A2 )
               => ( ( ord_le8861187494160871172list_a @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_607_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat @ X2 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X2 @ Xa )
                 => ( X2 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_608_x_Oorder__gt__0__iff__finite,axiom,
    ( ( ord_less_nat @ zero_zero_nat @ ( order_3240872107759947550t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
    = ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.order_gt_0_iff_finite
thf(fact_609_order__gt__0__iff__finite,axiom,
    ( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
    = ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% order_gt_0_iff_finite
thf(fact_610_alg__mult__gt__zero__iff__is__root,axiom,
    ! [P: list_a,X: a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) )
        = ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).

% alg_mult_gt_zero_iff_is_root
thf(fact_611_ring__hom__cring_Ohom__finprod,axiom,
    ! [R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,H: a > a,F: a > a,A2: set_a] :
      ( ( ring_h661254511236296859_b_a_b @ R @ S2 @ H )
     => ( ( member_a_a @ F
          @ ( pi_a_a @ A2
            @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ R ) ) )
       => ( ( H @ ( finpro205304725090349623_a_b_a @ R @ F @ A2 ) )
          = ( finpro205304725090349623_a_b_a @ S2 @ ( comp_a_a_a @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finprod
thf(fact_612_ring__hom__cring_Ohom__finprod,axiom,
    ! [R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,H: a > a,F: nat > a,A2: set_nat] :
      ( ( ring_h661254511236296859_b_a_b @ R @ S2 @ H )
     => ( ( member_nat_a @ F
          @ ( pi_nat_a @ A2
            @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ R ) ) )
       => ( ( H @ ( finpro1280035270526425175_b_nat @ R @ F @ A2 ) )
          = ( finpro1280035270526425175_b_nat @ S2 @ ( comp_a_a_nat @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finprod
thf(fact_613_ring__hom__cring_Ohom__finprod,axiom,
    ! [R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,H: a > list_a,F: a > a,A2: set_a] :
      ( ( ring_h8279546866833948963t_unit @ R @ S2 @ H )
     => ( ( member_a_a @ F
          @ ( pi_a_a @ A2
            @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ R ) ) )
       => ( ( H @ ( finpro205304725090349623_a_b_a @ R @ F @ A2 ) )
          = ( finpro4329226410377213737unit_a @ S2 @ ( comp_a_list_a_a @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finprod
thf(fact_614_ring__hom__cring_Ohom__finprod,axiom,
    ! [R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,H: list_a > a,F: nat > list_a,A2: set_nat] :
      ( ( ring_h1547129875642963619it_a_b @ R @ S2 @ H )
     => ( ( member_nat_list_a @ F
          @ ( pi_nat_list_a @ A2
            @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ R ) ) )
       => ( ( H @ ( finpro1918104735009086181it_nat @ R @ F @ A2 ) )
          = ( finpro1280035270526425175_b_nat @ S2 @ ( comp_list_a_a_nat @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finprod
thf(fact_615_ring__hom__cring_Ohom__finprod,axiom,
    ! [R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,H: list_a > a,F: a > list_a,A2: set_a] :
      ( ( ring_h1547129875642963619it_a_b @ R @ S2 @ H )
     => ( ( member_a_list_a @ F
          @ ( pi_a_list_a @ A2
            @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ R ) ) )
       => ( ( H @ ( finpro4329226410377213737unit_a @ R @ F @ A2 ) )
          = ( finpro205304725090349623_a_b_a @ S2 @ ( comp_list_a_a_a @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finprod
thf(fact_616_ring__hom__cring_Ohom__finprod,axiom,
    ! [R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,H: a > a,F: ( a > a ) > a,A2: set_a_a] :
      ( ( ring_h661254511236296859_b_a_b @ R @ S2 @ H )
     => ( ( member_a_a_a @ F
          @ ( pi_a_a_a @ A2
            @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ R ) ) )
       => ( ( H @ ( finpro3012607322079259884_b_a_a @ R @ F @ A2 ) )
          = ( finpro3012607322079259884_b_a_a @ S2 @ ( comp_a_a_a_a @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finprod
thf(fact_617_ring__hom__cring_Ohom__finprod,axiom,
    ! [R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,H: a > a,F: ( nat > a ) > a,A2: set_nat_a] :
      ( ( ring_h661254511236296859_b_a_b @ R @ S2 @ H )
     => ( ( member_nat_a_a @ F
          @ ( pi_nat_a_a @ A2
            @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ R ) ) )
       => ( ( H @ ( finpro5839458686994656414_nat_a @ R @ F @ A2 ) )
          = ( finpro5839458686994656414_nat_a @ S2 @ ( comp_a_a_nat_a @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finprod
thf(fact_618_ring__hom__cring_Ohom__finprod,axiom,
    ! [R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,H: list_a > list_a,F: a > list_a,A2: set_a] :
      ( ( ring_h8282015026914974507t_unit @ R @ S2 @ H )
     => ( ( member_a_list_a @ F
          @ ( pi_a_list_a @ A2
            @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ R ) ) )
       => ( ( H @ ( finpro4329226410377213737unit_a @ R @ F @ A2 ) )
          = ( finpro4329226410377213737unit_a @ S2 @ ( comp_list_a_list_a_a @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finprod
thf(fact_619_ring__hom__cring_Ohom__finprod,axiom,
    ! [R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,H: list_list_a > a,F: a > list_list_a,A2: set_a] :
      ( ( ring_h3216284694433613853it_a_b @ R @ S2 @ H )
     => ( ( member_a_list_list_a @ F
          @ ( pi_a_list_list_a @ A2
            @ ^ [Uu: a] : ( partia2464479390973590831t_unit @ R ) ) )
       => ( ( H @ ( finpro5596966875920909993unit_a @ R @ F @ A2 ) )
          = ( finpro205304725090349623_a_b_a @ S2 @ ( comp_list_list_a_a_a @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finprod
thf(fact_620_ring__hom__cring_Ohom__finprod,axiom,
    ! [R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,H: list_list_a > a,F: nat > list_list_a,A2: set_nat] :
      ( ( ring_h3216284694433613853it_a_b @ R @ S2 @ H )
     => ( ( member8650753269014980122list_a @ F
          @ ( pi_nat_list_list_a @ A2
            @ ^ [Uu: nat] : ( partia2464479390973590831t_unit @ R ) ) )
       => ( ( H @ ( finpro4561275463894985573it_nat @ R @ F @ A2 ) )
          = ( finpro1280035270526425175_b_nat @ S2 @ ( comp_l4331022603867890727_a_nat @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finprod
thf(fact_621_x_Ofinsum__Un__disjoint,axiom,
    ! [A2: set_nat,B: set_nat,G: nat > list_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( ( inf_inf_set_nat @ A2 @ B )
            = bot_bot_set_nat )
         => ( ( member_nat_list_a @ G
              @ ( pi_nat_list_a @ A2
                @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( member_nat_list_a @ G
                @ ( pi_nat_list_a @ B
                  @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_nat @ A2 @ B ) )
                = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ) ).

% x.finsum_Un_disjoint
thf(fact_622_x_Ofinsum__Un__disjoint,axiom,
    ! [A2: set_a,B: set_a,G: a > list_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ( ( inf_inf_set_a @ A2 @ B )
            = bot_bot_set_a )
         => ( ( member_a_list_a @ G
              @ ( pi_a_list_a @ A2
                @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( member_a_list_a @ G
                @ ( pi_a_list_a @ B
                  @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_a @ A2 @ B ) )
                = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ) ).

% x.finsum_Un_disjoint
thf(fact_623_x_Ofinsum__Un__disjoint,axiom,
    ! [A2: set_list_a,B: set_list_a,G: list_a > list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_list_a @ B )
       => ( ( ( inf_inf_set_list_a @ A2 @ B )
            = bot_bot_set_list_a )
         => ( ( member_list_a_list_a @ G
              @ ( pi_list_a_list_a @ A2
                @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( member_list_a_list_a @ G
                @ ( pi_list_a_list_a @ B
                  @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
             => ( ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_list_a @ A2 @ B ) )
                = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ) ).

% x.finsum_Un_disjoint
thf(fact_624_x_Oring_Oeval__hom_H,axiom,
    ! [A: list_a,P: list_list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( eval_a_b @ r @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ x )
          = ( eval_a_b @ r
            @ ( map_list_a_a
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
              @ P )
            @ ( eval_a_b @ r @ A @ x ) ) ) ) ) ).

% x.ring.eval_hom'
thf(fact_625_x_Ofinsum__Un__Int,axiom,
    ! [A2: set_nat,B: set_nat,G: nat > list_a] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B )
       => ( ( member_nat_list_a @ G
            @ ( pi_nat_list_a @ A2
              @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( member_nat_list_a @ G
              @ ( pi_nat_list_a @ B
                @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_nat @ A2 @ B ) ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( inf_inf_set_nat @ A2 @ B ) ) )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ).

% x.finsum_Un_Int
thf(fact_626_x_Ofinsum__Un__Int,axiom,
    ! [A2: set_a,B: set_a,G: a > list_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( finite_finite_a @ B )
       => ( ( member_a_list_a @ G
            @ ( pi_a_list_a @ A2
              @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( member_a_list_a @ G
              @ ( pi_a_list_a @ B
                @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_a @ A2 @ B ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( inf_inf_set_a @ A2 @ B ) ) )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ).

% x.finsum_Un_Int
thf(fact_627_x_Ofinsum__Un__Int,axiom,
    ! [A2: set_list_a,B: set_list_a,G: list_a > list_a] :
      ( ( finite_finite_list_a @ A2 )
     => ( ( finite_finite_list_a @ B )
       => ( ( member_list_a_list_a @ G
            @ ( pi_list_a_list_a @ A2
              @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
         => ( ( member_list_a_list_a @ G
              @ ( pi_list_a_list_a @ B
                @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
           => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( sup_sup_set_list_a @ A2 @ B ) ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( inf_inf_set_list_a @ A2 @ B ) ) )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) @ ( finsum8721804980556663006list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ B ) ) ) ) ) ) ) ).

% x.finsum_Un_Int
thf(fact_628_x_Oline__extension__smult__closed,axiom,
    ! [K: set_list_a,E: set_list_a,A: list_a,K2: list_a,U: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ! [K3: list_a,V3: list_a] :
            ( ( member_list_a @ K3 @ K )
           => ( ( member_list_a @ V3 @ E )
             => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ V3 ) @ E ) ) )
       => ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( member_list_a @ K2 @ K )
             => ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) )
               => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) ) ) ) ) ) ) ) ).

% x.line_extension_smult_closed
thf(fact_629_x_Oadd_Or__cancel,axiom,
    ! [A: list_a,C3: list_a,B3: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C3 )
        = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ C3 ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( A = B3 ) ) ) ) ) ).

% x.add.r_cancel
thf(fact_630_x_Oadd_Om__lcomm,axiom,
    ! [X: list_a,Y: list_a,Z: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).

% x.add.m_lcomm
thf(fact_631_x_Oadd_Om__comm,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
          = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).

% x.add.m_comm
thf(fact_632_x_Oadd_Om__assoc,axiom,
    ! [X: list_a,Y: list_a,Z: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).

% x.add.m_assoc
thf(fact_633_x_Oadd_Ol__cancel,axiom,
    ! [C3: list_a,A: list_a,B3: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ A )
        = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 @ B3 ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( A = B3 ) ) ) ) ) ).

% x.add.l_cancel
thf(fact_634_x_Osubring__props_I7_J,axiom,
    ! [K: set_list_a,H12: list_a,H23: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_list_a @ H12 @ K )
       => ( ( member_list_a @ H23 @ K )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H23 ) @ K ) ) ) ) ).

% x.subring_props(7)
thf(fact_635_x_Oadd_Oinj__on__multc,axiom,
    ! [C3: list_a] :
      ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( inj_on_list_a_list_a
        @ ^ [X3: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ C3 )
        @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.inj_on_multc
thf(fact_636_x_Oadd_Oinj__on__cmult,axiom,
    ! [C3: list_a] :
      ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( inj_on_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.inj_on_cmult
thf(fact_637_x_Oup__add__closed,axiom,
    ! [P: nat > list_a,Q: nat > list_a] :
      ( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_nat_list_a @ Q @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member_nat_list_a
          @ ^ [I2: nat] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( P @ I2 ) @ ( Q @ I2 ) )
          @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.up_add_closed
thf(fact_638_x_Ominus__unique,axiom,
    ! [Y: list_a,X: list_a,Y2: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y2 )
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
             => ( Y = Y2 ) ) ) ) ) ) ).

% x.minus_unique
thf(fact_639_x_Oadd_Or__inv__ex,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ? [X2: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
          & ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X2 )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.add.r_inv_ex
thf(fact_640_x_Oadd_Oone__unique,axiom,
    ! [U: list_a] :
      ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ! [X2: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
              = X2 ) )
       => ( U
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.add.one_unique
thf(fact_641_x_Oadd_Ol__inv__ex,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ? [X2: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
          & ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.add.l_inv_ex
thf(fact_642_x_Oadd_Oinv__comm,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).

% x.add.inv_comm
thf(fact_643_x_Or__distr,axiom,
    ! [X: list_a,Y: list_a,Z: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ Y ) ) ) ) ) ) ).

% x.r_distr
thf(fact_644_x_Ol__distr,axiom,
    ! [X: list_a,Y: list_a,Z: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).

% x.l_distr
thf(fact_645_x_Oline__extension__in__carrier,axiom,
    ! [K: set_list_a,A: list_a,E: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).

% x.line_extension_in_carrier
thf(fact_646_x_Oline__extension__mem__iff,axiom,
    ! [U: list_a,K: set_list_a,A: list_a,E: set_list_a] :
      ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) )
      = ( ? [X3: list_a] :
            ( ( member_list_a @ X3 @ K )
            & ? [Y3: list_a] :
                ( ( member_list_a @ Y3 @ E )
                & ( U
                  = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ A ) @ Y3 ) ) ) ) ) ) ).

% x.line_extension_mem_iff
thf(fact_647_x_Oadd_Oinj__on__g,axiom,
    ! [H2: set_list_a,A: list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( inj_on_list_a_list_a
          @ ^ [Y3: list_a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y3 @ A )
          @ H2 ) ) ) ).

% x.add.inj_on_g
thf(fact_648_x_Oeval__in__carrier,axiom,
    ! [P: list_list_a,X: list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.eval_in_carrier
thf(fact_649_x_Oa__lcos__m__assoc,axiom,
    ! [M: set_list_a,G: list_a,H: list_a] :
      ( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ M ) )
            = ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ H ) @ M ) ) ) ) ) ).

% x.a_lcos_m_assoc
thf(fact_650_x_Oadd_Oright__cancel,axiom,
    ! [X: list_a,Y: list_a,Z: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) )
            = ( Y = Z ) ) ) ) ) ).

% x.add.right_cancel
thf(fact_651_x_Oadd_Om__closed,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.add.m_closed
thf(fact_652_x_Or__zero,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        = X ) ) ).

% x.r_zero
thf(fact_653_x_Ol__zero,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
        = X ) ) ).

% x.l_zero
thf(fact_654_x_Oadd_Or__cancel__one_H,axiom,
    ! [X: list_a,A: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( X
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X ) )
          = ( A
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).

% x.add.r_cancel_one'
thf(fact_655_x_Oadd_Or__cancel__one,axiom,
    ! [X: list_a,A: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X )
            = X )
          = ( A
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).

% x.add.r_cancel_one
thf(fact_656_x_Oadd_Ol__cancel__one_H,axiom,
    ! [X: list_a,A: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( X
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A ) )
          = ( A
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).

% x.add.l_cancel_one'
thf(fact_657_x_Oadd_Ol__cancel__one,axiom,
    ! [X: list_a,A: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A )
            = X )
          = ( A
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).

% x.add.l_cancel_one
thf(fact_658_x_Ofinsum__addf,axiom,
    ! [F: nat > list_a,A2: set_nat,G: nat > list_a] :
      ( ( member_nat_list_a @ F
        @ ( pi_nat_list_a @ A2
          @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( member_nat_list_a @ G
          @ ( pi_nat_list_a @ A2
            @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
            @ ^ [X3: nat] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ X3 ) @ ( G @ X3 ) )
            @ A2 )
          = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ ( finsum3497512462216549110it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) ) ) ) ) ).

% x.finsum_addf
thf(fact_659_x_Ofinsum__addf,axiom,
    ! [F: a > list_a,A2: set_a,G: a > list_a] :
      ( ( member_a_list_a @ F
        @ ( pi_a_list_a @ A2
          @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
     => ( ( member_a_list_a @ G
          @ ( pi_a_list_a @ A2
            @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
            @ ^ [X3: a] : ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ X3 ) @ ( G @ X3 ) )
            @ A2 )
          = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A2 ) @ ( finsum7322697649718157656unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ A2 ) ) ) ) ) ).

% x.finsum_addf
thf(fact_660_ring_Oalg__mult_Ocong,axiom,
    polyno4422430861927485590lt_a_b = polyno4422430861927485590lt_a_b ).

% ring.alg_mult.cong
thf(fact_661_ring__hom__add,axiom,
    ! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( H @ ( add_a_b @ R @ X @ Y ) )
            = ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_add
thf(fact_662_ring__hom__add,axiom,
    ! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a,Y: a] :
      ( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( H @ ( add_a_b @ R @ X @ Y ) )
            = ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_add
thf(fact_663_ring__hom__add,axiom,
    ! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
      ( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
            = ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_add
thf(fact_664_ring__hom__add,axiom,
    ! [H: a > list_list_a,R: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a,Y: a] :
      ( ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ R @ S2 ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( H @ ( add_a_b @ R @ X @ Y ) )
            = ( add_li174743652000525320t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_add
thf(fact_665_ring__hom__add,axiom,
    ! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
            = ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_add
thf(fact_666_ring__hom__add,axiom,
    ! [H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
      ( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( H @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
            = ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_add
thf(fact_667_ring__hom__add,axiom,
    ! [H: set_list_a > a,R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,X: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( H @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) )
            = ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_add
thf(fact_668_ring__hom__add,axiom,
    ! [H: list_a > list_list_a,R: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a,Y: list_a] :
      ( ( member6714375691612171394list_a @ H @ ( ring_h8002040739877300486t_unit @ R @ S2 ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
            = ( add_li174743652000525320t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_add
thf(fact_669_ring__hom__add,axiom,
    ! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
      ( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S2 ) )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( H @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
            = ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_add
thf(fact_670_ring__hom__add,axiom,
    ! [H: set_list_a > list_a,R: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit,X: set_list_a,Y: set_list_a] :
      ( ( member5910328476188217884list_a @ H @ ( ring_h8038483918290310060t_unit @ R @ S2 ) )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( H @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) )
            = ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_add
thf(fact_671_abelian__monoidE_I5_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_monoid_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ X @ Y )
            = ( add_a_b @ R @ Y @ X ) ) ) ) ) ).

% abelian_monoidE(5)
thf(fact_672_abelian__monoidE_I5_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia226231641709521465t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
            = ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).

% abelian_monoidE(5)
thf(fact_673_abelian__monoidE_I5_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ X @ Y )
            = ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).

% abelian_monoidE(5)
thf(fact_674_abelian__monoidE_I5_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( add_se2486902527185523630t_unit @ R @ X @ Y )
            = ( add_se2486902527185523630t_unit @ R @ Y @ X ) ) ) ) ) ).

% abelian_monoidE(5)
thf(fact_675_abelian__monoidE_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( abelian_monoid_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoidE(3)
thf(fact_676_abelian__monoidE_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( abelia226231641709521465t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoidE(3)
thf(fact_677_abelian__monoidE_I3_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoidE(3)
thf(fact_678_abelian__monoidE_I3_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
           => ( ( add_se2486902527185523630t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) @ Z )
              = ( add_se2486902527185523630t_unit @ R @ X @ ( add_se2486902527185523630t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoidE(3)
thf(fact_679_abelian__monoidE_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_monoid_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% abelian_monoidE(1)
thf(fact_680_abelian__monoidE_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia226231641709521465t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% abelian_monoidE(1)
thf(fact_681_abelian__monoidE_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% abelian_monoidE(1)
thf(fact_682_abelian__monoidE_I1_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ).

% abelian_monoidE(1)
thf(fact_683_abelian__monoid_Oa__comm,axiom,
    ! [G2: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
         => ( ( add_a_b @ G2 @ X @ Y )
            = ( add_a_b @ G2 @ Y @ X ) ) ) ) ) ).

% abelian_monoid.a_comm
thf(fact_684_abelian__monoid_Oa__comm,axiom,
    ! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
         => ( ( add_li7652885771158616974t_unit @ G2 @ X @ Y )
            = ( add_li7652885771158616974t_unit @ G2 @ Y @ X ) ) ) ) ) ).

% abelian_monoid.a_comm
thf(fact_685_abelian__monoid_Oa__comm,axiom,
    ! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
         => ( ( add_li174743652000525320t_unit @ G2 @ X @ Y )
            = ( add_li174743652000525320t_unit @ G2 @ Y @ X ) ) ) ) ) ).

% abelian_monoid.a_comm
thf(fact_686_abelian__monoid_Oa__comm,axiom,
    ! [G2: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
         => ( ( add_se2486902527185523630t_unit @ G2 @ X @ Y )
            = ( add_se2486902527185523630t_unit @ G2 @ Y @ X ) ) ) ) ) ).

% abelian_monoid.a_comm
thf(fact_687_abelian__monoid_Oa__assoc,axiom,
    ! [G2: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G2 ) )
           => ( ( add_a_b @ G2 @ ( add_a_b @ G2 @ X @ Y ) @ Z )
              = ( add_a_b @ G2 @ X @ ( add_a_b @ G2 @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_assoc
thf(fact_688_abelian__monoid_Oa__assoc,axiom,
    ! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G2 ) )
           => ( ( add_li7652885771158616974t_unit @ G2 @ ( add_li7652885771158616974t_unit @ G2 @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ G2 @ X @ ( add_li7652885771158616974t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_assoc
thf(fact_689_abelian__monoid_Oa__assoc,axiom,
    ! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G2 ) )
           => ( ( add_li174743652000525320t_unit @ G2 @ ( add_li174743652000525320t_unit @ G2 @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ G2 @ X @ ( add_li174743652000525320t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_assoc
thf(fact_690_abelian__monoid_Oa__assoc,axiom,
    ! [G2: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
         => ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ G2 ) )
           => ( ( add_se2486902527185523630t_unit @ G2 @ ( add_se2486902527185523630t_unit @ G2 @ X @ Y ) @ Z )
              = ( add_se2486902527185523630t_unit @ G2 @ X @ ( add_se2486902527185523630t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_assoc
thf(fact_691_abelian__monoid_Oa__lcomm,axiom,
    ! [G2: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ G2 ) )
           => ( ( add_a_b @ G2 @ X @ ( add_a_b @ G2 @ Y @ Z ) )
              = ( add_a_b @ G2 @ Y @ ( add_a_b @ G2 @ X @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_lcomm
thf(fact_692_abelian__monoid_Oa__lcomm,axiom,
    ! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G2 ) )
           => ( ( add_li7652885771158616974t_unit @ G2 @ X @ ( add_li7652885771158616974t_unit @ G2 @ Y @ Z ) )
              = ( add_li7652885771158616974t_unit @ G2 @ Y @ ( add_li7652885771158616974t_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_lcomm
thf(fact_693_abelian__monoid_Oa__lcomm,axiom,
    ! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G2 ) )
           => ( ( add_li174743652000525320t_unit @ G2 @ X @ ( add_li174743652000525320t_unit @ G2 @ Y @ Z ) )
              = ( add_li174743652000525320t_unit @ G2 @ Y @ ( add_li174743652000525320t_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_lcomm
thf(fact_694_abelian__monoid_Oa__lcomm,axiom,
    ! [G2: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
         => ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ G2 ) )
           => ( ( add_se2486902527185523630t_unit @ G2 @ X @ ( add_se2486902527185523630t_unit @ G2 @ Y @ Z ) )
              = ( add_se2486902527185523630t_unit @ G2 @ Y @ ( add_se2486902527185523630t_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).

% abelian_monoid.a_lcomm
thf(fact_695_abelian__monoid_Oa__closed,axiom,
    ! [G2: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
         => ( member_a @ ( add_a_b @ G2 @ X @ Y ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ) ).

% abelian_monoid.a_closed
thf(fact_696_abelian__monoid_Oa__closed,axiom,
    ! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ G2 @ X @ Y ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) ) ) ).

