TPTP Problem File: SLH0047^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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_00184_006718__17273992_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1573 ( 235 unt; 294 typ; 0 def)
% Number of atoms : 4701 (1309 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 19000 ( 366 ~; 74 |; 146 &;15498 @)
% ( 0 <=>;2916 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 39 ( 38 usr)
% Number of type conns : 571 ( 571 >; 0 *; 0 +; 0 <<)
% Number of symbols : 257 ( 256 usr; 17 con; 0-4 aty)
% Number of variables : 3307 ( 40 ^;3199 !; 68 ?;3307 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:37:42.794
%------------------------------------------------------------------------------
% Could-be-implicit typings (38)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia4960592913263135132t_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_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_I_062_It__Nat__Onat_Mt__Int__Oint_J_J_Mt__List__Olist_I_062_It__Nat__Onat_Mt__Int__Oint_J_J_J,type,
produc4919227360403178295at_int: $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_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__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc1186641810826059865st_int: $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__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_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_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_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_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__List__Olist_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
list_nat_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
set_nat_int: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
multiset_list_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__Multiset__Omultiset_Itf__a_J,type,
multiset_a: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $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_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (256)
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_001tf__a_001tf__b,type,
a_l_coset_a_b: partia2175431115845679010xt_a_b > a > set_a > set_a ).
thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
set_add_a_b: partia2175431115845679010xt_a_b > set_a > set_a > set_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_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_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_Oassociated_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,
associ5603075271488036121t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > $o ).
thf(sy_c_Divisibility_Oassociated_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,
associ8249012953061539097t_unit: partia4556295656693239580t_unit > list_set_list_a > list_set_list_a > $o ).
thf(sy_c_Divisibility_Oassociated_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
associ8407585678920448409t_unit: partia2670972154091845814t_unit > list_a > list_a > $o ).
thf(sy_c_Divisibility_Oassociated_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
associ5860276527279195403xt_a_b: partia2175431115845679010xt_a_b > a > a > $o ).
thf(sy_c_Divisibility_Ofactors_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
factor5638265376665762323xt_a_b: partia2175431115845679010xt_a_b > list_a > a > $o ).
thf(sy_c_Divisibility_Omonoid__cancel_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
monoid5798828371819920185xt_a_b: partia2175431115845679010xt_a_b > $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_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,
prime_5738381090551951334t_unit: partia7496981018696276118t_unit > set_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_001tf__a_001tf__b,type,
embedded_Span_a_b: partia2175431115845679010xt_a_b > set_a > list_a > set_a ).
thf(sy_c_Embedded__Algebras_Oring_Ocombine_001tf__a_001tf__b,type,
embedded_combine_a_b: partia2175431115845679010xt_a_b > list_a > list_a > a ).
thf(sy_c_Embedded__Algebras_Oring_Odimension_001tf__a_001tf__b,type,
embedd2795209813406577254on_a_b: partia2175431115845679010xt_a_b > nat > set_a > set_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_Oindependent_001tf__a_001tf__b,type,
embedd5208550302661555450nt_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
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_001tf__a_001tf__b,type,
embedd9027525575939734154ra_a_b: set_a > set_a > partia2175431115845679010xt_a_b > $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_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_Group_OUnits_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,
units_4903515905731149798t_unit: partia2956882679547061052t_unit > set_list_list_a ).
thf(sy_c_Group_OUnits_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
units_2932844235741507942t_unit: partia2670972154091845814t_unit > set_list_a ).
thf(sy_c_Group_OUnits_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,
units_5837875185506529638t_unit: partia7496981018696276118t_unit > set_set_list_a ).
thf(sy_c_Group_OUnits_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
units_a_ring_ext_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Group_Om__inv_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
m_inv_2802811658206063947t_unit: partia2670972154091845814t_unit > list_a > list_a ).
thf(sy_c_Group_Om__inv_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,
m_inv_7863988576679134539t_unit: partia7496981018696276118t_unit > set_list_a > set_list_a ).
thf(sy_c_Group_Om__inv_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
m_inv_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > 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__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,
one_li1622763072977731901t_unit: partia4556295656693239580t_unit > list_set_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_Group_Opow_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,
pow_li1142815632869257134it_nat: partia2670972154091845814t_unit > list_a > nat > list_a ).
thf(sy_c_Group_Opow_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
pow_a_1026414303147256608_b_nat: partia2175431115845679010xt_a_b > a > nat > 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_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
minus_4782336368215558443list_a: set_set_list_a > set_set_list_a > set_set_list_a ).
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_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Ideal_Ocgenideal_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,
cgenid24865672677839267t_unit: partia2956882679547061052t_unit > list_list_a > set_list_list_a ).
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_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,
cgenid9032708300698165283t_unit: partia7496981018696276118t_unit > set_list_a > set_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_Ogenideal_001tf__a_001tf__b,type,
genideal_a_b: partia2175431115845679010xt_a_b > set_a > set_a ).
thf(sy_c_Ideal_Oideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ideal_8896367198367571637t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Ideal_Oideal_001tf__a_001tf__b,type,
ideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
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_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
maxima3875439991530298004t_unit: set_set_list_a > partia7496981018696276118t_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_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
primei7796083425553868872t_unit: set_set_list_a > partia7496981018696276118t_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_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_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_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_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_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
append_set_list_a: list_set_list_a > list_set_list_a > list_set_list_a ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Ofoldr_001tf__a_001t__Set__Oset_Itf__a_J,type,
foldr_a_set_a: ( a > set_a > set_a ) > list_a > set_a > set_a ).
thf(sy_c_List_Olist_OCons_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
cons_nat_int: ( nat > int ) > list_nat_int > list_nat_int ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
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_OCons_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
cons_set_list_a: set_list_a > list_set_list_a > list_set_list_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
nil_nat_int: list_nat_int ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
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_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
nil_set_list_a: list_set_list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Oset_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
set_nat_int2: list_nat_int > set_nat_int ).
thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
set_int2: list_int > set_int ).
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_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_Multiset_Omultiset_Ocount_001tf__a,type,
count_a: multiset_a > a > nat ).
thf(sy_c_Multiset_Oset__mset_001tf__a,type,
set_mset_a: multiset_a > set_a ).
thf(sy_c_Multiset_Osubseteq__mset_001t__List__Olist_Itf__a_J,type,
subseteq_mset_list_a: multiset_list_a > multiset_list_a > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
subseteq_mset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
size_s5718426915756887103at_int: list_nat_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
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__Int__Oint_J_J,type,
bot_bot_set_nat_int: set_nat_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
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_Itf__a_J,type,
ord_less_set_a: set_a > set_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_It__Nat__Onat_Mt__Int__Oint_J_J,type,
ord_le6569500216720880561at_int: set_nat_int > set_nat_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $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_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Polynomial__Divisibility_Opdivides_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno8016796738000020810t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Opdivides_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
polyno9075941895896075626t_unit: partia7496981018696276118t_unit > list_set_list_a > list_set_list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Opdivides_001tf__a_001tf__b,type,
polyno5814909790663948098es_a_b: partia2175431115845679010xt_a_b > list_a > list_a > $o ).
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_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_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_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno5893782122288709345t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a > list_list_a ).
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_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno1727750685288865234t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a > list_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_Oroots_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno7858422826990252003t_unit: partia2670972154091845814t_unit > list_list_a > multiset_list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Oroots_001tf__a_001tf__b,type,
polynomial_roots_a_b: partia2175431115845679010xt_a_b > list_a > multiset_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_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno859807163042199155t_unit: partia2670972154091845814t_unit > set_list_a > list_list_a > partia4960592913263135132t_unit ).
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_Opolynomial_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno1315193887021588240t_unit: partia2670972154091845814t_unit > set_list_a > list_list_a > $o ).
thf(sy_c_Polynomials_Opolynomial_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
polyno3115169382166032176t_unit: partia7496981018696276118t_unit > set_set_list_a > list_set_list_a > $o ).
thf(sy_c_Polynomials_Opolynomial_001tf__a_001tf__b,type,
polynomial_a_b: partia2175431115845679010xt_a_b > set_a > list_a > $o ).
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_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_Omonom_001tf__a_001tf__b,type,
monom_a_b: partia2175431115845679010xt_a_b > a > nat > list_a ).
thf(sy_c_Polynomials_Oring_Opoly__add_001tf__a_001tf__b,type,
poly_add_a_b: partia2175431115845679010xt_a_b > list_a > list_a > 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_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_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
var_se6008125447796440765t_unit: partia7496981018696276118t_unit > list_set_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_I_062_It__Nat__Onat_Mt__Int__Oint_J_J_001t__List__Olist_I_062_It__Nat__Onat_Mt__Int__Oint_J_J,type,
produc1393590784410523119at_int: list_nat_int > list_nat_int > produc4919227360403178295at_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).
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_RingHom_Oring__hom__ring_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_h4589914651911841480t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > ( list_list_a > list_a ) > $o ).
thf(sy_c_RingHom_Oring__hom__ring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h7848885096329822662it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > ( list_a > a ) > $o ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
a_inv_7033018035630854991t_unit: partia2956882679547061052t_unit > list_list_a > list_list_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_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
a_inv_5715216516650856053t_unit: partia7496981018696276118t_unit > set_list_a > set_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_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_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_Ocring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
cring_5991999922451032090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Ocring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
cring_3148771470849435808t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Ocring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
cring_3470013030684506304t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Ocring_001tf__a_001tf__b,type,
cring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Odomain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
domain7810152921033798211t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
domain6553523120543210313t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Odomain_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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
field_1861437471013600865t_unit: partia2956882679547061052t_unit > $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_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
field_1540243473349940225t_unit: partia4960592913263135132t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
field_26233345952514695t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
field_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_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_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_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_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_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_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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $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_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5115406448772830318t_unit: partia7496981018696276118t_unit > set_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5437400583859147359t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r1091214237498979717t_unit: partia7496981018696276118t_unit > set_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Int__Oint_J,type,
insert_nat_int: ( nat > int ) > set_nat_int > set_nat_int ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
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_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
insert_set_list_a: set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subcri7783154434480317835t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subdom3220114454046903646t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subfie4339374749748326226t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_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_001tf__a,type,
bound_a: a > nat > ( nat > 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__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_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_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 ).
% Relevant facts (1278)
thf(fact_0_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_2_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_3_univ__poly__is__cring,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_cring
thf(fact_4_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_5_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_6_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_7_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_8_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_9_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_10_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_11_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_12_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_13_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_14_subcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subcring_a_b @ H @ r ) ) ).
% subcringI'
thf(fact_15_cgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_16_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_17_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_18_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_19_domain_Ouniv__poly__is__cring,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( cring_5991999922451032090t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_cring
thf(fact_20_domain_Ouniv__poly__is__cring,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_cring
thf(fact_21_a__lcos__m__assoc,axiom,
! [M: set_a,G: a,H2: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H2 @ M ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H2 ) @ M ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_22_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_23_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_24_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_25_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_26_subdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subdomain_a_b @ H @ r ) ) ).
% subdomainI'
thf(fact_27_cring_Ocgenideal__is__principalideal,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ I ) @ R ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_28_cring_Ocgenideal__is__principalideal,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ R @ I ) @ R ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_29_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_30_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_31_a__l__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_32_subset__Idl__subset,axiom,
! [I2: set_a,H: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_33_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_34_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_35_is__cring,axiom,
cring_a_b @ r ).
% is_cring
thf(fact_36_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_37_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_38_subsetI,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_39_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_40_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_41_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_42_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_43_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_44_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_45_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_46_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_47_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_48_carrier__polynomial__shell,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_49_lagrange__aux__poly,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% lagrange_aux_poly
thf(fact_50_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_51_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_52_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_55_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_56_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_57_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_58_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_59_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A3 )
=> ( member_list_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_60_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_61_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_62_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_63_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A3 )
=> ( member_list_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_64_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_65_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_66_subsetD,axiom,
! [A2: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_67_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_68_in__mono,axiom,
! [A2: set_list_a,B2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( member_list_a @ X @ A2 )
=> ( member_list_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_69_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_70_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_71_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_72_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_73_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y3: list_a] :
~ ( member_list_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_74_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_75_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_76_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_77_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_78_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_79_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_80_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_81_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q2: list_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q2 )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q2 ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_82_cgenideal__eq__genideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I )
= ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_83_Idl__subset__ideal_H,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_84_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_85_a__lcos__mult__one,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
= M ) ) ).
% a_lcos_mult_one
thf(fact_86_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_87_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_88_poly__of__const__in__carrier,axiom,
! [S2: a] :
( ( member_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_89_domain_Olagrange__aux__poly,axiom,
! [R: partia2670972154091845814t_unit,S: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_list_a @ S )
=> ( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( lagran3534788790333317459t_unit @ R @ S ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_90_domain_Olagrange__aux__poly,axiom,
! [R: partia2175431115845679010xt_a_b,S: set_a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ R @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_91_domain_OsubdomainI_H,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( subdom7821232466298058046t_unit @ H @ R ) ) ) ).
% domain.subdomainI'
thf(fact_92_domain_OsubdomainI_H,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( subdomain_a_b @ H @ R ) ) ) ).
% domain.subdomainI'
thf(fact_93_cring_OsubcringI_H,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( subcri7763218559781929323t_unit @ H @ R ) ) ) ).
% cring.subcringI'
thf(fact_94_cring_OsubcringI_H,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( cring_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( subcring_a_b @ H @ R ) ) ) ).
% cring.subcringI'
thf(fact_95_local_Ominus__unique,axiom,
! [Y: a,X: a,Y4: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y4 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y4 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_96_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_97_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_98_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_99_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_100_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_101_insert__absorb2,axiom,
! [X: list_a,A2: set_list_a] :
( ( insert_list_a @ X @ ( insert_list_a @ X @ A2 ) )
= ( insert_list_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_102_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_103_insert__iff,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_104_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_105_insertCI,axiom,
! [A: list_a,B2: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A @ B2 )
=> ( A = B ) )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_106_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_107_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_108_insert__subset,axiom,
! [X: list_a,A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X @ A2 ) @ B2 )
= ( ( member_list_a @ X @ B2 )
& ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_109_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_110_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_111_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_112_singleton__insert__inj__eq,axiom,
! [B: list_a,A: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_113_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_114_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A2: set_list_a,B: list_a] :
( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_115_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_116_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_117_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_118_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_119_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_120_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_121_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_122_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_123_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B4: set_a] :
( ( A2
= ( insert_a @ A @ B4 ) )
& ~ ( member_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_124_mk__disjoint__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ? [B4: set_list_a] :
( ( A2
= ( insert_list_a @ A @ B4 ) )
& ~ ( member_list_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_125_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_126_insert__commute,axiom,
! [X: list_a,Y: list_a,A2: set_list_a] :
( ( insert_list_a @ X @ ( insert_list_a @ Y @ A2 ) )
= ( insert_list_a @ Y @ ( insert_list_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_127_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B2
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_128_insert__eq__iff,axiom,
! [A: list_a,A2: set_list_a,B: list_a,B2: set_list_a] :
( ~ ( member_list_a @ A @ A2 )
=> ( ~ ( member_list_a @ B @ B2 )
=> ( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_list_a] :
( ( A2
= ( insert_list_a @ B @ C3 ) )
& ~ ( member_list_a @ B @ C3 )
& ( B2
= ( insert_list_a @ A @ C3 ) )
& ~ ( member_list_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_129_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_130_insert__absorb,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_131_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_132_insert__ident,axiom,
! [X: list_a,A2: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ~ ( member_list_a @ X @ B2 )
=> ( ( ( insert_list_a @ X @ A2 )
= ( insert_list_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_133_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B4: set_a] :
( ( A2
= ( insert_a @ X @ B4 ) )
=> ( member_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_134_Set_Oset__insert,axiom,
! [X: list_a,A2: set_list_a] :
( ( member_list_a @ X @ A2 )
=> ~ ! [B4: set_list_a] :
( ( A2
= ( insert_list_a @ X @ B4 ) )
=> ( member_list_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_135_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_136_insertI2,axiom,
! [A: list_a,B2: set_list_a,B: list_a] :
( ( member_list_a @ A @ B2 )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_137_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_138_insertI1,axiom,
! [A: list_a,B2: set_list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ B2 ) ) ).
% insertI1
thf(fact_139_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_140_insertE,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_list_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_141_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_142_subringE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subringE(2)
thf(fact_143_subringE_I2_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subrin5643252653130547402t_unit @ H @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ H ) ) ).
% subringE(2)
thf(fact_144_subringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subringE(2)
thf(fact_145_subcringE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_146_subcringE_I2_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subcri7783154434480317835t_unit @ H @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_147_subcringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_148_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_149_subdomainE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_150_subdomainE_I2_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_151_subdomainE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_152_insert__mono,axiom,
! [C2: set_list_a,D: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ D )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ A @ C2 ) @ ( insert_list_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_153_insert__mono,axiom,
! [C2: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_154_subset__insert,axiom,
! [X: list_a,A2: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B2 ) )
= ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_155_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_156_subset__insertI,axiom,
! [B2: set_list_a,A: list_a] : ( ord_le8861187494160871172list_a @ B2 @ ( insert_list_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_157_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_158_subset__insertI2,axiom,
! [A2: set_list_a,B2: set_list_a,B: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_159_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_160_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_161_singletonD,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_162_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_163_singleton__iff,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_164_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_165_doubleton__eq__iff,axiom,
! [A: list_a,B: list_a,C: list_a,D2: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D2 @ bot_bot_set_list_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_166_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_167_insert__not__empty,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ A2 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_168_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_169_singleton__inject,axiom,
! [A: list_a,B: list_a] :
( ( ( insert_list_a @ A @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_170_subset__singletonD,axiom,
! [A2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
=> ( ( A2 = bot_bot_set_list_a )
| ( A2
= ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_171_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_172_subset__singleton__iff,axiom,
! [X4: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( ( X4 = bot_bot_set_list_a )
| ( X4
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_173_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_174_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_175_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_176_principalideal_Ogenerate,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_177_principalideal_Ogenerate,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( princi8786919440553033881t_unit @ I2 @ R )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
& ( I2
= ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_178_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_179_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_180_cring_Ocgenideal__eq__genideal,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( cgenid547466209912283029xt_a_b @ R @ I )
= ( genideal_a_b @ R @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ) ).
% cring.cgenideal_eq_genideal
thf(fact_181_cring_Ocgenideal__eq__genideal,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( cgenid9131348535277946915t_unit @ R @ I )
= ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ) ).
% cring.cgenideal_eq_genideal
thf(fact_182_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q2: list_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ Q2 )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q2 ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_183_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q2: list_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 @ Q2 )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ Q2 ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_184_subringE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_185_subringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_186_subringE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subringE(4)
thf(fact_187_subcringE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_188_subcringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_189_subcringE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subcringE(4)
thf(fact_190_subcring_Oaxioms_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( subring_a_b @ H @ R ) ) ).
% subcring.axioms(1)
thf(fact_191_subdomainE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_192_subdomainE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_193_subdomainE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subdomainE(4)
thf(fact_194_subdomain_Oaxioms_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( subcring_a_b @ H @ R ) ) ).
% subdomain.axioms(1)
thf(fact_195_subringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subringE(1)
thf(fact_196_subringE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_197_subcringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subcringE(1)
thf(fact_198_subcringE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_199_subdomainE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subdomainE(1)
thf(fact_200_subdomainE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_201_cring_Ocarrier__is__subcring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( subcring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_202_cring_Ocarrier__is__subcring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_203_ring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( P2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_204_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_205_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_206_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_207_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_208_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_209_finite__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) )
= ( finite_finite_list_a @ A2 ) ) ).