% abelian_monoid.a_closed
thf(fact_697_abelian__monoid_Oa__closed,axiom,
    ! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
         => ( member_list_list_a @ ( add_li174743652000525320t_unit @ G2 @ X @ Y ) @ ( partia2464479390973590831t_unit @ G2 ) ) ) ) ) ).

% abelian_monoid.a_closed
thf(fact_698_abelian__monoid_Oa__closed,axiom,
    ! [G2: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
         => ( member_set_list_a @ ( add_se2486902527185523630t_unit @ G2 @ X @ Y ) @ ( partia141011252114345353t_unit @ G2 ) ) ) ) ) ).

% abelian_monoid.a_closed
thf(fact_699_semiring_Osemiring__simprules_I12_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
              = ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(12)
thf(fact_700_semiring_Osemiring__simprules_I12_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
              = ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(12)
thf(fact_701_semiring_Osemiring__simprules_I12_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) )
              = ( add_li174743652000525320t_unit @ R @ Y @ ( add_li174743652000525320t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(12)
thf(fact_702_semiring_Osemiring__simprules_I12_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
           => ( ( add_se2486902527185523630t_unit @ R @ X @ ( add_se2486902527185523630t_unit @ R @ Y @ Z ) )
              = ( add_se2486902527185523630t_unit @ R @ Y @ ( add_se2486902527185523630t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(12)
thf(fact_703_semiring_Osemiring__simprules_I7_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( add_a_b @ R @ X @ Y )
            = ( add_a_b @ R @ Y @ X ) ) ) ) ) ).

% semiring.semiring_simprules(7)
thf(fact_704_semiring_Osemiring__simprules_I7_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
            = ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).

% semiring.semiring_simprules(7)
thf(fact_705_semiring_Osemiring__simprules_I7_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( add_li174743652000525320t_unit @ R @ X @ Y )
            = ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).

% semiring.semiring_simprules(7)
thf(fact_706_semiring_Osemiring__simprules_I7_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( add_se2486902527185523630t_unit @ R @ X @ Y )
            = ( add_se2486902527185523630t_unit @ R @ Y @ X ) ) ) ) ) ).

% semiring.semiring_simprules(7)
thf(fact_707_semiring_Osemiring__simprules_I5_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(5)
thf(fact_708_semiring_Osemiring__simprules_I5_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(5)
thf(fact_709_semiring_Osemiring__simprules_I5_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(5)
thf(fact_710_semiring_Osemiring__simprules_I5_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
           => ( ( add_se2486902527185523630t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) @ Z )
              = ( add_se2486902527185523630t_unit @ R @ X @ ( add_se2486902527185523630t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(5)
thf(fact_711_semiring_Osemiring__simprules_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% semiring.semiring_simprules(1)
thf(fact_712_semiring_Osemiring__simprules_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% semiring.semiring_simprules(1)
thf(fact_713_semiring_Osemiring__simprules_I1_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% semiring.semiring_simprules(1)
thf(fact_714_semiring_Osemiring__simprules_I1_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ).

% semiring.semiring_simprules(1)
thf(fact_715_abelian__monoidE_I4_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_monoid_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
          = X ) ) ) ).

% abelian_monoidE(4)
thf(fact_716_abelian__monoidE_I4_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia226231641709521465t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = X ) ) ) ).

% abelian_monoidE(4)
thf(fact_717_abelian__monoidE_I4_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = X ) ) ) ).

% abelian_monoidE(4)
thf(fact_718_abelian__monoidE_I4_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
          = X ) ) ) ).

% abelian_monoidE(4)
thf(fact_719_abelian__monoid_Ol__zero,axiom,
    ! [G2: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
       => ( ( add_a_b @ G2 @ ( zero_a_b @ G2 ) @ X )
          = X ) ) ) ).

% abelian_monoid.l_zero
thf(fact_720_abelian__monoid_Ol__zero,axiom,
    ! [G2: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
       => ( ( add_li7652885771158616974t_unit @ G2 @ ( zero_l4142658623432671053t_unit @ G2 ) @ X )
          = X ) ) ) ).

% abelian_monoid.l_zero
thf(fact_721_abelian__monoid_Ol__zero,axiom,
    ! [G2: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
       => ( ( add_li174743652000525320t_unit @ G2 @ ( zero_l347298301471573063t_unit @ G2 ) @ X )
          = X ) ) ) ).

% abelian_monoid.l_zero
thf(fact_722_abelian__monoid_Ol__zero,axiom,
    ! [G2: partia7496981018696276118t_unit,X: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
       => ( ( add_se2486902527185523630t_unit @ G2 @ ( zero_s2910681146719230829t_unit @ G2 ) @ X )
          = X ) ) ) ).

% abelian_monoid.l_zero
thf(fact_723_abelian__monoid_Or__zero,axiom,
    ! [G2: partia2175431115845679010xt_a_b,X: a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
       => ( ( add_a_b @ G2 @ X @ ( zero_a_b @ G2 ) )
          = X ) ) ) ).

% abelian_monoid.r_zero
thf(fact_724_abelian__monoid_Or__zero,axiom,
    ! [G2: partia2670972154091845814t_unit,X: list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
       => ( ( add_li7652885771158616974t_unit @ G2 @ X @ ( zero_l4142658623432671053t_unit @ G2 ) )
          = X ) ) ) ).

% abelian_monoid.r_zero
thf(fact_725_abelian__monoid_Or__zero,axiom,
    ! [G2: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
       => ( ( add_li174743652000525320t_unit @ G2 @ X @ ( zero_l347298301471573063t_unit @ G2 ) )
          = X ) ) ) ).

% abelian_monoid.r_zero
thf(fact_726_abelian__monoid_Or__zero,axiom,
    ! [G2: partia7496981018696276118t_unit,X: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
       => ( ( add_se2486902527185523630t_unit @ G2 @ X @ ( zero_s2910681146719230829t_unit @ G2 ) )
          = X ) ) ) ).

% abelian_monoid.r_zero
thf(fact_727_abelian__monoid_Ominus__unique,axiom,
    ! [G2: partia2175431115845679010xt_a_b,Y: a,X: a,Y2: a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( ( add_a_b @ G2 @ Y @ X )
          = ( zero_a_b @ G2 ) )
       => ( ( ( add_a_b @ G2 @ X @ Y2 )
            = ( zero_a_b @ G2 ) )
         => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
           => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
             => ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G2 ) )
               => ( Y = Y2 ) ) ) ) ) ) ) ).

% abelian_monoid.minus_unique
thf(fact_728_abelian__monoid_Ominus__unique,axiom,
    ! [G2: partia2670972154091845814t_unit,Y: list_a,X: list_a,Y2: list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( ( add_li7652885771158616974t_unit @ G2 @ Y @ X )
          = ( zero_l4142658623432671053t_unit @ G2 ) )
       => ( ( ( add_li7652885771158616974t_unit @ G2 @ X @ Y2 )
            = ( zero_l4142658623432671053t_unit @ G2 ) )
         => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
           => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
             => ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G2 ) )
               => ( Y = Y2 ) ) ) ) ) ) ) ).

% abelian_monoid.minus_unique
thf(fact_729_abelian__monoid_Ominus__unique,axiom,
    ! [G2: partia2956882679547061052t_unit,Y: list_list_a,X: list_list_a,Y2: list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( ( add_li174743652000525320t_unit @ G2 @ Y @ X )
          = ( zero_l347298301471573063t_unit @ G2 ) )
       => ( ( ( add_li174743652000525320t_unit @ G2 @ X @ Y2 )
            = ( zero_l347298301471573063t_unit @ G2 ) )
         => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
           => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
             => ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G2 ) )
               => ( Y = Y2 ) ) ) ) ) ) ) ).

% abelian_monoid.minus_unique
thf(fact_730_abelian__monoid_Ominus__unique,axiom,
    ! [G2: partia7496981018696276118t_unit,Y: set_list_a,X: set_list_a,Y2: set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( ( add_se2486902527185523630t_unit @ G2 @ Y @ X )
          = ( zero_s2910681146719230829t_unit @ G2 ) )
       => ( ( ( add_se2486902527185523630t_unit @ G2 @ X @ Y2 )
            = ( zero_s2910681146719230829t_unit @ G2 ) )
         => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
           => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
             => ( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ G2 ) )
               => ( Y = Y2 ) ) ) ) ) ) ) ).

% abelian_monoid.minus_unique
thf(fact_731_abelian__monoidI,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ! [X2: a,Y4: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
           => ( member_a @ ( add_a_b @ R @ X2 @ Y4 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
     => ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ! [X2: a,Y4: a,Z3: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R ) )
                 => ( ( add_a_b @ R @ ( add_a_b @ R @ X2 @ Y4 ) @ Z3 )
                    = ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y4 @ Z3 ) ) ) ) ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X2 )
                  = X2 ) )
           => ( ! [X2: a,Y4: a] :
                  ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
                 => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
                   => ( ( add_a_b @ R @ X2 @ Y4 )
                      = ( add_a_b @ R @ Y4 @ X2 ) ) ) )
             => ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).

% abelian_monoidI
thf(fact_732_abelian__monoidI,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ! [X2: list_a,Y4: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
           => ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
     => ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
       => ( ! [X2: list_a,Y4: list_a,Z3: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ R ) )
                 => ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 ) @ Z3 )
                    = ( add_li7652885771158616974t_unit @ R @ X2 @ ( add_li7652885771158616974t_unit @ R @ Y4 @ Z3 ) ) ) ) ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X2 )
                  = X2 ) )
           => ( ! [X2: list_a,Y4: list_a] :
                  ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
                 => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
                   => ( ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 )
                      = ( add_li7652885771158616974t_unit @ R @ Y4 @ X2 ) ) ) )
             => ( abelia226231641709521465t_unit @ R ) ) ) ) ) ) ).

% abelian_monoidI
thf(fact_733_abelian__monoidI,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ! [X2: list_list_a,Y4: list_list_a] :
          ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
           => ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X2 @ Y4 ) @ ( partia2464479390973590831t_unit @ R ) ) ) )
     => ( ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) )
       => ( ! [X2: list_list_a,Y4: list_list_a,Z3: list_list_a] :
              ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
               => ( ( member_list_list_a @ Z3 @ ( partia2464479390973590831t_unit @ R ) )
                 => ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X2 @ Y4 ) @ Z3 )
                    = ( add_li174743652000525320t_unit @ R @ X2 @ ( add_li174743652000525320t_unit @ R @ Y4 @ Z3 ) ) ) ) ) )
         => ( ! [X2: list_list_a] :
                ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
               => ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X2 )
                  = X2 ) )
           => ( ! [X2: list_list_a,Y4: list_list_a] :
                  ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
                 => ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
                   => ( ( add_li174743652000525320t_unit @ R @ X2 @ Y4 )
                      = ( add_li174743652000525320t_unit @ R @ Y4 @ X2 ) ) ) )
             => ( abelia3641329199688042803t_unit @ R ) ) ) ) ) ) ).

% abelian_monoidI
thf(fact_734_abelian__monoidI,axiom,
    ! [R: partia7496981018696276118t_unit] :
      ( ! [X2: set_list_a,Y4: set_list_a] :
          ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
           => ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ X2 @ Y4 ) @ ( partia141011252114345353t_unit @ R ) ) ) )
     => ( ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) )
       => ( ! [X2: set_list_a,Y4: set_list_a,Z3: set_list_a] :
              ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
             => ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
               => ( ( member_set_list_a @ Z3 @ ( partia141011252114345353t_unit @ R ) )
                 => ( ( add_se2486902527185523630t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X2 @ Y4 ) @ Z3 )
                    = ( add_se2486902527185523630t_unit @ R @ X2 @ ( add_se2486902527185523630t_unit @ R @ Y4 @ Z3 ) ) ) ) ) )
         => ( ! [X2: set_list_a] :
                ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
               => ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X2 )
                  = X2 ) )
           => ( ! [X2: set_list_a,Y4: set_list_a] :
                  ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
                 => ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
                   => ( ( add_se2486902527185523630t_unit @ R @ X2 @ Y4 )
                      = ( add_se2486902527185523630t_unit @ R @ Y4 @ X2 ) ) ) )
             => ( abelia3322010900105369177t_unit @ R ) ) ) ) ) ) ).

% abelian_monoidI
thf(fact_735_semiring_Osemiring__simprules_I11_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
          = X ) ) ) ).

% semiring.semiring_simprules(11)
thf(fact_736_semiring_Osemiring__simprules_I11_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
          = X ) ) ) ).

% semiring.semiring_simprules(11)
thf(fact_737_semiring_Osemiring__simprules_I11_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
          = X ) ) ) ).

% semiring.semiring_simprules(11)
thf(fact_738_semiring_Osemiring__simprules_I11_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( add_se2486902527185523630t_unit @ R @ X @ ( zero_s2910681146719230829t_unit @ R ) )
          = X ) ) ) ).

% semiring.semiring_simprules(11)
thf(fact_739_semiring_Osemiring__simprules_I6_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(6)
thf(fact_740_semiring_Osemiring__simprules_I6_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(6)
thf(fact_741_semiring_Osemiring__simprules_I6_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(6)
thf(fact_742_semiring_Osemiring__simprules_I6_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(6)
thf(fact_743_semiring_Ol__distr,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
              = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.l_distr
thf(fact_744_semiring_Ol__distr,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
              = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.l_distr
thf(fact_745_semiring_Ol__distr,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
              = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Z ) @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.l_distr
thf(fact_746_semiring_Ol__distr,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
           => ( ( mult_s7802724872828879953t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) @ Z )
              = ( add_se2486902527185523630t_unit @ R @ ( mult_s7802724872828879953t_unit @ R @ X @ Z ) @ ( mult_s7802724872828879953t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.l_distr
thf(fact_747_semiring_Or__distr,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
              = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).

% semiring.r_distr
thf(fact_748_semiring_Or__distr,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
              = ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).

% semiring.r_distr
thf(fact_749_semiring_Or__distr,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ Z @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
              = ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ Z @ X ) @ ( mult_l4853965630390486993t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).

% semiring.r_distr
thf(fact_750_semiring_Or__distr,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
           => ( ( mult_s7802724872828879953t_unit @ R @ Z @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) )
              = ( add_se2486902527185523630t_unit @ R @ ( mult_s7802724872828879953t_unit @ R @ Z @ X ) @ ( mult_s7802724872828879953t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).

% semiring.r_distr
thf(fact_751_ring__hom__cring_Ohom__add,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( H @ ( add_a_b @ R @ X @ Y ) )
              = ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_add
thf(fact_752_ring__hom__cring_Ohom__add,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a,Y: a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( H @ ( add_a_b @ R @ X @ Y ) )
              = ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_add
thf(fact_753_ring__hom__cring_Ohom__add,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
              = ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_add
thf(fact_754_ring__hom__cring_Ohom__add,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > list_list_a,R: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a,Y: a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ R @ S2 ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( H @ ( add_a_b @ R @ X @ Y ) )
              = ( add_li174743652000525320t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_add
thf(fact_755_ring__hom__cring_Ohom__add,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
              = ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_add
thf(fact_756_ring__hom__cring_Ohom__add,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( H @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
              = ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_add
thf(fact_757_ring__hom__cring_Ohom__add,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: set_list_a > a,R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,X: set_list_a,Y: set_list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) )
       => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
           => ( ( H @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) )
              = ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_add
thf(fact_758_ring__hom__cring_Ohom__add,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > list_list_a,R: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a,Y: list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member6714375691612171394list_a @ H @ ( ring_h8002040739877300486t_unit @ R @ S2 ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
              = ( add_li174743652000525320t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_add
thf(fact_759_ring__hom__cring_Ohom__add,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S2 ) )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( H @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
              = ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_add
thf(fact_760_ring__hom__cring_Ohom__add,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: set_list_a > list_a,R: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit,X: set_list_a,Y: set_list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member5910328476188217884list_a @ H @ ( ring_h8038483918290310060t_unit @ R @ S2 ) )
       => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
           => ( ( H @ ( add_se2486902527185523630t_unit @ R @ X @ Y ) )
              = ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_add
thf(fact_761_ring__hom__memI,axiom,
    ! [R: partia2175431115845679010xt_a_b,H: a > a,S2: partia2175431115845679010xt_a_b] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S2 ) ) )
     => ( ! [X2: a,Y4: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( H @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y4 ) )
                = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
       => ( ! [X2: a,Y4: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( H @ ( add_a_b @ R @ X2 @ Y4 ) )
                  = ( add_a_b @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
         => ( ( ( H @ ( one_a_ring_ext_a_b @ R ) )
              = ( one_a_ring_ext_a_b @ S2 ) )
           => ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) ) ) ) ) ) ).

% ring_hom_memI
thf(fact_762_ring__hom__memI,axiom,
    ! [R: partia2175431115845679010xt_a_b,H: a > list_a,S2: partia2670972154091845814t_unit] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_list_a @ ( H @ X2 ) @ ( partia5361259788508890537t_unit @ S2 ) ) )
     => ( ! [X2: a,Y4: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( H @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y4 ) )
                = ( mult_l7073676228092353617t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
       => ( ! [X2: a,Y4: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( H @ ( add_a_b @ R @ X2 @ Y4 ) )
                  = ( add_li7652885771158616974t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
         => ( ( ( H @ ( one_a_ring_ext_a_b @ R ) )
              = ( one_li8328186300101108157t_unit @ S2 ) )
           => ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) ) ) ) ) ) ).

% ring_hom_memI
thf(fact_763_ring__hom__memI,axiom,
    ! [R: partia2670972154091845814t_unit,H: list_a > a,S2: partia2175431115845679010xt_a_b] :
      ( ! [X2: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S2 ) ) )
     => ( ! [X2: list_a,Y4: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y4 ) )
                = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
       => ( ! [X2: list_a,Y4: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( H @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 ) )
                  = ( add_a_b @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
         => ( ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
              = ( one_a_ring_ext_a_b @ S2 ) )
           => ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) ) ) ) ) ) ).

% ring_hom_memI
thf(fact_764_ring__hom__memI,axiom,
    ! [R: partia2175431115845679010xt_a_b,H: a > list_list_a,S2: partia2956882679547061052t_unit] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_list_list_a @ ( H @ X2 ) @ ( partia2464479390973590831t_unit @ S2 ) ) )
     => ( ! [X2: a,Y4: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( H @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y4 ) )
                = ( mult_l4853965630390486993t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
       => ( ! [X2: a,Y4: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( H @ ( add_a_b @ R @ X2 @ Y4 ) )
                  = ( add_li174743652000525320t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
         => ( ( ( H @ ( one_a_ring_ext_a_b @ R ) )
              = ( one_li8234411390022467901t_unit @ S2 ) )
           => ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ R @ S2 ) ) ) ) ) ) ).

% ring_hom_memI
thf(fact_765_ring__hom__memI,axiom,
    ! [R: partia2175431115845679010xt_a_b,H: a > set_list_a,S2: partia7496981018696276118t_unit] :
      ( ! [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_set_list_a @ ( H @ X2 ) @ ( partia141011252114345353t_unit @ S2 ) ) )
     => ( ! [X2: a,Y4: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( H @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y4 ) )
                = ( mult_s7802724872828879953t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
       => ( ! [X2: a,Y4: a] :
              ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ R ) )
               => ( ( H @ ( add_a_b @ R @ X2 @ Y4 ) )
                  = ( add_se2486902527185523630t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
         => ( ( ( H @ ( one_a_ring_ext_a_b @ R ) )
              = ( one_se1127990129394575805t_unit @ S2 ) )
           => ( member_a_set_list_a @ H @ ( ring_h6109298854714515236t_unit @ R @ S2 ) ) ) ) ) ) ).

% ring_hom_memI
thf(fact_766_ring__hom__memI,axiom,
    ! [R: partia2670972154091845814t_unit,H: list_a > list_a,S2: partia2670972154091845814t_unit] :
      ( ! [X2: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( H @ X2 ) @ ( partia5361259788508890537t_unit @ S2 ) ) )
     => ( ! [X2: list_a,Y4: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y4 ) )
                = ( mult_l7073676228092353617t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
       => ( ! [X2: list_a,Y4: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( H @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 ) )
                  = ( add_li7652885771158616974t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
         => ( ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
              = ( one_li8328186300101108157t_unit @ S2 ) )
           => ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) ) ) ) ) ) ).

% ring_hom_memI
thf(fact_767_ring__hom__memI,axiom,
    ! [R: partia2956882679547061052t_unit,H: list_list_a > a,S2: partia2175431115845679010xt_a_b] :
      ( ! [X2: list_list_a] :
          ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S2 ) ) )
     => ( ! [X2: list_list_a,Y4: list_list_a] :
            ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( H @ ( mult_l4853965630390486993t_unit @ R @ X2 @ Y4 ) )
                = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
       => ( ! [X2: list_list_a,Y4: list_list_a] :
              ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( member_list_list_a @ Y4 @ ( partia2464479390973590831t_unit @ R ) )
               => ( ( H @ ( add_li174743652000525320t_unit @ R @ X2 @ Y4 ) )
                  = ( add_a_b @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
         => ( ( ( H @ ( one_li8234411390022467901t_unit @ R ) )
              = ( one_a_ring_ext_a_b @ S2 ) )
           => ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) ) ) ) ) ) ).

% ring_hom_memI
thf(fact_768_ring__hom__memI,axiom,
    ! [R: partia7496981018696276118t_unit,H: set_list_a > a,S2: partia2175431115845679010xt_a_b] :
      ( ! [X2: set_list_a] :
          ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
         => ( member_a @ ( H @ X2 ) @ ( partia707051561876973205xt_a_b @ S2 ) ) )
     => ( ! [X2: set_list_a,Y4: set_list_a] :
            ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
           => ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
             => ( ( H @ ( mult_s7802724872828879953t_unit @ R @ X2 @ Y4 ) )
                = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
       => ( ! [X2: set_list_a,Y4: set_list_a] :
              ( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
             => ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ R ) )
               => ( ( H @ ( add_se2486902527185523630t_unit @ R @ X2 @ Y4 ) )
                  = ( add_a_b @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
         => ( ( ( H @ ( one_se1127990129394575805t_unit @ R ) )
              = ( one_a_ring_ext_a_b @ S2 ) )
           => ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) ) ) ) ) ) ).

% ring_hom_memI
thf(fact_769_ring__hom__memI,axiom,
    ! [R: partia2670972154091845814t_unit,H: list_a > list_list_a,S2: partia2956882679547061052t_unit] :
      ( ! [X2: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_list_a @ ( H @ X2 ) @ ( partia2464479390973590831t_unit @ S2 ) ) )
     => ( ! [X2: list_a,Y4: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y4 ) )
                = ( mult_l4853965630390486993t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
       => ( ! [X2: list_a,Y4: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( H @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 ) )
                  = ( add_li174743652000525320t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
         => ( ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
              = ( one_li8234411390022467901t_unit @ S2 ) )
           => ( member6714375691612171394list_a @ H @ ( ring_h8002040739877300486t_unit @ R @ S2 ) ) ) ) ) ) ).

% ring_hom_memI
thf(fact_770_ring__hom__memI,axiom,
    ! [R: partia2670972154091845814t_unit,H: list_a > set_list_a,S2: partia7496981018696276118t_unit] :
      ( ! [X2: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_set_list_a @ ( H @ X2 ) @ ( partia141011252114345353t_unit @ S2 ) ) )
     => ( ! [X2: list_a,Y4: list_a] :
            ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y4 ) )
                = ( mult_s7802724872828879953t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
       => ( ! [X2: list_a,Y4: list_a] :
              ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ R ) )
               => ( ( H @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y4 ) )
                  = ( add_se2486902527185523630t_unit @ S2 @ ( H @ X2 ) @ ( H @ Y4 ) ) ) ) )
         => ( ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
              = ( one_se1127990129394575805t_unit @ S2 ) )
           => ( member4263473470251683292list_a @ H @ ( ring_h6188449271506562988t_unit @ R @ S2 ) ) ) ) ) ) ).

% ring_hom_memI
thf(fact_771_abelian__monoid_Ofinsum__addf,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: nat > a,A2: set_nat,G: nat > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_nat_a @ F
          @ ( pi_nat_a @ A2
            @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( ( member_nat_a @ G
            @ ( pi_nat_a @ A2
              @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ G2 ) ) )
         => ( ( finsum_a_b_nat @ G2
              @ ^ [X3: nat] : ( add_a_b @ G2 @ ( F @ X3 ) @ ( G @ X3 ) )
              @ A2 )
            = ( add_a_b @ G2 @ ( finsum_a_b_nat @ G2 @ F @ A2 ) @ ( finsum_a_b_nat @ G2 @ G @ A2 ) ) ) ) ) ) ).