% finite_insert
thf(fact_210_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_211_genideal__one,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) )
= ( partia707051561876973205xt_a_b @ r ) ) ).
% genideal_one
thf(fact_212_add_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ H ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H )
=> ! [Xa: a] :
( ( member_a @ Xa @ H )
=> ( member_a @ ( add_a_b @ r @ X2 @ Xa ) @ H ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% add.one_in_subset
thf(fact_213_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_214_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_215_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_216_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_217_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_218_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_219_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_220_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_221_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_ring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).
% ring_primeE(3)
thf(fact_231_ring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI
thf(fact_232_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_233_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_234_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_235_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_236_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_237_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_238_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_239_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_240_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_241_primeideal_Oprimeideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( primeideal_a_b @ I2 @ R ) ) ).
% primeideal.primeideal
thf(fact_242_ring__hom__one,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_243_ring__hom__one,axiom,
! [H2: a > set_list_a,R: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit] :
( ( member_a_set_list_a @ H2 @ ( ring_h6109298854714515236t_unit @ R @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_244_ring__hom__one,axiom,
! [H2: set_list_a > a,R: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b] :
( ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R @ S ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) ) ) ).
% ring_hom_one
thf(fact_245_ring__hom__one,axiom,
! [H2: set_list_a > set_list_a,R: partia7496981018696276118t_unit,S: partia7496981018696276118t_unit] :
( ( member5068272912271824380list_a @ H2 @ ( ring_h6076331213207892940t_unit @ R @ S ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_se1127990129394575805t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_246_subringE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H2: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H2 ) @ H ) ) ) ).
% subringE(5)
thf(fact_247_subringE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H2: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H2 ) @ H ) ) ) ).
% subringE(5)
thf(fact_248_subringE_I3_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subrin5643252653130547402t_unit @ H @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ H ) ) ).
% subringE(3)
thf(fact_249_subringE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subringE(3)
thf(fact_250_subcringE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H2: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H2 ) @ H ) ) ) ).
% subcringE(5)
thf(fact_251_subcringE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H2: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H2 ) @ H ) ) ) ).
% subcringE(5)
thf(fact_252_subdomainE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H2: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H2 ) @ H ) ) ) ).
% subdomainE(5)
thf(fact_253_subdomainE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H2: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H2 ) @ H ) ) ) ).
% subdomainE(5)
thf(fact_254_cring_Ocring__simprules_I21_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X ) )
= X ) ) ) ).
% cring.cring_simprules(21)
thf(fact_255_cring_Ocring__simprules_I21_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
= X ) ) ) ).
% cring.cring_simprules(21)
thf(fact_256_cring_Ocring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_inv_a_b @ R @ X ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% cring.cring_simprules(3)
thf(fact_257_cring_Ocring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% cring.cring_simprules(3)
thf(fact_258_subcringE_I3_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subcri7783154434480317835t_unit @ H @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ H ) ) ).
% subcringE(3)
thf(fact_259_subcringE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subcringE(3)
thf(fact_260_cring_Ocring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( a_inv_a_b @ R @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ).
% cring.cring_simprules(22)
thf(fact_261_cring_Ocring__simprules_I22_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% cring.cring_simprules(22)
thf(fact_262_cring_Ocring__simprules_I22_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( a_inv_5715216516650856053t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% cring.cring_simprules(22)
thf(fact_263_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,X3: a,Y5: a] : ( add_a_b @ R3 @ X3 @ ( a_inv_a_b @ R3 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_264_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R3: partia2670972154091845814t_unit,X3: list_a,Y5: list_a] : ( add_li7652885771158616974t_unit @ R3 @ X3 @ ( a_inv_8944721093294617173t_unit @ R3 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_265_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_266_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_267_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_268_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_269_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_270_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_271_subdomainE_I3_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ H ) ) ).
% subdomainE(3)
thf(fact_272_subdomainE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subdomainE(3)
thf(fact_273_cring_Ocring__simprules_I6_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% cring.cring_simprules(6)
thf(fact_274_cring_Ocring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% cring.cring_simprules(6)
thf(fact_275_cring_Ocring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% cring.cring_simprules(6)
thf(fact_276_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_277_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_278_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_279_primeideal_OI__notcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_280_primeideal_OI__notcarr,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( ( partia5361259788508890537t_unit @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_281_primeideal_Oaxioms_I2_J,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( cring_3148771470849435808t_unit @ R ) ) ).
% primeideal.axioms(2)
thf(fact_282_primeideal_Oaxioms_I2_J,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( cring_a_b @ R ) ) ).
% primeideal.axioms(2)
thf(fact_283_cring_Ocring__simprules_I20_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(20)
thf(fact_284_cring_Ocring__simprules_I20_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(20)
thf(fact_285_cring_Ocring__simprules_I19_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ ( add_a_b @ R @ X @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(19)
thf(fact_286_cring_Ocring__simprules_I19_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(19)
thf(fact_287_cring_Ocring__simprules_I18_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(18)
thf(fact_288_cring_Ocring__simprules_I18_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y ) )
= Y ) ) ) ) ).
% cring.cring_simprules(18)
thf(fact_289_cring_Ocring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( a_minus_a_b @ R @ X @ Y )
= ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) ) ) ) ).
% cring.cring_simprules(15)
thf(fact_290_cring_Ocring__simprules_I15_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( a_minu3984020753470702548t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ).
% cring.cring_simprules(15)
thf(fact_291_subdomain_Osub__one__not__zero,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_292_subdomain_Osub__one__not__zero,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_293_subdomain_Osub__one__not__zero,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_294_cring_Ocring__simprules_I17_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X @ ( a_inv_5715216516650856053t_unit @ R @ X ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% cring.cring_simprules(17)
thf(fact_295_cring_Ocring__simprules_I17_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( a_inv_a_b @ R @ X ) )
= ( zero_a_b @ R ) ) ) ) ).
% cring.cring_simprules(17)
thf(fact_296_cring_Ocring__simprules_I17_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ X ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% cring.cring_simprules(17)
thf(fact_297_cring_Ocring__simprules_I9_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ X )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% cring.cring_simprules(9)
thf(fact_298_cring_Ocring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% cring.cring_simprules(9)
thf(fact_299_cring_Ocring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% cring.cring_simprules(9)
thf(fact_300_cring_Osum__zero__eq__neg,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( add_se2486902527185523630t_unit @ R @ X @ Y )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( X
= ( a_inv_5715216516650856053t_unit @ R @ Y ) ) ) ) ) ) ).
% cring.sum_zero_eq_neg
thf(fact_301_cring_Osum__zero__eq__neg,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( add_a_b @ R @ X @ Y )
= ( zero_a_b @ R ) )
=> ( X
= ( a_inv_a_b @ R @ Y ) ) ) ) ) ) ).
% cring.sum_zero_eq_neg
thf(fact_302_cring_Osum__zero__eq__neg,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( X
= ( a_inv_8944721093294617173t_unit @ R @ Y ) ) ) ) ) ) ).
% cring.sum_zero_eq_neg
thf(fact_303_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( ( partia141011252114345353t_unit @ R )
!= ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) )
= ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_304_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_305_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_306_semiring_Ocarrier__one__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( ( partia141011252114345353t_unit @ R )
= ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) )
= ( ( one_se1127990129394575805t_unit @ R )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_307_semiring_Ocarrier__one__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_308_semiring_Ocarrier__one__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_309_semiring_Oone__zeroI,axiom,
! [R: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( ( partia141011252114345353t_unit @ R )
= ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) )
=> ( ( one_se1127990129394575805t_unit @ R )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_310_semiring_Oone__zeroI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_311_semiring_Oone__zeroI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_312_semiring_Oone__zeroD,axiom,
! [R: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( ( one_se1127990129394575805t_unit @ R )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( partia141011252114345353t_unit @ R )
= ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_313_semiring_Oone__zeroD,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( 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 ) ) ) ) ).
% semiring.one_zeroD
thf(fact_314_semiring_Oone__zeroD,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_315_cring_Ois__cring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( cring_3148771470849435808t_unit @ R ) ) ).
% cring.is_cring
thf(fact_316_cring_Ois__cring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( cring_a_b @ R ) ) ).
% cring.is_cring
thf(fact_317_domain_Ozeroprimeideal,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( primei7796083425553868872t_unit @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_318_domain_Ozeroprimeideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_319_domain_Ozeroprimeideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_320_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 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_321_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 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_322_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 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_323_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 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_324_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_325_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_326_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_327_cring_Odomain__eq__zeroprimeideal,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( domain1617769409708967785t_unit @ R )
= ( primei7796083425553868872t_unit @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) @ R ) ) ) ).
% cring.domain_eq_zeroprimeideal
thf(fact_328_cring_Odomain__eq__zeroprimeideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( domain_a_b @ R )
= ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ) ).
% cring.domain_eq_zeroprimeideal
thf(fact_329_cring_Odomain__eq__zeroprimeideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( domain6553523120543210313t_unit @ R )
= ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ) ).
% cring.domain_eq_zeroprimeideal
thf(fact_330_cring_Ozeroprimeideal__domainI,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( primei7796083425553868872t_unit @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) @ R )
=> ( domain1617769409708967785t_unit @ R ) ) ) ).
% cring.zeroprimeideal_domainI
thf(fact_331_cring_Ozeroprimeideal__domainI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R )
=> ( domain_a_b @ R ) ) ) ).
% cring.zeroprimeideal_domainI
thf(fact_332_cring_Ozeroprimeideal__domainI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ) ).
% cring.zeroprimeideal_domainI
thf(fact_333_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_334_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_335_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_336_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_337_finite_OinsertI,axiom,
! [A2: set_list_a,A: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_338_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_339_ring__hom__closed,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_340_ring__hom__closed,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_341_ring__hom__closed,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_342_ring__hom__closed,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_343_domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( cring_a_b @ R ) ) ).
% domain.axioms(1)
thf(fact_344_domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( cring_3148771470849435808t_unit @ R ) ) ).
% domain.axioms(1)
thf(fact_345_semiring_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( abelia226231641709521465t_unit @ R ) ) ).
% semiring.axioms(1)
thf(fact_346_semiring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( abelian_monoid_a_b @ R ) ) ).
% semiring.axioms(1)
thf(fact_347_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 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_348_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_349_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 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_350_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_351_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A4: set_a] :
( ? [A5: a] :
( A
= ( insert_a @ A5 @ A4 ) )
=> ~ ( finite_finite_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_352_finite_Ocases,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( A != bot_bot_set_list_a )
=> ~ ! [A4: set_list_a] :
( ? [A5: list_a] :
( A
= ( insert_list_a @ A5 @ A4 ) )
=> ~ ( finite_finite_list_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_353_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A6: set_a] :
( ( A6 = bot_bot_set_a )
| ? [A3: set_a,B5: a] :
( ( A6
= ( insert_a @ B5 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_354_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A6: set_list_a] :
( ( A6 = bot_bot_set_list_a )
| ? [A3: set_list_a,B5: list_a] :
( ( A6
= ( insert_list_a @ B5 @ A3 ) )
& ( finite_finite_list_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_355_finite__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X2 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_356_finite__induct,axiom,
! [F: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X2: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ~ ( member_list_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ X2 @ F2 ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_induct
thf(fact_357_finite__ne__induct,axiom,
! [F: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( F != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X2 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_358_finite__ne__induct,axiom,
! [F: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( F != bot_bot_set_list_a )
=> ( ! [X2: list_a] : ( P @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
=> ( ! [X2: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( F2 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ X2 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_ne_induct
thf(fact_359_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A4: set_a] :
( ~ ( finite_finite_a @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ~ ( member_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ X2 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_360_infinite__finite__induct,axiom,
! [P: set_list_a > $o,A2: set_list_a] :
( ! [A4: set_list_a] :
( ~ ( finite_finite_list_a @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X2: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ~ ( member_list_a @ X2 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ X2 @ F2 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_361_cring_Ocring__simprules_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% cring.cring_simprules(2)
thf(fact_362_cring_Ocring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% cring.cring_simprules(2)
thf(fact_363_cring_Ocring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% cring.cring_simprules(2)
thf(fact_364_cring_Ocring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(23)
thf(fact_365_cring_Ocring__simprules_I23_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(23)
thf(fact_366_cring_Ocring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% cring.cring_simprules(10)
thf(fact_367_cring_Ocring__simprules_I10_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% cring.cring_simprules(10)
thf(fact_368_cring_Ocring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(7)
thf(fact_369_cring_Ocring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(7)
thf(fact_370_cring_Ocring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% cring.cring_simprules(1)
thf(fact_371_cring_Ocring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% cring.cring_simprules(1)
thf(fact_372_ring__hom__add,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_373_ring__hom__add,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_374_ring__hom__add,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_375_ring__hom__add,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_376_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_377_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_378_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_379_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_380_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_381_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_382_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_383_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_384_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_385_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_386_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_387_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_388_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_389_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_390_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_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_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_401_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_402_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_403_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_404_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_405_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_406_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_407_cring_Ocring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_minus_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% cring.cring_simprules(4)
thf(fact_408_cring_Ocring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% cring.cring_simprules(4)
thf(fact_409_finite__subset__induct_H,axiom,
! [F: set_list_a,A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( ord_le8861187494160871172list_a @ F @ A2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A5: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( member_list_a @ A5 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A2 )
=> ( ~ ( member_list_a @ A5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ A5 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_410_finite__subset__induct_H,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ~ ( member_a @ A5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A5 @ F2 ) ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_411_finite__subset__induct,axiom,
! [F: set_list_a,A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F )
=> ( ( ord_le8861187494160871172list_a @ F @ A2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A5: list_a,F2: set_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( ( member_list_a @ A5 @ A2 )
=> ( ~ ( member_list_a @ A5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_list_a @ A5 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_412_finite__subset__induct,axiom,
! [F: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F )
=> ( ( ord_less_eq_set_a @ F @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( member_a @ A5 @ A2 )
=> ( ~ ( member_a @ A5 @ F2 )
=> ( ( P @ F2 )
=> ( P @ ( insert_a @ A5 @ F2 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ) ).
% finite_subset_induct
thf(fact_413_cring_Ocring__simprules_I16_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X @ ( zero_s2910681146719230829t_unit @ R ) )
= X ) ) ) ).
% cring.cring_simprules(16)
thf(fact_414_cring_Ocring__simprules_I16_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% cring.cring_simprules(16)
thf(fact_415_cring_Ocring__simprules_I16_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% cring.cring_simprules(16)
thf(fact_416_cring_Ocring__simprules_I8_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(8)
thf(fact_417_cring_Ocring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(8)
thf(fact_418_cring_Ocring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(8)
thf(fact_419_abelian__monoidI,axiom,
! [R: partia7496981018696276118t_unit] :
( ! [X2: set_list_a,Y3: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ X2 @ Y3 ) @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ! [X2: set_list_a,Y3: set_list_a,Z3: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Z3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X2 @ Y3 ) @ Z3 )
= ( add_se2486902527185523630t_unit @ R @ X2 @ ( add_se2486902527185523630t_unit @ R @ Y3 @ 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,Y3: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X2 @ Y3 )
= ( add_se2486902527185523630t_unit @ R @ Y3 @ X2 ) ) ) )
=> ( abelia3322010900105369177t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_420_abelian__monoidI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X2 @ Y3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X2: a,Y3: a,Z3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X2 @ Y3 ) @ Z3 )
= ( add_a_b @ R @ X2 @ ( add_a_b @ R @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X2 @ Y3 )
= ( add_a_b @ R @ Y3 @ X2 ) ) ) )
=> ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_421_abelian__monoidI,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ! [X2: list_a,Y3: list_a,Z3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) @ Z3 )
= ( add_li7652885771158616974t_unit @ R @ X2 @ ( add_li7652885771158616974t_unit @ R @ Y3 @ Z3 ) ) ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X2 )
= X2 ) )
=> ( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 )
= ( add_li7652885771158616974t_unit @ R @ Y3 @ X2 ) ) ) )
=> ( abelia226231641709521465t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_422_abelian__monoid_Ominus__unique,axiom,
! [G2: partia7496981018696276118t_unit,Y: set_list_a,X: set_list_a,Y4: set_list_a] :
( ( abelia3322010900105369177t_unit @ G2 )
=> ( ( ( add_se2486902527185523630t_unit @ G2 @ Y @ X )
= ( zero_s2910681146719230829t_unit @ G2 ) )
=> ( ( ( add_se2486902527185523630t_unit @ G2 @ X @ Y4 )
= ( zero_s2910681146719230829t_unit @ G2 ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ G2 ) )
=> ( ( member_set_list_a @ Y4 @ ( partia141011252114345353t_unit @ G2 ) )
=> ( Y = Y4 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_423_abelian__monoid_Ominus__unique,axiom,
! [G2: partia2175431115845679010xt_a_b,Y: a,X: a,Y4: a] :
( ( abelian_monoid_a_b @ G2 )
=> ( ( ( add_a_b @ G2 @ Y @ X )
= ( zero_a_b @ G2 ) )
=> ( ( ( add_a_b @ G2 @ X @ Y4 )
= ( zero_a_b @ G2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ G2 ) )
=> ( Y = Y4 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_424_abelian__monoid_Ominus__unique,axiom,
! [G2: partia2670972154091845814t_unit,Y: list_a,X: list_a,Y4: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( ( add_li7652885771158616974t_unit @ G2 @ Y @ X )
= ( zero_l4142658623432671053t_unit @ G2 ) )
=> ( ( ( add_li7652885771158616974t_unit @ G2 @ X @ Y4 )
= ( zero_l4142658623432671053t_unit @ G2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( Y = Y4 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_425_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_426_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_427_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_428_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_429_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_430_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_431_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_432_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_433_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_434_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_435_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_436_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_437_maximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ r )
=> ( primeideal_a_b @ I2 @ r ) ) ).
% maximalideal_prime
thf(fact_438_primeideal__iff__prime,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeideal_iff_prime
thf(fact_439_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_440_subringI,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ H ) ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ H ) ) )
=> ( subring_a_b @ H @ r ) ) ) ) ) ) ).
% subringI
thf(fact_441_subdomainI,axiom,
! [H: set_a] :
( ( subcring_a_b @ H @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
= ( zero_a_b @ r ) )
=> ( ( H12
= ( zero_a_b @ r ) )
| ( H23
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H @ r ) ) ) ) ).
% subdomainI
thf(fact_442_const__term__simprules__shell_I3_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q2 ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P2 ) @ ( const_term_a_b @ r @ Q2 ) ) ) ) ) ) ).
% const_term_simprules_shell(3)
thf(fact_443_domain_Oring__primeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P2 )
=> ( prime_a_ring_ext_a_b @ R @ P2 ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_444_domain_Oring__primeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P2 )
=> ( prime_2011924034616061926t_unit @ R @ P2 ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_445_domain_Oring__primeE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,P2: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r1091214237498979717t_unit @ R @ P2 )
=> ( P2
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_446_domain_Oring__primeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P2 )
=> ( P2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_447_domain_Oring__primeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P2 )
=> ( P2
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_448_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,R2: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ R2 )
=> ( R2
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_449_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( R2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_450_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ( R2
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_451_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_452_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_453_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_454_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_455_m__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_456_m__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_457_integral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_458_local_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_459_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_460_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_461_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_462_inv__unique,axiom,
! [Y: a,X: a,Y4: 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 @ Y4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y4 ) ) ) ) ) ) ).