% abelian_monoid.finsum_addf
thf(fact_772_abelian__monoid_Ofinsum__addf,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: a > a,A2: set_a,G: a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a_a @ F
          @ ( pi_a_a @ A2
            @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( ( member_a_a @ G
            @ ( pi_a_a @ A2
              @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
         => ( ( finsum_a_b_a @ G2
              @ ^ [X3: a] : ( add_a_b @ G2 @ ( F @ X3 ) @ ( G @ X3 ) )
              @ A2 )
            = ( add_a_b @ G2 @ ( finsum_a_b_a @ G2 @ F @ A2 ) @ ( finsum_a_b_a @ G2 @ G @ A2 ) ) ) ) ) ) ).

% abelian_monoid.finsum_addf
thf(fact_773_abelian__monoid_Ofinsum__addf,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: ( a > a ) > a,A2: set_a_a,G: ( a > a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a_a_a @ F
          @ ( pi_a_a_a @ A2
            @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( ( member_a_a_a @ G
            @ ( pi_a_a_a @ A2
              @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
         => ( ( finsum_a_b_a_a @ G2
              @ ^ [X3: a > a] : ( add_a_b @ G2 @ ( F @ X3 ) @ ( G @ X3 ) )
              @ A2 )
            = ( add_a_b @ G2 @ ( finsum_a_b_a_a @ G2 @ F @ A2 ) @ ( finsum_a_b_a_a @ G2 @ G @ A2 ) ) ) ) ) ) ).

% abelian_monoid.finsum_addf
thf(fact_774_abelian__monoid_Ofinsum__addf,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: ( a > list_a ) > a,A2: set_a_list_a,G: ( a > list_a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a_list_a_a @ F
          @ ( pi_a_list_a_a @ A2
            @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( ( member_a_list_a_a @ G
            @ ( pi_a_list_a_a @ A2
              @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
         => ( ( finsum_a_b_a_list_a @ G2
              @ ^ [X3: a > list_a] : ( add_a_b @ G2 @ ( F @ X3 ) @ ( G @ X3 ) )
              @ A2 )
            = ( add_a_b @ G2 @ ( finsum_a_b_a_list_a @ G2 @ F @ A2 ) @ ( finsum_a_b_a_list_a @ G2 @ G @ A2 ) ) ) ) ) ) ).

% abelian_monoid.finsum_addf
thf(fact_775_abelian__monoid_Ofinsum__addf,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: ( nat > a ) > a,A2: set_nat_a,G: ( nat > a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_nat_a_a @ F
          @ ( pi_nat_a_a @ A2
            @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( ( member_nat_a_a @ G
            @ ( pi_nat_a_a @ A2
              @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
         => ( ( finsum_a_b_nat_a @ G2
              @ ^ [X3: nat > a] : ( add_a_b @ G2 @ ( F @ X3 ) @ ( G @ X3 ) )
              @ A2 )
            = ( add_a_b @ G2 @ ( finsum_a_b_nat_a @ G2 @ F @ A2 ) @ ( finsum_a_b_nat_a @ G2 @ G @ A2 ) ) ) ) ) ) ).

% abelian_monoid.finsum_addf
thf(fact_776_abelian__monoid_Ofinsum__addf,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: ( nat > list_a ) > a,A2: set_nat_list_a,G: ( nat > list_a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_nat_list_a_a @ F
          @ ( pi_nat_list_a_a @ A2
            @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( ( member_nat_list_a_a @ G
            @ ( pi_nat_list_a_a @ A2
              @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
         => ( ( finsum1341700292807219277list_a @ G2
              @ ^ [X3: nat > list_a] : ( add_a_b @ G2 @ ( F @ X3 ) @ ( G @ X3 ) )
              @ A2 )
            = ( add_a_b @ G2 @ ( finsum1341700292807219277list_a @ G2 @ F @ A2 ) @ ( finsum1341700292807219277list_a @ G2 @ G @ A2 ) ) ) ) ) ) ).

% abelian_monoid.finsum_addf
thf(fact_777_abelian__monoid_Ofinsum__addf,axiom,
    ! [G2: partia2670972154091845814t_unit,F: nat > list_a,A2: set_nat,G: nat > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_nat_list_a @ F
          @ ( pi_nat_list_a @ A2
            @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ G2 ) ) )
       => ( ( member_nat_list_a @ G
            @ ( pi_nat_list_a @ A2
              @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ G2 ) ) )
         => ( ( finsum3497512462216549110it_nat @ G2
              @ ^ [X3: nat] : ( add_li7652885771158616974t_unit @ G2 @ ( F @ X3 ) @ ( G @ X3 ) )
              @ A2 )
            = ( add_li7652885771158616974t_unit @ G2 @ ( finsum3497512462216549110it_nat @ G2 @ F @ A2 ) @ ( finsum3497512462216549110it_nat @ G2 @ G @ A2 ) ) ) ) ) ) ).

% abelian_monoid.finsum_addf
thf(fact_778_abelian__monoid_Ofinsum__addf,axiom,
    ! [G2: partia2670972154091845814t_unit,F: a > list_a,A2: set_a,G: a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_a_list_a @ F
          @ ( pi_a_list_a @ A2
            @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
       => ( ( member_a_list_a @ G
            @ ( pi_a_list_a @ A2
              @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
         => ( ( finsum7322697649718157656unit_a @ G2
              @ ^ [X3: a] : ( add_li7652885771158616974t_unit @ G2 @ ( F @ X3 ) @ ( G @ X3 ) )
              @ A2 )
            = ( add_li7652885771158616974t_unit @ G2 @ ( finsum7322697649718157656unit_a @ G2 @ F @ A2 ) @ ( finsum7322697649718157656unit_a @ G2 @ G @ A2 ) ) ) ) ) ) ).

% abelian_monoid.finsum_addf
thf(fact_779_abelian__monoid_Ofinsum__Un__Int,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_nat,B: set_nat,G: nat > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( finite_finite_nat @ B )
         => ( ( member_nat_a @ G
              @ ( pi_nat_a @ A2
                @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ G2 ) ) )
           => ( ( member_nat_a @ G
                @ ( pi_nat_a @ B
                  @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ G2 ) ) )
             => ( ( add_a_b @ G2 @ ( finsum_a_b_nat @ G2 @ G @ ( sup_sup_set_nat @ A2 @ B ) ) @ ( finsum_a_b_nat @ G2 @ G @ ( inf_inf_set_nat @ A2 @ B ) ) )
                = ( add_a_b @ G2 @ ( finsum_a_b_nat @ G2 @ G @ A2 ) @ ( finsum_a_b_nat @ G2 @ G @ B ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_Int
thf(fact_780_abelian__monoid_Ofinsum__Un__Int,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_a,B: set_a,G: a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( finite_finite_a @ B )
         => ( ( member_a_a @ G
              @ ( pi_a_a @ A2
                @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
           => ( ( member_a_a @ G
                @ ( pi_a_a @ B
                  @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
             => ( ( add_a_b @ G2 @ ( finsum_a_b_a @ G2 @ G @ ( sup_sup_set_a @ A2 @ B ) ) @ ( finsum_a_b_a @ G2 @ G @ ( inf_inf_set_a @ A2 @ B ) ) )
                = ( add_a_b @ G2 @ ( finsum_a_b_a @ G2 @ G @ A2 ) @ ( finsum_a_b_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_Int
thf(fact_781_abelian__monoid_Ofinsum__Un__Int,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_list_a,B: set_list_a,G: list_a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( finite_finite_list_a @ B )
         => ( ( member_list_a_a @ G
              @ ( pi_list_a_a @ A2
                @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
           => ( ( member_list_a_a @ G
                @ ( pi_list_a_a @ B
                  @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
             => ( ( add_a_b @ G2 @ ( finsum_a_b_list_a @ G2 @ G @ ( sup_sup_set_list_a @ A2 @ B ) ) @ ( finsum_a_b_list_a @ G2 @ G @ ( inf_inf_set_list_a @ A2 @ B ) ) )
                = ( add_a_b @ G2 @ ( finsum_a_b_list_a @ G2 @ G @ A2 ) @ ( finsum_a_b_list_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_Int
thf(fact_782_abelian__monoid_Ofinsum__Un__Int,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_nat,B: set_nat,G: nat > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( finite_finite_nat @ B )
         => ( ( member_nat_list_a @ G
              @ ( pi_nat_list_a @ A2
                @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ G2 ) ) )
           => ( ( member_nat_list_a @ G
                @ ( pi_nat_list_a @ B
                  @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ G2 ) ) )
             => ( ( add_li7652885771158616974t_unit @ G2 @ ( finsum3497512462216549110it_nat @ G2 @ G @ ( sup_sup_set_nat @ A2 @ B ) ) @ ( finsum3497512462216549110it_nat @ G2 @ G @ ( inf_inf_set_nat @ A2 @ B ) ) )
                = ( add_li7652885771158616974t_unit @ G2 @ ( finsum3497512462216549110it_nat @ G2 @ G @ A2 ) @ ( finsum3497512462216549110it_nat @ G2 @ G @ B ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_Int
thf(fact_783_abelian__monoid_Ofinsum__Un__Int,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_a,B: set_a,G: a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( finite_finite_a @ B )
         => ( ( member_a_list_a @ G
              @ ( pi_a_list_a @ A2
                @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
           => ( ( member_a_list_a @ G
                @ ( pi_a_list_a @ B
                  @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
             => ( ( add_li7652885771158616974t_unit @ G2 @ ( finsum7322697649718157656unit_a @ G2 @ G @ ( sup_sup_set_a @ A2 @ B ) ) @ ( finsum7322697649718157656unit_a @ G2 @ G @ ( inf_inf_set_a @ A2 @ B ) ) )
                = ( add_li7652885771158616974t_unit @ G2 @ ( finsum7322697649718157656unit_a @ G2 @ G @ A2 ) @ ( finsum7322697649718157656unit_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_Int
thf(fact_784_abelian__monoid_Ofinsum__Un__Int,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_a_a,B: set_a_a,G: ( a > a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finite_finite_a_a @ A2 )
       => ( ( finite_finite_a_a @ B )
         => ( ( member_a_a_a @ G
              @ ( pi_a_a_a @ A2
                @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
           => ( ( member_a_a_a @ G
                @ ( pi_a_a_a @ B
                  @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
             => ( ( add_a_b @ G2 @ ( finsum_a_b_a_a @ G2 @ G @ ( sup_sup_set_a_a @ A2 @ B ) ) @ ( finsum_a_b_a_a @ G2 @ G @ ( inf_inf_set_a_a @ A2 @ B ) ) )
                = ( add_a_b @ G2 @ ( finsum_a_b_a_a @ G2 @ G @ A2 ) @ ( finsum_a_b_a_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_Int
thf(fact_785_abelian__monoid_Ofinsum__Un__Int,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_nat_a,B: set_nat_a,G: ( nat > a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ( finite_finite_nat_a @ B )
         => ( ( member_nat_a_a @ G
              @ ( pi_nat_a_a @ A2
                @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
           => ( ( member_nat_a_a @ G
                @ ( pi_nat_a_a @ B
                  @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
             => ( ( add_a_b @ G2 @ ( finsum_a_b_nat_a @ G2 @ G @ ( sup_sup_set_nat_a @ A2 @ B ) ) @ ( finsum_a_b_nat_a @ G2 @ G @ ( inf_inf_set_nat_a @ A2 @ B ) ) )
                = ( add_a_b @ G2 @ ( finsum_a_b_nat_a @ G2 @ G @ A2 ) @ ( finsum_a_b_nat_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_Int
thf(fact_786_abelian__monoid_Ofinsum__Un__Int,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_list_a,B: set_list_a,G: list_a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( finite_finite_list_a @ B )
         => ( ( member_list_a_list_a @ G
              @ ( pi_list_a_list_a @ A2
                @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
           => ( ( member_list_a_list_a @ G
                @ ( pi_list_a_list_a @ B
                  @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
             => ( ( add_li7652885771158616974t_unit @ G2 @ ( finsum8721804980556663006list_a @ G2 @ G @ ( sup_sup_set_list_a @ A2 @ B ) ) @ ( finsum8721804980556663006list_a @ G2 @ G @ ( inf_inf_set_list_a @ A2 @ B ) ) )
                = ( add_li7652885771158616974t_unit @ G2 @ ( finsum8721804980556663006list_a @ G2 @ G @ A2 ) @ ( finsum8721804980556663006list_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_Int
thf(fact_787_abelian__monoid_Ofinsum__Un__Int,axiom,
    ! [G2: partia2956882679547061052t_unit,A2: set_nat,B: set_nat,G: nat > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( finite_finite_nat @ B )
         => ( ( member8650753269014980122list_a @ G
              @ ( pi_nat_list_list_a @ A2
                @ ^ [Uu: nat] : ( partia2464479390973590831t_unit @ G2 ) ) )
           => ( ( member8650753269014980122list_a @ G
                @ ( pi_nat_list_list_a @ B
                  @ ^ [Uu: nat] : ( partia2464479390973590831t_unit @ G2 ) ) )
             => ( ( add_li174743652000525320t_unit @ G2 @ ( finsum3990971441743328240it_nat @ G2 @ G @ ( sup_sup_set_nat @ A2 @ B ) ) @ ( finsum3990971441743328240it_nat @ G2 @ G @ ( inf_inf_set_nat @ A2 @ B ) ) )
                = ( add_li174743652000525320t_unit @ G2 @ ( finsum3990971441743328240it_nat @ G2 @ G @ A2 ) @ ( finsum3990971441743328240it_nat @ G2 @ G @ B ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_Int
thf(fact_788_abelian__monoid_Ofinsum__Un__Int,axiom,
    ! [G2: partia2956882679547061052t_unit,A2: set_a,B: set_a,G: a > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( finite_finite_a @ B )
         => ( ( member_a_list_list_a @ G
              @ ( pi_a_list_list_a @ A2
                @ ^ [Uu: a] : ( partia2464479390973590831t_unit @ G2 ) ) )
           => ( ( member_a_list_list_a @ G
                @ ( pi_a_list_list_a @ B
                  @ ^ [Uu: a] : ( partia2464479390973590831t_unit @ G2 ) ) )
             => ( ( add_li174743652000525320t_unit @ G2 @ ( finsum463596448938265310unit_a @ G2 @ G @ ( sup_sup_set_a @ A2 @ B ) ) @ ( finsum463596448938265310unit_a @ G2 @ G @ ( inf_inf_set_a @ A2 @ B ) ) )
                = ( add_li174743652000525320t_unit @ G2 @ ( finsum463596448938265310unit_a @ G2 @ G @ A2 ) @ ( finsum463596448938265310unit_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_Int
thf(fact_789_abelian__monoid_Ofinsum__Un__disjoint,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_nat,B: set_nat,G: nat > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( finite_finite_nat @ B )
         => ( ( ( inf_inf_set_nat @ A2 @ B )
              = bot_bot_set_nat )
           => ( ( member_nat_a @ G
                @ ( pi_nat_a @ A2
                  @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ G2 ) ) )
             => ( ( member_nat_a @ G
                  @ ( pi_nat_a @ B
                    @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ G2 ) ) )
               => ( ( finsum_a_b_nat @ G2 @ G @ ( sup_sup_set_nat @ A2 @ B ) )
                  = ( add_a_b @ G2 @ ( finsum_a_b_nat @ G2 @ G @ A2 ) @ ( finsum_a_b_nat @ G2 @ G @ B ) ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_disjoint
thf(fact_790_abelian__monoid_Ofinsum__Un__disjoint,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_a,B: set_a,G: a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( finite_finite_a @ B )
         => ( ( ( inf_inf_set_a @ A2 @ B )
              = bot_bot_set_a )
           => ( ( member_a_a @ G
                @ ( pi_a_a @ A2
                  @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
             => ( ( member_a_a @ G
                  @ ( pi_a_a @ B
                    @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
               => ( ( finsum_a_b_a @ G2 @ G @ ( sup_sup_set_a @ A2 @ B ) )
                  = ( add_a_b @ G2 @ ( finsum_a_b_a @ G2 @ G @ A2 ) @ ( finsum_a_b_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_disjoint
thf(fact_791_abelian__monoid_Ofinsum__Un__disjoint,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_list_a,B: set_list_a,G: list_a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( finite_finite_list_a @ B )
         => ( ( ( inf_inf_set_list_a @ A2 @ B )
              = bot_bot_set_list_a )
           => ( ( member_list_a_a @ G
                @ ( pi_list_a_a @ A2
                  @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
             => ( ( member_list_a_a @ G
                  @ ( pi_list_a_a @ B
                    @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
               => ( ( finsum_a_b_list_a @ G2 @ G @ ( sup_sup_set_list_a @ A2 @ B ) )
                  = ( add_a_b @ G2 @ ( finsum_a_b_list_a @ G2 @ G @ A2 ) @ ( finsum_a_b_list_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_disjoint
thf(fact_792_abelian__monoid_Ofinsum__Un__disjoint,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_nat,B: set_nat,G: nat > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( finite_finite_nat @ B )
         => ( ( ( inf_inf_set_nat @ A2 @ B )
              = bot_bot_set_nat )
           => ( ( member_nat_list_a @ G
                @ ( pi_nat_list_a @ A2
                  @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ G2 ) ) )
             => ( ( member_nat_list_a @ G
                  @ ( pi_nat_list_a @ B
                    @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ G2 ) ) )
               => ( ( finsum3497512462216549110it_nat @ G2 @ G @ ( sup_sup_set_nat @ A2 @ B ) )
                  = ( add_li7652885771158616974t_unit @ G2 @ ( finsum3497512462216549110it_nat @ G2 @ G @ A2 ) @ ( finsum3497512462216549110it_nat @ G2 @ G @ B ) ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_disjoint
thf(fact_793_abelian__monoid_Ofinsum__Un__disjoint,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_a,B: set_a,G: a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( finite_finite_a @ B )
         => ( ( ( inf_inf_set_a @ A2 @ B )
              = bot_bot_set_a )
           => ( ( member_a_list_a @ G
                @ ( pi_a_list_a @ A2
                  @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
             => ( ( member_a_list_a @ G
                  @ ( pi_a_list_a @ B
                    @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
               => ( ( finsum7322697649718157656unit_a @ G2 @ G @ ( sup_sup_set_a @ A2 @ B ) )
                  = ( add_li7652885771158616974t_unit @ G2 @ ( finsum7322697649718157656unit_a @ G2 @ G @ A2 ) @ ( finsum7322697649718157656unit_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_disjoint
thf(fact_794_abelian__monoid_Ofinsum__Un__disjoint,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_a_a,B: set_a_a,G: ( a > a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finite_finite_a_a @ A2 )
       => ( ( finite_finite_a_a @ B )
         => ( ( ( inf_inf_set_a_a @ A2 @ B )
              = bot_bot_set_a_a )
           => ( ( member_a_a_a @ G
                @ ( pi_a_a_a @ A2
                  @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
             => ( ( member_a_a_a @ G
                  @ ( pi_a_a_a @ B
                    @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
               => ( ( finsum_a_b_a_a @ G2 @ G @ ( sup_sup_set_a_a @ A2 @ B ) )
                  = ( add_a_b @ G2 @ ( finsum_a_b_a_a @ G2 @ G @ A2 ) @ ( finsum_a_b_a_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_disjoint
thf(fact_795_abelian__monoid_Ofinsum__Un__disjoint,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_nat_a,B: set_nat_a,G: ( nat > a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finite_finite_nat_a @ A2 )
       => ( ( finite_finite_nat_a @ B )
         => ( ( ( inf_inf_set_nat_a @ A2 @ B )
              = bot_bot_set_nat_a )
           => ( ( member_nat_a_a @ G
                @ ( pi_nat_a_a @ A2
                  @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
             => ( ( member_nat_a_a @ G
                  @ ( pi_nat_a_a @ B
                    @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
               => ( ( finsum_a_b_nat_a @ G2 @ G @ ( sup_sup_set_nat_a @ A2 @ B ) )
                  = ( add_a_b @ G2 @ ( finsum_a_b_nat_a @ G2 @ G @ A2 ) @ ( finsum_a_b_nat_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_disjoint
thf(fact_796_abelian__monoid_Ofinsum__Un__disjoint,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_list_a,B: set_list_a,G: list_a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( finite_finite_list_a @ B )
         => ( ( ( inf_inf_set_list_a @ A2 @ B )
              = bot_bot_set_list_a )
           => ( ( member_list_a_list_a @ G
                @ ( pi_list_a_list_a @ A2
                  @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
             => ( ( member_list_a_list_a @ G
                  @ ( pi_list_a_list_a @ B
                    @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
               => ( ( finsum8721804980556663006list_a @ G2 @ G @ ( sup_sup_set_list_a @ A2 @ B ) )
                  = ( add_li7652885771158616974t_unit @ G2 @ ( finsum8721804980556663006list_a @ G2 @ G @ A2 ) @ ( finsum8721804980556663006list_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_disjoint
thf(fact_797_abelian__monoid_Ofinsum__Un__disjoint,axiom,
    ! [G2: partia2956882679547061052t_unit,A2: set_nat,B: set_nat,G: nat > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( finite_finite_nat @ B )
         => ( ( ( inf_inf_set_nat @ A2 @ B )
              = bot_bot_set_nat )
           => ( ( member8650753269014980122list_a @ G
                @ ( pi_nat_list_list_a @ A2
                  @ ^ [Uu: nat] : ( partia2464479390973590831t_unit @ G2 ) ) )
             => ( ( member8650753269014980122list_a @ G
                  @ ( pi_nat_list_list_a @ B
                    @ ^ [Uu: nat] : ( partia2464479390973590831t_unit @ G2 ) ) )
               => ( ( finsum3990971441743328240it_nat @ G2 @ G @ ( sup_sup_set_nat @ A2 @ B ) )
                  = ( add_li174743652000525320t_unit @ G2 @ ( finsum3990971441743328240it_nat @ G2 @ G @ A2 ) @ ( finsum3990971441743328240it_nat @ G2 @ G @ B ) ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_disjoint
thf(fact_798_abelian__monoid_Ofinsum__Un__disjoint,axiom,
    ! [G2: partia2956882679547061052t_unit,A2: set_a,B: set_a,G: a > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( finite_finite_a @ A2 )
       => ( ( finite_finite_a @ B )
         => ( ( ( inf_inf_set_a @ A2 @ B )
              = bot_bot_set_a )
           => ( ( member_a_list_list_a @ G
                @ ( pi_a_list_list_a @ A2
                  @ ^ [Uu: a] : ( partia2464479390973590831t_unit @ G2 ) ) )
             => ( ( member_a_list_list_a @ G
                  @ ( pi_a_list_list_a @ B
                    @ ^ [Uu: a] : ( partia2464479390973590831t_unit @ G2 ) ) )
               => ( ( finsum463596448938265310unit_a @ G2 @ G @ ( sup_sup_set_a @ A2 @ B ) )
                  = ( add_li174743652000525320t_unit @ G2 @ ( finsum463596448938265310unit_a @ G2 @ G @ A2 ) @ ( finsum463596448938265310unit_a @ G2 @ G @ B ) ) ) ) ) ) ) ) ) ).

% abelian_monoid.finsum_Un_disjoint
thf(fact_799_ring__hom__closed,axiom,
    ! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a] :
      ( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).

% ring_hom_closed
thf(fact_800_ring__hom__closed,axiom,
    ! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a] :
      ( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).

% ring_hom_closed
thf(fact_801_ring__hom__closed,axiom,
    ! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a] :
      ( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).

% ring_hom_closed
thf(fact_802_ring__hom__closed,axiom,
    ! [H: a > list_list_a,R: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a] :
      ( ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ R @ S2 ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ).

% ring_hom_closed
thf(fact_803_ring__hom__closed,axiom,
    ! [H: a > set_list_a,R: partia2175431115845679010xt_a_b,S2: partia7496981018696276118t_unit,X: a] :
      ( ( member_a_set_list_a @ H @ ( ring_h6109298854714515236t_unit @ R @ S2 ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( member_set_list_a @ ( H @ X ) @ ( partia141011252114345353t_unit @ S2 ) ) ) ) ).

% ring_hom_closed
thf(fact_804_ring__hom__closed,axiom,
    ! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a] :
      ( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).

% ring_hom_closed
thf(fact_805_ring__hom__closed,axiom,
    ! [H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a] :
      ( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).

% ring_hom_closed
thf(fact_806_ring__hom__closed,axiom,
    ! [H: set_list_a > a,R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,X: set_list_a] :
      ( ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).

% ring_hom_closed
thf(fact_807_ring__hom__closed,axiom,
    ! [H: list_a > list_list_a,R: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a] :
      ( ( member6714375691612171394list_a @ H @ ( ring_h8002040739877300486t_unit @ R @ S2 ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ).

% ring_hom_closed
thf(fact_808_ring__hom__closed,axiom,
    ! [H: list_a > set_list_a,R: partia2670972154091845814t_unit,S2: partia7496981018696276118t_unit,X: list_a] :
      ( ( member4263473470251683292list_a @ H @ ( ring_h6188449271506562988t_unit @ R @ S2 ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( member_set_list_a @ ( H @ X ) @ ( partia141011252114345353t_unit @ S2 ) ) ) ) ).

% ring_hom_closed
thf(fact_809_ring__hom__trans,axiom,
    ! [F: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,G: a > list_a,T: partia2670972154091845814t_unit] :
      ( ( member_a_a @ F @ ( ring_hom_a_b_a_b @ R @ S2 ) )
     => ( ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ S2 @ T ) )
       => ( member_a_list_a @ ( comp_a_list_a_a @ G @ F ) @ ( ring_h405018892823518980t_unit @ R @ T ) ) ) ) ).

% ring_hom_trans
thf(fact_810_ring__hom__trans,axiom,
    ! [F: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,G: a > a,T: partia2175431115845679010xt_a_b] :
      ( ( member_list_a_a @ F @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
     => ( ( member_a_a @ G @ ( ring_hom_a_b_a_b @ S2 @ T ) )
       => ( member_list_a_a @ ( comp_a_a_list_a @ G @ F ) @ ( ring_h2895973938487309444it_a_b @ R @ T ) ) ) ) ).

% ring_hom_trans
thf(fact_811_ring__hom__trans,axiom,
    ! [F: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,G: list_a > a,T: partia2175431115845679010xt_a_b] :
      ( ( member_a_list_a @ F @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
     => ( ( member_list_a_a @ G @ ( ring_h2895973938487309444it_a_b @ S2 @ T ) )
       => ( member_a_a @ ( comp_list_a_a_a @ G @ F ) @ ( ring_hom_a_b_a_b @ R @ T ) ) ) ) ).

% ring_hom_trans
thf(fact_812_ring__hom__trans,axiom,
    ! [F: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,G: list_a > a,T: partia2175431115845679010xt_a_b] :
      ( ( member_list_a_list_a @ F @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
     => ( ( member_list_a_a @ G @ ( ring_h2895973938487309444it_a_b @ S2 @ T ) )
       => ( member_list_a_a @ ( comp_list_a_a_list_a @ G @ F ) @ ( ring_h2895973938487309444it_a_b @ R @ T ) ) ) ) ).

% ring_hom_trans
thf(fact_813_ring__hom__trans,axiom,
    ! [F: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,G: a > list_a,T: partia2670972154091845814t_unit] :
      ( ( member_list_a_a @ F @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
     => ( ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ S2 @ T ) )
       => ( member_list_a_list_a @ ( comp_a_list_a_list_a @ G @ F ) @ ( ring_h7399960747407462284t_unit @ R @ T ) ) ) ) ).

% ring_hom_trans
thf(fact_814_ring__hom__trans,axiom,
    ! [F: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,G: list_a > list_a,T: partia2670972154091845814t_unit] :
      ( ( member_a_list_a @ F @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
     => ( ( member_list_a_list_a @ G @ ( ring_h7399960747407462284t_unit @ S2 @ T ) )
       => ( member_a_list_a @ ( comp_list_a_list_a_a @ G @ F ) @ ( ring_h405018892823518980t_unit @ R @ T ) ) ) ) ).

% ring_hom_trans
thf(fact_815_ring__hom__trans,axiom,
    ! [F: set_list_a > a,R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,G: a > a,T: partia2175431115845679010xt_a_b] :
      ( ( member_set_list_a_a @ F @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) )
     => ( ( member_a_a @ G @ ( ring_hom_a_b_a_b @ S2 @ T ) )
       => ( member_set_list_a_a @ ( comp_a_a_set_list_a @ G @ F ) @ ( ring_h8906680420194085028it_a_b @ R @ T ) ) ) ) ).

% ring_hom_trans
thf(fact_816_ring__hom__trans,axiom,
    ! [F: set_list_a > list_a,R: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit,G: list_a > a,T: partia2175431115845679010xt_a_b] :
      ( ( member5910328476188217884list_a @ F @ ( ring_h8038483918290310060t_unit @ R @ S2 ) )
     => ( ( member_list_a_a @ G @ ( ring_h2895973938487309444it_a_b @ S2 @ T ) )
       => ( member_set_list_a_a @ ( comp_l2163758027525145287list_a @ G @ F ) @ ( ring_h8906680420194085028it_a_b @ R @ T ) ) ) ) ).

% ring_hom_trans
thf(fact_817_ring__hom__trans,axiom,
    ! [F: a > list_list_a,R: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,G: list_list_a > list_a,T: partia2670972154091845814t_unit] :
      ( ( member_a_list_list_a @ F @ ( ring_h6858658657455840382t_unit @ R @ S2 ) )
     => ( ( member7168557129179038582list_a @ G @ ( ring_h5031276006722532742t_unit @ S2 @ T ) )
       => ( member_a_list_a @ ( comp_l6744607643581155169st_a_a @ G @ F ) @ ( ring_h405018892823518980t_unit @ R @ T ) ) ) ) ).

% ring_hom_trans
thf(fact_818_ring__hom__trans,axiom,
    ! [F: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,G: a > list_a,T: partia2670972154091845814t_unit] :
      ( ( member_list_list_a_a @ F @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) )
     => ( ( member_a_list_a @ G @ ( ring_h405018892823518980t_unit @ S2 @ T ) )
       => ( member7168557129179038582list_a @ ( comp_a5642107814311143777list_a @ G @ F ) @ ( ring_h5031276006722532742t_unit @ R @ T ) ) ) ) ).

% ring_hom_trans
thf(fact_819_ring__hom__one,axiom,
    ! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit] :
      ( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S2 ) )
     => ( ( H @ ( one_li8234411390022467901t_unit @ R ) )
        = ( one_li8328186300101108157t_unit @ S2 ) ) ) ).

% ring_hom_one
thf(fact_820_ring__hom__one,axiom,
    ! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b] :
      ( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
     => ( ( H @ ( one_a_ring_ext_a_b @ R ) )
        = ( one_a_ring_ext_a_b @ S2 ) ) ) ).

% ring_hom_one
thf(fact_821_ring__hom__one,axiom,
    ! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit] :
      ( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
     => ( ( H @ ( one_a_ring_ext_a_b @ R ) )
        = ( one_li8328186300101108157t_unit @ S2 ) ) ) ).

% ring_hom_one
thf(fact_822_ring__hom__one,axiom,
    ! [H: a > set_list_a,R: partia2175431115845679010xt_a_b,S2: partia7496981018696276118t_unit] :
      ( ( member_a_set_list_a @ H @ ( ring_h6109298854714515236t_unit @ R @ S2 ) )
     => ( ( H @ ( one_a_ring_ext_a_b @ R ) )
        = ( one_se1127990129394575805t_unit @ S2 ) ) ) ).

% ring_hom_one
thf(fact_823_ring__hom__one,axiom,
    ! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b] :
      ( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
     => ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
        = ( one_a_ring_ext_a_b @ S2 ) ) ) ).

% ring_hom_one
thf(fact_824_ring__hom__one,axiom,
    ! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit] :
      ( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
     => ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
        = ( one_li8328186300101108157t_unit @ S2 ) ) ) ).

% ring_hom_one
thf(fact_825_ring__hom__one,axiom,
    ! [H: list_a > set_list_a,R: partia2670972154091845814t_unit,S2: partia7496981018696276118t_unit] :
      ( ( member4263473470251683292list_a @ H @ ( ring_h6188449271506562988t_unit @ R @ S2 ) )
     => ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
        = ( one_se1127990129394575805t_unit @ S2 ) ) ) ).

% ring_hom_one
thf(fact_826_ring__hom__one,axiom,
    ! [H: set_list_a > a,R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b] :
      ( ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) )
     => ( ( H @ ( one_se1127990129394575805t_unit @ R ) )
        = ( one_a_ring_ext_a_b @ S2 ) ) ) ).

% ring_hom_one
thf(fact_827_ring__hom__one,axiom,
    ! [H: set_list_a > list_a,R: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit] :
      ( ( member5910328476188217884list_a @ H @ ( ring_h8038483918290310060t_unit @ R @ S2 ) )
     => ( ( H @ ( one_se1127990129394575805t_unit @ R ) )
        = ( one_li8328186300101108157t_unit @ S2 ) ) ) ).

% ring_hom_one
thf(fact_828_ring__hom__one,axiom,
    ! [H: set_list_a > set_list_a,R: partia7496981018696276118t_unit,S2: partia7496981018696276118t_unit] :
      ( ( member5068272912271824380list_a @ H @ ( ring_h6076331213207892940t_unit @ R @ S2 ) )
     => ( ( H @ ( one_se1127990129394575805t_unit @ R ) )
        = ( one_se1127990129394575805t_unit @ S2 ) ) ) ).

% ring_hom_one
thf(fact_829_ring__hom__cring_Ohom__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,H: a > a] :
      ( ( ring_h661254511236296859_b_a_b @ R @ S2 @ H )
     => ( ( H @ ( zero_a_b @ R ) )
        = ( zero_a_b @ S2 ) ) ) ).

% ring_hom_cring.hom_zero
thf(fact_830_ring__hom__cring_Ohom__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,H: a > list_a] :
      ( ( ring_h8279546866833948963t_unit @ R @ S2 @ H )
     => ( ( H @ ( zero_a_b @ R ) )
        = ( zero_l4142658623432671053t_unit @ S2 ) ) ) ).

% ring_hom_cring.hom_zero
thf(fact_831_ring__hom__cring_Ohom__zero,axiom,
    ! [R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,H: list_a > a] :
      ( ( ring_h1547129875642963619it_a_b @ R @ S2 @ H )
     => ( ( H @ ( zero_l4142658623432671053t_unit @ R ) )
        = ( zero_a_b @ S2 ) ) ) ).

% ring_hom_cring.hom_zero
thf(fact_832_ring__hom__cring_Ohom__zero,axiom,
    ! [R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,H: list_a > list_a] :
      ( ( ring_h8282015026914974507t_unit @ R @ S2 @ H )
     => ( ( H @ ( zero_l4142658623432671053t_unit @ R ) )
        = ( zero_l4142658623432671053t_unit @ S2 ) ) ) ).

% ring_hom_cring.hom_zero
thf(fact_833_ring__hom__cring_Ohom__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,H: a > list_list_a] :
      ( ( ring_h1996671968938926877t_unit @ R @ S2 @ H )
     => ( ( H @ ( zero_a_b @ R ) )
        = ( zero_l347298301471573063t_unit @ S2 ) ) ) ).

% ring_hom_cring.hom_zero
thf(fact_834_ring__hom__cring_Ohom__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b,S2: partia7496981018696276118t_unit,H: a > set_list_a] :
      ( ( ring_h7527734465757070659t_unit @ R @ S2 @ H )
     => ( ( H @ ( zero_a_b @ R ) )
        = ( zero_s2910681146719230829t_unit @ S2 ) ) ) ).

% ring_hom_cring.hom_zero
thf(fact_835_ring__hom__cring_Ohom__zero,axiom,
    ! [R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,H: list_list_a > a] :
      ( ( ring_h3216284694433613853it_a_b @ R @ S2 @ H )
     => ( ( H @ ( zero_l347298301471573063t_unit @ R ) )
        = ( zero_a_b @ S2 ) ) ) ).

% ring_hom_cring.hom_zero
thf(fact_836_ring__hom__cring_Ohom__zero,axiom,
    ! [R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,H: set_list_a > a] :
      ( ( ring_h1101743994381864643it_a_b @ R @ S2 @ H )
     => ( ( H @ ( zero_s2910681146719230829t_unit @ R ) )
        = ( zero_a_b @ S2 ) ) ) ).

% ring_hom_cring.hom_zero
thf(fact_837_ring__hom__cring_Ohom__zero,axiom,
    ! [R: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,H: list_a > list_list_a] :
      ( ( ring_h3424142382897944869t_unit @ R @ S2 @ H )
     => ( ( H @ ( zero_l4142658623432671053t_unit @ R ) )
        = ( zero_l347298301471573063t_unit @ S2 ) ) ) ).

% ring_hom_cring.hom_zero
thf(fact_838_ring__hom__cring_Ohom__zero,axiom,
    ! [R: partia2670972154091845814t_unit,S2: partia7496981018696276118t_unit,H: list_a > set_list_a] :
      ( ( ring_h5296475915237130059t_unit @ R @ S2 @ H )
     => ( ( H @ ( zero_l4142658623432671053t_unit @ R ) )
        = ( zero_s2910681146719230829t_unit @ S2 ) ) ) ).

% ring_hom_cring.hom_zero
thf(fact_839_ring__hom__cring_Ohomh,axiom,
    ! [R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,H: list_list_a > list_a] :
      ( ( ring_h453377649743177125t_unit @ R @ S2 @ H )
     => ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S2 ) ) ) ).

% ring_hom_cring.homh
thf(fact_840_ring__hom__cring_Ohomh,axiom,
    ! [R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,H: a > list_a] :
      ( ( ring_h8279546866833948963t_unit @ R @ S2 @ H )
     => ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) ) ) ).

% ring_hom_cring.homh
thf(fact_841_ring__hom__cring_Ohomh,axiom,
    ! [R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,H: set_list_a > a] :
      ( ( ring_h1101743994381864643it_a_b @ R @ S2 @ H )
     => ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) ) ) ).

% ring_hom_cring.homh
thf(fact_842_ring__hom__cring_Ohomh,axiom,
    ! [R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,H: list_a > a] :
      ( ( ring_h1547129875642963619it_a_b @ R @ S2 @ H )
     => ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) ) ) ).

% ring_hom_cring.homh
thf(fact_843_semiring_Oaxioms_I1_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( semiring_a_b @ R )
     => ( abelian_monoid_a_b @ R ) ) ).

% semiring.axioms(1)
thf(fact_844_semiring_Oaxioms_I1_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( abelia226231641709521465t_unit @ R ) ) ).

% semiring.axioms(1)
thf(fact_845_domain_Oone__not__zero,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( one_li8234411390022467901t_unit @ R )
       != ( zero_l347298301471573063t_unit @ R ) ) ) ).

% domain.one_not_zero
thf(fact_846_domain_Oone__not__zero,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( domain_a_b @ R )
     => ( ( one_a_ring_ext_a_b @ R )
       != ( zero_a_b @ R ) ) ) ).

% domain.one_not_zero
thf(fact_847_domain_Oone__not__zero,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( one_li8328186300101108157t_unit @ R )
       != ( zero_l4142658623432671053t_unit @ R ) ) ) ).

% domain.one_not_zero
thf(fact_848_domain_Oone__not__zero,axiom,
    ! [R: partia7496981018696276118t_unit] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( one_se1127990129394575805t_unit @ R )
       != ( zero_s2910681146719230829t_unit @ R ) ) ) ).

% domain.one_not_zero
thf(fact_849_domain_Ozero__not__one,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( zero_l347298301471573063t_unit @ R )
       != ( one_li8234411390022467901t_unit @ R ) ) ) ).

% domain.zero_not_one
thf(fact_850_domain_Ozero__not__one,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( domain_a_b @ R )
     => ( ( zero_a_b @ R )
       != ( one_a_ring_ext_a_b @ R ) ) ) ).

% domain.zero_not_one
thf(fact_851_domain_Ozero__not__one,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( zero_l4142658623432671053t_unit @ R )
       != ( one_li8328186300101108157t_unit @ R ) ) ) ).

% domain.zero_not_one
thf(fact_852_domain_Ozero__not__one,axiom,
    ! [R: partia7496981018696276118t_unit] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( zero_s2910681146719230829t_unit @ R )
       != ( one_se1127990129394575805t_unit @ R ) ) ) ).

% domain.zero_not_one
thf(fact_853_ring__hom__mult,axiom,
    ! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
            = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_mult
thf(fact_854_ring__hom__mult,axiom,
    ! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a,Y: a] :
      ( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
            = ( mult_l7073676228092353617t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_mult
thf(fact_855_ring__hom__mult,axiom,
    ! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
      ( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
            = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_mult
thf(fact_856_ring__hom__mult,axiom,
    ! [H: a > list_list_a,R: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a,Y: a] :
      ( ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ R @ S2 ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
            = ( mult_l4853965630390486993t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_mult
thf(fact_857_ring__hom__mult,axiom,
    ! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
            = ( mult_l7073676228092353617t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_mult
thf(fact_858_ring__hom__mult,axiom,
    ! [H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
      ( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( H @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
            = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_mult
thf(fact_859_ring__hom__mult,axiom,
    ! [H: set_list_a > a,R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,X: set_list_a,Y: set_list_a] :
      ( ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( H @ ( mult_s7802724872828879953t_unit @ R @ X @ Y ) )
            = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_mult
thf(fact_860_ring__hom__mult,axiom,
    ! [H: list_a > list_list_a,R: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a,Y: list_a] :
      ( ( member6714375691612171394list_a @ H @ ( ring_h8002040739877300486t_unit @ R @ S2 ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
            = ( mult_l4853965630390486993t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_mult
thf(fact_861_ring__hom__mult,axiom,
    ! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
      ( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S2 ) )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( H @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
            = ( mult_l7073676228092353617t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_mult
thf(fact_862_ring__hom__mult,axiom,
    ! [H: set_list_a > list_a,R: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit,X: set_list_a,Y: set_list_a] :
      ( ( member5910328476188217884list_a @ H @ ( ring_h8038483918290310060t_unit @ R @ S2 ) )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( H @ ( mult_s7802724872828879953t_unit @ R @ X @ Y ) )
            = ( mult_l7073676228092353617t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).

% ring_hom_mult
thf(fact_863_abelian__monoidE_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( abelian_monoid_a_b @ R )
     => ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% abelian_monoidE(2)
thf(fact_864_abelian__monoidE_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( abelia226231641709521465t_unit @ R )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% abelian_monoidE(2)
thf(fact_865_abelian__monoidE_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( abelia3641329199688042803t_unit @ R )
     => ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% abelian_monoidE(2)
thf(fact_866_abelian__monoidE_I2_J,axiom,
    ! [R: partia7496981018696276118t_unit] :
      ( ( abelia3322010900105369177t_unit @ R )
     => ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).

% abelian_monoidE(2)
thf(fact_867_abelian__monoid_Ozero__closed,axiom,
    ! [G2: partia2175431115845679010xt_a_b] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( member_a @ ( zero_a_b @ G2 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ).

% abelian_monoid.zero_closed
thf(fact_868_abelian__monoid_Ozero__closed,axiom,
    ! [G2: partia2670972154091845814t_unit] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ G2 ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) ).

% abelian_monoid.zero_closed
thf(fact_869_abelian__monoid_Ozero__closed,axiom,
    ! [G2: partia2956882679547061052t_unit] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( member_list_list_a @ ( zero_l347298301471573063t_unit @ G2 ) @ ( partia2464479390973590831t_unit @ G2 ) ) ) ).

% abelian_monoid.zero_closed
thf(fact_870_abelian__monoid_Ozero__closed,axiom,
    ! [G2: partia7496981018696276118t_unit] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ G2 ) @ ( partia141011252114345353t_unit @ G2 ) ) ) ).

% abelian_monoid.zero_closed
thf(fact_871_semiring_Osemiring__simprules_I2_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( semiring_a_b @ R )
     => ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% semiring.semiring_simprules(2)
thf(fact_872_semiring_Osemiring__simprules_I2_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% semiring.semiring_simprules(2)
thf(fact_873_semiring_Osemiring__simprules_I2_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% semiring.semiring_simprules(2)
thf(fact_874_semiring_Osemiring__simprules_I2_J,axiom,
    ! [R: partia7496981018696276118t_unit] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).

% semiring.semiring_simprules(2)
thf(fact_875_semiring_Osemiring__simprules_I8_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
              = ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(8)
thf(fact_876_semiring_Osemiring__simprules_I8_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z )
              = ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(8)
thf(fact_877_semiring_Osemiring__simprules_I8_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( mult_l4853965630390486993t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ Z )
              = ( mult_l4853965630390486993t_unit @ R @ X @ ( mult_l4853965630390486993t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(8)
thf(fact_878_semiring_Osemiring__simprules_I8_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R ) )
           => ( ( mult_s7802724872828879953t_unit @ R @ ( mult_s7802724872828879953t_unit @ R @ X @ Y ) @ Z )
              = ( mult_s7802724872828879953t_unit @ R @ X @ ( mult_s7802724872828879953t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).

% semiring.semiring_simprules(8)
thf(fact_879_semiring_Osemiring__simprules_I3_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).

% semiring.semiring_simprules(3)
thf(fact_880_semiring_Osemiring__simprules_I3_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).

% semiring.semiring_simprules(3)
thf(fact_881_semiring_Osemiring__simprules_I3_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).

% semiring.semiring_simprules(3)
thf(fact_882_semiring_Osemiring__simprules_I3_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
         => ( member_set_list_a @ ( mult_s7802724872828879953t_unit @ R @ X @ Y ) @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ).

% semiring.semiring_simprules(3)
thf(fact_883_semiring_Osemiring__simprules_I4_J,axiom,
    ! [R: partia2175431115845679010xt_a_b] :
      ( ( semiring_a_b @ R )
     => ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).

% semiring.semiring_simprules(4)
thf(fact_884_semiring_Osemiring__simprules_I4_J,axiom,
    ! [R: partia2670972154091845814t_unit] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).

% semiring.semiring_simprules(4)
thf(fact_885_semiring_Osemiring__simprules_I4_J,axiom,
    ! [R: partia2956882679547061052t_unit] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).

% semiring.semiring_simprules(4)
thf(fact_886_semiring_Osemiring__simprules_I4_J,axiom,
    ! [R: partia7496981018696276118t_unit] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).

% semiring.semiring_simprules(4)
thf(fact_887_ring__hom__cring_Ohom__closed,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ) ).

% ring_hom_cring.hom_closed
thf(fact_888_ring__hom__cring_Ohom__closed,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ) ).

% ring_hom_cring.hom_closed
thf(fact_889_ring__hom__cring_Ohom__closed,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ) ).

% ring_hom_cring.hom_closed
thf(fact_890_ring__hom__cring_Ohom__closed,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > list_list_a,R: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ R @ S2 ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ) ).

% ring_hom_cring.hom_closed
thf(fact_891_ring__hom__cring_Ohom__closed,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > set_list_a,R: partia2175431115845679010xt_a_b,S2: partia7496981018696276118t_unit,X: a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_set_list_a @ H @ ( ring_h6109298854714515236t_unit @ R @ S2 ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( member_set_list_a @ ( H @ X ) @ ( partia141011252114345353t_unit @ S2 ) ) ) ) ) ).

% ring_hom_cring.hom_closed
thf(fact_892_ring__hom__cring_Ohom__closed,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ) ).

% ring_hom_cring.hom_closed
thf(fact_893_ring__hom__cring_Ohom__closed,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
         => ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ) ).

% ring_hom_cring.hom_closed
thf(fact_894_ring__hom__cring_Ohom__closed,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: set_list_a > a,R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,X: set_list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) )
       => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
         => ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ) ).

% ring_hom_cring.hom_closed
thf(fact_895_ring__hom__cring_Ohom__closed,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > list_list_a,R: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member6714375691612171394list_a @ H @ ( ring_h8002040739877300486t_unit @ R @ S2 ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ) ).

% ring_hom_cring.hom_closed
thf(fact_896_ring__hom__cring_Ohom__closed,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > set_list_a,R: partia2670972154091845814t_unit,S2: partia7496981018696276118t_unit,X: list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member4263473470251683292list_a @ H @ ( ring_h6188449271506562988t_unit @ R @ S2 ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( member_set_list_a @ ( H @ X ) @ ( partia141011252114345353t_unit @ S2 ) ) ) ) ) ).