% inv_unique
thf(fact_463_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_464_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_465_Diff__idemp,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_466_Diff__iff,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
= ( ( member_list_a @ C @ A2 )
& ~ ( member_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_467_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_468_DiffI,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_469_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_470_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 )
& ? [Y5: a] :
( ( member_a @ Y5 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ A ) @ Y5 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_471_subcringI,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( mult_a_ring_ext_a_b @ r @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ r @ H23 @ H12 ) ) ) )
=> ( subcring_a_b @ H @ r ) ) ) ).
% subcringI
thf(fact_472_univ__poly__a__inv__consistent,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_473_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_474_const__term__simprules__shell_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_a @ ( const_term_a_b @ r @ P2 ) @ K ) ) ) ).
% const_term_simprules_shell(1)
thf(fact_475_const__term__simprules__shell_I4_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) )
= ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P2 ) ) ) ) ) ).
% const_term_simprules_shell(4)
thf(fact_476_Diff__empty,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Diff_empty
thf(fact_477_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_478_empty__Diff,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A2 )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_479_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_480_Diff__cancel,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ A2 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_481_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_482_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_483_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_484_insert__Diff1,axiom,
! [X: list_a,B2: set_list_a,A2: set_list_a] :
( ( member_list_a @ X @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ B2 )
= ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_485_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_486_Diff__insert0,axiom,
! [X: list_a,A2: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ B2 ) )
= ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_487_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_488_Diff__eq__empty__iff,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( minus_646659088055828811list_a @ A2 @ B2 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_489_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_490_insert__Diff__single,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_491_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_492_finite__Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B2: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B2 ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_493_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_494_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_495_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_496_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_497_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_498_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_499_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_500_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_501_DiffD2,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ~ ( member_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_502_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_503_DiffD1,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ( member_list_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_504_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_505_DiffE,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ~ ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_506_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_507_maximalideal_Ois__maximalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( maximalideal_a_b @ I2 @ R ) ) ).
% maximalideal.is_maximalideal
thf(fact_508_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_509_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C2 @ D ) ) ) ) ).
% Diff_mono
thf(fact_510_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_511_double__diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_512_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_513_insert__Diff__if,axiom,
! [X: list_a,B2: set_list_a,A2: set_list_a] :
( ( ( member_list_a @ X @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ B2 )
= ( minus_646659088055828811list_a @ A2 @ B2 ) ) )
& ( ~ ( member_list_a @ X @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ B2 )
= ( insert_list_a @ X @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_514_insert__Diff__if,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_515_subringE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_516_subringE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_517_subcringE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_518_subcringE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_519_subcring_Osub__m__comm,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 )
= ( mult_l7073676228092353617t_unit @ R @ H22 @ H1 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_520_subcring_Osub__m__comm,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ R @ H22 @ H1 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_521_subdomainE_I8_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 )
= ( mult_l7073676228092353617t_unit @ R @ H22 @ H1 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_522_subdomainE_I8_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ R @ H22 @ H1 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_523_subdomainE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_524_subdomainE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H22 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_525_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ Q2 ) )
= ( mult_l7073676228092353617t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P2 ) @ ( const_6738166269504826821t_unit @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_526_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 @ Q2 ) )
= ( mult_a_ring_ext_a_b @ R @ ( const_term_a_b @ R @ P2 ) @ ( const_term_a_b @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_527_maximalideal_OI__notcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_528_maximalideal_OI__notcarr,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( maxima6585700282301356660t_unit @ I2 @ R )
=> ( ( partia5361259788508890537t_unit @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_529_subset__Diff__insert,axiom,
! [A2: set_list_a,B2: set_list_a,X: list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B2 @ ( insert_list_a @ X @ C2 ) ) )
= ( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B2 @ C2 ) )
& ~ ( member_list_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_530_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_531_Diff__insert__absorb,axiom,
! [X: list_a,A2: set_list_a] :
( ~ ( member_list_a @ X @ A2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X @ A2 ) @ ( insert_list_a @ X @ bot_bot_set_list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_532_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_533_Diff__insert2,axiom,
! [A2: set_list_a,A: list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_534_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_535_insert__Diff,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_536_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_537_Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_538_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_539_cring_Ocring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% cring.cring_simprules(5)
thf(fact_540_cring_Ocring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% cring.cring_simprules(5)
thf(fact_541_cring_Ocring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(11)
thf(fact_542_cring_Ocring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(11)
thf(fact_543_cring_Ocring__simprules_I14_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ Y )
= ( mult_a_ring_ext_a_b @ R @ Y @ X ) ) ) ) ) ).
% cring.cring_simprules(14)
thf(fact_544_cring_Ocring__simprules_I14_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ Y )
= ( mult_l7073676228092353617t_unit @ R @ Y @ X ) ) ) ) ) ).
% cring.cring_simprules(14)
thf(fact_545_cring_Ocring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ R @ Y @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(24)
thf(fact_546_cring_Ocring__simprules_I24_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ R @ Y @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(24)
thf(fact_547_ring__hom__mult,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_548_ring__hom__mult,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_549_ring__hom__mult,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_550_ring__hom__mult,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_551_primeideal_OI__prime,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( primeideal_a_b @ I2 @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ A @ B ) @ I2 )
=> ( ( member_a @ A @ I2 )
| ( member_a @ B @ I2 ) ) ) ) ) ) ).
% primeideal.I_prime
thf(fact_552_primeideal_OI__prime,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ A @ B ) @ I2 )
=> ( ( member_list_a @ A @ I2 )
| ( member_list_a @ B @ I2 ) ) ) ) ) ) ).
% primeideal.I_prime
thf(fact_553_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_554_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_555_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_556_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_557_subdomain_Osubintegral,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit,H1: set_list_a,H22: set_list_a] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( ( member_set_list_a @ H1 @ H )
=> ( ( member_set_list_a @ H22 @ H )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ H1 @ H22 )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( H1
= ( zero_s2910681146719230829t_unit @ R ) )
| ( H22
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_558_subdomain_Osubintegral,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H1: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( H1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( H22
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_559_subdomain_Osubintegral,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H1: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( zero_a_b @ R ) )
=> ( ( H1
= ( zero_a_b @ R ) )
| ( H22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_560_domain_Ointegral,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_561_domain_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_562_domain_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_563_domain_Om__lcancel,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( mult_s7802724872828879953t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_564_domain_Om__lcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( mult_a_ring_ext_a_b @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_565_domain_Om__lcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( mult_l7073676228092353617t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_566_domain_Om__rcancel,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ B @ A )
= ( mult_s7802724872828879953t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_567_domain_Om__rcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ B @ A )
= ( mult_a_ring_ext_a_b @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_568_domain_Om__rcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ B @ A )
= ( mult_l7073676228092353617t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_569_domain_Ointegral__iff,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
= ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_570_domain_Ointegral__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
= ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_571_domain_Ointegral__iff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_572_subset__insert__iff,axiom,
! [A2: set_list_a,X: list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B2 ) )
= ( ( ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B2 ) )
& ( ~ ( member_list_a @ X @ A2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_573_subset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_574_Diff__single__insert,axiom,
! [A2: set_list_a,X: list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_575_Diff__single__insert,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_576_finite__empty__induct,axiom,
! [A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: list_a,A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ( member_list_a @ A5 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ A5 @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_577_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: a,A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A5 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_578_infinite__coinduct,axiom,
! [X4: set_list_a > $o,A2: set_list_a] :
( ( X4 @ A2 )
=> ( ! [A4: set_list_a] :
( ( X4 @ A4 )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ A4 )
& ( ( X4 @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_579_infinite__coinduct,axiom,
! [X4: set_a > $o,A2: set_a] :
( ( X4 @ A2 )
=> ( ! [A4: set_a] :
( ( X4 @ A4 )
=> ? [X5: a] :
( ( member_a @ X5 @ A4 )
& ( ( X4 @ ( minus_minus_set_a @ A4 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_580_infinite__remove,axiom,
! [S: set_list_a,A: list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_581_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_582_cring_Ocring__simprules_I26_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_583_cring_Ocring__simprules_I26_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
= ( zero_a_b @ R ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_584_cring_Ocring__simprules_I26_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% cring.cring_simprules(26)
thf(fact_585_cring_Ocring__simprules_I27_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ X @ ( zero_s2910681146719230829t_unit @ R ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_586_cring_Ocring__simprules_I27_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_587_cring_Ocring__simprules_I27_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% cring.cring_simprules(27)
thf(fact_588_cring_Ocring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(13)
thf(fact_589_cring_Ocring__simprules_I13_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% cring.cring_simprules(13)
thf(fact_590_cring_Ocring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% cring.cring_simprules(25)
thf(fact_591_cring_Ocring__simprules_I25_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ R @ Z @ Y ) ) ) ) ) ) ) ).
% cring.cring_simprules(25)
thf(fact_592_cring_Ocring__simprules_I12_J,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(12)
thf(fact_593_cring_Ocring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(12)
thf(fact_594_cring_Ocring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X )
= X ) ) ) ).
% cring.cring_simprules(12)
thf(fact_595_cring_Ocring__simprules_I28_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ X ) @ Y )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(28)
thf(fact_596_cring_Ocring__simprules_I28_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ Y )
= ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(28)
thf(fact_597_cring_Ocring__simprules_I29_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X @ ( a_inv_a_b @ R @ Y ) )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(29)
thf(fact_598_cring_Ocring__simprules_I29_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X @ ( a_inv_8944721093294617173t_unit @ R @ Y ) )
= ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X @ Y ) ) ) ) ) ) ).
% cring.cring_simprules(29)
thf(fact_599_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_600_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_601_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_602_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_603_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_604_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_605_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_606_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_607_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_608_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_609_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_610_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_611_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_612_cring_Omaximalideal__prime,axiom,
! [R: partia2670972154091845814t_unit,I2: set_list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( maxima6585700282301356660t_unit @ I2 @ R )
=> ( primei6309817859076077608t_unit @ I2 @ R ) ) ) ).
% cring.maximalideal_prime
thf(fact_613_cring_Omaximalideal__prime,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a] :
( ( cring_a_b @ R )
=> ( ( maximalideal_a_b @ I2 @ R )
=> ( primeideal_a_b @ I2 @ R ) ) ) ).
% cring.maximalideal_prime
thf(fact_614_remove__induct,axiom,
! [P: set_list_a > $o,B2: set_list_a] :
( ( P @ bot_bot_set_list_a )
=> ( ( ~ ( finite_finite_list_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ( A4 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A4 @ B2 )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A4 )
=> ( P @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_615_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_616_finite__remove__induct,axiom,
! [B2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ B2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ( A4 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A4 @ B2 )
=> ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A4 )
=> ( P @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_617_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_618_domain_Osquare__eq__one,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ X @ X )
= ( one_se1127990129394575805t_unit @ R ) )
=> ( ( X
= ( one_se1127990129394575805t_unit @ R ) )
| ( X
= ( a_inv_5715216516650856053t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_619_domain_Osquare__eq__one,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ X @ X )
= ( one_a_ring_ext_a_b @ R ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ R ) )
| ( X
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_620_domain_Osquare__eq__one,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ X @ X )
= ( one_li8328186300101108157t_unit @ R ) )
=> ( ( X
= ( one_li8328186300101108157t_unit @ R ) )
| ( X
= ( a_inv_8944721093294617173t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_621_ring__hom__memI,axiom,
! [R: partia7496981018696276118t_unit,H2: set_list_a > set_list_a,S: partia7496981018696276118t_unit] :
( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_set_list_a @ ( H2 @ X2 ) @ ( partia141011252114345353t_unit @ S ) ) )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H2 @ ( mult_s7802724872828879953t_unit @ R @ X2 @ Y3 ) )
= ( mult_s7802724872828879953t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H2 @ ( add_se2486902527185523630t_unit @ R @ X2 @ Y3 ) )
= ( add_se2486902527185523630t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_se1127990129394575805t_unit @ S ) )
=> ( member5068272912271824380list_a @ H2 @ ( ring_h6076331213207892940t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_622_ring__hom__memI,axiom,
! [R: partia7496981018696276118t_unit,H2: set_list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_a @ ( H2 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H2 @ ( mult_s7802724872828879953t_unit @ R @ X2 @ Y3 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H2 @ ( add_se2486902527185523630t_unit @ R @ X2 @ Y3 ) )
= ( add_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_623_ring__hom__memI,axiom,
! [R: partia7496981018696276118t_unit,H2: set_list_a > list_a,S: partia2670972154091845814t_unit] :
( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H2 @ ( mult_s7802724872828879953t_unit @ R @ X2 @ Y3 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ! [X2: set_list_a,Y3: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H2 @ ( add_se2486902527185523630t_unit @ R @ X2 @ Y3 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ( ( H2 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member5910328476188217884list_a @ H2 @ ( ring_h8038483918290310060t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_624_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > set_list_a,S: partia7496981018696276118t_unit] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_set_list_a @ ( H2 @ X2 ) @ ( partia141011252114345353t_unit @ S ) ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y3 ) )
= ( mult_s7802724872828879953t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X2 @ Y3 ) )
= ( add_se2486902527185523630t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_se1127990129394575805t_unit @ S ) )
=> ( member_a_set_list_a @ H2 @ ( ring_h6109298854714515236t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_625_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > a,S: partia2175431115845679010xt_a_b] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H2 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y3 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X2 @ Y3 ) )
= ( add_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_626_ring__hom__memI,axiom,
! [R: partia2175431115845679010xt_a_b,H2: a > list_a,S: partia2670972154091845814t_unit] :
( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H2 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y3 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X2 @ Y3 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ( ( H2 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_627_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > set_list_a,S: partia7496981018696276118t_unit] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_set_list_a @ ( H2 @ X2 ) @ ( partia141011252114345353t_unit @ S ) ) )
=> ( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y3 ) )
= ( mult_s7802724872828879953t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) )
= ( add_se2486902527185523630t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ( ( H2 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_se1127990129394575805t_unit @ S ) )
=> ( member4263473470251683292list_a @ H2 @ ( ring_h6188449271506562988t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_628_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > a,S: partia2175431115845679010xt_a_b] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H2 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y3 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) )
= ( add_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ( ( H2 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_629_ring__hom__memI,axiom,
! [R: partia2670972154091845814t_unit,H2: list_a > list_a,S: partia2670972154091845814t_unit] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y3 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y3 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y3 ) ) ) ) )
=> ( ( ( H2 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R @ S ) ) ) ) ) ) ).
% ring_hom_memI
thf(fact_630_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_631_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_632_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ R @ P2 ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_633_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_a @ ( const_term_a_b @ R @ P2 ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_634_domain_Oprimeideal__iff__prime,axiom,
! [R: partia7496981018696276118t_unit,P2: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P2 @ ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
=> ( ( primei7796083425553868872t_unit @ ( cgenid9032708300698165283t_unit @ R @ P2 ) @ R )
= ( ring_r1091214237498979717t_unit @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_635_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ P2 ) @ R )
= ( ring_ring_prime_a_b @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_636_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( primei6309817859076077608t_unit @ ( cgenid9131348535277946915t_unit @ R @ P2 ) @ R )
= ( ring_r6430282645014804837t_unit @ R @ P2 ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_637_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ Q2 ) )
= ( add_li7652885771158616974t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P2 ) @ ( const_6738166269504826821t_unit @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_638_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 @ Q2 ) )
= ( add_a_b @ R @ ( const_term_a_b @ R @ P2 ) @ ( const_term_a_b @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_639_domain_Oconst__term__simprules__shell_I4_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) )
= ( a_inv_8944721093294617173t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P2 ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(4)
thf(fact_640_domain_Oconst__term__simprules__shell_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) )
= ( a_inv_a_b @ R @ ( const_term_a_b @ R @ P2 ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(4)
thf(fact_641_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( prime_2011924034616061926t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_642_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( prime_5738381090551951334t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_643_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_644_ring__prime__def,axiom,
( ring_r1091214237498979717t_unit
= ( ^ [R3: partia7496981018696276118t_unit,A6: set_list_a] :
( ( A6
!= ( zero_s2910681146719230829t_unit @ R3 ) )
& ( prime_5738381090551951334t_unit @ R3 @ A6 ) ) ) ) ).
% ring_prime_def
thf(fact_645_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A6: a] :
( ( A6
!= ( zero_a_b @ R3 ) )
& ( prime_a_ring_ext_a_b @ R3 @ A6 ) ) ) ) ).
% ring_prime_def
thf(fact_646_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R3: partia2670972154091845814t_unit,A6: list_a] :
( ( A6
!= ( zero_l4142658623432671053t_unit @ R3 ) )
& ( prime_2011924034616061926t_unit @ R3 @ A6 ) ) ) ) ).
% ring_prime_def
thf(fact_647_domainI,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ! [A5: set_list_a,B6: set_list_a] :
( ( ( mult_s7802724872828879953t_unit @ R @ A5 @ B6 )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B6 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A5
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B6
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) )
=> ( domain1617769409708967785t_unit @ R ) ) ) ) ).
% domainI
thf(fact_648_domainI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) )
=> ( ! [A5: a,B6: a] :
( ( ( mult_a_ring_ext_a_b @ R @ A5 @ B6 )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A5
= ( zero_a_b @ R ) )
| ( B6
= ( zero_a_b @ R ) ) ) ) ) )
=> ( domain_a_b @ R ) ) ) ) ).
% domainI
thf(fact_649_domainI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ! [A5: list_a,B6: list_a] :
( ( ( mult_l7073676228092353617t_unit @ R @ A5 @ B6 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A5
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B6
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) )
=> ( domain6553523120543210313t_unit @ R ) ) ) ) ).
% domainI
thf(fact_650_ring__irreducibleI,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A5: a,B6: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A5 @ B6 ) )
=> ( ( member_a @ A5 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B6 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R2 ) ) ) ) ).
% ring_irreducibleI
thf(fact_651_monoid__cancelI,axiom,
( ! [A5: a,B6: a,C4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C4 @ A5 )
= ( mult_a_ring_ext_a_b @ r @ C4 @ B6 ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A5 = B6 ) ) ) ) )
=> ( ! [A5: a,B6: a,C4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A5 @ C4 )
= ( mult_a_ring_ext_a_b @ r @ B6 @ C4 ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A5 = B6 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_652_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_653_const__term__simprules__shell_I2_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P2 ) @ ( const_term_a_b @ r @ Q2 ) ) ) ) ) ) ).
% const_term_simprules_shell(2)
thf(fact_654_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_655_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_656_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_657_subring__props_I7_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_658_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_659_subring__props_I6_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_660_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_661_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_662_subring__props_I5_J,axiom,
! [K: set_a,H2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( a_inv_a_b @ r @ H2 ) @ K ) ) ) ).
% subring_props(5)
thf(fact_663_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_664_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_665_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_666_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_667_pprime__iff__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_668_Units__inv__comm,axiom,
! [X: a,Y: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_669_ideal__eq__carrier__iff,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_670_ring__irreducibleE_I4_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_671_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_672_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_673_ring__irreducibleE_I5_J,axiom,
! [R2: a,A: a,B: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_674_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K3: a,V2: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V2 ) @ 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_675_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A5
!= ( zero_a_b @ r ) )
=> ? [X5: a] :
( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A5 @ X5 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_676_cring__fieldI,axiom,
( ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ r ) ) ).
% cring_fieldI
thf(fact_677_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_678_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_679_Units__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_one_closed
thf(fact_680_Units__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_681_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_682_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_683_field_Ocarrier__is__subfield,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( subfie4339374749748326226t_unit @ ( partia141011252114345353t_unit @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_684_field_Ocarrier__is__subfield,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( subfield_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_685_field_Ocarrier__is__subfield,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( subfie1779122896746047282t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% field.carrier_is_subfield
thf(fact_686_subfieldE_I4_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_a @ K1 @ K )
=> ( ( member_list_a @ K22 @ K )
=> ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
= ( mult_l7073676228092353617t_unit @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_687_subfieldE_I4_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K @ R )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K22 @ K )
=> ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( mult_a_ring_ext_a_b @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_688_subfieldE_I1_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subring_a_b @ K @ R ) ) ).
% subfieldE(1)
thf(fact_689_subfieldE_I2_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subcring_a_b @ K @ R ) ) ).
% subfieldE(2)
thf(fact_690_subfield_Oaxioms_I1_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subdomain_a_b @ K @ R ) ) ).
% subfield.axioms(1)
thf(fact_691_field_Oaxioms_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( domain1617769409708967785t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_692_field_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% field.axioms(1)
thf(fact_693_field_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_694_fieldE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( cring_a_b @ R ) ) ).
% fieldE(1)
thf(fact_695_fieldE_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( cring_3148771470849435808t_unit @ R ) ) ).
% fieldE(1)
thf(fact_696_fieldE_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( cring_3470013030684506304t_unit @ R ) ) ).
% fieldE(1)
thf(fact_697_subfieldE_I3_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subfieldE(3)
thf(fact_698_subfieldE_I3_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_699_subfieldE_I5_J,axiom,
! [K: set_set_list_a,R: partia7496981018696276118t_unit,K1: set_list_a,K22: set_list_a] :
( ( subfie4339374749748326226t_unit @ K @ R )
=> ( ( member_set_list_a @ K1 @ K )
=> ( ( member_set_list_a @ K22 @ K )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ K1 @ K22 )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( K1
= ( zero_s2910681146719230829t_unit @ R ) )
| ( K22
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_700_subfieldE_I5_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_a @ K1 @ K )
=> ( ( member_list_a @ K22 @ K )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( K1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( K22
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_701_subfieldE_I5_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K @ R )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K22 @ K )
=> ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( zero_a_b @ R ) )
=> ( ( K1
= ( zero_a_b @ R ) )
| ( K22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_702_subfieldE_I6_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% subfieldE(6)
thf(fact_703_subfieldE_I6_J,axiom,
! [K: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subfie4339374749748326226t_unit @ K @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% subfieldE(6)
thf(fact_704_subfieldE_I6_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subfieldE(6)
thf(fact_705_Ring_Oone__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_706_Ring_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_707_Ring_Oone__not__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_708_field__Units,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( units_5837875185506529638t_unit @ R )
= ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) ) ) ).
% field_Units
thf(fact_709_field__Units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( units_a_ring_ext_a_b @ R )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% field_Units
thf(fact_710_field__Units,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( units_2932844235741507942t_unit @ R )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ).
% field_Units
thf(fact_711_cring_Ocring__fieldI,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( ( units_5837875185506529638t_unit @ R )
= ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
=> ( field_26233345952514695t_unit @ R ) ) ) ).
% cring.cring_fieldI
thf(fact_712_cring_Ocring__fieldI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( ( units_a_ring_ext_a_b @ R )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ R ) ) ) ).
% cring.cring_fieldI
thf(fact_713_cring_Ocring__fieldI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ( units_2932844235741507942t_unit @ R )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( field_6388047844668329575t_unit @ R ) ) ) ).
% cring.cring_fieldI
thf(fact_714_cring_Oideal__eq__carrier__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( cgenid547466209912283029xt_a_b @ R @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ).
% cring.ideal_eq_carrier_iff
thf(fact_715_cring_Oideal__eq__carrier__iff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( cgenid9131348535277946915t_unit @ R @ A ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ).
% cring.ideal_eq_carrier_iff
thf(fact_716_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_717_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ~ ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_718_Ring_Ointegral,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_719_Ring_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( field_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_720_Ring_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_721_cring_Ofield__intro2,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( ( zero_s2910681146719230829t_unit @ R )
!= ( one_se1127990129394575805t_unit @ R ) )
=> ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
=> ( member_set_list_a @ X2 @ ( units_5837875185506529638t_unit @ R ) ) )
=> ( field_26233345952514695t_unit @ R ) ) ) ) ).
% cring.field_intro2
thf(fact_722_cring_Ofield__intro2,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( ( zero_a_b @ R )
!= ( one_a_ring_ext_a_b @ R ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( member_a @ X2 @ ( units_a_ring_ext_a_b @ R ) ) )
=> ( field_a_b @ R ) ) ) ) ).
% cring.field_intro2
thf(fact_723_cring_Ofield__intro2,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ( zero_l4142658623432671053t_unit @ R )
!= ( one_li8328186300101108157t_unit @ R ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ R ) ) )
=> ( field_6388047844668329575t_unit @ R ) ) ) ) ).
% cring.field_intro2
thf(fact_724_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ R @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_725_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( R2
= ( mult_l7073676228092353617t_unit @ R @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_726_field_Ozeromaximalideal,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( maxima3875439991530298004t_unit @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_727_field_Ozeromaximalideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_728_field_Ozeromaximalideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_729_cring_Ocring__fieldI2,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( ( zero_s2910681146719230829t_unit @ R )
!= ( one_se1127990129394575805t_unit @ R ) )
=> ( ! [A5: set_list_a] :
( ( member_set_list_a @ A5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A5
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ? [X5: set_list_a] :
( ( member_set_list_a @ X5 @ ( partia141011252114345353t_unit @ R ) )
& ( ( mult_s7802724872828879953t_unit @ R @ A5 @ X5 )
= ( one_se1127990129394575805t_unit @ R ) ) ) ) )
=> ( field_26233345952514695t_unit @ R ) ) ) ) ).
% cring.cring_fieldI2
thf(fact_730_cring_Ocring__fieldI2,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( ( zero_a_b @ R )
!= ( one_a_ring_ext_a_b @ R ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A5
!= ( zero_a_b @ R ) )
=> ? [X5: a] :
( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( mult_a_ring_ext_a_b @ R @ A5 @ X5 )
= ( one_a_ring_ext_a_b @ R ) ) ) ) )
=> ( field_a_b @ R ) ) ) ) ).
% cring.cring_fieldI2
thf(fact_731_cring_Ocring__fieldI2,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ( zero_l4142658623432671053t_unit @ R )
!= ( one_li8328186300101108157t_unit @ R ) )
=> ( ! [A5: list_a] :
( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A5
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ ( partia5361259788508890537t_unit @ R ) )
& ( ( mult_l7073676228092353617t_unit @ R @ A5 @ X5 )
= ( one_li8328186300101108157t_unit @ R ) ) ) ) )
=> ( field_6388047844668329575t_unit @ R ) ) ) ) ).
% cring.cring_fieldI2
thf(fact_732_cring_Ozeromaximalideal__fieldI,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( maxima3875439991530298004t_unit @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) @ R )
=> ( field_26233345952514695t_unit @ R ) ) ) ).
% cring.zeromaximalideal_fieldI
thf(fact_733_cring_Ozeromaximalideal__fieldI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R )
=> ( field_a_b @ R ) ) ) ).
% cring.zeromaximalideal_fieldI
thf(fact_734_cring_Ozeromaximalideal__fieldI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R )
=> ( field_6388047844668329575t_unit @ R ) ) ) ).
% cring.zeromaximalideal_fieldI
thf(fact_735_cring_Ozeromaximalideal__eq__field,axiom,
! [R: partia7496981018696276118t_unit] :
( ( cring_3470013030684506304t_unit @ R )
=> ( ( maxima3875439991530298004t_unit @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) @ R )
= ( field_26233345952514695t_unit @ R ) ) ) ).
% cring.zeromaximalideal_eq_field
thf(fact_736_cring_Ozeromaximalideal__eq__field,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R )
= ( field_a_b @ R ) ) ) ).
% cring.zeromaximalideal_eq_field
thf(fact_737_cring_Ozeromaximalideal__eq__field,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R )
= ( field_6388047844668329575t_unit @ R ) ) ) ).
% cring.zeromaximalideal_eq_field
thf(fact_738_domain_Oring__irreducibleI,axiom,
! [R: partia7496981018696276118t_unit,R2: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R2 @ ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
=> ( ~ ( member_set_list_a @ R2 @ ( units_5837875185506529638t_unit @ R ) )
=> ( ! [A5: set_list_a,B6: set_list_a] :
( ( member_set_list_a @ A5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B6 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( R2
= ( mult_s7802724872828879953t_unit @ R @ A5 @ B6 ) )
=> ( ( member_set_list_a @ A5 @ ( units_5837875185506529638t_unit @ R ) )
| ( member_set_list_a @ B6 @ ( units_5837875185506529638t_unit @ R ) ) ) ) ) )
=> ( ring_r5115406448772830318t_unit @ R @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_739_domain_Oring__irreducibleI,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ R ) )
=> ( ! [A5: a,B6: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ R @ A5 @ B6 ) )
=> ( ( member_a @ A5 @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B6 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ R @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_740_domain_Oring__irreducibleI,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ~ ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ R ) )
=> ( ! [A5: list_a,B6: list_a] :
( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( R2
= ( mult_l7073676228092353617t_unit @ R @ A5 @ B6 ) )
=> ( ( member_list_a @ A5 @ ( units_2932844235741507942t_unit @ R ) )
| ( member_list_a @ B6 @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ R @ R2 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_741_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_742_long__division__a__inv_I1_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_743_long__division__add_I1_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ A @ Q2 ) @ ( polynomial_pdiv_a_b @ r @ B @ Q2 ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_744_is__root__poly__mult__imp__is__root,axiom,
! [P2: list_a,Q2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q2 @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_745_long__division__closed_I1_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_746_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_747_pirreducibleE_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_748_pprimeE_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_749_pirreducibleE_I3_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R2 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ) ) ) ).
% pirreducibleE(3)
thf(fact_750_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_751_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_752_domain_Ouniv__poly__is__principal,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_753_domain_Ouniv__poly__is__principal,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_754_principal__domain_Oprimeness__condition,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P2 )
= ( ring_ring_prime_a_b @ R @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_755_principal__domain_Oprimeness__condition,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P2 )
= ( ring_r6430282645014804837t_unit @ R @ P2 ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_756_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2175431115845679010xt_a_b,P2: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P2 )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ P2 ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_757_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2670972154091845814t_unit,P2: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P2 )
=> ( maxima6585700282301356660t_unit @ ( cgenid9131348535277946915t_unit @ R @ P2 ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_758_rupture__is__field__iff__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_759_pdiv__pmod,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_760_domain_Opprime__iff__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_761_domain_Opprime__iff__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_762_domain_Olong__division__a__inv_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) @ Q2 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_763_domain_Olong__division__a__inv_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pdiv_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_764_domain_Olong__division__add_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) @ Q2 )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno5893782122288709345t_unit @ R @ A @ Q2 ) @ ( polyno5893782122288709345t_unit @ R @ B @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_765_domain_Olong__division__add_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pdiv_a_b @ R @ A @ Q2 ) @ ( polynomial_pdiv_a_b @ R @ B @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_766_long__division__closed_I2_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_767_long__division__add_I2_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ A @ Q2 ) @ ( polynomial_pmod_a_b @ r @ B @ Q2 ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_768_long__division__add__iff,axiom,
! [K: set_a,A: list_a,B: list_a,C: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
= ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ C ) @ Q2 )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_769_long__division__a__inv_I2_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_770_domain_Olong__division__closed_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_771_domain_Olong__division__closed_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_772_domain_Olong__division__add_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) @ Q2 )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno1727750685288865234t_unit @ R @ A @ Q2 ) @ ( polyno1727750685288865234t_unit @ R @ B @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_773_domain_Olong__division__add_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pmod_a_b @ R @ A @ Q2 ) @ ( polynomial_pmod_a_b @ R @ B @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_774_domain_Olong__division__add__iff,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ A @ Q2 )
= ( polyno1727750685288865234t_unit @ R @ B @ Q2 ) )
= ( ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ C ) @ Q2 )
= ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_775_domain_Olong__division__add__iff,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,C: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ R @ A @ Q2 )
= ( polynomial_pmod_a_b @ R @ B @ Q2 ) )
= ( ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ C ) @ Q2 )
= ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_776_domain_Olong__division__a__inv_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) @ Q2 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(2)
thf(fact_777_domain_Olong__division__a__inv_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pmod_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(2)
thf(fact_778_domain_Orupture__is__field__iff__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( field_1540243473349940225t_unit @ ( polyno859807163042199155t_unit @ R @ K @ P2 ) )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) ) ) ) ) ).
% domain.rupture_is_field_iff_pirreducible
thf(fact_779_domain_Orupture__is__field__iff__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ R @ K @ P2 ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) ) ) ) ).
% domain.rupture_is_field_iff_pirreducible
thf(fact_780_domain_Opdiv__pmod,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( P2
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q2 @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 ) ) @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_781_domain_Opdiv__pmod,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q2 @ ( polynomial_pdiv_a_b @ R @ P2 @ Q2 ) ) @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_782_domain_Ouniv__poly__not__field,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ~ ( field_1861437471013600865t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_not_field
thf(fact_783_domain_Ouniv__poly__not__field,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_not_field
thf(fact_784_domain_OpirreducibleE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
=> ~ ( member_list_list_a @ P2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_785_domain_OpirreducibleE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_786_domain_Olong__division__closed_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_787_domain_Olong__division__closed_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ R @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_788_domain_OpprimeE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
=> ~ ( member_list_list_a @ P2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_789_domain_OpprimeE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_790_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno6951661231331188332t_unit @ R @ P2 @ X )
| ( polyno6951661231331188332t_unit @ R @ Q2 @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_791_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ R @ Q2 @ X ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_792_domain_OpirreducibleE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P2
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q2 @ R2 ) )
=> ( ( member_list_list_a @ Q2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
| ( member_list_list_a @ R2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_793_domain_OpirreducibleE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q2 @ R2 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
| ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_794_pirreducibleI,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R4 ) )
=> ( ( member_list_a @ Q3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_795_pprimeE_I3_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
| ( polyno5814909790663948098es_a_b @ r @ P2 @ R2 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_796_same__pmod__iff__pdivides,axiom,
! [K: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
= ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
= ( polyno5814909790663948098es_a_b @ r @ Q2 @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_797_subfield__m__inv_I2_J,axiom,
! [K: set_a,K2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K2 @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(2)
thf(fact_798_subfield__m__inv_I3_J,axiom,
! [K: set_a,K2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) @ K2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% subfield_m_inv(3)
thf(fact_799_zero__pdivides,axiom,
! [P2: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P2 )
= ( P2 = nil_a ) ) ).
% zero_pdivides
thf(fact_800_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_801_inv__eq__imp__eq,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( m_inv_a_ring_ext_a_b @ r @ Y ) )
=> ( X = Y ) ) ) ) ).
% inv_eq_imp_eq
thf(fact_802_const__term__not__zero,axiom,
! [P2: list_a] :
( ( ( const_term_a_b @ r @ P2 )
!= ( zero_a_b @ r ) )
=> ( P2 != nil_a ) ) ).
% const_term_not_zero
thf(fact_803_inv__eq__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( one_a_ring_ext_a_b @ r ) )
= ( X
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% inv_eq_one_eq
thf(fact_804_inv__unique_H,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 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( Y
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ) ) ) ).
% inv_unique'
thf(fact_805_inv__char,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 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ) ).
% inv_char
thf(fact_806_comm__inv__char,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 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ X )
= Y ) ) ) ) ).
% comm_inv_char
thf(fact_807_inv__eq__neg__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% inv_eq_neg_one_eq
thf(fact_808_inv__eq__self,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( X
= ( m_inv_a_ring_ext_a_b @ r @ X ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ r ) )
| ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% inv_eq_self
thf(fact_809_long__division__zero_I2_J,axiom,
! [K: set_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q2 )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_810_pdivides__zero,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P2 @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_811_pirreducibleE_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_812_long__division__zero_I1_J,axiom,
! [K: set_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q2 )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_813_pprimeE_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_814_subfield__m__inv_I1_J,axiom,
! [K: set_a,K2: 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 @ ( m_inv_a_ring_ext_a_b @ r @ K2 ) @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% subfield_m_inv(1)
thf(fact_815_pmod__zero__iff__pdivides,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P2 @ Q2 )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q2 @ P2 ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_816_pprimeI,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q3 )
| ( polyno5814909790663948098es_a_b @ r @ P2 @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ) ) ).
% pprimeI
thf(fact_817_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_818_inv__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% inv_one
thf(fact_819_Units__inv__Units,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_inv_Units
thf(fact_820_Units__inv__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( m_inv_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= X ) ) ).
% Units_inv_inv
thf(fact_821_Units__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_inv_closed
thf(fact_822_inv__neg__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ).
% inv_neg_one
thf(fact_823_Units__l__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ X ) @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_l_inv
thf(fact_824_Units__r__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( m_inv_a_ring_ext_a_b @ r @ X ) )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% Units_r_inv
thf(fact_825_domain_Ozero__pdivides,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( polyno8016796738000020810t_unit @ R @ nil_list_a @ P2 )
= ( P2 = nil_list_a ) ) ) ).
% domain.zero_pdivides
thf(fact_826_domain_Ozero__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( polyno5814909790663948098es_a_b @ R @ nil_a @ P2 )
= ( P2 = nil_a ) ) ) ).
% domain.zero_pdivides
thf(fact_827_domain_Ozero__pdivides__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( polyno8016796738000020810t_unit @ R @ nil_list_a @ nil_list_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_828_domain_Ozero__pdivides__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( polyno5814909790663948098es_a_b @ R @ nil_a @ nil_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_829_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_830_domain_Opdivides__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( polyno8016796738000020810t_unit @ R @ P2 @ nil_list_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_831_domain_Opdivides__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ R @ P2 @ nil_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_832_domain_Opmod__zero__iff__pdivides,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ R @ Q2 @ P2 ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_833_domain_Opmod__zero__iff__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ R @ P2 @ Q2 )
= nil_a )
= ( polyno5814909790663948098es_a_b @ R @ Q2 @ P2 ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_834_domain_Oinv__eq__self,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ R ) )
=> ( ( X
= ( m_inv_2802811658206063947t_unit @ R @ X ) )
=> ( ( X
= ( one_li8328186300101108157t_unit @ R ) )
| ( X
= ( a_inv_8944721093294617173t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) ) ) ) ) ) ) ).
% domain.inv_eq_self
thf(fact_835_domain_Oinv__eq__self,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_b @ R ) )
=> ( ( X
= ( m_inv_a_ring_ext_a_b @ R @ X ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ R ) )
| ( X
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ).
% domain.inv_eq_self
thf(fact_836_domain_Oinv__eq__self,axiom,
! [R: partia7496981018696276118t_unit,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X @ ( units_5837875185506529638t_unit @ R ) )
=> ( ( X
= ( m_inv_7863988576679134539t_unit @ R @ X ) )
=> ( ( X
= ( one_se1127990129394575805t_unit @ R ) )
| ( X
= ( a_inv_5715216516650856053t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) ) ) ) ) ) ) ).