% ring_hom_cring.hom_closed
thf(fact_897_ring__hom__cring_Ohom__one,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S2 ) )
       => ( ( H @ ( one_li8234411390022467901t_unit @ R ) )
          = ( one_li8328186300101108157t_unit @ S2 ) ) ) ) ).

% ring_hom_cring.hom_one
thf(fact_898_ring__hom__cring_Ohom__one,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
       => ( ( H @ ( one_a_ring_ext_a_b @ R ) )
          = ( one_a_ring_ext_a_b @ S2 ) ) ) ) ).

% ring_hom_cring.hom_one
thf(fact_899_ring__hom__cring_Ohom__one,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
       => ( ( H @ ( one_a_ring_ext_a_b @ R ) )
          = ( one_li8328186300101108157t_unit @ S2 ) ) ) ) ).

% ring_hom_cring.hom_one
thf(fact_900_ring__hom__cring_Ohom__one,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > set_list_a,R: partia2175431115845679010xt_a_b,S2: partia7496981018696276118t_unit] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_set_list_a @ H @ ( ring_h6109298854714515236t_unit @ R @ S2 ) )
       => ( ( H @ ( one_a_ring_ext_a_b @ R ) )
          = ( one_se1127990129394575805t_unit @ S2 ) ) ) ) ).

% ring_hom_cring.hom_one
thf(fact_901_ring__hom__cring_Ohom__one,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
       => ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
          = ( one_a_ring_ext_a_b @ S2 ) ) ) ) ).

% ring_hom_cring.hom_one
thf(fact_902_ring__hom__cring_Ohom__one,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
       => ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
          = ( one_li8328186300101108157t_unit @ S2 ) ) ) ) ).

% ring_hom_cring.hom_one
thf(fact_903_ring__hom__cring_Ohom__one,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > set_list_a,R: partia2670972154091845814t_unit,S2: partia7496981018696276118t_unit] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member4263473470251683292list_a @ H @ ( ring_h6188449271506562988t_unit @ R @ S2 ) )
       => ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
          = ( one_se1127990129394575805t_unit @ S2 ) ) ) ) ).

% ring_hom_cring.hom_one
thf(fact_904_ring__hom__cring_Ohom__one,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: set_list_a > a,R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) )
       => ( ( H @ ( one_se1127990129394575805t_unit @ R ) )
          = ( one_a_ring_ext_a_b @ S2 ) ) ) ) ).

% ring_hom_cring.hom_one
thf(fact_905_ring__hom__cring_Ohom__one,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: set_list_a > list_a,R: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member5910328476188217884list_a @ H @ ( ring_h8038483918290310060t_unit @ R @ S2 ) )
       => ( ( H @ ( one_se1127990129394575805t_unit @ R ) )
          = ( one_li8328186300101108157t_unit @ S2 ) ) ) ) ).

% ring_hom_cring.hom_one
thf(fact_906_ring__hom__cring_Ohom__one,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: set_list_a > set_list_a,R: partia7496981018696276118t_unit,S2: partia7496981018696276118t_unit] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member5068272912271824380list_a @ H @ ( ring_h6076331213207892940t_unit @ R @ S2 ) )
       => ( ( H @ ( one_se1127990129394575805t_unit @ R ) )
          = ( one_se1127990129394575805t_unit @ S2 ) ) ) ) ).

% ring_hom_cring.hom_one
thf(fact_907_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [R: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( polyno1672195411705137432t_unit @ R @ P @ X ) )
          = ( polyno5142720416380192742t_unit @ R @ P @ X ) ) ) ) ).

% domain.alg_mult_gt_zero_iff_is_root
thf(fact_908_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [R: partia7496981018696276118t_unit,P: list_set_list_a,X: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( polyno1088517687229135038t_unit @ R @ P @ X ) )
          = ( polyno4320237611291262604t_unit @ R @ P @ X ) ) ) ) ).

% domain.alg_mult_gt_zero_iff_is_root
thf(fact_909_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [R: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( polyno4259638811958763678t_unit @ R @ P @ X ) )
          = ( polyno6951661231331188332t_unit @ R @ P @ X ) ) ) ) ).

% domain.alg_mult_gt_zero_iff_is_root
thf(fact_910_domain_Oalg__mult__gt__zero__iff__is__root,axiom,
    ! [R: partia2175431115845679010xt_a_b,P: list_a,X: a] :
      ( ( domain_a_b @ R )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ R @ P @ X ) )
          = ( polyno4133073214067823460ot_a_b @ R @ P @ X ) ) ) ) ).

% domain.alg_mult_gt_zero_iff_is_root
thf(fact_911_domain_Ointegral,axiom,
    ! [R: partia2175431115845679010xt_a_b,A: a,B3: a] :
      ( ( domain_a_b @ R )
     => ( ( ( mult_a_ring_ext_a_b @ R @ A @ B3 )
          = ( zero_a_b @ R ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( A
                = ( zero_a_b @ R ) )
              | ( B3
                = ( zero_a_b @ R ) ) ) ) ) ) ) ).

% domain.integral
thf(fact_912_domain_Ointegral,axiom,
    ! [R: partia2670972154091845814t_unit,A: list_a,B3: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B3 )
          = ( zero_l4142658623432671053t_unit @ R ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( A
                = ( zero_l4142658623432671053t_unit @ R ) )
              | ( B3
                = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).

% domain.integral
thf(fact_913_domain_Ointegral,axiom,
    ! [R: partia2956882679547061052t_unit,A: list_list_a,B3: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B3 )
          = ( zero_l347298301471573063t_unit @ R ) )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( A
                = ( zero_l347298301471573063t_unit @ R ) )
              | ( B3
                = ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).

% domain.integral
thf(fact_914_domain_Ointegral,axiom,
    ! [R: partia7496981018696276118t_unit,A: set_list_a,B3: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B3 )
          = ( zero_s2910681146719230829t_unit @ R ) )
       => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ B3 @ ( partia141011252114345353t_unit @ R ) )
           => ( ( A
                = ( zero_s2910681146719230829t_unit @ R ) )
              | ( B3
                = ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).

% domain.integral
thf(fact_915_domain_Om__lcancel,axiom,
    ! [R: partia2175431115845679010xt_a_b,A: a,B3: a,C3: a] :
      ( ( domain_a_b @ R )
     => ( ( A
         != ( zero_a_b @ R ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( ( mult_a_ring_ext_a_b @ R @ A @ B3 )
                  = ( mult_a_ring_ext_a_b @ R @ A @ C3 ) )
                = ( B3 = C3 ) ) ) ) ) ) ) ).

% domain.m_lcancel
thf(fact_916_domain_Om__lcancel,axiom,
    ! [R: partia2670972154091845814t_unit,A: list_a,B3: list_a,C3: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( A
         != ( zero_l4142658623432671053t_unit @ R ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B3 )
                  = ( mult_l7073676228092353617t_unit @ R @ A @ C3 ) )
                = ( B3 = C3 ) ) ) ) ) ) ) ).

% domain.m_lcancel
thf(fact_917_domain_Om__lcancel,axiom,
    ! [R: partia2956882679547061052t_unit,A: list_list_a,B3: list_list_a,C3: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( A
         != ( zero_l347298301471573063t_unit @ R ) )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( member_list_list_a @ C3 @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B3 )
                  = ( mult_l4853965630390486993t_unit @ R @ A @ C3 ) )
                = ( B3 = C3 ) ) ) ) ) ) ) ).

% domain.m_lcancel
thf(fact_918_domain_Om__lcancel,axiom,
    ! [R: partia7496981018696276118t_unit,A: set_list_a,B3: set_list_a,C3: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( A
         != ( zero_s2910681146719230829t_unit @ R ) )
       => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ B3 @ ( partia141011252114345353t_unit @ R ) )
           => ( ( member_set_list_a @ C3 @ ( partia141011252114345353t_unit @ R ) )
             => ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B3 )
                  = ( mult_s7802724872828879953t_unit @ R @ A @ C3 ) )
                = ( B3 = C3 ) ) ) ) ) ) ) ).

% domain.m_lcancel
thf(fact_919_domain_Om__rcancel,axiom,
    ! [R: partia2175431115845679010xt_a_b,A: a,B3: a,C3: a] :
      ( ( domain_a_b @ R )
     => ( ( A
         != ( zero_a_b @ R ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ R ) )
             => ( ( ( mult_a_ring_ext_a_b @ R @ B3 @ A )
                  = ( mult_a_ring_ext_a_b @ R @ C3 @ A ) )
                = ( B3 = C3 ) ) ) ) ) ) ) ).

% domain.m_rcancel
thf(fact_920_domain_Om__rcancel,axiom,
    ! [R: partia2670972154091845814t_unit,A: list_a,B3: list_a,C3: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( A
         != ( zero_l4142658623432671053t_unit @ R ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ R ) )
             => ( ( ( mult_l7073676228092353617t_unit @ R @ B3 @ A )
                  = ( mult_l7073676228092353617t_unit @ R @ C3 @ A ) )
                = ( B3 = C3 ) ) ) ) ) ) ) ).

% domain.m_rcancel
thf(fact_921_domain_Om__rcancel,axiom,
    ! [R: partia2956882679547061052t_unit,A: list_list_a,B3: list_list_a,C3: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( A
         != ( zero_l347298301471573063t_unit @ R ) )
       => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( member_list_list_a @ C3 @ ( partia2464479390973590831t_unit @ R ) )
             => ( ( ( mult_l4853965630390486993t_unit @ R @ B3 @ A )
                  = ( mult_l4853965630390486993t_unit @ R @ C3 @ A ) )
                = ( B3 = C3 ) ) ) ) ) ) ) ).

% domain.m_rcancel
thf(fact_922_domain_Om__rcancel,axiom,
    ! [R: partia7496981018696276118t_unit,A: set_list_a,B3: set_list_a,C3: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( A
         != ( zero_s2910681146719230829t_unit @ R ) )
       => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ B3 @ ( partia141011252114345353t_unit @ R ) )
           => ( ( member_set_list_a @ C3 @ ( partia141011252114345353t_unit @ R ) )
             => ( ( ( mult_s7802724872828879953t_unit @ R @ B3 @ A )
                  = ( mult_s7802724872828879953t_unit @ R @ C3 @ A ) )
                = ( B3 = C3 ) ) ) ) ) ) ) ).

% domain.m_rcancel
thf(fact_923_domain_Ointegral__iff,axiom,
    ! [R: partia2175431115845679010xt_a_b,A: a,B3: a] :
      ( ( domain_a_b @ R )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( ( mult_a_ring_ext_a_b @ R @ A @ B3 )
              = ( zero_a_b @ R ) )
            = ( ( A
                = ( zero_a_b @ R ) )
              | ( B3
                = ( zero_a_b @ R ) ) ) ) ) ) ) ).

% domain.integral_iff
thf(fact_924_domain_Ointegral__iff,axiom,
    ! [R: partia2670972154091845814t_unit,A: list_a,B3: list_a] :
      ( ( domain6553523120543210313t_unit @ R )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B3 )
              = ( zero_l4142658623432671053t_unit @ R ) )
            = ( ( A
                = ( zero_l4142658623432671053t_unit @ R ) )
              | ( B3
                = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).

% domain.integral_iff
thf(fact_925_domain_Ointegral__iff,axiom,
    ! [R: partia2956882679547061052t_unit,A: list_list_a,B3: list_list_a] :
      ( ( domain7810152921033798211t_unit @ R )
     => ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B3 )
              = ( zero_l347298301471573063t_unit @ R ) )
            = ( ( A
                = ( zero_l347298301471573063t_unit @ R ) )
              | ( B3
                = ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).

% domain.integral_iff
thf(fact_926_domain_Ointegral__iff,axiom,
    ! [R: partia7496981018696276118t_unit,A: set_list_a,B3: set_list_a] :
      ( ( domain1617769409708967785t_unit @ R )
     => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
       => ( ( member_set_list_a @ B3 @ ( partia141011252114345353t_unit @ R ) )
         => ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B3 )
              = ( zero_s2910681146719230829t_unit @ R ) )
            = ( ( A
                = ( zero_s2910681146719230829t_unit @ R ) )
              | ( B3
                = ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).

% domain.integral_iff
thf(fact_927_semiring_Ol__null,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
          = ( zero_a_b @ R ) ) ) ) ).

% semiring.l_null
thf(fact_928_semiring_Ol__null,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% semiring.l_null
thf(fact_929_semiring_Ol__null,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% semiring.l_null
thf(fact_930_semiring_Ol__null,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( mult_s7802724872828879953t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
          = ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).

% semiring.l_null
thf(fact_931_semiring_Or__null,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
          = ( zero_a_b @ R ) ) ) ) ).

% semiring.r_null
thf(fact_932_semiring_Or__null,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
          = ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).

% semiring.r_null
thf(fact_933_semiring_Or__null,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
          = ( zero_l347298301471573063t_unit @ R ) ) ) ) ).

% semiring.r_null
thf(fact_934_semiring_Or__null,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( mult_s7802724872828879953t_unit @ R @ X @ ( zero_s2910681146719230829t_unit @ R ) )
          = ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).

% semiring.r_null
thf(fact_935_abelian__monoid_Ofinsum__empty,axiom,
    ! [G2: partia2956882679547061052t_unit,F: a > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( finsum463596448938265310unit_a @ G2 @ F @ bot_bot_set_a )
        = ( zero_l347298301471573063t_unit @ G2 ) ) ) ).

% abelian_monoid.finsum_empty
thf(fact_936_abelian__monoid_Ofinsum__empty,axiom,
    ! [G2: partia7496981018696276118t_unit,F: a > set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( finsum8105788327582199096unit_a @ G2 @ F @ bot_bot_set_a )
        = ( zero_s2910681146719230829t_unit @ G2 ) ) ) ).

% abelian_monoid.finsum_empty
thf(fact_937_abelian__monoid_Ofinsum__empty,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finsum_a_b_a @ G2 @ F @ bot_bot_set_a )
        = ( zero_a_b @ G2 ) ) ) ).

% abelian_monoid.finsum_empty
thf(fact_938_abelian__monoid_Ofinsum__empty,axiom,
    ! [G2: partia2670972154091845814t_unit,F: a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( finsum7322697649718157656unit_a @ G2 @ F @ bot_bot_set_a )
        = ( zero_l4142658623432671053t_unit @ G2 ) ) ) ).

% abelian_monoid.finsum_empty
thf(fact_939_abelian__monoid_Ofinsum__empty,axiom,
    ! [G2: partia2956882679547061052t_unit,F: list_a > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( finsum159916373282764644list_a @ G2 @ F @ bot_bot_set_list_a )
        = ( zero_l347298301471573063t_unit @ G2 ) ) ) ).

% abelian_monoid.finsum_empty
thf(fact_940_abelian__monoid_Ofinsum__empty,axiom,
    ! [G2: partia7496981018696276118t_unit,F: list_a > set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( finsum3353672621075604670list_a @ G2 @ F @ bot_bot_set_list_a )
        = ( zero_s2910681146719230829t_unit @ G2 ) ) ) ).

% abelian_monoid.finsum_empty
thf(fact_941_abelian__monoid_Ofinsum__empty,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: list_a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( finsum_a_b_list_a @ G2 @ F @ bot_bot_set_list_a )
        = ( zero_a_b @ G2 ) ) ) ).

% abelian_monoid.finsum_empty
thf(fact_942_abelian__monoid_Ofinsum__empty,axiom,
    ! [G2: partia2670972154091845814t_unit,F: list_a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( finsum8721804980556663006list_a @ G2 @ F @ bot_bot_set_list_a )
        = ( zero_l4142658623432671053t_unit @ G2 ) ) ) ).

% abelian_monoid.finsum_empty
thf(fact_943_abelian__monoid_Ofinsum__infinite,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_a,F: a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ~ ( finite_finite_a @ A2 )
       => ( ( finsum_a_b_a @ G2 @ F @ A2 )
          = ( zero_a_b @ G2 ) ) ) ) ).

% abelian_monoid.finsum_infinite
thf(fact_944_abelian__monoid_Ofinsum__infinite,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_nat,F: nat > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ~ ( finite_finite_nat @ A2 )
       => ( ( finsum_a_b_nat @ G2 @ F @ A2 )
          = ( zero_a_b @ G2 ) ) ) ) ).

% abelian_monoid.finsum_infinite
thf(fact_945_abelian__monoid_Ofinsum__infinite,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_list_a,F: list_a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ~ ( finite_finite_list_a @ A2 )
       => ( ( finsum_a_b_list_a @ G2 @ F @ A2 )
          = ( zero_a_b @ G2 ) ) ) ) ).

% abelian_monoid.finsum_infinite
thf(fact_946_abelian__monoid_Ofinsum__infinite,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_a,F: a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ~ ( finite_finite_a @ A2 )
       => ( ( finsum7322697649718157656unit_a @ G2 @ F @ A2 )
          = ( zero_l4142658623432671053t_unit @ G2 ) ) ) ) ).

% abelian_monoid.finsum_infinite
thf(fact_947_abelian__monoid_Ofinsum__infinite,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_nat,F: nat > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ~ ( finite_finite_nat @ A2 )
       => ( ( finsum3497512462216549110it_nat @ G2 @ F @ A2 )
          = ( zero_l4142658623432671053t_unit @ G2 ) ) ) ) ).

% abelian_monoid.finsum_infinite
thf(fact_948_abelian__monoid_Ofinsum__infinite,axiom,
    ! [G2: partia2956882679547061052t_unit,A2: set_a,F: a > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ~ ( finite_finite_a @ A2 )
       => ( ( finsum463596448938265310unit_a @ G2 @ F @ A2 )
          = ( zero_l347298301471573063t_unit @ G2 ) ) ) ) ).

% abelian_monoid.finsum_infinite
thf(fact_949_abelian__monoid_Ofinsum__infinite,axiom,
    ! [G2: partia7496981018696276118t_unit,A2: set_a,F: a > set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ~ ( finite_finite_a @ A2 )
       => ( ( finsum8105788327582199096unit_a @ G2 @ F @ A2 )
          = ( zero_s2910681146719230829t_unit @ G2 ) ) ) ) ).

% abelian_monoid.finsum_infinite
thf(fact_950_abelian__monoid_Ofinsum__infinite,axiom,
    ! [G2: partia2956882679547061052t_unit,A2: set_nat,F: nat > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ~ ( finite_finite_nat @ A2 )
       => ( ( finsum3990971441743328240it_nat @ G2 @ F @ A2 )
          = ( zero_l347298301471573063t_unit @ G2 ) ) ) ) ).

% abelian_monoid.finsum_infinite
thf(fact_951_abelian__monoid_Ofinsum__infinite,axiom,
    ! [G2: partia7496981018696276118t_unit,A2: set_nat,F: nat > set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ~ ( finite_finite_nat @ A2 )
       => ( ( finsum1644613019690897174it_nat @ G2 @ F @ A2 )
          = ( zero_s2910681146719230829t_unit @ G2 ) ) ) ) ).

% abelian_monoid.finsum_infinite
thf(fact_952_abelian__monoid_Ofinsum__infinite,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_list_a,F: list_a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ~ ( finite_finite_list_a @ A2 )
       => ( ( finsum8721804980556663006list_a @ G2 @ F @ A2 )
          = ( zero_l4142658623432671053t_unit @ G2 ) ) ) ) ).

% abelian_monoid.finsum_infinite
thf(fact_953_semiring_Osemiring__simprules_I9_J,axiom,
    ! [R: partia2175431115845679010xt_a_b,X: a] :
      ( ( semiring_a_b @ R )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
       => ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(9)
thf(fact_954_semiring_Osemiring__simprules_I9_J,axiom,
    ! [R: partia2670972154091845814t_unit,X: list_a] :
      ( ( semiri2871908745932252451t_unit @ R )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
       => ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(9)
thf(fact_955_semiring_Osemiring__simprules_I9_J,axiom,
    ! [R: partia2956882679547061052t_unit,X: list_list_a] :
      ( ( semiri2265994252334843677t_unit @ R )
     => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
       => ( ( mult_l4853965630390486993t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(9)
thf(fact_956_semiring_Osemiring__simprules_I9_J,axiom,
    ! [R: partia7496981018696276118t_unit,X: set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
       => ( ( mult_s7802724872828879953t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) @ X )
          = X ) ) ) ).

% semiring.semiring_simprules(9)
thf(fact_957_ring__hom__cring_Ohom__mult,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a,Y: a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
              = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_mult
thf(fact_958_ring__hom__cring_Ohom__mult,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a,Y: a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
              = ( mult_l7073676228092353617t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_mult
thf(fact_959_ring__hom__cring_Ohom__mult,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
              = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_mult
thf(fact_960_ring__hom__cring_Ohom__mult,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: a > list_list_a,R: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a,Y: a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ R @ S2 ) )
       => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
         => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
           => ( ( H @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
              = ( mult_l4853965630390486993t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_mult
thf(fact_961_ring__hom__cring_Ohom__mult,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
              = ( mult_l7073676228092353617t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_mult
thf(fact_962_ring__hom__cring_Ohom__mult,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( H @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
              = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_mult
thf(fact_963_ring__hom__cring_Ohom__mult,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: set_list_a > a,R: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,X: set_list_a,Y: set_list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member_set_list_a_a @ H @ ( ring_h8906680420194085028it_a_b @ R @ S2 ) )
       => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
           => ( ( H @ ( mult_s7802724872828879953t_unit @ R @ X @ Y ) )
              = ( mult_a_ring_ext_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_mult
thf(fact_964_ring__hom__cring_Ohom__mult,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_a > list_list_a,R: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a,Y: list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member6714375691612171394list_a @ H @ ( ring_h8002040739877300486t_unit @ R @ S2 ) )
       => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
         => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
           => ( ( H @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
              = ( mult_l4853965630390486993t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_mult
thf(fact_965_ring__hom__cring_Ohom__mult,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S2 ) )
       => ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
         => ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
           => ( ( H @ ( mult_l4853965630390486993t_unit @ R @ X @ Y ) )
              = ( mult_l7073676228092353617t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_mult
thf(fact_966_ring__hom__cring_Ohom__mult,axiom,
    ! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H: set_list_a > list_a,R: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit,X: set_list_a,Y: set_list_a] :
      ( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
     => ( ( member5910328476188217884list_a @ H @ ( ring_h8038483918290310060t_unit @ R @ S2 ) )
       => ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
           => ( ( H @ ( mult_s7802724872828879953t_unit @ R @ X @ Y ) )
              = ( mult_l7073676228092353617t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ) ).

% ring_hom_cring.hom_mult
thf(fact_967_abelian__monoid_Ofinsum__cong_H,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_nat,B: set_nat,G: nat > a,F: nat > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( A2 = B )
       => ( ( member_nat_a @ G
            @ ( pi_nat_a @ B
              @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ G2 ) ) )
         => ( ! [I: nat] :
                ( ( member_nat @ I @ B )
               => ( ( F @ I )
                  = ( G @ I ) ) )
           => ( ( finsum_a_b_nat @ G2 @ F @ A2 )
              = ( finsum_a_b_nat @ G2 @ G @ B ) ) ) ) ) ) ).

% abelian_monoid.finsum_cong'
thf(fact_968_abelian__monoid_Ofinsum__cong_H,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_a,B: set_a,G: a > a,F: a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( A2 = B )
       => ( ( member_a_a @ G
            @ ( pi_a_a @ B
              @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
         => ( ! [I: a] :
                ( ( member_a @ I @ B )
               => ( ( F @ I )
                  = ( G @ I ) ) )
           => ( ( finsum_a_b_a @ G2 @ F @ A2 )
              = ( finsum_a_b_a @ G2 @ G @ B ) ) ) ) ) ) ).

% abelian_monoid.finsum_cong'
thf(fact_969_abelian__monoid_Ofinsum__cong_H,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_nat,B: set_nat,G: nat > list_a,F: nat > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( A2 = B )
       => ( ( member_nat_list_a @ G
            @ ( pi_nat_list_a @ B
              @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ G2 ) ) )
         => ( ! [I: nat] :
                ( ( member_nat @ I @ B )
               => ( ( F @ I )
                  = ( G @ I ) ) )
           => ( ( finsum3497512462216549110it_nat @ G2 @ F @ A2 )
              = ( finsum3497512462216549110it_nat @ G2 @ G @ B ) ) ) ) ) ) ).