% domain.inv_eq_self
thf(fact_837_domain_OpprimeI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P2 != nil_list_a )
=> ( ~ ( member_list_list_a @ P2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ! [Q3: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q3 @ R4 ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ Q3 )
| ( polyno8016796738000020810t_unit @ R @ P2 @ R4 ) ) ) ) )
=> ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_838_domain_OpprimeI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ! [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q3 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ Q3 )
| ( polyno5814909790663948098es_a_b @ R @ P2 @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_839_domain_Olong__division__zero_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ nil_list_a @ Q2 )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_840_domain_Olong__division__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pmod_a_b @ R @ nil_a @ Q2 )
= nil_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_841_domain_OpirreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
=> ( P2 != nil_list_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_842_domain_OpirreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_843_domain_Olong__division__zero_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ nil_list_a @ Q2 )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_844_domain_Olong__division__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ R @ nil_a @ Q2 )
= nil_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_845_domain_OpprimeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
=> ( P2 != nil_list_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_846_domain_OpprimeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_847_field_OsubfieldI_H,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( ! [K3: set_list_a] :
( ( member_set_list_a @ K3 @ ( minus_4782336368215558443list_a @ K @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
=> ( member_set_list_a @ ( m_inv_7863988576679134539t_unit @ R @ K3 ) @ K ) )
=> ( subfie4339374749748326226t_unit @ K @ R ) ) ) ) ).
% field.subfieldI'
thf(fact_848_field_OsubfieldI_H,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ! [K3: list_a] :
( ( member_list_a @ K3 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ R @ K3 ) @ K ) )
=> ( subfie1779122896746047282t_unit @ K @ R ) ) ) ) ).
% field.subfieldI'
thf(fact_849_field_OsubfieldI_H,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( field_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ! [K3: a] :
( ( member_a @ K3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ R @ K3 ) @ K ) )
=> ( subfield_a_b @ K @ R ) ) ) ) ).
% field.subfieldI'
thf(fact_850_domain_Osame__pmod__iff__pdivides,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ A @ Q2 )
= ( polyno1727750685288865234t_unit @ R @ B @ Q2 ) )
= ( polyno8016796738000020810t_unit @ R @ Q2 @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_851_domain_Osame__pmod__iff__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ R @ A @ Q2 )
= ( polynomial_pmod_a_b @ R @ B @ Q2 ) )
= ( polyno5814909790663948098es_a_b @ R @ Q2 @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_852_domain_OpprimeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q2 @ R2 ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ Q2 )
| ( polyno8016796738000020810t_unit @ R @ P2 @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_853_domain_OpprimeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q2 @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ Q2 )
| ( polyno5814909790663948098es_a_b @ R @ P2 @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_854_domain_OpirreducibleI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P2 != nil_list_a )
=> ( ~ ( member_list_list_a @ P2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ! [Q3: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P2
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q3 @ R4 ) )
=> ( ( member_list_list_a @ Q3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
| ( member_list_list_a @ R4 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_855_domain_OpirreducibleI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ! [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q3 @ R4 ) )
=> ( ( member_list_a @ Q3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
| ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_856_exists__unique__long__division,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ? [X2: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ X2 )
& ! [Y6: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ Y6 )
=> ( Y6 = X2 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_857_is__root__imp__pdivides,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 ) ) ) ).
% is_root_imp_pdivides
thf(fact_858_pdivides__imp__is__root,axiom,
! [P2: list_a,X: a] :
( ( P2 != nil_a )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 )
=> ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_859_monic__degree__one__root__condition,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) @ B )
= ( A = B ) ) ) ).
% monic_degree_one_root_condition
thf(fact_860_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V2: a,Va: list_a] :
( X
!= ( cons_a @ V2 @ Va ) ) ) ).
% normalize.cases
thf(fact_861_univ__poly__one,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
( ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) )
= ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) ) ).
% univ_poly_one
thf(fact_862_univ__poly__one,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ K ) )
= ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_863_var__def,axiom,
( var_li8453953174693405341t_unit
= ( ^ [R3: partia2670972154091845814t_unit] : ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ nil_list_a ) ) ) ) ).
% var_def
thf(fact_864_var__def,axiom,
( var_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( zero_a_b @ R3 ) @ nil_a ) ) ) ) ).
% var_def
thf(fact_865_var__def,axiom,
( var_se6008125447796440765t_unit
= ( ^ [R3: partia7496981018696276118t_unit] : ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ nil_set_list_a ) ) ) ) ).
% var_def
thf(fact_866_domain_Opdivides__imp__is__root,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( P2 != nil_set_list_a )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R ) )
=> ( ( polyno9075941895896075626t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ P2 )
=> ( polyno4320237611291262604t_unit @ R @ P2 @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_867_domain_Opdivides__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( P2 != nil_list_a )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ P2 )
=> ( polyno6951661231331188332t_unit @ R @ P2 @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_868_domain_Opdivides__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( P2 != nil_a )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X ) @ nil_a ) ) @ P2 )
=> ( polyno4133073214067823460ot_a_b @ R @ P2 @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_869_domain_Ois__root__imp__pdivides,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a,X: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( polyno4320237611291262604t_unit @ R @ P2 @ X )
=> ( polyno9075941895896075626t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X ) @ nil_set_list_a ) ) @ P2 ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_870_domain_Ois__root__imp__pdivides,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R @ P2 @ X )
=> ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X ) @ nil_list_a ) ) @ P2 ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_871_domain_Ois__root__imp__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P2 @ X )
=> ( polyno5814909790663948098es_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X ) @ nil_a ) ) @ P2 ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_872_domain_Oexists__unique__long__division,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q2 != nil_list_a )
=> ? [X2: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R @ P2 @ Q2 @ X2 )
& ! [Y6: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R @ P2 @ Q2 @ Y6 )
=> ( Y6 = X2 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_873_domain_Oexists__unique__long__division,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q2 != nil_a )
=> ? [X2: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R @ P2 @ Q2 @ X2 )
& ! [Y6: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R @ P2 @ Q2 @ Y6 )
=> ( Y6 = X2 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_874_long__divisionE,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_875_long__divisionI,axiom,
! [K: set_a,P2: list_a,Q2: list_a,B: list_a,R2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_876_associated__polynomials__iff,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q2 )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ X3 @ nil_a ) @ Q2 ) ) ) ) ) ) ) ) ).
% associated_polynomials_iff
thf(fact_877_poly__mult__var,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( P2 = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ ( var_a_b @ r ) )
= nil_a ) )
& ( ( P2 != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ ( var_a_b @ r ) )
= ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).
% poly_mult_var
thf(fact_878_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_879_poly__mult_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V2: a,Va: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V2 @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_880_combine_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [K3: a,Ks: list_a,U2: a,Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ K3 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_881_associated__polynomials__imp__same__is__root,axiom,
! [P2: list_a,Q2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q2 )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
= ( polyno4133073214067823460ot_a_b @ r @ Q2 @ X ) ) ) ) ) ).
% associated_polynomials_imp_same_is_root
thf(fact_882_cgenideal__pirreducible,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 )
=> ( ( member_list_a @ Q2 @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q2 ) ) ) ) ) ) ).
% cgenideal_pirreducible
thf(fact_883_exists__long__division,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ~ ! [B6: list_a] :
( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B6 @ R4 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_884_subring__degree__one__associatedI,axiom,
! [K: set_a,A: a,A7: a,B: a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ K )
=> ( ( member_a @ A7 @ K )
=> ( ( member_a @ B @ K )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A7 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ r @ A7 @ B ) @ nil_a ) ) ) ) ) ) ) ) ).
% subring_degree_one_associatedI
thf(fact_885_cring_Oassociated__iff__same__ideal,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( associ5860276527279195403xt_a_b @ R @ A @ B )
= ( ( cgenid547466209912283029xt_a_b @ R @ A )
= ( cgenid547466209912283029xt_a_b @ R @ B ) ) ) ) ) ) ).
% cring.associated_iff_same_ideal
thf(fact_886_cring_Oassociated__iff__same__ideal,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( associ8407585678920448409t_unit @ R @ A @ B )
= ( ( cgenid9131348535277946915t_unit @ R @ A )
= ( cgenid9131348535277946915t_unit @ R @ B ) ) ) ) ) ) ).
% cring.associated_iff_same_ideal
thf(fact_887_domain_Oring__associated__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( associ5860276527279195403xt_a_b @ R @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ R ) )
& ( A
= ( mult_a_ring_ext_a_b @ R @ X3 @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_888_domain_Oring__associated__iff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( associ8407585678920448409t_unit @ R @ A @ B )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ R ) )
& ( A
= ( mult_l7073676228092353617t_unit @ R @ X3 @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_889_domain_Oassociated__polynomials__imp__same__is__root,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q2 )
=> ( ( polyno6951661231331188332t_unit @ R @ P2 @ X )
= ( polyno6951661231331188332t_unit @ R @ Q2 @ X ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_is_root
thf(fact_890_domain_Oassociated__polynomials__imp__same__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a,X: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ Q2 )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P2 @ X )
= ( polyno4133073214067823460ot_a_b @ R @ Q2 @ X ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_is_root
thf(fact_891_domain_Ocgenideal__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q2 )
=> ( ( member_list_list_a @ Q2 @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) )
=> ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.cgenideal_pirreducible
thf(fact_892_domain_Ocgenideal__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ Q2 )
=> ( ( member_list_a @ Q2 @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 @ Q2 ) ) ) ) ) ) ) ).
% domain.cgenideal_pirreducible
thf(fact_893_domain_Oexists__long__division,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q2 != nil_list_a )
=> ~ ! [B6: list_list_a] :
( ( member_list_list_a @ B6 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ! [R4: list_list_a] :
( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ~ ( polyno6947042923167803568t_unit @ R @ P2 @ Q2 @ ( produc8696003437204565271list_a @ B6 @ R4 ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_894_domain_Oexists__long__division,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q2 != nil_a )
=> ~ ! [B6: list_a] :
( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ R @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B6 @ R4 ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_895_domain_Osubring__degree__one__associatedI,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a,A: set_list_a,A7: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( ( member_set_list_a @ A @ K )
=> ( ( member_set_list_a @ A7 @ K )
=> ( ( member_set_list_a @ B @ K )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ A7 )
= ( one_se1127990129394575805t_unit @ R ) )
=> ( associ8249012953061539097t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( cons_set_list_a @ A @ ( cons_set_list_a @ B @ nil_set_list_a ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( mult_s7802724872828879953t_unit @ R @ A7 @ B ) @ nil_set_list_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_896_domain_Osubring__degree__one__associatedI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_a,A7: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_a @ A @ K )
=> ( ( member_list_a @ A7 @ K )
=> ( ( member_list_a @ B @ K )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ A7 )
= ( one_li8328186300101108157t_unit @ R ) )
=> ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( mult_l7073676228092353617t_unit @ R @ A7 @ B ) @ nil_list_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_897_domain_Osubring__degree__one__associatedI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: a,A7: a,B: a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_a @ A @ K )
=> ( ( member_a @ A7 @ K )
=> ( ( member_a @ B @ K )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ A7 )
= ( one_a_ring_ext_a_b @ R ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ R @ A7 @ B ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_898_domain_Olong__divisionE,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q2 != nil_list_a )
=> ( polyno6947042923167803568t_unit @ R @ P2 @ Q2 @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 ) @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_899_domain_Olong__divisionE,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q2 != nil_a )
=> ( polyno2806191415236617128es_a_b @ R @ P2 @ Q2 @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P2 @ Q2 ) @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_900_domain_Olong__divisionI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a,B: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ R @ P2 @ Q2 @ ( produc8696003437204565271list_a @ B @ R2 ) )
=> ( ( produc8696003437204565271list_a @ B @ R2 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P2 @ Q2 ) @ ( polyno1727750685288865234t_unit @ R @ P2 @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_901_domain_Olong__divisionI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a,B: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ R @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P2 @ Q2 ) @ ( polynomial_pmod_a_b @ R @ P2 @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_902_domain_Opoly__mult__var,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( P2 = nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ ( var_li8453953174693405341t_unit @ R ) )
= nil_list_a ) )
& ( ( P2 != nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ ( var_li8453953174693405341t_unit @ R ) )
= ( append_list_a @ P2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_903_domain_Opoly__mult__var,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
=> ( ( ( P2 = nil_set_list_a )
=> ( ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 @ ( var_se6008125447796440765t_unit @ R ) )
= nil_set_list_a ) )
& ( ( P2 != nil_set_list_a )
=> ( ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 @ ( var_se6008125447796440765t_unit @ R ) )
= ( append_set_list_a @ P2 @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_904_domain_Opoly__mult__var,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( P2 = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 @ ( var_a_b @ R ) )
= nil_a ) )
& ( ( P2 != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 @ ( var_a_b @ R ) )
= ( append_a @ P2 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_905_domain_Oassociated__polynomials__iff,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a,P2: list_set_list_a,Q2: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subfie4339374749748326226t_unit @ K @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
=> ( ( member5524387281408368019list_a @ Q2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
=> ( ( associ8249012953061539097t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P2 @ Q2 )
= ( ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ ( minus_4782336368215558443list_a @ K @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
& ( P2
= ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( cons_set_list_a @ X3 @ nil_set_list_a ) @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.associated_polynomials_iff
thf(fact_906_domain_Oassociated__polynomials__iff,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ Q2 )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
& ( P2
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( cons_list_a @ X3 @ nil_list_a ) @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.associated_polynomials_iff
thf(fact_907_domain_Oassociated__polynomials__iff,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 @ Q2 )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
& ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cons_a @ X3 @ nil_a ) @ Q2 ) ) ) ) ) ) ) ) ) ).
% domain.associated_polynomials_iff
thf(fact_908_const__term__eq__last,axiom,
! [P2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P2 @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ).
% const_term_eq_last
thf(fact_909_const__term__explicit,axiom,
! [P2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P2 != nil_a )
=> ( ( ( const_term_a_b @ r @ P2 )
= A )
=> ~ ! [P3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P2
!= ( append_a @ P3 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_910_lead__coeff__in__carrier,axiom,
! [K: set_a,A: a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ ( cons_a @ A @ P2 ) )
=> ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ).
% lead_coeff_in_carrier
thf(fact_911_associated__polynomials__imp__same__roots,axiom,
! [P2: list_a,Q2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q2 )
=> ( ( polynomial_roots_a_b @ r @ P2 )
= ( polynomial_roots_a_b @ r @ Q2 ) ) ) ) ) ).
% associated_polynomials_imp_same_roots
thf(fact_912_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_913_assoc__subst,axiom,
! [A: a,B: a,F3: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A5: a,B6: a] :
( ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A5 @ B6 ) )
=> ( ( member_a @ ( F3 @ A5 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F3 @ B6 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F3 @ A5 ) @ ( F3 @ B6 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F3 @ A ) @ ( F3 @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_914_associated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_915_Units__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ).
% Units_assoc
thf(fact_916_mult__cong__l,axiom,
! [A: a,A7: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ A7 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A7 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_917_mult__cong__r,axiom,
! [B: a,B7: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B7 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B7 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_918_Units__cong,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_cong
thf(fact_919_associated__iff__same__ideal,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ( cgenid547466209912283029xt_a_b @ r @ A )
= ( cgenid547466209912283029xt_a_b @ r @ B ) ) ) ) ) ).
% associated_iff_same_ideal
thf(fact_920_associatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_921_associatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_922_ring__associated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ X3 @ B ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_923_polynomial__incl,axiom,
! [K: set_a,P2: list_a] :
( ( polynomial_a_b @ r @ K @ P2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K ) ) ).
% polynomial_incl
thf(fact_924_var__closed_I2_J,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( polynomial_a_b @ r @ K @ ( var_a_b @ r ) ) ) ).
% var_closed(2)
thf(fact_925_const__term__simprules_I1_J,axiom,
! [P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( const_term_a_b @ r @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% const_term_simprules(1)
thf(fact_926_lead__coeff__not__zero,axiom,
! [K: set_a,A: a,P2: list_a] :
( ( polynomial_a_b @ r @ K @ ( cons_a @ A @ P2 ) )
=> ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% lead_coeff_not_zero
thf(fact_927_const__term__zero,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ( P2 != nil_a )
=> ( ( ( const_term_a_b @ r @ P2 )
= ( zero_a_b @ r ) )
=> ~ ! [P3: list_a] :
( ( polynomial_a_b @ r @ K @ P3 )
=> ( ( P3 != nil_a )
=> ( P2
!= ( append_a @ P3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_928_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_929_zero__is__polynomial,axiom,
! [K: set_a] : ( polynomial_a_b @ r @ K @ nil_a ) ).
% zero_is_polynomial
thf(fact_930_carrier__polynomial,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P2 ) ) ) ).
% carrier_polynomial
thf(fact_931_polynomial__in__carrier,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% polynomial_in_carrier
thf(fact_932_one__is__polynomial,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( polynomial_a_b @ r @ K @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) ) ) ).
% one_is_polynomial
thf(fact_933_const__is__polynomial,axiom,
! [A: a,K: set_a] :
( ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K @ ( cons_a @ A @ nil_a ) ) ) ).
% const_is_polynomial
thf(fact_934_univ__poly__carrier,axiom,
( polynomial_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,K4: set_a,P4: list_a] : ( member_list_a @ P4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K4 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_935_domain_Ovar__closed_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( polyno1315193887021588240t_unit @ R @ K @ ( var_li8453953174693405341t_unit @ R ) ) ) ) ).
% domain.var_closed(2)
thf(fact_936_domain_Ovar__closed_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( polynomial_a_b @ R @ K @ ( var_a_b @ R ) ) ) ) ).
% domain.var_closed(2)
thf(fact_937_cring_Oconst__term__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( cring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( const_term_a_b @ R @ P2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% cring.const_term_simprules(1)
thf(fact_938_cring_Oconst__term__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ R @ P2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% cring.const_term_simprules(1)
thf(fact_939_domain_Oone__is__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( polyno1315193887021588240t_unit @ R @ K @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_940_domain_Oone__is__polynomial,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( polyno3115169382166032176t_unit @ R @ K @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_941_domain_Oone__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( polynomial_a_b @ R @ K @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_942_domain_Oassociated__polynomials__imp__same__roots,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q2 )
=> ( ( polyno7858422826990252003t_unit @ R @ P2 )
= ( polyno7858422826990252003t_unit @ R @ Q2 ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_roots
thf(fact_943_domain_Oassociated__polynomials__imp__same__roots,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ Q2 )
=> ( ( polynomial_roots_a_b @ R @ P2 )
= ( polynomial_roots_a_b @ R @ Q2 ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_roots
thf(fact_944_exp__base__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_945_list_Osimps_I15_J,axiom,
! [X21: list_a,X22: list_list_a] :
( ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) )
= ( insert_list_a @ X21 @ ( set_list_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_946_list_Osimps_I15_J,axiom,
! [X21: a,X22: list_a] :
( ( set_a2 @ ( cons_a @ X21 @ X22 ) )
= ( insert_a @ X21 @ ( set_a2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_947_list_Osimps_I15_J,axiom,
! [X21: int,X22: list_int] :
( ( set_int2 @ ( cons_int @ X21 @ X22 ) )
= ( insert_int @ X21 @ ( set_int2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_948_list_Osimps_I15_J,axiom,
! [X21: nat > int,X22: list_nat_int] :
( ( set_nat_int2 @ ( cons_nat_int @ X21 @ X22 ) )
= ( insert_nat_int @ X21 @ ( set_nat_int2 @ X22 ) ) ) ).
% list.simps(15)
thf(fact_949_set__empty2,axiom,
! [Xs: list_int] :
( ( bot_bot_set_int
= ( set_int2 @ Xs ) )
= ( Xs = nil_int ) ) ).
% set_empty2
thf(fact_950_set__empty2,axiom,
! [Xs: list_nat_int] :
( ( bot_bot_set_nat_int
= ( set_nat_int2 @ Xs ) )
= ( Xs = nil_nat_int ) ) ).
% set_empty2
thf(fact_951_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_952_set__empty2,axiom,
! [Xs: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% set_empty2
thf(fact_953_set__empty,axiom,
! [Xs: list_int] :
( ( ( set_int2 @ Xs )
= bot_bot_set_int )
= ( Xs = nil_int ) ) ).
% set_empty
thf(fact_954_set__empty,axiom,
! [Xs: list_nat_int] :
( ( ( set_nat_int2 @ Xs )
= bot_bot_set_nat_int )
= ( Xs = nil_nat_int ) ) ).
% set_empty
thf(fact_955_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_956_set__empty,axiom,
! [Xs: list_list_a] :
( ( ( set_list_a2 @ Xs )
= bot_bot_set_list_a )
= ( Xs = nil_list_a ) ) ).
% set_empty
thf(fact_957_List_Ofinite__set,axiom,
! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_958_subset__code_I1_J,axiom,
! [Xs: list_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B2 )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_959_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( member_a @ X3 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_960_finite__list,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ? [Xs2: list_a] :
( ( set_a2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_961_set__subset__Cons,axiom,
! [Xs: list_int,X: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_962_set__subset__Cons,axiom,
! [Xs: list_nat_int,X: nat > int] : ( ord_le6569500216720880561at_int @ ( set_nat_int2 @ Xs ) @ ( set_nat_int2 @ ( cons_nat_int @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_963_set__subset__Cons,axiom,
! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_964_empty__set,axiom,
( bot_bot_set_int
= ( set_int2 @ nil_int ) ) ).
% empty_set
thf(fact_965_empty__set,axiom,
( bot_bot_set_nat_int
= ( set_nat_int2 @ nil_nat_int ) ) ).
% empty_set
thf(fact_966_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_967_empty__set,axiom,
( bot_bot_set_list_a
= ( set_list_a2 @ nil_list_a ) ) ).
% empty_set
thf(fact_968_Span__m__inv__simprule,axiom,
! [K: set_a,Us2: list_a,K2: a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ 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 ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
=> ( member_a @ A @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ) ).
% Span_m_inv_simprule
thf(fact_969_factors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% factors_mult
thf(fact_970_pdivides__imp__roots__incl,axiom,
! [P2: list_a,Q2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ ( polynomial_roots_a_b @ r @ Q2 ) ) ) ) ) ) ).
% pdivides_imp_roots_incl
thf(fact_971_eval__append__aux,axiom,
! [P2: list_a,B: a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P2 @ ( cons_a @ B @ nil_a ) ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P2 @ A ) @ A ) @ B ) ) ) ) ) ).
% eval_append_aux
thf(fact_972_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_973_const__term__def,axiom,
! [P2: list_a] :
( ( const_term_a_b @ r @ P2 )
= ( eval_a_b @ r @ P2 @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_974_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_975_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_976_Span__in__carrier,axiom,
! [K: set_a,Us2: list_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_in_carrier
thf(fact_977_eval__in__carrier,axiom,
! [P2: list_a,X: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_978_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_979_eval__poly__in__carrier,axiom,
! [K: set_a,P2: list_a,X: a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_980_factors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_981_Span__subgroup__props_I1_J,axiom,
! [K: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% Span_subgroup_props(1)
thf(fact_982_Span__base__incl,axiom,
! [K: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ).
% Span_base_incl
thf(fact_983_Span__same__set,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( set_a2 @ Us2 )
= ( set_a2 @ Vs ) )
=> ( ( embedded_Span_a_b @ r @ K @ Us2 )
= ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ).
% Span_same_set
thf(fact_984_mono__Span__sublist,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( set_a2 @ Vs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ).
% mono_Span_sublist
thf(fact_985_mono__Span__subset,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ).
% mono_Span_subset
thf(fact_986_is__root__def,axiom,
! [P2: list_a,X: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P2 @ X )
= ( zero_a_b @ r ) )
& ( P2 != nil_a ) ) ) ).
% is_root_def
thf(fact_987_subalgebra__Span__incl,axiom,
! [K: set_a,V: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ V )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ V ) ) ) ) ).
% subalgebra_Span_incl
thf(fact_988_Span__subalgebraI,axiom,
! [K: set_a,E: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K @ E @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ E )
=> ( ! [V3: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V3 @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ V3 )
=> ( ord_less_eq_set_a @ E @ V3 ) ) )
=> ( E
= ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ).
% Span_subalgebraI
thf(fact_989_Span__subgroup__props_I2_J,axiom,
! [K: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( zero_a_b @ r ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ).
% Span_subgroup_props(2)
thf(fact_990_Span__subgroup__props_I3_J,axiom,
! [K: set_a,Us2: list_a,V1: a,V22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V1 @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
=> ( ( member_a @ V22 @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ).
% Span_subgroup_props(3)
thf(fact_991_Span__smult__closed,axiom,
! [K: set_a,Us2: list_a,K2: a,V4: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ V4 @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ V4 ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ).
% Span_smult_closed
thf(fact_992_mono__Span,axiom,
! [K: set_a,Us2: list_a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ ( cons_a @ U @ Us2 ) ) ) ) ) ) ).
% mono_Span
thf(fact_993_Span__subgroup__props_I4_J,axiom,
! [K: set_a,Us2: list_a,V4: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ V4 @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
=> ( member_a @ ( a_inv_a_b @ r @ V4 ) @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ).
% Span_subgroup_props(4)
thf(fact_994_mono__Span__append_I2_J,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ ( append_a @ Vs @ Us2 ) ) ) ) ) ) ).
% mono_Span_append(2)
thf(fact_995_mono__Span__append_I1_J,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) ) ) ) ) ) ).
% mono_Span_append(1)
thf(fact_996_pdivides__imp__root__sharing,axiom,
! [P2: list_a,Q2: list_a,A: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( eval_a_b @ r @ P2 @ A )
= ( zero_a_b @ r ) )
=> ( ( eval_a_b @ r @ Q2 @ A )
= ( zero_a_b @ r ) ) ) ) ) ) ).
% pdivides_imp_root_sharing
thf(fact_997_Span__is__subalgebra,axiom,
! [K: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ r ) ) ) ).
% Span_is_subalgebra
thf(fact_998_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_999_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia7496981018696276118t_unit,P2: list_set_list_a,Q2: list_set_list_a,A: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( polyno9075941895896075626t_unit @ R @ P2 @ Q2 )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( eval_s5133945360527818456t_unit @ R @ P2 @ A )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( eval_s5133945360527818456t_unit @ R @ Q2 @ A )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_1000_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a,A: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ Q2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( eval_l34571156754992824t_unit @ R @ P2 @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ Q2 @ A )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_1001_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a,A: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ Q2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( eval_a_b @ R @ P2 @ A )
= ( zero_a_b @ R ) )
=> ( ( eval_a_b @ R @ Q2 @ A )
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_1002_domain_Opdivides__imp__roots__incl,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ Q2 )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ R @ P2 ) @ ( polyno7858422826990252003t_unit @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.pdivides_imp_roots_incl
thf(fact_1003_domain_Opdivides__imp__roots__incl,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ Q2 )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ R @ P2 ) @ ( polynomial_roots_a_b @ R @ Q2 ) ) ) ) ) ) ) ).
% domain.pdivides_imp_roots_incl
thf(fact_1004_Span__mem__iff,axiom,
! [K: set_a,Us2: list_a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ? [Ks2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ K )
& ( ( embedded_combine_a_b @ r @ ( cons_a @ X3 @ Ks2 ) @ ( cons_a @ A @ Us2 ) )
= ( zero_a_b @ r ) ) ) ) ) ) ) ) ) ).
% Span_mem_iff
thf(fact_1005_Span__finite__dimension,axiom,
! [K: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd8708762675212832759on_a_b @ r @ K @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ).
% Span_finite_dimension
thf(fact_1006_Span__append__eq__set__add,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_Span_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) )
= ( set_add_a_b @ r @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) ) ) ) ) ) ).
% Span_append_eq_set_add
thf(fact_1007_telescopic__base__dim_I1_J,axiom,
! [K: set_a,F: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ F )
=> ( ( embedd8708762675212832759on_a_b @ r @ F @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ) ) ) ).
% telescopic_base_dim(1)
thf(fact_1008_combine_Osimps_I2_J,axiom,
! [Us2: list_a] :
( ( embedded_combine_a_b @ r @ nil_a @ Us2 )
= ( zero_a_b @ r ) ) ).
% combine.simps(2)
thf(fact_1009_combine_Osimps_I3_J,axiom,
! [Ks3: list_a] :
( ( embedded_combine_a_b @ r @ Ks3 @ nil_a )
= ( zero_a_b @ r ) ) ).
% combine.simps(3)
thf(fact_1010_setadd__subset__G,axiom,
! [H: set_a,K: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_1011_set__add__comm,axiom,
! [I2: set_a,J: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I2 @ J )
= ( set_add_a_b @ r @ J @ I2 ) ) ) ) ).
% set_add_comm
thf(fact_1012_set__add__closed,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_1013_sum__space__dim_I1_J,axiom,
! [K: set_a,E: set_a,F: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ F )
=> ( embedd8708762675212832759on_a_b @ r @ K @ ( set_add_a_b @ r @ E @ F ) ) ) ) ) ).
% sum_space_dim(1)
thf(fact_1014_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_1015_combine_Osimps_I1_J,axiom,
! [K2: a,Ks3: list_a,U: a,Us2: list_a] :
( ( embedded_combine_a_b @ r @ ( cons_a @ K2 @ Ks3 ) @ ( cons_a @ U @ Us2 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedded_combine_a_b @ r @ Ks3 @ Us2 ) ) ) ).
% combine.simps(1)
thf(fact_1016_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_1017_combine_Oelims,axiom,
! [X: list_a,Xa3: list_a,Y: a] :
( ( ( embedded_combine_a_b @ r @ X @ Xa3 )
= Y )
=> ( ! [K3: a,Ks: list_a] :
( ( X
= ( cons_a @ K3 @ Ks ) )
=> ! [U2: a,Us: list_a] :
( ( Xa3
= ( cons_a @ U2 @ Us ) )
=> ( Y
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K3 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks @ Us ) ) ) ) )
=> ( ( ( X = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa3 = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_1018_combine__append__zero,axiom,
! [Us2: list_a,Ks3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks3 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Us2 )
= ( embedded_combine_a_b @ r @ Ks3 @ Us2 ) ) ) ).
% combine_append_zero
thf(fact_1019_combine__in__carrier,axiom,
! [Ks3: list_a,Us2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( embedded_combine_a_b @ r @ Ks3 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% combine_in_carrier
thf(fact_1020_Span__mem__imp__non__trivial__combine,axiom,
! [K: set_a,Us2: list_a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
=> ~ ! [K3: a] :
( ( member_a @ K3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ! [Ks: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ K )
=> ( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us2 ) )
=> ( ( embedded_combine_a_b @ r @ ( cons_a @ K3 @ Ks ) @ ( cons_a @ A @ Us2 ) )
!= ( zero_a_b @ r ) ) ) ) ) ) ) ) ).
% Span_mem_imp_non_trivial_combine
thf(fact_1021_eval__as__unique__hom,axiom,
! [K: set_a,X: a,H2: list_a > a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H2 )
=> ( ! [K3: a] :
( ( member_a @ K3 @ K )
=> ( ( H2 @ ( cons_a @ K3 @ nil_a ) )
= K3 ) )
=> ( ( ( H2 @ ( var_a_b @ r ) )
= X )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( H2 @ P2 )
= ( eval_a_b @ r @ P2 @ X ) ) ) ) ) ) ) ) ).
% eval_as_unique_hom
thf(fact_1022_Span__mem__iff__length__version,axiom,
! [K: set_a,Us2: list_a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
= ( ? [Ks2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ K )
& ( ( size_size_list_a @ Ks2 )
= ( size_size_list_a @ Us2 ) )
& ( A
= ( embedded_combine_a_b @ r @ Ks2 @ Us2 ) ) ) ) ) ) ) ).
% Span_mem_iff_length_version
thf(fact_1023_combine__eq__eval,axiom,
! [Ks3: list_a,X: a] :
( ( embedded_combine_a_b @ r @ Ks3 @ ( polyno2922411391617481336se_a_b @ r @ X @ ( size_size_list_a @ Ks3 ) ) )
= ( eval_a_b @ r @ Ks3 @ X ) ) ).
% combine_eq_eval
thf(fact_1024_univ__poly__a__inv__length,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) )
= ( size_size_list_a @ P2 ) ) ) ) ).
% univ_poly_a_inv_length
thf(fact_1025_associated__polynomials__imp__same__length,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q2 )
=> ( ( size_size_list_a @ P2 )
= ( size_size_list_a @ Q2 ) ) ) ) ) ) ).
% associated_polynomials_imp_same_length
thf(fact_1026_combine__append,axiom,
! [Ks3: list_a,Us2: list_a,Ks4: list_a,Vs: list_a] :
( ( ( size_size_list_a @ Ks3 )
= ( size_size_list_a @ Us2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( embedded_combine_a_b @ r @ Ks3 @ Us2 ) @ ( embedded_combine_a_b @ r @ Ks4 @ Vs ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks3 @ Ks4 ) @ ( append_a @ Us2 @ Vs ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_1027_determination__of__hom,axiom,
! [K: set_a,A2: partia2175431115845679010xt_a_b,H2: list_a > a,G: list_a > a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ A2 @ H2 )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ A2 @ G )
=> ( ! [K3: a] :
( ( member_a @ K3 @ K )
=> ( ( H2 @ ( cons_a @ K3 @ nil_a ) )
= ( G @ ( cons_a @ K3 @ nil_a ) ) ) )
=> ( ( ( H2 @ ( var_a_b @ r ) )
= ( G @ ( var_a_b @ r ) ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( H2 @ P2 )
= ( G @ P2 ) ) ) ) ) ) ) ) ).
% determination_of_hom
thf(fact_1028_impossible__Cons,axiom,
! [Xs: list_int,Ys: list_int,X: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
=> ( Xs
!= ( cons_int @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1029_impossible__Cons,axiom,
! [Xs: list_nat_int,Ys: list_nat_int,X: nat > int] :
( ( ord_less_eq_nat @ ( size_s5718426915756887103at_int @ Xs ) @ ( size_s5718426915756887103at_int @ Ys ) )
=> ( Xs
!= ( cons_nat_int @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1030_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_1031_domain_Ouniv__poly__a__inv__length,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 ) )
= ( size_s349497388124573686list_a @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_length
thf(fact_1032_domain_Ouniv__poly__a__inv__length,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 ) )
= ( size_size_list_a @ P2 ) ) ) ) ) ).
% domain.univ_poly_a_inv_length
thf(fact_1033_domain_Oassociated__polynomials__imp__same__length,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P2: list_list_a,Q2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P2 @ Q2 )
=> ( ( size_s349497388124573686list_a @ P2 )
= ( size_s349497388124573686list_a @ Q2 ) ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_length
thf(fact_1034_domain_Oassociated__polynomials__imp__same__length,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P2: list_a,Q2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ K ) @ P2 @ Q2 )
=> ( ( size_size_list_a @ P2 )
= ( size_size_list_a @ Q2 ) ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_length
thf(fact_1035_domain_Odetermination__of__hom,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A2: partia2175431115845679010xt_a_b,H2: list_a > a,G: list_a > a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ A2 @ H2 )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ A2 @ G )
=> ( ! [K3: a] :
( ( member_a @ K3 @ K )
=> ( ( H2 @ ( cons_a @ K3 @ nil_a ) )
= ( G @ ( cons_a @ K3 @ nil_a ) ) ) )
=> ( ( ( H2 @ ( var_a_b @ R ) )
= ( G @ ( var_a_b @ R ) ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( H2 @ P2 )
= ( G @ P2 ) ) ) ) ) ) ) ) ) ).
% domain.determination_of_hom
thf(fact_1036_domain_Oeval__as__unique__hom,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,X: a,H2: list_a > a,P2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ R @ H2 )
=> ( ! [K3: a] :
( ( member_a @ K3 @ K )
=> ( ( H2 @ ( cons_a @ K3 @ nil_a ) )
= K3 ) )
=> ( ( ( H2 @ ( var_a_b @ R ) )
= X )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( H2 @ P2 )
= ( eval_a_b @ R @ P2 @ X ) ) ) ) ) ) ) ) ) ).
% domain.eval_as_unique_hom
thf(fact_1037_domain_Oeval__as__unique__hom,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,X: list_a,H2: list_list_a > list_a,P2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_h4589914651911841480t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ R @ H2 )
=> ( ! [K3: list_a] :
( ( member_list_a @ K3 @ K )
=> ( ( H2 @ ( cons_list_a @ K3 @ nil_list_a ) )
= K3 ) )
=> ( ( ( H2 @ ( var_li8453953174693405341t_unit @ R ) )
= X )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( H2 @ P2 )
= ( eval_l34571156754992824t_unit @ R @ P2 @ X ) ) ) ) ) ) ) ) ) ).
% domain.eval_as_unique_hom
thf(fact_1038_dependent__imp__non__trivial__combine,axiom,
! [K: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ~ ! [Ks: list_a] :
( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us2 ) )
=> ( ( ( embedded_combine_a_b @ r @ Ks @ Us2 )
= ( zero_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ K )
=> ( ( set_a2 @ Ks )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ) ) ) ).
% dependent_imp_non_trivial_combine
thf(fact_1039_eval__append,axiom,
! [P2: list_a,Q2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P2 @ Q2 ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P2 @ A ) @ ( pow_a_1026414303147256608_b_nat @ r @ A @ ( size_size_list_a @ Q2 ) ) ) @ ( eval_a_b @ r @ Q2 @ A ) ) ) ) ) ) ).
% eval_append
thf(fact_1040_Span_Osimps,axiom,
! [K: set_a,Us2: list_a] :
( ( embedded_Span_a_b @ r @ K @ Us2 )
= ( foldr_a_set_a @ ( embedd971793762689825387on_a_b @ r @ K ) @ Us2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% Span.simps
thf(fact_1041_independent__backwards_I2_J,axiom,
! [K: set_a,U: a,Us2: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ Us2 ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 ) ) ).
% independent_backwards(2)
thf(fact_1042_li__Nil,axiom,
! [K: set_a] : ( embedd5208550302661555450nt_a_b @ r @ K @ nil_a ) ).
% li_Nil
thf(fact_1043_Units__pow__closed,axiom,
! [X: a,D2: nat] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ D2 ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_1044_independent__backwards_I3_J,axiom,
! [K: set_a,U: a,Us2: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ Us2 ) )
=> ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% independent_backwards(3)
thf(fact_1045_pow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_1046_nat__pow__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_1047_nat__pow__comm,axiom,
! [X: a,N: nat,M2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_1048_group__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_1049_independent__backwards_I1_J,axiom,
! [K: set_a,U: a,Us2: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ Us2 ) )
=> ~ ( member_a @ U @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ).
% independent_backwards(1)
thf(fact_1050_independent__split_I2_J,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 ) ) ) ).
% independent_split(2)
thf(fact_1051_independent__split_I1_J,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K @ Vs ) ) ) ).
% independent_split(1)
thf(fact_1052_independent__in__carrier,axiom,
! [K: set_a,Us2: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% independent_in_carrier
thf(fact_1053_li__Cons,axiom,
! [U: a,K: set_a,Us2: list_a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ U @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ Us2 ) ) ) ) ) ).