% abelian_monoid.finsum_cong'
thf(fact_970_abelian__monoid_Ofinsum__cong_H,axiom,
    ! [G2: partia2670972154091845814t_unit,A2: set_a,B: set_a,G: a > list_a,F: a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( A2 = B )
       => ( ( member_a_list_a @ G
            @ ( pi_a_list_a @ B
              @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
         => ( ! [I: a] :
                ( ( member_a @ I @ B )
               => ( ( F @ I )
                  = ( G @ I ) ) )
           => ( ( finsum7322697649718157656unit_a @ G2 @ F @ A2 )
              = ( finsum7322697649718157656unit_a @ G2 @ G @ B ) ) ) ) ) ) ).

% abelian_monoid.finsum_cong'
thf(fact_971_abelian__monoid_Ofinsum__cong_H,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_a_a,B: set_a_a,G: ( a > a ) > a,F: ( a > a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( A2 = B )
       => ( ( member_a_a_a @ G
            @ ( pi_a_a_a @ B
              @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
         => ( ! [I: a > a] :
                ( ( member_a_a @ I @ B )
               => ( ( F @ I )
                  = ( G @ I ) ) )
           => ( ( finsum_a_b_a_a @ G2 @ F @ A2 )
              = ( finsum_a_b_a_a @ G2 @ G @ B ) ) ) ) ) ) ).

% abelian_monoid.finsum_cong'
thf(fact_972_abelian__monoid_Ofinsum__cong_H,axiom,
    ! [G2: partia2175431115845679010xt_a_b,A2: set_nat_a,B: set_nat_a,G: ( nat > a ) > a,F: ( nat > a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( A2 = B )
       => ( ( member_nat_a_a @ G
            @ ( pi_nat_a_a @ B
              @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
         => ( ! [I: nat > a] :
                ( ( member_nat_a @ I @ B )
               => ( ( F @ I )
                  = ( G @ I ) ) )
           => ( ( finsum_a_b_nat_a @ G2 @ F @ A2 )
              = ( finsum_a_b_nat_a @ G2 @ G @ B ) ) ) ) ) ) ).

% abelian_monoid.finsum_cong'
thf(fact_973_abelian__monoid_Ofinsum__cong_H,axiom,
    ! [G2: partia2956882679547061052t_unit,A2: set_a,B: set_a,G: a > list_list_a,F: a > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( A2 = B )
       => ( ( member_a_list_list_a @ G
            @ ( pi_a_list_list_a @ B
              @ ^ [Uu: a] : ( partia2464479390973590831t_unit @ G2 ) ) )
         => ( ! [I: a] :
                ( ( member_a @ I @ B )
               => ( ( F @ I )
                  = ( G @ I ) ) )
           => ( ( finsum463596448938265310unit_a @ G2 @ F @ A2 )
              = ( finsum463596448938265310unit_a @ G2 @ G @ B ) ) ) ) ) ) ).

% abelian_monoid.finsum_cong'
thf(fact_974_abelian__monoid_Ofinsum__cong_H,axiom,
    ! [G2: partia2956882679547061052t_unit,A2: set_nat,B: set_nat,G: nat > list_list_a,F: nat > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( A2 = B )
       => ( ( member8650753269014980122list_a @ G
            @ ( pi_nat_list_list_a @ B
              @ ^ [Uu: nat] : ( partia2464479390973590831t_unit @ G2 ) ) )
         => ( ! [I: nat] :
                ( ( member_nat @ I @ B )
               => ( ( F @ I )
                  = ( G @ I ) ) )
           => ( ( finsum3990971441743328240it_nat @ G2 @ F @ A2 )
              = ( finsum3990971441743328240it_nat @ G2 @ G @ B ) ) ) ) ) ) ).

% abelian_monoid.finsum_cong'
thf(fact_975_abelian__monoid_Ofinsum__cong_H,axiom,
    ! [G2: partia7496981018696276118t_unit,A2: set_a,B: set_a,G: a > set_list_a,F: a > set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( A2 = B )
       => ( ( member_a_set_list_a @ G
            @ ( pi_a_set_list_a @ B
              @ ^ [Uu: a] : ( partia141011252114345353t_unit @ G2 ) ) )
         => ( ! [I: a] :
                ( ( member_a @ I @ B )
               => ( ( F @ I )
                  = ( G @ I ) ) )
           => ( ( finsum8105788327582199096unit_a @ G2 @ F @ A2 )
              = ( finsum8105788327582199096unit_a @ G2 @ G @ B ) ) ) ) ) ) ).

% abelian_monoid.finsum_cong'
thf(fact_976_abelian__monoid_Ofinsum__cong_H,axiom,
    ! [G2: partia7496981018696276118t_unit,A2: set_nat,B: set_nat,G: nat > set_list_a,F: nat > set_list_a] :
      ( ( abelia3322010900105369177t_unit @ G2 )
     => ( ( A2 = B )
       => ( ( member491565700723299188list_a @ G
            @ ( pi_nat_set_list_a @ B
              @ ^ [Uu: nat] : ( partia141011252114345353t_unit @ G2 ) ) )
         => ( ! [I: nat] :
                ( ( member_nat @ I @ B )
               => ( ( F @ I )
                  = ( G @ I ) ) )
           => ( ( finsum1644613019690897174it_nat @ G2 @ F @ A2 )
              = ( finsum1644613019690897174it_nat @ G2 @ G @ B ) ) ) ) ) ) ).

% abelian_monoid.finsum_cong'
thf(fact_977_abelian__monoid_Ofinsum__closed,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: nat > a,A2: set_nat] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_nat_a @ F
          @ ( pi_nat_a @ A2
            @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( member_a @ ( finsum_a_b_nat @ G2 @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ).

% abelian_monoid.finsum_closed
thf(fact_978_abelian__monoid_Ofinsum__closed,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: a > a,A2: set_a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a_a @ F
          @ ( pi_a_a @ A2
            @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( member_a @ ( finsum_a_b_a @ G2 @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ).

% abelian_monoid.finsum_closed
thf(fact_979_abelian__monoid_Ofinsum__closed,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: ( a > a ) > a,A2: set_a_a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a_a_a @ F
          @ ( pi_a_a_a @ A2
            @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( member_a @ ( finsum_a_b_a_a @ G2 @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ).

% abelian_monoid.finsum_closed
thf(fact_980_abelian__monoid_Ofinsum__closed,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: ( a > list_a ) > a,A2: set_a_list_a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a_list_a_a @ F
          @ ( pi_a_list_a_a @ A2
            @ ^ [Uu: a > list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( member_a @ ( finsum_a_b_a_list_a @ G2 @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ).

% abelian_monoid.finsum_closed
thf(fact_981_abelian__monoid_Ofinsum__closed,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: ( nat > a ) > a,A2: set_nat_a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_nat_a_a @ F
          @ ( pi_nat_a_a @ A2
            @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( member_a @ ( finsum_a_b_nat_a @ G2 @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ).

% abelian_monoid.finsum_closed
thf(fact_982_abelian__monoid_Ofinsum__closed,axiom,
    ! [G2: partia2175431115845679010xt_a_b,F: ( nat > list_a ) > a,A2: set_nat_list_a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_nat_list_a_a @ F
          @ ( pi_nat_list_a_a @ A2
            @ ^ [Uu: nat > list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
       => ( member_a @ ( finsum1341700292807219277list_a @ G2 @ F @ A2 ) @ ( partia707051561876973205xt_a_b @ G2 ) ) ) ) ).

% abelian_monoid.finsum_closed
thf(fact_983_abelian__monoid_Ofinsum__closed,axiom,
    ! [G2: partia2670972154091845814t_unit,F: nat > list_a,A2: set_nat] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_nat_list_a @ F
          @ ( pi_nat_list_a @ A2
            @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ G2 ) ) )
       => ( member_list_a @ ( finsum3497512462216549110it_nat @ G2 @ F @ A2 ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) ) ).

% abelian_monoid.finsum_closed
thf(fact_984_abelian__monoid_Ofinsum__closed,axiom,
    ! [G2: partia2670972154091845814t_unit,F: a > list_a,A2: set_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_a_list_a @ F
          @ ( pi_a_list_a @ A2
            @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
       => ( member_list_a @ ( finsum7322697649718157656unit_a @ G2 @ F @ A2 ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) ) ).

% abelian_monoid.finsum_closed
thf(fact_985_ring__hom__cring_Ohom__finsum,axiom,
    ! [R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,H: list_a > a,F: nat > list_a,A2: set_nat] :
      ( ( ring_h1547129875642963619it_a_b @ R @ S2 @ H )
     => ( ( member_nat_list_a @ F
          @ ( pi_nat_list_a @ A2
            @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ R ) ) )
       => ( ( H @ ( finsum3497512462216549110it_nat @ R @ F @ A2 ) )
          = ( finsum_a_b_nat @ S2 @ ( comp_list_a_a_nat @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finsum
thf(fact_986_ring__hom__cring_Ohom__finsum,axiom,
    ! [R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,H: list_a > a,F: a > list_a,A2: set_a] :
      ( ( ring_h1547129875642963619it_a_b @ R @ S2 @ H )
     => ( ( member_a_list_a @ F
          @ ( pi_a_list_a @ A2
            @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ R ) ) )
       => ( ( H @ ( finsum7322697649718157656unit_a @ R @ F @ A2 ) )
          = ( finsum_a_b_a @ S2 @ ( comp_list_a_a_a @ H @ F ) @ A2 ) ) ) ) ).

% ring_hom_cring.hom_finsum
thf(fact_987_abelian__monoid_Ofinsum__singleton,axiom,
    ! [G2: partia2175431115845679010xt_a_b,I3: a,A2: set_a,F: a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a @ I3 @ A2 )
       => ( ( finite_finite_a @ A2 )
         => ( ( member_a_a @ F
              @ ( pi_a_a @ A2
                @ ^ [Uu: a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
           => ( ( finsum_a_b_a @ G2
                @ ^ [J: a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_a_b @ G2 ) )
                @ A2 )
              = ( F @ I3 ) ) ) ) ) ) ).

% abelian_monoid.finsum_singleton
thf(fact_988_abelian__monoid_Ofinsum__singleton,axiom,
    ! [G2: partia2175431115845679010xt_a_b,I3: nat,A2: set_nat,F: nat > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_nat @ I3 @ A2 )
       => ( ( finite_finite_nat @ A2 )
         => ( ( member_nat_a @ F
              @ ( pi_nat_a @ A2
                @ ^ [Uu: nat] : ( partia707051561876973205xt_a_b @ G2 ) ) )
           => ( ( finsum_a_b_nat @ G2
                @ ^ [J: nat] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_a_b @ G2 ) )
                @ A2 )
              = ( F @ I3 ) ) ) ) ) ) ).

% abelian_monoid.finsum_singleton
thf(fact_989_abelian__monoid_Ofinsum__singleton,axiom,
    ! [G2: partia2175431115845679010xt_a_b,I3: list_a,A2: set_list_a,F: list_a > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_list_a @ I3 @ A2 )
       => ( ( finite_finite_list_a @ A2 )
         => ( ( member_list_a_a @ F
              @ ( pi_list_a_a @ A2
                @ ^ [Uu: list_a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
           => ( ( finsum_a_b_list_a @ G2
                @ ^ [J: list_a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_a_b @ G2 ) )
                @ A2 )
              = ( F @ I3 ) ) ) ) ) ) ).

% abelian_monoid.finsum_singleton
thf(fact_990_abelian__monoid_Ofinsum__singleton,axiom,
    ! [G2: partia2670972154091845814t_unit,I3: a,A2: set_a,F: a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_a @ I3 @ A2 )
       => ( ( finite_finite_a @ A2 )
         => ( ( member_a_list_a @ F
              @ ( pi_a_list_a @ A2
                @ ^ [Uu: a] : ( partia5361259788508890537t_unit @ G2 ) ) )
           => ( ( finsum7322697649718157656unit_a @ G2
                @ ^ [J: a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ G2 ) )
                @ A2 )
              = ( F @ I3 ) ) ) ) ) ) ).

% abelian_monoid.finsum_singleton
thf(fact_991_abelian__monoid_Ofinsum__singleton,axiom,
    ! [G2: partia2670972154091845814t_unit,I3: nat,A2: set_nat,F: nat > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_nat @ I3 @ A2 )
       => ( ( finite_finite_nat @ A2 )
         => ( ( member_nat_list_a @ F
              @ ( pi_nat_list_a @ A2
                @ ^ [Uu: nat] : ( partia5361259788508890537t_unit @ G2 ) ) )
           => ( ( finsum3497512462216549110it_nat @ G2
                @ ^ [J: nat] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ G2 ) )
                @ A2 )
              = ( F @ I3 ) ) ) ) ) ) ).

% abelian_monoid.finsum_singleton
thf(fact_992_abelian__monoid_Ofinsum__singleton,axiom,
    ! [G2: partia2175431115845679010xt_a_b,I3: nat > a,A2: set_nat_a,F: ( nat > a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_nat_a @ I3 @ A2 )
       => ( ( finite_finite_nat_a @ A2 )
         => ( ( member_nat_a_a @ F
              @ ( pi_nat_a_a @ A2
                @ ^ [Uu: nat > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
           => ( ( finsum_a_b_nat_a @ G2
                @ ^ [J: nat > a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_a_b @ G2 ) )
                @ A2 )
              = ( F @ I3 ) ) ) ) ) ) ).

% abelian_monoid.finsum_singleton
thf(fact_993_abelian__monoid_Ofinsum__singleton,axiom,
    ! [G2: partia2175431115845679010xt_a_b,I3: a > a,A2: set_a_a,F: ( a > a ) > a] :
      ( ( abelian_monoid_a_b @ G2 )
     => ( ( member_a_a @ I3 @ A2 )
       => ( ( finite_finite_a_a @ A2 )
         => ( ( member_a_a_a @ F
              @ ( pi_a_a_a @ A2
                @ ^ [Uu: a > a] : ( partia707051561876973205xt_a_b @ G2 ) ) )
           => ( ( finsum_a_b_a_a @ G2
                @ ^ [J: a > a] : ( if_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_a_b @ G2 ) )
                @ A2 )
              = ( F @ I3 ) ) ) ) ) ) ).

% abelian_monoid.finsum_singleton
thf(fact_994_abelian__monoid_Ofinsum__singleton,axiom,
    ! [G2: partia2670972154091845814t_unit,I3: list_a,A2: set_list_a,F: list_a > list_a] :
      ( ( abelia226231641709521465t_unit @ G2 )
     => ( ( member_list_a @ I3 @ A2 )
       => ( ( finite_finite_list_a @ A2 )
         => ( ( member_list_a_list_a @ F
              @ ( pi_list_a_list_a @ A2
                @ ^ [Uu: list_a] : ( partia5361259788508890537t_unit @ G2 ) ) )
           => ( ( finsum8721804980556663006list_a @ G2
                @ ^ [J: list_a] : ( if_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l4142658623432671053t_unit @ G2 ) )
                @ A2 )
              = ( F @ I3 ) ) ) ) ) ) ).

% abelian_monoid.finsum_singleton
thf(fact_995_abelian__monoid_Ofinsum__singleton,axiom,
    ! [G2: partia2956882679547061052t_unit,I3: a,A2: set_a,F: a > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( member_a @ I3 @ A2 )
       => ( ( finite_finite_a @ A2 )
         => ( ( member_a_list_list_a @ F
              @ ( pi_a_list_list_a @ A2
                @ ^ [Uu: a] : ( partia2464479390973590831t_unit @ G2 ) ) )
           => ( ( finsum463596448938265310unit_a @ G2
                @ ^ [J: a] : ( if_list_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l347298301471573063t_unit @ G2 ) )
                @ A2 )
              = ( F @ I3 ) ) ) ) ) ) ).

% abelian_monoid.finsum_singleton
thf(fact_996_abelian__monoid_Ofinsum__singleton,axiom,
    ! [G2: partia2956882679547061052t_unit,I3: nat,A2: set_nat,F: nat > list_list_a] :
      ( ( abelia3641329199688042803t_unit @ G2 )
     => ( ( member_nat @ I3 @ A2 )
       => ( ( finite_finite_nat @ A2 )
         => ( ( member8650753269014980122list_a @ F
              @ ( pi_nat_list_list_a @ A2
                @ ^ [Uu: nat] : ( partia2464479390973590831t_unit @ G2 ) ) )
           => ( ( finsum3990971441743328240it_nat @ G2
                @ ^ [J: nat] : ( if_list_list_a @ ( I3 = J ) @ ( F @ J ) @ ( zero_l347298301471573063t_unit @ G2 ) )
                @ A2 )
              = ( F @ I3 ) ) ) ) ) ) ).

% abelian_monoid.finsum_singleton
thf(fact_997_semiring_Ofinsum__ldistr,axiom,
    ! [R: partia7496981018696276118t_unit,A2: set_list_a,A: set_list_a,F: list_a > set_list_a] :
      ( ( semiri4000464634269493571t_unit @ R )
     => ( ( finite_finite_list_a @ A2 )
       => ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
         => ( ( member4263473470251683292list_a @ F
              @ ( pi_list_a_set_list_a @ A2
                @ ^ [Uu: list_a] : ( partia141011252114345353t_unit @ R ) ) )
           => ( ( mult_s7802724872828879953t_unit @ R @ ( finsum3353672621075604670list_a @ R @ F @ A2 ) @ A )
              = ( finsum3353672621075604670list_a @ R
                @ ^ [I2: list_a] : ( mult_s7802724872828879953t_unit @ R @ ( F @ I2 ) @ A )
                @ A2 ) ) ) ) ) ) ).

% semiring.finsum_ldistr
thf(fact_998_x_Oexp__base__closed,axiom,
    ! [X: list_a,N3: nat] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N3 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.exp_base_closed
thf(fact_999_x_Oee__trans,axiom,
    ! [As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
      ( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
     => ( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ Cs )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Cs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
             => ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Cs ) ) ) ) ) ) ).

% x.ee_trans
thf(fact_1000_x_Oee__sym,axiom,
    ! [As: list_list_a,Bs: list_list_a] :
      ( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ As ) ) ) ) ).

% x.ee_sym
thf(fact_1001_add_Ol__cancel,axiom,
    ! [C3: a,A: a,B3: a] :
      ( ( ( add_a_b @ r @ C3 @ A )
        = ( add_a_b @ r @ C3 @ B3 ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( A = B3 ) ) ) ) ) ).

% add.l_cancel
thf(fact_1002_add_Or__cancel,axiom,
    ! [A: a,C3: a,B3: a] :
      ( ( ( add_a_b @ r @ A @ C3 )
        = ( add_a_b @ r @ B3 @ C3 ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( A = B3 ) ) ) ) ) ).

% add.r_cancel
thf(fact_1003_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_1004_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_1005_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_1006_add_Oinj__on__cmult,axiom,
    ! [C3: a] :
      ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( inj_on_a_a @ ( add_a_b @ r @ C3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% add.inj_on_cmult
thf(fact_1007_add_Oinj__on__multc,axiom,
    ! [C3: a] :
      ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( inj_on_a_a
        @ ^ [X3: a] : ( add_a_b @ r @ X3 @ C3 )
        @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% add.inj_on_multc
thf(fact_1008_up__add__closed,axiom,
    ! [P: nat > a,Q: nat > a] :
      ( ( member_nat_a @ P @ ( up_a_b @ r ) )
     => ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
       => ( member_nat_a
          @ ^ [I2: nat] : ( add_a_b @ r @ ( P @ I2 ) @ ( Q @ I2 ) )
          @ ( up_a_b @ r ) ) ) ) ).

% up_add_closed
thf(fact_1009_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_1010_add_Ol__inv__ex,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ? [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( add_a_b @ r @ X2 @ X )
            = ( zero_a_b @ r ) ) ) ) ).

% add.l_inv_ex
thf(fact_1011_add_Oone__unique,axiom,
    ! [U: a] :
      ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ! [X2: a] :
            ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( add_a_b @ r @ U @ X2 )
              = X2 ) )
       => ( U
          = ( zero_a_b @ r ) ) ) ) ).

% add.one_unique
thf(fact_1012_add_Or__inv__ex,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ? [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( add_a_b @ r @ X @ X2 )
            = ( zero_a_b @ r ) ) ) ) ).

% add.r_inv_ex
thf(fact_1013_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_1014_l__distr,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).

% l_distr
thf(fact_1015_r__distr,axiom,
    ! [X: a,Y: a,Z: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
            = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).

% r_distr
thf(fact_1016_add_Oinj__on__g,axiom,
    ! [H2: set_a,A: a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
       => ( inj_on_a_a
          @ ^ [Y3: a] : ( add_a_b @ r @ Y3 @ A )
          @ H2 ) ) ) ).

% add.inj_on_g
thf(fact_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_x_Oee__refl,axiom,
    ! [As: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ As ) ) ).

% x.ee_refl
thf(fact_1026_x_Oring_Ohom__add,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( eval_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
          = ( add_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).

% x.ring.hom_add
thf(fact_1027_x_Oconst__term__simprules_I1_J,axiom,
    ! [P: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.const_term_simprules(1)
thf(fact_1028_x_Osubfield__m__inv__simprule,axiom,
    ! [K: set_list_a,K2: list_a,A: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
       => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A ) @ K )
           => ( member_list_a @ A @ K ) ) ) ) ) ).

% x.subfield_m_inv_simprule
thf(fact_1029_x_Osubalbegra__incl__imp__finite__dimension,axiom,
    ! [K: set_list_a,E: set_list_a,V: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
       => ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
         => ( ( ord_le8861187494160871172list_a @ V @ E )
           => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ V ) ) ) ) ) ).

% x.subalbegra_incl_imp_finite_dimension
thf(fact_1030_x_Ofactors__closed,axiom,
    ! [Fs: list_list_a,A: list_a] :
      ( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.factors_closed
thf(fact_1031_x_OsubringI,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ H2 )
       => ( ! [H3: list_a] :
              ( ( member_list_a @ H3 @ H2 )
             => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H3 ) @ H2 ) )
         => ( ! [H1: list_a,H22: list_a] :
                ( ( member_list_a @ H1 @ H2 )
               => ( ( member_list_a @ H22 @ H2 )
                 => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 ) @ H2 ) ) )
           => ( ! [H1: list_a,H22: list_a] :
                  ( ( member_list_a @ H1 @ H2 )
                 => ( ( member_list_a @ H22 @ H2 )
                   => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 ) @ H2 ) ) )
             => ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).

% x.subringI
thf(fact_1032_x_Osubring__props_I5_J,axiom,
    ! [K: set_list_a,H: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_list_a @ H @ K )
       => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ K ) ) ) ).

% x.subring_props(5)
thf(fact_1033_x_Otelescopic__base__dim_I1_J,axiom,
    ! [K: set_list_a,F2: set_list_a,E: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ F2 )
         => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ E )
           => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E ) ) ) ) ) ).

% x.telescopic_base_dim(1)
thf(fact_1034_x_Oup__a__inv__closed,axiom,
    ! [P: nat > list_a] :
      ( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( member_nat_list_a
        @ ^ [I2: nat] : ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( P @ I2 ) )
        @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.up_a_inv_closed
thf(fact_1035_x_Or__neg2,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y ) )
          = Y ) ) ) ).

% x.r_neg2
thf(fact_1036_x_Or__neg1,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
          = Y ) ) ) ).

% x.r_neg1
thf(fact_1037_x_Ominus__add,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
          = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ) ) ).

% x.minus_add
thf(fact_1038_x_Oadd_Oinv__solve__right_H,axiom,
    ! [A: list_a,B3: list_a,C3: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 ) )
              = A )
            = ( B3
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C3 ) ) ) ) ) ) ).

% x.add.inv_solve_right'
thf(fact_1039_x_Oadd_Oinv__solve__right,axiom,
    ! [A: list_a,B3: list_a,C3: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( A
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C3 ) ) )
            = ( B3
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C3 ) ) ) ) ) ) ).

% x.add.inv_solve_right
thf(fact_1040_x_Oadd_Oinv__solve__left_H,axiom,
    ! [A: list_a,B3: list_a,C3: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 ) @ C3 )
              = A )
            = ( C3
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ A ) ) ) ) ) ) ).

% x.add.inv_solve_left'
thf(fact_1041_x_Oadd_Oinv__solve__left,axiom,
    ! [A: list_a,B3: list_a,C3: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( A
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 ) @ C3 ) )
            = ( C3
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ A ) ) ) ) ) ) ).

% x.add.inv_solve_left
thf(fact_1042_x_Oadd_Oinv__mult__group,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
          = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) ) ) ) ) ).

% x.add.inv_mult_group
thf(fact_1043_x_Oa__transpose__inv,axiom,
    ! [X: list_a,Y: list_a,Z: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
        = Z )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Z )
              = Y ) ) ) ) ) ).

% x.a_transpose_inv
thf(fact_1044_x_Or__minus,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) ) ) ) ).

% x.r_minus
thf(fact_1045_x_Ol__minus,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) ) ) ) ).

% x.l_minus
thf(fact_1046_univ__poly__a__inv__consistent,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
          = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ).

% univ_poly_a_inv_consistent
thf(fact_1047_x_Ominus__eq,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
      = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ).

% x.minus_eq
thf(fact_1048_x_Oconst__term__def,axiom,
    ! [P: list_list_a] :
      ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
      = ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.const_term_def
thf(fact_1049_x_Oadd_Oinv__inj,axiom,
    inj_on_list_a_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% x.add.inv_inj
thf(fact_1050_x_Ofinite__dimension__imp__subalgebra,axiom,
    ! [K: set_list_a,E: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
       => ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.finite_dimension_imp_subalgebra
thf(fact_1051_x_Ol__neg,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ X )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.l_neg
thf(fact_1052_x_Ominus__equality,axiom,
    ! [Y: list_a,X: list_a] :
      ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
            = Y ) ) ) ) ).

% x.minus_equality
thf(fact_1053_x_Or__neg,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
        = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.r_neg
thf(fact_1054_x_Osum__zero__eq__neg,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
            = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( X
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ) ) ).

% x.sum_zero_eq_neg
thf(fact_1055_x_Ogenideal__self_H,axiom,
    ! [I3: list_a] :
      ( ( member_list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( member_list_a @ I3 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I3 @ bot_bot_set_list_a ) ) ) ) ).

% x.genideal_self'
thf(fact_1056_x_Ogenideal__zero,axiom,
    ( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
    = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).

% x.genideal_zero
thf(fact_1057_x_Oring_Oa__inv__closed,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( member_a @ ( eval_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% x.ring.a_inv_closed
thf(fact_1058_x_Ozeropideal,axiom,
    princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).

% x.zeropideal
thf(fact_1059_x_Ocarrier__one__not__zero,axiom,
    ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     != ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
    = ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.carrier_one_not_zero
thf(fact_1060_x_Ocarrier__one__zero,axiom,
    ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
      = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
    = ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.carrier_one_zero
thf(fact_1061_x_Oone__zeroD,axiom,
    ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
   => ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
      = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).

% x.one_zeroD
thf(fact_1062_x_Oone__zeroI,axiom,
    ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
      = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
   => ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
      = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.one_zeroI
thf(fact_1063_x_OIdl__subset__ideal_H,axiom,
    ! [A: list_a,B3: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B3 @ bot_bot_set_list_a ) ) )
          = ( member_list_a @ A @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B3 @ bot_bot_set_list_a ) ) ) ) ) ) ).

% x.Idl_subset_ideal'
thf(fact_1064_x_Ogenideal__one,axiom,
    ( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
    = ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.genideal_one
thf(fact_1065_x_Oadd_Oone__in__subset,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( H2 != bot_bot_set_list_a )
       => ( ! [X2: list_a] :
              ( ( member_list_a @ X2 @ H2 )
             => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 ) @ H2 ) )
         => ( ! [X2: list_a] :
                ( ( member_list_a @ X2 @ H2 )
               => ! [Xa2: list_a] :
                    ( ( member_list_a @ Xa2 @ H2 )
                   => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Xa2 ) @ H2 ) ) )
           => ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ H2 ) ) ) ) ) ).

% x.add.one_in_subset
thf(fact_1066_x_Ominus__minus,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
        = X ) ) ).

% x.minus_minus
thf(fact_1067_x_Oadd_Oinv__closed,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.inv_closed
thf(fact_1068_x_Ominus__zero,axiom,
    ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
    = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.minus_zero
thf(fact_1069_x_Oadd_Oinv__eq__1__iff,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        = ( X
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.add.inv_eq_1_iff
thf(fact_1070_x_Odomain__eq__zeroprimeideal,axiom,
    ( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
    = ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.domain_eq_zeroprimeideal
thf(fact_1071_x_Ozeroprimeideal__domainI,axiom,
    ( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
   => ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.zeroprimeideal_domainI
thf(fact_1072_x_OSpan__m__inv__simprule,axiom,
    ! [K: set_list_a,Us: list_list_a,K2: list_a,A: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
         => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) )
             => ( member_list_a @ A @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ) ) ).

% x.Span_m_inv_simprule
thf(fact_1073_x_Oring_Otrivial__ker__imp__inj,axiom,
    ( ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
        @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) )
      = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
   => ( inj_on_list_a_a
      @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
      @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.ring.trivial_ker_imp_inj
thf(fact_1074_up__a__inv__closed,axiom,
    ! [P: nat > a] :
      ( ( member_nat_a @ P @ ( up_a_b @ r ) )
     => ( member_nat_a
        @ ^ [I2: nat] : ( a_inv_a_b @ r @ ( P @ I2 ) )
        @ ( up_a_b @ r ) ) ) ).

% up_a_inv_closed
thf(fact_1075_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_1076_add_Oinv__solve__left,axiom,
    ! [A: a,B3: a,C3: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( A
              = ( add_a_b @ r @ ( a_inv_a_b @ r @ B3 ) @ C3 ) )
            = ( C3
              = ( add_a_b @ r @ B3 @ A ) ) ) ) ) ) ).

% add.inv_solve_left
thf(fact_1077_add_Oinv__solve__left_H,axiom,
    ! [A: a,B3: a,C3: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B3 ) @ C3 )
              = A )
            = ( C3
              = ( add_a_b @ r @ B3 @ A ) ) ) ) ) ) ).

% add.inv_solve_left'
thf(fact_1078_add_Oinv__solve__right,axiom,
    ! [A: a,B3: a,C3: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( A
              = ( add_a_b @ r @ B3 @ ( a_inv_a_b @ r @ C3 ) ) )
            = ( B3
              = ( add_a_b @ r @ A @ C3 ) ) ) ) ) ) ).

% add.inv_solve_right
thf(fact_1079_add_Oinv__solve__right_H,axiom,
    ! [A: a,B3: a,C3: a] :
      ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( member_a @ C3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( ( add_a_b @ r @ B3 @ ( a_inv_a_b @ r @ C3 ) )
              = A )
            = ( B3
              = ( add_a_b @ r @ A @ C3 ) ) ) ) ) ) ).

% add.inv_solve_right'
thf(fact_1080_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_1081_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_1082_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_1083_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_1084_l__minus,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
          = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).

% l_minus
thf(fact_1085_r__minus,axiom,
    ! [X: a,Y: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
       => ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
          = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).

% r_minus
thf(fact_1086_const__term__def,axiom,
    ! [P: list_a] :
      ( ( const_term_a_b @ r @ P )
      = ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).

% const_term_def
thf(fact_1087_a__inv__inj,axiom,
    inj_on_a_a @ ( a_inv_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).

% a_inv_inj
thf(fact_1088_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_1089_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_1090_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_1091_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_1092_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_1093_const__term__simprules__shell_I1_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_a @ ( const_term_a_b @ r @ P ) @ K ) ) ) ).

% const_term_simprules_shell(1)
thf(fact_1094_square__eq__one,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( ( mult_a_ring_ext_a_b @ r @ X @ X )
          = ( one_a_ring_ext_a_b @ r ) )
       => ( ( X
            = ( one_a_ring_ext_a_b @ r ) )
          | ( X
            = ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).

% square_eq_one
thf(fact_1095_zeropideal,axiom,
    principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).

% zeropideal
thf(fact_1096_one__zeroI,axiom,
    ( ( ( partia707051561876973205xt_a_b @ r )
      = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
   => ( ( one_a_ring_ext_a_b @ r )
      = ( zero_a_b @ r ) ) ) ).

% one_zeroI
thf(fact_1097_one__zeroD,axiom,
    ( ( ( one_a_ring_ext_a_b @ r )
      = ( zero_a_b @ r ) )
   => ( ( partia707051561876973205xt_a_b @ r )
      = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).

% one_zeroD
thf(fact_1098_carrier__one__zero,axiom,
    ( ( ( partia707051561876973205xt_a_b @ r )
      = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
    = ( ( one_a_ring_ext_a_b @ r )
      = ( zero_a_b @ r ) ) ) ).

% carrier_one_zero
thf(fact_1099_carrier__one__not__zero,axiom,
    ( ( ( partia707051561876973205xt_a_b @ r )
     != ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
    = ( ( one_a_ring_ext_a_b @ r )
     != ( zero_a_b @ r ) ) ) ).

% carrier_one_not_zero
thf(fact_1100_const__term__simprules__shell_I4_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
          = ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P ) ) ) ) ) ).

% const_term_simprules_shell(4)
thf(fact_1101_univ__poly__a__inv__def_H,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
          = ( map_a_a @ ( a_inv_a_b @ r ) @ P ) ) ) ) ).

% univ_poly_a_inv_def'
thf(fact_1102_const__term__simprules__shell_I3_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
            = ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).

% const_term_simprules_shell(3)
thf(fact_1103_add_Oone__in__subset,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( H2 != bot_bot_set_a )
       => ( ! [X2: a] :
              ( ( member_a @ X2 @ H2 )
             => ( member_a @ ( a_inv_a_b @ r @ X2 ) @ H2 ) )
         => ( ! [X2: a] :
                ( ( member_a @ X2 @ H2 )
               => ! [Xa2: a] :
                    ( ( member_a @ Xa2 @ H2 )
                   => ( member_a @ ( add_a_b @ r @ X2 @ Xa2 ) @ H2 ) ) )
           => ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).

% add.one_in_subset
thf(fact_1104_const__term__simprules__shell_I2_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
            = ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).

% const_term_simprules_shell(2)
thf(fact_1105_x_OSpan__in__carrier,axiom,
    ! [K: set_list_a,Us: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.Span_in_carrier
thf(fact_1106_subringI,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H2 )
       => ( ! [H3: a] :
              ( ( member_a @ H3 @ H2 )
             => ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H2 ) )
         => ( ! [H1: a,H22: a] :
                ( ( member_a @ H1 @ H2 )
               => ( ( member_a @ H22 @ H2 )
                 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ H2 ) ) )
           => ( ! [H1: a,H22: a] :
                  ( ( member_a @ H1 @ H2 )
                 => ( ( member_a @ H22 @ H2 )
                   => ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ H2 ) ) )
             => ( subring_a_b @ H2 @ r ) ) ) ) ) ) ).

% subringI
thf(fact_1107_eval__rewrite,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( P
          = ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ K ) @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( var_a_b @ r ) ) ) ) ) ).

% eval_rewrite
thf(fact_1108_map__norm__in__poly__ring__carrier,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ).

% map_norm_in_poly_ring_carrier
thf(fact_1109_x_OSpan__strict__incl,axiom,
    ! [K: set_list_a,Us: list_list_a,Vs: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( ord_less_set_list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
           => ? [X2: list_a] :
                ( ( member_list_a @ X2 @ ( set_list_a2 @ Vs ) )
                & ~ ( member_list_a @ X2 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ) ) ).

% x.Span_strict_incl
thf(fact_1110_x_Omono__Span__subset,axiom,
    ! [K: set_list_a,Us: list_list_a,Vs: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).

% x.mono_Span_subset
thf(fact_1111_x_Omono__Span__sublist,axiom,
    ! [K: set_list_a,Us: list_list_a,Vs: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( set_list_a2 @ Vs ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).

% x.mono_Span_sublist
thf(fact_1112_x_OSpan__same__set,axiom,
    ! [K: set_list_a,Us: list_list_a,Vs: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ( set_list_a2 @ Us )
            = ( set_list_a2 @ Vs ) )
         => ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us )
            = ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).

% x.Span_same_set
thf(fact_1113_x_OSpan__base__incl,axiom,
    ! [K: set_list_a,Us: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ).

% x.Span_base_incl
thf(fact_1114_x_OSpan__subgroup__props_I1_J,axiom,
    ! [K: set_list_a,Us: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.Span_subgroup_props(1)
thf(fact_1115_x_Osubalgebra__Span__incl,axiom,
    ! [K: set_list_a,V: set_list_a,Us: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ V )
         => ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) @ V ) ) ) ) ).

% x.subalgebra_Span_incl
thf(fact_1116_x_OSpan__subalgebraI,axiom,
    ! [K: set_list_a,E: set_list_a,Us: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ E )
         => ( ! [V4: set_list_a] :
                ( ( embedd1768981623711841426t_unit @ K @ V4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
               => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ V4 )
                 => ( ord_le8861187494160871172list_a @ E @ V4 ) ) )
           => ( E
              = ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ) ).

% x.Span_subalgebraI
thf(fact_1117_x_OSpan__subgroup__props_I2_J,axiom,
    ! [K: set_list_a,Us: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ).

% x.Span_subgroup_props(2)
thf(fact_1118_x_OSpan__subgroup__props_I3_J,axiom,
    ! [K: set_list_a,Us: list_list_a,V1: list_a,V22: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ V1 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) )
         => ( ( member_list_a @ V22 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) )
           => ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ V1 @ V22 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ) ).

% x.Span_subgroup_props(3)
thf(fact_1119_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_1120_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_1121_x_OSpan__smult__closed,axiom,
    ! [K: set_list_a,Us: list_list_a,K2: list_a,V5: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ K2 @ K )
         => ( ( member_list_a @ V5 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) )
           => ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ V5 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ) ).

% x.Span_smult_closed
thf(fact_1122_x_OSpan__subgroup__props_I4_J,axiom,
    ! [K: set_list_a,Us: list_list_a,V5: list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( member_list_a @ V5 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) )
         => ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ V5 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ).

% x.Span_subgroup_props(4)
thf(fact_1123_local_Ominus__zero,axiom,
    ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
    = ( zero_a_b @ r ) ) ).

% local.minus_zero
thf(fact_1124_x_OSpan__finite__dimension,axiom,
    ! [K: set_list_a,Us: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ).

% x.Span_finite_dimension
thf(fact_1125_x_OSpan__is__subalgebra,axiom,
    ! [K: set_list_a,Us: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( embedd1768981623711841426t_unit @ K @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.Span_is_subalgebra
thf(fact_1126_x_Oring_Oinj__iff__trivial__ker,axiom,
    ( ( inj_on_list_a_a
      @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
      @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
    = ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
        @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) )
      = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).

% x.ring.inj_iff_trivial_ker
thf(fact_1127_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_1128_x_Oring_Ohom__a__inv,axiom,
    ! [X: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( eval_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ x )
        = ( a_inv_a_b @ r @ ( eval_a_b @ r @ X @ x ) ) ) ) ).

% x.ring.hom_a_inv
thf(fact_1129_euclidean__domainI,axiom,
    ! [Phi: a > nat] :
      ( ! [A3: a,B4: a] :
          ( ( member_a @ A3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
         => ( ( member_a @ B4 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
           => ? [Q3: a,R4: a] :
                ( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ r ) )
                & ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ r ) )
                & ( A3
                  = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B4 @ Q3 ) @ R4 ) )
                & ( ( R4
                    = ( zero_a_b @ r ) )
                  | ( ord_less_nat @ ( Phi @ R4 ) @ ( Phi @ B4 ) ) ) ) ) )
     => ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).

% euclidean_domainI
thf(fact_1130_x_Omaximalideal__prime,axiom,
    ! [I4: set_list_a] :
      ( ( maxima6585700282301356660t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( primei6309817859076077608t_unit @ I4 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.maximalideal_prime
thf(fact_1131_x_Oring_Oadditive__subgroup__a__kernel,axiom,
    ( additi4714453376129182166t_unit
    @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
      @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) )
    @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% x.ring.additive_subgroup_a_kernel
thf(fact_1132_x_Oring_Oinj__on__Span__iff__trivial__ker,axiom,
    ! [K: set_list_a,Us: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( inj_on_list_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) )
          = ( ( a_kern7116238624728830086it_a_b
              @ ( partia9041243232023819264t_unit
                @ ^ [Uu: set_list_a] : ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us )
                @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
              @ r
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) )
            = ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ).

% x.ring.inj_on_Span_iff_trivial_ker
thf(fact_1133_x_OSpan__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( embedd4402942584324845940t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        = ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.Span_consistent
thf(fact_1134_x_Ouniv__poly__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( univ_p7953238456130426574t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        = ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.univ_poly_consistent
thf(fact_1135_x_Oline__extension__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( embedd5150658419831591667t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        = ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.line_extension_consistent
thf(fact_1136_x_Opoly__of__const__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( poly_o8716471131768098070t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        = ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.poly_of_const_consistent
thf(fact_1137_x_Osubdomain__is__domain,axiom,
    ! [H2: set_list_a] :
      ( ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( domain6553523120543210313t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : H2
          @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.subdomain_is_domain
thf(fact_1138_x_Osubdomain__iff,axiom,
    ! [H2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
        = ( domain6553523120543210313t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : H2
            @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.subdomain_iff
thf(fact_1139_x_Oeval__consistent,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( eval_l34571156754992824t_unit
          @ ( partia9041243232023819264t_unit
            @ ^ [Uu: set_list_a] : K
            @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        = ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.eval_consistent
thf(fact_1140_map__in__poly__ring__carrier,axiom,
    ! [P: list_a,F: a > list_a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ! [A3: a] :
            ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
           => ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
       => ( ! [A3: a] :
              ( ( A3
               != ( zero_a_b @ r ) )
             => ( ( F @ A3 )
               != nil_a ) )
         => ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).

% map_in_poly_ring_carrier
thf(fact_1141_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_1142_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_1143_subring__is__domain,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( domain_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ r ) ) ) ).

% subring_is_domain
thf(fact_1144_eval_Osimps_I1_J,axiom,
    ( ( eval_a_b @ r @ nil_a )
    = ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).

% eval.simps(1)
thf(fact_1145_const__term__not__zero,axiom,
    ! [P: list_a] :
      ( ( ( const_term_a_b @ r @ P )
       != ( zero_a_b @ r ) )
     => ( P != nil_a ) ) ).

% const_term_not_zero
thf(fact_1146_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_1147_is__root__def,axiom,
    ! [P: list_a,X: a] :
      ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
      = ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
        & ( ( eval_a_b @ r @ P @ X )
          = ( zero_a_b @ r ) )
        & ( P != nil_a ) ) ) ).

% is_root_def
thf(fact_1148_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_1149_subdomain__iff,axiom,
    ! [H2: set_a] :
      ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( subdomain_a_b @ H2 @ r )
        = ( domain_a_b
          @ ( partia8674076737563717228xt_a_b
            @ ^ [Uu: set_a] : H2
            @ r ) ) ) ) ).

% subdomain_iff
thf(fact_1150_x_Oring_Otrivial__hom__iff,axiom,
    ( ( ( image_list_a_a
        @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
        @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
      = ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
    = ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
        @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) )
      = ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.ring.trivial_hom_iff
thf(fact_1151_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_1152_subdomainI_H,axiom,
    ! [H2: set_a] :
      ( ( subring_a_b @ H2 @ r )
     => ( subdomain_a_b @ H2 @ r ) ) ).

% subdomainI'
thf(fact_1153_x_Oring__primeI,axiom,
    ! [P: list_a] :
      ( ( P
       != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
       => ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).

% x.ring_primeI
thf(fact_1154_x_Oeval_Osimps_I1_J,axiom,
    ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
    = ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.eval.simps(1)
thf(fact_1155_pirreducibleE_I1_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
         => ( P != nil_a ) ) ) ) ).

% pirreducibleE(1)
thf(fact_1156_x_Oconst__term__not__zero,axiom,
    ! [P: list_list_a] :
      ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
       != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( P != nil_list_a ) ) ).

% x.const_term_not_zero
thf(fact_1157_zeroprimeideal,axiom,
    primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).

% zeroprimeideal
thf(fact_1158_subdomain__is__domain,axiom,
    ! [H2: set_a] :
      ( ( subdomain_a_b @ H2 @ r )
     => ( domain_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : H2
          @ r ) ) ) ).

% subdomain_is_domain
thf(fact_1159_x_Ois__root__def,axiom,
    ! [P: list_list_a,X: list_a] :
      ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X )
      = ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        & ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X )
          = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        & ( P != nil_list_a ) ) ) ).

% x.is_root_def
thf(fact_1160_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_1161_x_Oring_Oimg__is__subring,axiom,
    ! [K: set_list_a] :
      ( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( subring_a_b
        @ ( image_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
          @ K )
        @ r ) ) ).

% x.ring.img_is_subring
thf(fact_1162_x_Oring_Oimg__is__domain,axiom,
    ( ( domain_a_b @ r )
   => ( domain_a_b
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] :
            ( image_list_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        @ r ) ) ) ).

% x.ring.img_is_domain
thf(fact_1163_x_Oring_Oinfinite__dimension__hom,axiom,
    ! [K: set_list_a,E: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ( one_a_ring_ext_a_b @ r )
         != ( zero_a_b @ r ) )
       => ( ( inj_on_list_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ E )
         => ( ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
           => ( ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
             => ~ ( embedd8708762675212832759on_a_b @ r
                  @ ( image_list_a_a
                    @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
                    @ K )
                  @ ( image_list_a_a
                    @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
                    @ E ) ) ) ) ) ) ) ).

% x.ring.infinite_dimension_hom
thf(fact_1164_subring__props_I2_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( member_a @ ( zero_a_b @ r ) @ K ) ) ).

% subring_props(2)
thf(fact_1165_subring__props_I7_J,axiom,
    ! [K: set_a,H12: a,H23: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ H12 @ K )
       => ( ( member_a @ H23 @ K )
         => ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ K ) ) ) ) ).

% subring_props(7)
thf(fact_1166_subring__props_I6_J,axiom,
    ! [K: set_a,H12: a,H23: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ H12 @ K )
       => ( ( member_a @ H23 @ K )
         => ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ K ) ) ) ) ).

% subring_props(6)
thf(fact_1167_subring__props_I4_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( K != bot_bot_set_a ) ) ).

% subring_props(4)
thf(fact_1168_subring__props_I3_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K ) ) ).

% subring_props(3)
thf(fact_1169_subring__props_I5_J,axiom,
    ! [K: set_a,H: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ H @ K )
       => ( member_a @ ( a_inv_a_b @ r @ H ) @ K ) ) ) ).

% subring_props(5)
thf(fact_1170_telescopic__base__dim_I1_J,axiom,
    ! [K: set_a,F2: set_a,E: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( subfield_a_b @ F2 @ r )
       => ( ( embedd8708762675212832759on_a_b @ r @ K @ F2 )
         => ( ( embedd8708762675212832759on_a_b @ r @ F2 @ E )
           => ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ) ) ) ).

% telescopic_base_dim(1)
thf(fact_1171_subring__props_I1_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% subring_props(1)
thf(fact_1172_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_1173_univ__poly__subfield__of__consts,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_subfield_of_consts
thf(fact_1174_x_Oadd_Osurj__const__mult,axiom,
    ! [A: list_a] :
      ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( image_list_a_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        = ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add.surj_const_mult
thf(fact_1175_univ__poly__infinite__dimension,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).

% univ_poly_infinite_dimension
thf(fact_1176_pprimeE_I1_J,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
         => ( P != nil_a ) ) ) ) ).

% pprimeE(1)
thf(fact_1177_pprime__iff__pirreducible,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
          = ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).

% pprime_iff_pirreducible
thf(fact_1178_x_Ouniv__poly__subfield__of__consts,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).

% x.univ_poly_subfield_of_consts
thf(fact_1179_subfield__m__inv__simprule,axiom,
    ! [K: set_a,K2: a,A: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
       => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ A ) @ K )
           => ( member_a @ A @ K ) ) ) ) ) ).

% subfield_m_inv_simprule
thf(fact_1180_x_Oring_Oimg__is__subfield_I2_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( ( one_a_ring_ext_a_b @ r )
         != ( zero_a_b @ r ) )
       => ( subfield_a_b
          @ ( image_list_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ K )
          @ r ) ) ) ).

% x.ring.img_is_subfield(2)
thf(fact_1181_univ__poly__is__principal,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).

% univ_poly_is_principal
thf(fact_1182_exists__unique__long__division,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ? [X2: produc9164743771328383783list_a] :
                ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ X2 )
                & ! [Y5: produc9164743771328383783list_a] :
                    ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y5 )
                   => ( Y5 = X2 ) ) ) ) ) ) ) ).