% li_Cons
thf(fact_1054_independent__same__set,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ( set_a2 @ Us2 )
= ( set_a2 @ Vs ) )
=> ( ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Vs ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( embedd5208550302661555450nt_a_b @ r @ K @ Vs ) ) ) ) ) ).
% independent_same_set
thf(fact_1055_independent_Osimps,axiom,
! [A1: set_a,A22: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ A1 @ A22 )
= ( ? [K4: set_a] :
( ( A1 = K4 )
& ( A22 = nil_a ) )
| ? [U3: a,K4: set_a,Us3: list_a] :
( ( A1 = K4 )
& ( A22
= ( cons_a @ U3 @ Us3 ) )
& ( member_a @ U3 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ U3 @ ( embedded_Span_a_b @ r @ K4 @ Us3 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K4 @ Us3 ) ) ) ) ).
% independent.simps
thf(fact_1056_independent_Ocases,axiom,
! [A1: set_a,A22: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ A1 @ A22 )
=> ( ( A22 != nil_a )
=> ~ ! [U2: a,Us: list_a] :
( ( A22
= ( cons_a @ U2 @ Us ) )
=> ( ( member_a @ U2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ U2 @ ( embedded_Span_a_b @ r @ A1 @ Us ) )
=> ~ ( embedd5208550302661555450nt_a_b @ r @ A1 @ Us ) ) ) ) ) ) ).
% independent.cases
thf(fact_1057_independent__rotate1__aux,axiom,
! [K: set_a,U: a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ ( append_a @ Us2 @ Vs ) ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ ( append_a @ Us2 @ ( cons_a @ U @ nil_a ) ) @ Vs ) ) ) ) ).
% independent_rotate1_aux
thf(fact_1058_filter__base,axiom,
! [K: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ~ ! [Vs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Vs2 )
=> ( ( embedded_Span_a_b @ r @ K @ Vs2 )
!= ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ).
% filter_base
thf(fact_1059_independent__replacement,axiom,
! [K: set_a,U: a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ Us2 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Vs )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ ( cons_a @ U @ Us2 ) ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Vs ) )
& ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ X2 @ Us2 ) ) ) ) ) ) ) ).
% independent_replacement
thf(fact_1060_Span_Oelims,axiom,
! [X: set_a,Xa3: list_a,Y: set_a] :
( ( ( embedded_Span_a_b @ r @ X @ Xa3 )
= Y )
=> ( Y
= ( foldr_a_set_a @ ( embedd971793762689825387on_a_b @ r @ X ) @ Xa3 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% Span.elims
thf(fact_1061_independent__length__le,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Vs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Us2 ) @ ( size_size_list_a @ Vs ) ) ) ) ) ) ).
% independent_length_le
thf(fact_1062_replacement__theorem,axiom,
! [K: set_a,Us4: list_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ Us4 @ Us2 ) )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Vs )
=> ( ( ord_less_eq_set_a @ ( embedded_Span_a_b @ r @ K @ ( append_a @ Us4 @ Us2 ) ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
=> ? [Vs3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs3 ) @ ( set_a2 @ Vs ) )
& ( ( size_size_list_a @ Vs3 )
= ( size_size_list_a @ Us4 ) )
& ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ Vs3 @ Us2 ) ) ) ) ) ) ) ).
% replacement_theorem
thf(fact_1063_unique__decomposition,axiom,
! [K: set_a,Us2: list_a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( ( member_a @ A @ ( embedded_Span_a_b @ r @ K @ Us2 ) )
=> ? [X2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ X2 ) @ K )
& ( ( size_size_list_a @ X2 )
= ( size_size_list_a @ Us2 ) )
& ( A
= ( embedded_combine_a_b @ r @ X2 @ Us2 ) )
& ! [Y6: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ Y6 ) @ K )
& ( ( size_size_list_a @ Y6 )
= ( size_size_list_a @ Us2 ) )
& ( A
= ( embedded_combine_a_b @ r @ Y6 @ Us2 ) ) )
=> ( Y6 = X2 ) ) ) ) ) ) ).
% unique_decomposition
thf(fact_1064_nat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_1065_nat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( one_a_ring_ext_a_b @ r ) ) ).
% nat_pow_one
thf(fact_1066_independent__rotate1,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ ( rotate1_a @ Us2 ) @ Vs ) ) ) ) ).
% independent_rotate1
thf(fact_1067_trivial__combine__imp__independent,axiom,
! [K: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [Ks: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ K )
=> ( ( ( embedded_combine_a_b @ r @ Ks @ Us2 )
= ( zero_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us2 ) @ Ks ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 ) ) ) ) ).
% trivial_combine_imp_independent
thf(fact_1068_independent__split_I3_J,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) )
=> ( ( inf_inf_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% independent_split(3)
thf(fact_1069_subring__inter,axiom,
! [I2: set_a,J: set_a] :
( ( subring_a_b @ I2 @ r )
=> ( ( subring_a_b @ J @ r )
=> ( subring_a_b @ ( inf_inf_set_a @ I2 @ J ) @ r ) ) ) ).
% subring_inter
thf(fact_1070_subalgebra__inter,axiom,
! [K: set_a,V: set_a,V5: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V5 @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ ( inf_inf_set_a @ V @ V5 ) @ r ) ) ) ).
% subalgebra_inter
thf(fact_1071_subcring__inter,axiom,
! [I2: set_a,J: set_a] :
( ( subcring_a_b @ I2 @ r )
=> ( ( subcring_a_b @ J @ r )
=> ( subcring_a_b @ ( inf_inf_set_a @ I2 @ J ) @ r ) ) ) ).
% subcring_inter
thf(fact_1072_IntI,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_1073_IntI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_1074_Int__iff,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B2 ) )
= ( ( member_list_a @ C @ A2 )
& ( member_list_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1075_Int__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_1076_combine__take,axiom,
! [Us2: list_a,Ks3: list_a] :
( ( embedded_combine_a_b @ r @ ( take_a @ ( size_size_list_a @ Us2 ) @ Ks3 ) @ Us2 )
= ( embedded_combine_a_b @ r @ Ks3 @ Us2 ) ) ).
% combine_take
thf(fact_1077_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_1078_finite__Int,axiom,
! [F: set_a,G2: set_a] :
( ( ( finite_finite_a @ F )
| ( finite_finite_a @ G2 ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F @ G2 ) ) ) ).
% finite_Int
thf(fact_1079_Int__insert__right__if1,axiom,
! [A: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1080_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_1081_Int__insert__right__if0,axiom,
! [A: list_a,A2: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( inf_inf_set_list_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1082_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% Int_insert_right_if0
thf(fact_1083_insert__inter__insert,axiom,
! [A: list_a,A2: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ ( insert_list_a @ A @ B2 ) )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_1084_insert__inter__insert,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ).
% insert_inter_insert
thf(fact_1085_Int__insert__left__if1,axiom,
! [A: list_a,C2: set_list_a,B2: set_list_a] :
( ( member_list_a @ A @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B2 ) @ C2 )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1086_Int__insert__left__if1,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_1087_Int__insert__left__if0,axiom,
! [A: list_a,C2: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ A @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_list_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1088_Int__insert__left__if0,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_1089_independent__append,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Vs )
=> ( ( ( inf_inf_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ Us2 @ Vs ) ) ) ) ) ) ).
% independent_append
thf(fact_1090_disjoint__insert_I2_J,axiom,
! [A2: set_a,B: a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B @ B2 ) ) )
= ( ~ ( member_a @ B @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1091_disjoint__insert_I2_J,axiom,
! [A2: set_list_a,B: list_a,B2: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ B @ B2 ) ) )
= ( ~ ( member_list_a @ B @ A2 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ B2 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_1092_disjoint__insert_I1_J,axiom,
! [B2: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B2 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ B2 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1093_disjoint__insert_I1_J,axiom,
! [B2: set_list_a,A: list_a,A2: set_list_a] :
( ( ( inf_inf_set_list_a @ B2 @ ( insert_list_a @ A @ A2 ) )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A @ B2 )
& ( ( inf_inf_set_list_a @ B2 @ A2 )
= bot_bot_set_list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_1094_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_a @ A @ B2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1095_insert__disjoint_I2_J,axiom,
! [A: list_a,A2: set_list_a,B2: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ B2 ) )
= ( ~ ( member_list_a @ A @ B2 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ B2 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_1096_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B2 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B2 )
& ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1097_insert__disjoint_I1_J,axiom,
! [A: list_a,A2: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ B2 )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A @ B2 )
& ( ( inf_inf_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_1098_Diff__disjoint,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( minus_646659088055828811list_a @ B2 @ A2 ) )
= bot_bot_set_list_a ) ).
% Diff_disjoint
thf(fact_1099_Diff__disjoint,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_1100_take__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( take_a @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_1101_take__all__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ( take_a @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_1102_independent__imp__trivial__combine,axiom,
! [K: set_a,Us2: list_a,Ks3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K )
=> ( ( ( embedded_combine_a_b @ r @ Ks3 @ Us2 )
= ( zero_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us2 ) @ Ks3 ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ) ) ).
% independent_imp_trivial_combine
thf(fact_1103_non__trivial__combine__imp__dependent,axiom,
! [K: set_a,Ks3: list_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ K )
=> ( ( ( embedded_combine_a_b @ r @ Ks3 @ Us2 )
= ( zero_a_b @ r ) )
=> ( ~ ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us2 ) @ Ks3 ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ~ ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 ) ) ) ) ) ).
% non_trivial_combine_imp_dependent
thf(fact_1104_Int__Collect__mono,axiom,
! [A2: set_list_a,B2: set_list_a,P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B2 @ ( collect_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1105_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1106_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_1107_Int__absorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_1108_Int__absorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_1109_Int__lower2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_1110_Int__lower1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_1111_Int__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_1112_set__take__subset,axiom,
! [N: nat,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ N @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% set_take_subset
thf(fact_1113_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_1114_Int__emptyI,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ~ ( member_list_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_1115_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1116_disjoint__iff,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ~ ( member_list_a @ X3 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_1117_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_1118_Int__empty__left,axiom,
! [B2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B2 )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_1119_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_1120_Int__empty__right,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_1121_disjoint__iff__not__equal,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ! [Y5: a] :
( ( member_a @ Y5 @ B2 )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1122_disjoint__iff__not__equal,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ! [Y5: list_a] :
( ( member_list_a @ Y5 @ B2 )
=> ( X3 != Y5 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_1123_Int__insert__right,axiom,
! [A: list_a,A2: set_list_a,B2: set_list_a] :
( ( ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( inf_inf_set_list_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1124_Int__insert__right,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B2 ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_insert_right
thf(fact_1125_Int__insert__left,axiom,
! [A: list_a,C2: set_list_a,B2: set_list_a] :
( ( ( member_list_a @ A @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B2 ) @ C2 )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ B2 @ C2 ) ) ) )
& ( ~ ( member_list_a @ A @ C2 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_list_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1126_Int__insert__left,axiom,
! [A: a,C2: set_a,B2: set_a] :
( ( ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) )
& ( ~ ( member_a @ A @ C2 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_1127_Diff__Int__distrib2,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C2 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C2 ) @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Diff_Int_distrib2
thf(fact_1128_Diff__Int__distrib,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C2 @ A2 ) @ ( inf_inf_set_a @ C2 @ B2 ) ) ) ).
% Diff_Int_distrib
thf(fact_1129_Diff__Diff__Int,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Diff_Diff_Int
thf(fact_1130_Diff__Int2,axiom,
! [A2: set_a,C2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C2 ) @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C2 ) @ B2 ) ) ).
% Diff_Int2
thf(fact_1131_Int__Diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) ) ) ).
% Int_Diff
thf(fact_1132_IntE,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B2 ) )
=> ~ ( ( member_list_a @ C @ A2 )
=> ~ ( member_list_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1133_IntE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_1134_IntD1,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B2 ) )
=> ( member_list_a @ C @ A2 ) ) ).
% IntD1
thf(fact_1135_IntD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_1136_IntD2,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B2 ) )
=> ( member_list_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1137_IntD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_1138_Int__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_1139_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_1140_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_1141_Int__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_1142_Int__left__commute,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_1143_Int__Diff__disjoint,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B2 ) @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
= bot_bot_set_list_a ) ).
% Int_Diff_disjoint
thf(fact_1144_Int__Diff__disjoint,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ A2 @ B2 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_1145_Diff__triv,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a )
=> ( ( minus_646659088055828811list_a @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_1146_Diff__triv,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B2 )
= A2 ) ) ).
% Diff_triv
thf(fact_1147_set__take__subset__set__take,axiom,
! [M2: nat,N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ M2 @ Xs ) ) @ ( set_a2 @ ( take_a @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_1148_combine__normalize,axiom,
! [Ks3: list_a,Us2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( embedded_combine_a_b @ r @ Ks3 @ Us2 )
= A )
=> ~ ! [Ks5: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us2 ) @ Ks3 ) ) @ ( set_a2 @ Ks5 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks5 ) @ ( sup_sup_set_a @ ( set_a2 @ ( take_a @ ( size_size_list_a @ Us2 ) @ Ks3 ) ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( ( size_size_list_a @ Ks5 )
= ( size_size_list_a @ Us2 ) )
=> ( ( embedded_combine_a_b @ r @ Ks5 @ Us2 )
!= A ) ) ) ) ) ) ) ).
% combine_normalize
thf(fact_1149_eval__monom,axiom,
! [B: a,A: a,N: nat] :
( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( monom_a_b @ r @ B @ N ) @ A )
= ( mult_a_ring_ext_a_b @ r @ B @ ( pow_a_1026414303147256608_b_nat @ r @ A @ N ) ) ) ) ) ).
% eval_monom
thf(fact_1150_Span__strict__incl,axiom,
! [K: set_a,Us2: list_a,Vs: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ Vs ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Vs ) )
& ~ ( member_a @ X2 @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ) ) ) ) ).
% Span_strict_incl
thf(fact_1151_Un__iff,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A2 @ B2 ) )
= ( ( member_list_a @ C @ A2 )
| ( member_list_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_1152_Un__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_1153_UnCI,axiom,
! [C: list_a,B2: set_list_a,A2: set_list_a] :
( ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ A2 ) )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_1154_UnCI,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_1155_independent__strict__incl,axiom,
! [K: set_a,U: a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ ( cons_a @ U @ Us2 ) )
=> ( ord_less_set_a @ ( embedded_Span_a_b @ r @ K @ Us2 ) @ ( embedded_Span_a_b @ r @ K @ ( cons_a @ U @ Us2 ) ) ) ) ) ).
% independent_strict_incl
thf(fact_1156_Un__subset__iff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_1157_Un__empty,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_1158_Un__empty,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( sup_sup_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a )
= ( ( A2 = bot_bot_set_list_a )
& ( B2 = bot_bot_set_list_a ) ) ) ).
% Un_empty
thf(fact_1159_finite__Un,axiom,
! [F: set_a,G2: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F @ G2 ) )
= ( ( finite_finite_a @ F )
& ( finite_finite_a @ G2 ) ) ) ).
% finite_Un
thf(fact_1160_Un__insert__left,axiom,
! [A: list_a,B2: set_list_a,C2: set_list_a] :
( ( sup_sup_set_list_a @ ( insert_list_a @ A @ B2 ) @ C2 )
= ( insert_list_a @ A @ ( sup_sup_set_list_a @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1161_Un__insert__left,axiom,
! [A: a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_1162_Un__insert__right,axiom,
! [A2: set_list_a,A: list_a,B2: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( insert_list_a @ A @ ( sup_sup_set_list_a @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_1163_Un__insert__right,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_1164_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_1165_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_1166_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_1167_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_1168_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_1169_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_1170_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_1171_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_1172_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_1173_Un__Diff__cancel,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_1174_Un__Diff__cancel2,axiom,
! [B2: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_a @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_1175_monom__in__carrier,axiom,
! [A: a,N: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monom_in_carrier
thf(fact_1176_monom__is__polynomial,axiom,
! [K: set_a,A: a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K @ ( monom_a_b @ r @ A @ N ) ) ) ) ).
% monom_is_polynomial
thf(fact_1177_monom__eq__var__pow,axiom,
! [K: set_a,A: a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( monom_a_b @ r @ A @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A @ nil_a ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ) ) ).
% monom_eq_var_pow
thf(fact_1178_poly__add__monom,axiom,
! [P2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( poly_add_a_b @ r @ ( monom_a_b @ r @ A @ ( size_size_list_a @ P2 ) ) @ P2 )
= ( cons_a @ A @ P2 ) ) ) ) ).
% poly_add_monom
thf(fact_1179_complete__base,axiom,
! [K: set_a,N: nat,E: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ E )
=> ? [Vs2: list_a] :
( ( ( size_size_list_a @ ( append_a @ Vs2 @ Us2 ) )
= N )
& ( embedd5208550302661555450nt_a_b @ r @ K @ ( append_a @ Vs2 @ Us2 ) )
& ( ( embedded_Span_a_b @ r @ K @ ( append_a @ Vs2 @ Us2 ) )
= E ) ) ) ) ) ) ).
% complete_base
thf(fact_1180_dimension__is__inj,axiom,
! [K: set_a,N: nat,E: set_a,M2: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ E )
=> ( N = M2 ) ) ) ) ).
% dimension_is_inj
thf(fact_1181_finite__dimension__def,axiom,
! [K: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K @ E )
= ( ? [N2: nat] : ( embedd2795209813406577254on_a_b @ r @ N2 @ K @ E ) ) ) ).
% finite_dimension_def
thf(fact_1182_finite__dimensionI,axiom,
! [N: nat,K: set_a,E: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( embedd8708762675212832759on_a_b @ r @ K @ E ) ) ).
% finite_dimensionI
thf(fact_1183_finite__dimensionE_H,axiom,
! [K: set_a,E: set_a] :
( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ~ ! [N3: nat] :
~ ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E ) ) ).
% finite_dimensionE'
thf(fact_1184_space__subgroup__props_I2_J,axiom,
! [K: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( member_a @ ( zero_a_b @ r ) @ E ) ) ) ).
% space_subgroup_props(2)
thf(fact_1185_space__subgroup__props_I3_J,axiom,
! [K: set_a,N: nat,E: set_a,V1: a,V22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( member_a @ V1 @ E )
=> ( ( member_a @ V22 @ E )
=> ( member_a @ ( add_a_b @ r @ V1 @ V22 ) @ E ) ) ) ) ) ).
% space_subgroup_props(3)
thf(fact_1186_space__subgroup__props_I5_J,axiom,
! [K: set_a,N: nat,E: set_a,K2: a,V4: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ V4 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ V4 ) @ E ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_1187_space__subgroup__props_I4_J,axiom,
! [K: set_a,N: nat,E: set_a,V4: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( member_a @ V4 @ E )
=> ( member_a @ ( a_inv_a_b @ r @ V4 ) @ E ) ) ) ) ).
% space_subgroup_props(4)
thf(fact_1188_poly__add__closed,axiom,
! [K: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P12 )
=> ( ( polynomial_a_b @ r @ K @ P23 )
=> ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_add_closed
thf(fact_1189_unique__dimension,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd8708762675212832759on_a_b @ r @ K @ E )
=> ? [X2: nat] :
( ( embedd2795209813406577254on_a_b @ r @ X2 @ K @ E )
& ! [Y6: nat] :
( ( embedd2795209813406577254on_a_b @ r @ Y6 @ K @ E )
=> ( Y6 = X2 ) ) ) ) ) ).
% unique_dimension
thf(fact_1190_poly__add__comm,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P12 @ P23 )
= ( poly_add_a_b @ r @ P23 @ P12 ) ) ) ) ).
% poly_add_comm
thf(fact_1191_poly__add__in__carrier,axiom,
! [P12: list_a,P23: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P12 @ P23 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_add_in_carrier
thf(fact_1192_space__subgroup__props_I1_J,axiom,
! [K: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_1193_poly__add__zero_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ( poly_add_a_b @ r @ nil_a @ P2 )
= P2 ) ) ) ).
% poly_add_zero(2)
thf(fact_1194_poly__add__zero_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ( poly_add_a_b @ r @ P2 @ nil_a )
= P2 ) ) ) ).
% poly_add_zero(1)
thf(fact_1195_polynomial__pow__not__zero,axiom,
! [P2: list_a,N: nat] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_1196_poly__add__is__polynomial,axiom,
! [K: set_a,P12: list_a,P23: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P12 ) @ K )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P23 ) @ K )
=> ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P12 @ P23 ) ) ) ) ) ).
% poly_add_is_polynomial
thf(fact_1197_subring__polynomial__pow__not__zero,axiom,
! [K: set_a,P2: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P2 @ N )
!= nil_a ) ) ) ) ).
% subring_polynomial_pow_not_zero
thf(fact_1198_dimensionI,axiom,
! [K: set_a,Us2: list_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( ( ( embedded_Span_a_b @ r @ K @ Us2 )
= E )
=> ( embedd2795209813406577254on_a_b @ r @ ( size_size_list_a @ Us2 ) @ K @ E ) ) ) ) ).
% dimensionI
thf(fact_1199_var__pow__closed,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% var_pow_closed
thf(fact_1200_eval__poly__add,axiom,
! [P2: list_a,Q2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P2 @ Q2 ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P2 @ A ) @ ( eval_a_b @ r @ Q2 @ A ) ) ) ) ) ) ).
% eval_poly_add
thf(fact_1201_const__term__simprules_I3_J,axiom,
! [P2: list_a,Q2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_add_a_b @ r @ P2 @ Q2 ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P2 ) @ ( const_term_a_b @ r @ Q2 ) ) ) ) ) ).
% const_term_simprules(3)
thf(fact_1202_polynomial__pow__division,axiom,
! [P2: list_a,N: nat,M2: nat] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ M2 ) ) ) ) ).
% polynomial_pow_division
thf(fact_1203_unitary__monom__eq__var__pow,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( monom_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ).
% unitary_monom_eq_var_pow
thf(fact_1204_eval__poly__add__aux,axiom,
! [P2: list_a,Q2: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( size_size_list_a @ P2 )
= ( size_size_list_a @ Q2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P2 @ Q2 ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P2 @ A ) @ ( eval_a_b @ r @ Q2 @ A ) ) ) ) ) ) ) ).
% eval_poly_add_aux
thf(fact_1205_independent__length__le__dimension,axiom,
! [K: set_a,N: nat,E: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ E )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Us2 ) @ N ) ) ) ) ) ).
% independent_length_le_dimension
thf(fact_1206_independent__length__eq__dimension,axiom,
! [K: set_a,N: nat,E: set_a,Us2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ E )
=> ( ( ( size_size_list_a @ Us2 )
= N )
= ( ( embedded_Span_a_b @ r @ K @ Us2 )
= E ) ) ) ) ) ) ).
% independent_length_eq_dimension
thf(fact_1207_pirreducible__pow__pdivides__iff,axiom,
! [K: set_a,P2: list_a,Q2: list_a,R2: list_a,N: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P2 @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R2 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P2 @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_1208_space__subgroup__props_I6_J,axiom,
! [K: set_a,N: nat,E: set_a,K2: a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( 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 ) @ E )
=> ( member_a @ A @ E ) ) ) ) ) ) ).
% space_subgroup_props(6)
thf(fact_1209_exists__base,axiom,
! [K: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ? [Vs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( embedd5208550302661555450nt_a_b @ r @ K @ Vs2 )
& ( ( size_size_list_a @ Vs2 )
= N )
& ( ( embedded_Span_a_b @ r @ K @ Vs2 )
= E ) ) ) ) ).
% exists_base
thf(fact_1210_dimension__independent,axiom,
! [K: set_a,Us2: list_a] :
( ( embedd5208550302661555450nt_a_b @ r @ K @ Us2 )
=> ( embedd2795209813406577254on_a_b @ r @ ( size_size_list_a @ Us2 ) @ K @ ( embedded_Span_a_b @ r @ K @ Us2 ) ) ) ).
% dimension_independent
thf(fact_1211_roots__inclI,axiom,
! [P2: list_a,Q2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P2 != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A5 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ A5 ) ) @ Q2 ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ ( polynomial_roots_a_b @ r @ Q2 ) ) ) ) ) ) ).
% roots_inclI
thf(fact_1212_alg__multE_I2_J,axiom,
! [X: a,P2: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N ) @ P2 )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_1213_alg__multE_I1_J,axiom,
! [X: a,P2: list_a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) ) @ P2 ) ) ) ) ).
% alg_multE(1)
thf(fact_1214_le__alg__mult__imp__pdivides,axiom,
! [X: a,P2: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N ) @ P2 ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_1215_dimension__direct__sum__space,axiom,
! [K: set_a,N: nat,E: set_a,M2: nat,F: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ F )
=> ( ( ( inf_inf_set_a @ E @ F )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( plus_plus_nat @ N @ M2 ) @ K @ ( set_add_a_b @ r @ E @ F ) ) ) ) ) ) ).
% dimension_direct_sum_space
thf(fact_1216_roots__inclI_H,axiom,
! [P2: list_a,M2: multiset_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P2 != nil_a )
=> ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ A5 ) @ ( count_a @ M2 @ A5 ) ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ M2 ) ) ) ).
% roots_inclI'
thf(fact_1217_nat__pow__mult,axiom,
! [X: a,N: nat,M2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M2 ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( plus_plus_nat @ N @ M2 ) ) ) ) ).
% nat_pow_mult
thf(fact_1218_alg__mult__eq__count__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4422430861927485590lt_a_b @ r @ P2 )
= ( count_a @ ( polynomial_roots_a_b @ r @ P2 ) ) ) ) ).
% alg_mult_eq_count_roots
thf(fact_1219_dimension__sum__space,axiom,
! [K: set_a,N: nat,E: set_a,M2: nat,F: set_a,K2: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ F )
=> ( ( embedd2795209813406577254on_a_b @ r @ K2 @ K @ ( inf_inf_set_a @ E @ F ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ K2 ) @ K @ ( set_add_a_b @ r @ E @ F ) ) ) ) ) ) ).
% dimension_sum_space
thf(fact_1220_pdivides__imp__degree__le,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% pdivides_imp_degree_le
thf(fact_1221_poly__mult__degree__le,axiom,
! [X: list_a,Y: list_a,N: nat,M2: nat] :
( ( 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 ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ M2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ N @ M2 ) ) ) ) ) ) ).
% poly_mult_degree_le
thf(fact_1222_telescopic__base__aux,axiom,
! [K: set_a,F: set_a,N: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N @ K @ F )
=> ( ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ F @ E )
=> ( embedd2795209813406577254on_a_b @ r @ N @ K @ E ) ) ) ) ) ).
% telescopic_base_aux
thf(fact_1223_pmod__const_I2_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ r @ P2 @ Q2 )
= P2 ) ) ) ) ) ).
% pmod_const(2)
thf(fact_1224_degree__one__imp__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_1225_univ__poly__a__inv__degree,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% univ_poly_a_inv_degree
thf(fact_1226_degree__oneE,axiom,
! [P2: list_a,K: set_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A5: a] :
( ( member_a @ A5 @ K )
=> ( ( A5
!= ( zero_a_b @ r ) )
=> ! [B6: a] :
( ( member_a @ B6 @ K )
=> ( P2
!= ( cons_a @ A5 @ ( cons_a @ B6 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_1227_poly__add__degree__le,axiom,
! [X: list_a,N: 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 ) @ N )
=> ( ( 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 ) @ N )
=> ( 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 ) @ N ) ) ) ) ) ).
% poly_add_degree_le
thf(fact_1228_pmod__degree,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ( ( ( polynomial_pmod_a_b @ r @ P2 @ Q2 )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% pmod_degree
thf(fact_1229_pirreducible__degree,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_1230_pmod__const_I1_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ r @ P2 @ Q2 )
= nil_a ) ) ) ) ) ).
% pmod_const(1)
thf(fact_1231_poly__sub__degree__le,axiom,
! [X: list_a,N: 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 ) @ N )
=> ( ( 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 ) @ N )
=> ( 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 ) @ N ) ) ) ) ) ).
% poly_sub_degree_le
thf(fact_1232_subfield__long__division__theorem__shell,axiom,
! [K: set_a,P2: list_a,B: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ? [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ Q3 ) @ R4 ) )
& ( ( R4
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_1233_poly__mult__degree__le__1,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 ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) ) ) ) ) ).
% poly_mult_degree_le_1
thf(fact_1234_field__long__division__theorem,axiom,
! [K: set_a,P2: list_a,B: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ( polynomial_a_b @ r @ K @ B )
=> ( ( B != nil_a )
=> ? [Q3: list_a,R4: list_a] :
( ( polynomial_a_b @ r @ K @ Q3 )
& ( polynomial_a_b @ r @ K @ R4 )
& ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ Q3 ) @ R4 ) )
& ( ( R4 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% field_long_division_theorem
thf(fact_1235_long__dividesI,axiom,
! [B: list_a,R2: list_a,P2: list_a,Q2: list_a] :
( ( member_list_a @ B @ ( 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 ) ) ) )
=> ( ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ B ) @ 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 @ Q2 ) @ one_one_nat ) ) )
=> ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B @ R2 ) ) ) ) ) ) ).
% long_dividesI
thf(fact_1236_nat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ one_one_nat )
= X ) ) ).
% nat_pow_eone
thf(fact_1237_dimension__one,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ K @ K ) ) ).
% dimension_one
thf(fact_1238_euclidean__domainI,axiom,
! [Phi: a > nat] :
( ! [A5: a,B6: a] :
( ( member_a @ A5 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B6 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q4: a,R5: a] :
( ( member_a @ Q4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A5
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B6 @ Q4 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ r ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B6 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).
% euclidean_domainI
thf(fact_1239_boundD__carrier,axiom,
! [N: nat,F3: nat > a,M2: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F3 )
=> ( ( ord_less_nat @ N @ M2 )
=> ( member_a @ ( F3 @ M2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_1240_degree__one__associatedI,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( hd_a @ P2 ) ) @ ( const_term_a_b @ r @ P2 ) ) @ nil_a ) ) ) ) ) ) ).
% degree_one_associatedI
thf(fact_1241_univ__poly__units_H,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
= ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( P2 != nil_a )
& ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ).
% univ_poly_units'
thf(fact_1242_nat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_1243_zero__dim,axiom,
! [K: set_a] : ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% zero_dim
thf(fact_1244_nunit__factors,axiom,
! [A: a,As: list_a] :
( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( factor5638265376665762323xt_a_b @ r @ As @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ).
% nunit_factors
thf(fact_1245_dimension__zero,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ zero_zero_nat @ K @ E )
=> ( E
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% dimension_zero
thf(fact_1246_alg__mult__gt__zero__iff__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1247_degree__zero__imp__not__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_1248_rupture__one__not__zero,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) )
=> ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) )
!= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) ) ) ) ) ) ).
% rupture_one_not_zero
thf(fact_1249_degree__one__root_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( hd_a @ P2 ) ) @ ( const_term_a_b @ r @ P2 ) ) @ K ) ) ) ) ).
% degree_one_root(2)
thf(fact_1250_degree__one__root_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( ( eval_a_b @ r @ P2 @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( hd_a @ P2 ) ) @ ( const_term_a_b @ r @ P2 ) ) ) )
= ( zero_a_b @ r ) ) ) ) ) ).
% degree_one_root(1)
thf(fact_1251_local_Onat__pow__0,axiom,
! [X: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_1252_polynomialI,axiom,
! [P2: list_a,K: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K )
=> ( ( ( hd_a @ P2 )
!= ( zero_a_b @ r ) )
=> ( polynomial_a_b @ r @ K @ P2 ) ) ) ).
% polynomialI
thf(fact_1253_degree__zero__imp__splitted,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P2 ) ) ) ).
% degree_zero_imp_splitted
thf(fact_1254_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_1255_degree__zero__imp__empty__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ r @ P2 )
= zero_zero_multiset_a ) ) ) ).
% degree_zero_imp_empty_roots
thf(fact_1256_pirreducible__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ r @ P2 )
= zero_zero_multiset_a ) ) ) ) ).
% pirreducible_roots
thf(fact_1257_exists__unique__pirreducible__gen,axiom,
! [K: set_a,H2: list_a > a] :
( ( subfield_a_b @ K @ r )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H2 )
=> ( ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H2 )
!= ( insert_list_a @ nil_a @ bot_bot_set_list_a ) )
=> ( ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H2 )
!= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ X2 )
& ( ( hd_a @ X2 )
= ( one_a_ring_ext_a_b @ r ) )
& ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H2 )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ X2 ) )
& ! [Y6: list_a] :
( ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Y6 )
& ( ( hd_a @ Y6 )
= ( one_a_ring_ext_a_b @ r ) )
& ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H2 )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ Y6 ) ) )
=> ( Y6 = X2 ) ) ) ) ) ) ) ).
% exists_unique_pirreducible_gen
thf(fact_1258_eval_Oelims,axiom,
! [X: list_a,Y: a > a] :
( ( ( eval_a_b @ r @ X )
= Y )
=> ( ( ( X = nil_a )
=> ( Y
!= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) )
=> ~ ! [V2: a,Va: list_a] :
( ( X
= ( cons_a @ V2 @ Va ) )
=> ( Y
!= ( ^ [X3: a] : ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V2 @ Va ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V2 @ Va ) ) @ one_one_nat ) ) ) @ ( eval_a_b @ r @ ( tl_a @ ( cons_a @ V2 @ Va ) ) @ X3 ) ) ) ) ) ) ) ).
% eval.elims
thf(fact_1259_eval_Osimps_I2_J,axiom,
! [V4: a,Va2: list_a] :
( ( eval_a_b @ r @ ( cons_a @ V4 @ Va2 ) )
= ( ^ [X3: a] : ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( hd_a @ ( cons_a @ V4 @ Va2 ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ ( minus_minus_nat @ ( size_size_list_a @ ( cons_a @ V4 @ Va2 ) ) @ one_one_nat ) ) ) @ ( eval_a_b @ r @ ( tl_a @ ( cons_a @ V4 @ Va2 ) ) @ X3 ) ) ) ) ).
% eval.simps(2)
thf(fact_1260_not__empty__rootsE,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( polynomial_roots_a_b @ r @ P2 )
!= zero_zero_multiset_a )
=> ~ ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A5 @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) ) )
=> ( ( member_list_a @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A5 ) @ nil_a ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A5 ) @ nil_a ) ) @ P2 ) ) ) ) ) ) ).
% not_empty_rootsE
thf(fact_1261_exists__unique__gen,axiom,
! [K: set_a,I2: set_list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( ideal_8896367198367571637t_unit @ I2 @ ( univ_poly_a_b @ r @ K ) )
=> ( ( I2
!= ( insert_list_a @ nil_a @ bot_bot_set_list_a ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( ( hd_a @ X2 )
= ( one_a_ring_ext_a_b @ r ) )
& ( I2
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ X2 ) )
& ! [Y6: list_a] :
( ( ( member_list_a @ Y6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( ( hd_a @ Y6 )
= ( one_a_ring_ext_a_b @ r ) )
& ( I2
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ Y6 ) ) )
=> ( Y6 = X2 ) ) ) ) ) ) ).
% exists_unique_gen
thf(fact_1262_roots__mem__iff__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ X @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) ) )
= ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% roots_mem_iff_is_root
thf(fact_1263_oneideal,axiom,
ideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% oneideal
thf(fact_1264_add__ideals,axiom,
! [I2: set_a,J: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ideal_a_b @ J @ r )
=> ( ideal_a_b @ ( set_add_a_b @ r @ I2 @ J ) @ r ) ) ) ).
% add_ideals
thf(fact_1265_i__intersect,axiom,
! [I2: set_a,J: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ideal_a_b @ J @ r )
=> ( ideal_a_b @ ( inf_inf_set_a @ I2 @ J ) @ r ) ) ) ).
% i_intersect
thf(fact_1266_cgenideal__ideal,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ r ) ) ).
% cgenideal_ideal
thf(fact_1267_genideal__minimal,axiom,
! [I2: set_a,S: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ord_less_eq_set_a @ S @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ S ) @ I2 ) ) ) ).
% genideal_minimal
thf(fact_1268_cgenideal__minimal,axiom,
! [J: set_a,A: a] :
( ( ideal_a_b @ J @ r )
=> ( ( member_a @ A @ J )
=> ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ J ) ) ) ).
% cgenideal_minimal
thf(fact_1269_zeroideal,axiom,
ideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroideal
thf(fact_1270_Idl__subset__ideal,axiom,
! [I2: set_a,H: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ I2 )
= ( ord_less_eq_set_a @ H @ I2 ) ) ) ) ).
% Idl_subset_ideal
thf(fact_1271_genideal__ideal,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( genideal_a_b @ r @ S ) @ r ) ) ).
% genideal_ideal
thf(fact_1272_ideal__is__subalgebra,axiom,
! [K: set_a,I2: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ideal_a_b @ I2 @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ I2 @ r ) ) ) ).
% ideal_is_subalgebra
thf(fact_1273_union__genideal,axiom,
! [I2: set_a,J: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ideal_a_b @ J @ r )
=> ( ( genideal_a_b @ r @ ( sup_sup_set_a @ I2 @ J ) )
= ( set_add_a_b @ r @ I2 @ J ) ) ) ) ).
% union_genideal
thf(fact_1274_INTEG_OR_Odense__repr_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ~ ! [V2: int,Va: list_int] :
( X
!= ( cons_int @ V2 @ Va ) ) ) ).
% INTEG.R.dense_repr.cases
thf(fact_1275_INTEG_OP_Odense__repr_Ocases,axiom,
! [X: list_nat_int] :
( ( X != nil_nat_int )
=> ~ ! [V2: nat > int,Va: list_nat_int] :
( X
!= ( cons_nat_int @ V2 @ Va ) ) ) ).
% INTEG.P.dense_repr.cases
thf(fact_1276_INTEG_OP_Opoly__mult_Ocases,axiom,
! [X: produc4919227360403178295at_int] :
( ! [P22: list_nat_int] :
( X
!= ( produc1393590784410523119at_int @ nil_nat_int @ P22 ) )
=> ~ ! [V2: nat > int,Va: list_nat_int,P22: list_nat_int] :
( X
!= ( produc1393590784410523119at_int @ ( cons_nat_int @ V2 @ Va ) @ P22 ) ) ) ).
% INTEG.P.poly_mult.cases
thf(fact_1277_INTEG_OR_Opoly__mult_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [P22: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ P22 ) )
=> ~ ! [V2: int,Va: list_int,P22: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ V2 @ Va ) @ P22 ) ) ) ).
% INTEG.R.poly_mult.cases
% Conjectures (1)
thf(conj_0,conjecture,
cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
%------------------------------------------------------------------------------