% exists_unique_long_division
thf(fact_1183_x_Oring_Oimg__is__subalgebra,axiom,
    ! [K: set_list_a,V: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( embedd9027525575939734154ra_a_b
          @ ( image_list_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ K )
          @ ( image_list_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ V )
          @ r ) ) ) ).

% x.ring.img_is_subalgebra
thf(fact_1184_x_Oring_Oline__extension__hom,axiom,
    ! [K: set_list_a,A: list_a,E: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( embedd971793762689825387on_a_b @ r
              @ ( image_list_a_a
                @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
                @ K )
              @ ( eval_a_b @ r @ A @ x )
              @ ( image_list_a_a
                @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
                @ E ) )
            = ( image_list_a_a
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
              @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) ) ) ) ) ) ).

% x.ring.line_extension_hom
thf(fact_1185_subalgebra__inter,axiom,
    ! [K: set_a,V: set_a,V2: set_a] :
      ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
     => ( ( embedd9027525575939734154ra_a_b @ K @ V2 @ r )
       => ( embedd9027525575939734154ra_a_b @ K @ ( inf_inf_set_a @ V @ V2 ) @ r ) ) ) ).

% subalgebra_inter
thf(fact_1186_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_1187_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_1188_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_1189_line__extension__mem__iff,axiom,
    ! [U: a,K: set_a,A: a,E: set_a] :
      ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
      = ( ? [X3: a] :
            ( ( member_a @ X3 @ K )
            & ? [Y3: a] :
                ( ( member_a @ Y3 @ E )
                & ( U
                  = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y3 ) ) ) ) ) ) ).

% line_extension_mem_iff
thf(fact_1190_finite__dimension__imp__subalgebra,axiom,
    ! [K: set_a,E: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
       => ( embedd9027525575939734154ra_a_b @ K @ E @ r ) ) ) ).

% finite_dimension_imp_subalgebra
thf(fact_1191_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_1192_subalbegra__incl__imp__finite__dimension,axiom,
    ! [K: set_a,E: set_a,V: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
       => ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
         => ( ( ord_less_eq_set_a @ V @ E )
           => ( embedd8708762675212832759on_a_b @ r @ K @ V ) ) ) ) ) ).

% subalbegra_incl_imp_finite_dimension
thf(fact_1193_line__extension__smult__closed,axiom,
    ! [K: set_a,E: set_a,A: a,K2: a,U: a] :
      ( ( subfield_a_b @ K @ r )
     => ( ! [K3: a,V3: a] :
            ( ( member_a @ K3 @ K )
           => ( ( member_a @ V3 @ E )
             => ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V3 ) @ E ) ) )
       => ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
           => ( ( member_a @ K2 @ K )
             => ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
               => ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) ) ) ) ) ) ) ) ).

% line_extension_smult_closed
thf(fact_1194_x_Oring_Othe__elem__wf_H,axiom,
    ! [X4: set_list_a] :
      ( ( member_set_list_a @ X4
        @ ( partia141011252114345353t_unit
          @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
            @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) ) ) )
     => ? [X2: list_a] :
          ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
          & ( ( image_list_a_a
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
              @ X4 )
            = ( insert_a @ ( eval_a_b @ r @ X2 @ x ) @ bot_bot_set_a ) ) ) ) ).

% x.ring.the_elem_wf'
thf(fact_1195_x_Oring_Onon__trivial__field__hom__imp__inj,axiom,
    ( ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
   => ( ( ( image_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
          @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       != ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
     => ( inj_on_list_a_a
        @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
        @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.ring.non_trivial_field_hom_imp_inj
thf(fact_1196_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_1197_x_Osubfield__iff_I2_J,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( field_6388047844668329575t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : K
          @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.subfield_iff(2)
thf(fact_1198_x_Ocring__fieldI2,axiom,
    ( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     != ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
   => ( ! [A3: list_a] :
          ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( A3
             != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
           => ? [X5: list_a] :
                ( ( member_list_a @ X5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
                & ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ X5 )
                  = ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) )
     => ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.cring_fieldI2
thf(fact_1199_x_Osubfield__iff_I1_J,axiom,
    ! [K: set_list_a] :
      ( ( field_6388047844668329575t_unit
        @ ( partia9041243232023819264t_unit
          @ ^ [Uu: set_list_a] : K
          @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.subfield_iff(1)
thf(fact_1200_x_Ozeromaximalideal__eq__field,axiom,
    ( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
    = ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.zeromaximalideal_eq_field
thf(fact_1201_x_Ozeromaximalideal__fieldI,axiom,
    ( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
   => ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.zeromaximalideal_fieldI
thf(fact_1202_x_Oring_Othe__elem__wf,axiom,
    ! [X4: set_list_a] :
      ( ( member_set_list_a @ X4
        @ ( partia141011252114345353t_unit
          @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
            @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) ) ) )
     => ? [X2: a] :
          ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
          & ( ( image_list_a_a
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
              @ X4 )
            = ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).

% x.ring.the_elem_wf
thf(fact_1203_x_OFactRing__zeroideal_I2_J,axiom,
    is_rin2993610189962786360t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).

% x.FactRing_zeroideal(2)
thf(fact_1204_x_OFactRing__zeroideal_I1_J,axiom,
    is_rin4843644836746533432t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).

% x.FactRing_zeroideal(1)
thf(fact_1205_x_Oring_OFactRing__iso,axiom,
    ( ( ( image_list_a_a
        @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
        @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
      = ( partia707051561876973205xt_a_b @ r ) )
   => ( is_rin5597148638330396976it_a_b
      @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
        @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) )
      @ r ) ) ).

% x.ring.FactRing_iso
thf(fact_1206_x_Oring_Othe__elem__surj,axiom,
    ( ( image_set_list_a_a
      @ ^ [X6: set_list_a] :
          ( the_elem_a
          @ ( image_list_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ X6 ) )
      @ ( partia141011252114345353t_unit
        @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
          @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) ) ) )
    = ( image_list_a_a
      @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
      @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.ring.the_elem_surj
thf(fact_1207_x_Oring_Othe__elem__inj,axiom,
    ! [X4: set_list_a,Y6: set_list_a] :
      ( ( member_set_list_a @ X4
        @ ( partia141011252114345353t_unit
          @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
            @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) ) ) )
     => ( ( member_set_list_a @ Y6
          @ ( partia141011252114345353t_unit
            @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
              @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
                @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) ) ) )
       => ( ( ( the_elem_a
              @ ( image_list_a_a
                @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
                @ X4 ) )
            = ( the_elem_a
              @ ( image_list_a_a
                @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
                @ Y6 ) ) )
         => ( X4 = Y6 ) ) ) ) ).

% x.ring.the_elem_inj
thf(fact_1208_subfield__iff_I2_J,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( field_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : K
          @ r ) ) ) ).

% subfield_iff(2)
thf(fact_1209_cring__fieldI2,axiom,
    ( ( ( zero_a_b @ r )
     != ( one_a_ring_ext_a_b @ r ) )
   => ( ! [A3: a] :
          ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
         => ( ( A3
             != ( zero_a_b @ r ) )
           => ? [X5: a] :
                ( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ r ) )
                & ( ( mult_a_ring_ext_a_b @ r @ A3 @ X5 )
                  = ( one_a_ring_ext_a_b @ r ) ) ) ) )
     => ( field_a_b @ r ) ) ) ).

% cring_fieldI2
thf(fact_1210_subfield__iff_I1_J,axiom,
    ! [K: set_a] :
      ( ( field_a_b
        @ ( partia8674076737563717228xt_a_b
          @ ^ [Uu: set_a] : K
          @ r ) )
     => ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
       => ( subfield_a_b @ K @ r ) ) ) ).

% subfield_iff(1)
thf(fact_1211_x_Oring_Othe__elem__hom,axiom,
    ( member_set_list_a_a
    @ ^ [X6: set_list_a] :
        ( the_elem_a
        @ ( image_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
          @ X6 ) )
    @ ( ring_h8906680420194085028it_a_b
      @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
        @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) )
      @ r ) ) ).

% x.ring.the_elem_hom
thf(fact_1212_x_Oring_OFactRing__iso__set__aux,axiom,
    ( member_set_list_a_a
    @ ^ [X6: set_list_a] :
        ( the_elem_a
        @ ( image_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
          @ X6 ) )
    @ ( ring_i8122894263081988538it_a_b
      @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
        @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) )
      @ ( partia8674076737563717228xt_a_b
        @ ^ [Uu: set_a] :
            ( image_list_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
        @ r ) ) ) ).

% x.ring.FactRing_iso_set_aux
thf(fact_1213_x_Oring_OFactRing__iso__set,axiom,
    ( ( ( image_list_a_a
        @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
        @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
      = ( partia707051561876973205xt_a_b @ r ) )
   => ( member_set_list_a_a
      @ ^ [X6: set_list_a] :
          ( the_elem_a
          @ ( image_list_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ X6 ) )
      @ ( ring_i8122894263081988538it_a_b
        @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
          @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) )
        @ r ) ) ) ).

% x.ring.FactRing_iso_set
thf(fact_1214_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_1215_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_1216_maximalideal__prime,axiom,
    ! [I4: set_a] :
      ( ( maximalideal_a_b @ I4 @ r )
     => ( primeideal_a_b @ I4 @ r ) ) ).

% maximalideal_prime
thf(fact_1217_rupture__is__field__iff__pirreducible,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) )
          = ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).

% rupture_is_field_iff_pirreducible
thf(fact_1218_primeideal__iff__prime,axiom,
    ! [P: a] :
      ( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
     => ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r )
        = ( ring_ring_prime_a_b @ r @ P ) ) ) ).

% primeideal_iff_prime
thf(fact_1219_cgenideal__self,axiom,
    ! [I3: a] :
      ( ( member_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ I3 @ ( cgenid547466209912283029xt_a_b @ r @ I3 ) ) ) ).

% cgenideal_self
thf(fact_1220_cgenideal__is__principalideal,axiom,
    ! [I3: a] :
      ( ( member_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
     => ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I3 ) @ r ) ) ).

% cgenideal_is_principalideal
thf(fact_1221_Span__var__pow__base,axiom,
    ! [K: set_a,N3: nat] :
      ( ( subfield_a_b @ K @ r )
     => ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N3 ) )
        = ( collect_list_a
          @ ^ [Q4: list_a] :
              ( ( member_list_a @ Q4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
              & ( ord_less_eq_nat @ ( size_size_list_a @ Q4 ) @ N3 ) ) ) ) ) ).

% Span_var_pow_base
thf(fact_1222_long__division__a__inv_I1_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q )
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) ) ) ) ) ).

% long_division_a_inv(1)
thf(fact_1223_x_Ocgenideal__self,axiom,
    ! [I3: list_a] :
      ( ( member_list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( member_list_a @ I3 @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 ) ) ) ).

% x.cgenideal_self
thf(fact_1224_long__division__closed_I1_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% long_division_closed(1)
thf(fact_1225_x_Ocgenideal__is__principalideal,axiom,
    ! [I3: list_a] :
      ( ( member_list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% x.cgenideal_is_principalideal
thf(fact_1226_univ__poly__a__inv__length,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
          = ( size_size_list_a @ P ) ) ) ) ).

% univ_poly_a_inv_length
thf(fact_1227_long__division__add_I1_J,axiom,
    ! [K: set_a,A: list_a,B3: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B3 ) @ Q )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ A @ Q ) @ ( polynomial_pdiv_a_b @ r @ B3 @ Q ) ) ) ) ) ) ) ).

% long_division_add(1)
thf(fact_1228_long__division__zero_I1_J,axiom,
    ! [K: set_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q )
          = nil_a ) ) ) ).

% long_division_zero(1)
thf(fact_1229_x_Ocgenideal__eq__genideal,axiom,
    ! [I3: list_a] :
      ( ( member_list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 )
        = ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I3 @ bot_bot_set_list_a ) ) ) ) ).

% x.cgenideal_eq_genideal
thf(fact_1230_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_1231_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_1232_poly__of__const__over__subfield,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K )
        = ( collect_list_a
          @ ^ [P2: list_a] :
              ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
              & ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
                = zero_zero_nat ) ) ) ) ) ).

% poly_of_const_over_subfield
thf(fact_1233_x_Oee__length,axiom,
    ! [As: list_list_a,Bs: list_list_a] :
      ( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
     => ( ( size_s349497388124573686list_a @ As )
        = ( size_s349497388124573686list_a @ Bs ) ) ) ).

% x.ee_length
thf(fact_1234_univ__poly__is__euclidean,axiom,
    ! [K: set_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ring_e7478897652244013592t_unit @ ( univ_poly_a_b @ r @ K )
        @ ^ [P2: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).

% univ_poly_is_euclidean
thf(fact_1235_univ__poly__a__inv__degree,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subring_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( minus_minus_nat @ ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) @ one_one_nat )
          = ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).

% univ_poly_a_inv_degree
thf(fact_1236_degree__one__imp__pirreducible,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
            = one_one_nat )
         => ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).

% degree_one_imp_pirreducible
thf(fact_1237_poly__add__degree__le,axiom,
    ! [X: list_a,N3: nat,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N3 )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N3 )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ N3 ) ) ) ) ) ).

% poly_add_degree_le
thf(fact_1238_degree__zero__imp__not__is__root,axiom,
    ! [P: list_a,X: a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
          = zero_zero_nat )
       => ~ ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).

% degree_zero_imp_not_is_root
thf(fact_1239_poly__sub__degree__le,axiom,
    ! [X: list_a,N3: nat,Y: list_a] :
      ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N3 )
       => ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N3 )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ N3 ) ) ) ) ) ).

% poly_sub_degree_le
thf(fact_1240_pirreducible__degree,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
         => ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).

% pirreducible_degree
thf(fact_1241_x_Osubfield__long__division__theorem__shell,axiom,
    ! [K: set_list_a,P: list_list_a,B3: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
       => ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
         => ( ( B3
             != ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
           => ? [Q5: list_list_a,R3: list_list_a] :
                ( ( member_list_list_a @ Q5 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
                & ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
                & ( P
                  = ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ B3 @ Q5 ) @ R3 ) )
                & ( ( R3
                    = ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
                  | ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ B3 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% x.subfield_long_division_theorem_shell
thf(fact_1242_x_Opirreducible__degree,axiom,
    ! [K: set_list_a,P: list_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
       => ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
         => ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ) ).

% x.pirreducible_degree
thf(fact_1243_rupture__one__not__zero,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) )
         => ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) )
           != ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) ) ) ) ) ) ).

% rupture_one_not_zero
thf(fact_1244_pmod__const_I1_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
           => ( ( polynomial_pdiv_a_b @ r @ P @ Q )
              = nil_a ) ) ) ) ) ).

% pmod_const(1)
thf(fact_1245_subfield__long__division__theorem__shell,axiom,
    ! [K: set_a,P: list_a,B3: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( B3
             != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ? [Q5: list_a,R3: list_a] :
                ( ( member_list_a @ Q5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                & ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                & ( P
                  = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ B3 @ Q5 ) @ R3 ) )
                & ( ( R3
                    = ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                  | ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R3 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B3 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% subfield_long_division_theorem_shell
thf(fact_1246_x_Opoly__of__const__over__subfield,axiom,
    ! [K: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K )
        = ( collect_list_list_a
          @ ^ [P2: list_list_a] :
              ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
              & ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
                = zero_zero_nat ) ) ) ) ) ).

% x.poly_of_const_over_subfield
thf(fact_1247_degree__zero__imp__splitted,axiom,
    ! [P: list_a] :
      ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
          = zero_zero_nat )
       => ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).

% degree_zero_imp_splitted
thf(fact_1248_long__dividesI,axiom,
    ! [B3: list_a,R2: list_a,P: list_a,Q: list_a] :
      ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( P
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ B3 ) @ R2 ) )
         => ( ( ( R2 = nil_a )
              | ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) )
           => ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B3 @ R2 ) ) ) ) ) ) ).

% long_dividesI
thf(fact_1249_poly__add_Ocases,axiom,
    ! [X: produc9164743771328383783list_a] :
      ~ ! [P1: list_a,P22: list_a] :
          ( X
         != ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).

% poly_add.cases
thf(fact_1250_exp__base__closed,axiom,
    ! [X: a,N3: nat] :
      ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X @ N3 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% exp_base_closed
thf(fact_1251_eval__in__carrier,axiom,
    ! [P: list_a,X: a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
       => ( member_a @ ( eval_a_b @ r @ P @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).

% eval_in_carrier
thf(fact_1252_const__term__simprules_I1_J,axiom,
    ! [P: list_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
     => ( member_a @ ( const_term_a_b @ r @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).

% const_term_simprules(1)
thf(fact_1253_exists__long__division,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ~ ! [B4: list_a] :
                  ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                 => ! [R3: list_a] :
                      ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                     => ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B4 @ R3 ) ) ) ) ) ) ) ) ).

% exists_long_division
thf(fact_1254_x_Olong__dividesI,axiom,
    ! [B3: list_list_a,R2: list_list_a,P: list_list_a,Q: list_list_a] :
      ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
     => ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
       => ( ( P
            = ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ Q @ B3 ) @ R2 ) )
         => ( ( ( R2 = nil_list_a )
              | ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) ) )
           => ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q @ ( produc8696003437204565271list_a @ B3 @ R2 ) ) ) ) ) ) ).

% x.long_dividesI
thf(fact_1255_pmod__image__characterization,axiom,
    ! [K: set_a,P: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( P != nil_a )
         => ( ( image_list_a_list_a
              @ ^ [Q4: list_a] : ( polynomial_pmod_a_b @ r @ Q4 @ P )
              @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
            = ( collect_list_a
              @ ^ [Q4: list_a] :
                  ( ( member_list_a @ Q4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
                  & ( ord_less_eq_nat @ ( size_size_list_a @ Q4 ) @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% pmod_image_characterization
thf(fact_1256_x_Opoly__add_Ocases,axiom,
    ! [X: produc7709606177366032167list_a] :
      ~ ! [P1: list_list_a,P22: list_list_a] :
          ( X
         != ( produc8696003437204565271list_a @ P1 @ P22 ) ) ).

% x.poly_add.cases
thf(fact_1257_long__division__closed_I2_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).

% long_division_closed(2)
thf(fact_1258_long__division__add__iff,axiom,
    ! [K: set_a,A: list_a,B3: list_a,C3: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
             => ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
                  = ( polynomial_pmod_a_b @ r @ B3 @ Q ) )
                = ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ C3 ) @ Q )
                  = ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ B3 @ C3 ) @ Q ) ) ) ) ) ) ) ) ).

% long_division_add_iff
thf(fact_1259_long__division__add_I2_J,axiom,
    ! [K: set_a,A: list_a,B3: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
           => ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B3 ) @ Q )
              = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ A @ Q ) @ ( polynomial_pmod_a_b @ r @ B3 @ Q ) ) ) ) ) ) ) ).

% long_division_add(2)
thf(fact_1260_long__division__zero_I2_J,axiom,
    ! [K: set_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( polynomial_pmod_a_b @ r @ nil_a @ Q )
          = nil_a ) ) ) ).

% long_division_zero(2)
thf(fact_1261_long__division__a__inv_I2_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q )
            = ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).

% long_division_a_inv(2)
thf(fact_1262_pdiv__pmod,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( P
            = ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).

% pdiv_pmod
thf(fact_1263_pmod__const_I2_J,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
           => ( ( polynomial_pmod_a_b @ r @ P @ Q )
              = P ) ) ) ) ) ).

% pmod_const(2)
thf(fact_1264_pmod__degree,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
                = nil_a )
              | ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).

% pmod_degree
thf(fact_1265_long__divisionE,axiom,
    ! [K: set_a,P: list_a,Q: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ).

% long_divisionE
thf(fact_1266_long__divisionI,axiom,
    ! [K: set_a,P: list_a,Q: list_a,B3: list_a,R2: list_a] :
      ( ( subfield_a_b @ K @ r )
     => ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
       => ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
         => ( ( Q != nil_a )
           => ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B3 @ R2 ) )
             => ( ( produc6837034575241423639list_a @ B3 @ R2 )
                = ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ) ).

% long_divisionI
thf(fact_1267_x_Oring_Oset__add__ker__hom_I1_J,axiom,
    ! [I4: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( image_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
          @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4
            @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) ) ) )
        = ( image_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
          @ I4 ) ) ) ).

% x.ring.set_add_ker_hom(1)
thf(fact_1268_x_Oring_Oset__add__ker__hom_I2_J,axiom,
    ! [I4: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( image_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
          @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
            @ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x ) )
            @ I4 ) )
        = ( image_list_a_a
          @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
          @ I4 ) ) ) ).

% x.ring.set_add_ker_hom(2)
thf(fact_1269_x_Oadd__additive__subgroups,axiom,
    ! [H2: set_list_a,K: set_list_a] :
      ( ( additi4714453376129182166t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( additi4714453376129182166t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
       => ( additi4714453376129182166t_unit @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 @ K ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).

% x.add_additive_subgroups
thf(fact_1270_x_Oset__add__closed,axiom,
    ! [A2: set_list_a,B: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_le8861187494160871172list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.set_add_closed
thf(fact_1271_x_Oset__add__comm,axiom,
    ! [I4: set_list_a,J2: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ I4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_le8861187494160871172list_a @ J2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I4 @ J2 )
          = ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ J2 @ I4 ) ) ) ) ).

% x.set_add_comm
thf(fact_1272_x_Osetadd__subset__G,axiom,
    ! [H2: set_list_a,K: set_list_a] :
      ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).

% x.setadd_subset_G
thf(fact_1273_x_Osum__space__dim_I1_J,axiom,
    ! [K: set_list_a,E: set_list_a,F2: set_list_a] :
      ( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
     => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
       => ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ F2 )
         => ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ E @ F2 ) ) ) ) ) ).

% x.sum_space_dim(1)
thf(fact_1274_x_Odegree__oneE,axiom,
    ! [P: list_list_a,K: set_list_a] :
      ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
     => ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
          = one_one_nat )
       => ~ ! [A3: list_a] :
              ( ( member_list_a @ A3 @ K )
             => ( ( A3
                 != ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
               => ! [B4: list_a] :
                    ( ( member_list_a @ B4 @ K )
                   => ( P
                     != ( cons_list_a @ A3 @ ( cons_list_a @ B4 @ nil_list_a ) ) ) ) ) ) ) ) ).

% x.degree_oneE
thf(fact_1275_x_Oring_OSpan__hom,axiom,
    ! [K: set_list_a,Us: list_list_a] :
      ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
     => ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
       => ( ( embedded_Span_a_b @ r
            @ ( image_list_a_a
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
              @ K )
            @ ( map_list_a_a
              @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
              @ Us ) )
          = ( image_list_a_a
            @ ^ [P2: list_a] : ( eval_a_b @ r @ P2 @ x )
            @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us ) ) ) ) ) ).

% x.ring.Span_hom
thf(fact_1276_x_Onormalize_Ocases,axiom,
    ! [X: list_list_a] :
      ( ( X != nil_list_a )
     => ~ ! [V3: list_a,Va: list_list_a] :
            ( X
           != ( cons_list_a @ V3 @ Va ) ) ) ).

% x.normalize.cases
thf(fact_1277_x_Ocombine_Ocases,axiom,
    ! [X: produc7709606177366032167list_a] :
      ( ! [K3: list_a,Ks: list_list_a,U2: list_a,Us2: list_list_a] :
          ( X
         != ( produc8696003437204565271list_a @ ( cons_list_a @ K3 @ Ks ) @ ( cons_list_a @ U2 @ Us2 ) ) )
     => ( ! [Us2: list_list_a] :
            ( X
           != ( produc8696003437204565271list_a @ nil_list_a @ Us2 ) )
       => ~ ! [Ks: list_list_a] :
              ( X
             != ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).

% x.combine.cases

% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
    ! [X: a,Y: a] :
      ( ( if_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001tf__a_T,axiom,
    ! [X: a,Y: a] :
      ( ( if_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( if_list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( if_list_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [P3: $o] :
      ( ( P3 = $true )
      | ( P3 = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X: list_list_a,Y: list_list_a] :
      ( ( if_list_list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_Itf__a_J_J_T,axiom,
    ! [X: list_list_a,Y: list_list_a] :
      ( ( if_list_list_a @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( eval_a_b @ r @ ( lagran9092808442999052491ux_a_b @ r @ s ) @ x )
    = ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ).

%------------------------------------------------------------------------------