TPTP Problem File: SLH0417^1.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_00276_010885__17402812_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1629 ( 421 unt; 343 typ; 0 def)
% Number of atoms : 3885 (1244 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 15883 ( 467 ~; 53 |; 264 &;13005 @)
% ( 0 <=>;2094 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 35 ( 34 usr)
% Number of type conns : 1520 (1520 >; 0 *; 0 +; 0 <<)
% Number of symbols : 312 ( 309 usr; 20 con; 0-4 aty)
% Number of variables : 3449 ( 314 ^;3010 !; 125 ?;3449 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:38:34.872
%------------------------------------------------------------------------------
% Could-be-implicit typings (34)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2956882679547061052t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia7496981018696276118t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_Itf__a_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2670972154091845814t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_Itf__a_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_Itf__a_J_Mt__Product____Type__Ounit_J_J_J,type,
partia6043505979758434576t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
partia2175431115845679010xt_a_b: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
set_se1917860372504128155list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_li6773872926390105121list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_li3422455791611400469list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_se5067313844698916539list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
set_li1071299071675007611list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_nat_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
set_nat_set_list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_a_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_list_a_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
set_list_list_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
set_a_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
set_set_list_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
set_nat_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
set_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
set_a_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
set_list_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
set_nat_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
set_a_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (309)
thf(sy_c_AbelCoset_OA__RCOSETS_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_RCOS6220190738183020281t_unit: partia2670972154091845814t_unit > set_list_a > set_set_list_a ).
thf(sy_c_AbelCoset_OA__RCOSETS_001tf__a_001tf__b,type,
a_RCOSETS_a_b: partia2175431115845679010xt_a_b > set_a > set_set_a ).
thf(sy_c_AbelCoset_Oa__kernel_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
a_kern7116238624728830086it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > ( list_a > a ) > set_list_a ).
thf(sy_c_AbelCoset_Oa__l__coset_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_l_co7008843373686234386t_unit: partia2670972154091845814t_unit > list_a > set_list_a > set_list_a ).
thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
a_l_coset_a_b: partia2175431115845679010xt_a_b > a > set_a > set_a ).
thf(sy_c_AbelCoset_Oabelian__group__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
abelia8217020544048703197it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > ( list_a > a ) > $o ).
thf(sy_c_AbelCoset_Oabelian__subgroup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
abelia6695205329122750356t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_AbelCoset_Oadditive__subgroup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
additi4714453376129182166t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_AbelCoset_Oset__add_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
set_ad92425877771022410t_unit: partia2670972154091845814t_unit > set_list_a > set_list_a > set_list_a ).
thf(sy_c_AbelCoset_Oset__add_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_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_001t__Set__Oset_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
partia5907974310037520643t_unit: partia6043505979758434576t_unit > set_set_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
partia707051561876973205xt_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001t__List__Olist_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
partia9041243232023819264t_unit: ( set_list_a > set_list_a ) > partia2670972154091845814t_unit > partia2670972154091845814t_unit ).
thf(sy_c_Congruence_Opartial__object_Ocarrier__update_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
partia8674076737563717228xt_a_b: ( set_a > set_a ) > partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b ).
thf(sy_c_Divisibility_Oirreducible_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
irredu4230924414530676029t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Divisibility_Oirreducible_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
irredu6211895646901577903xt_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Divisibility_Omonoid__cancel_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
monoid4303264861975686087t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Divisibility_Oprime_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
prime_2011924034616061926t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Divisibility_Oprime_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
prime_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Embedded__Algebras_Oring_Odimension_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
embedd3793949463769647726t_unit: partia2670972154091845814t_unit > nat > set_list_a > set_list_a > $o ).
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_Oline__extension_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
embedd5150658419831591667t_unit: partia2670972154091845814t_unit > set_list_a > list_a > set_list_a > set_list_a ).
thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001tf__a_001tf__b,type,
embedd971793762689825387on_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a > set_a ).
thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_001t__List__Olist_Itf__a_J,type,
finpro3417560807142560175list_a: partia2956882679547061052t_unit > ( list_a > list_list_a ) > set_list_a > list_list_a ).
thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001t__List__Olist_Itf__a_J,type,
finpro738134188688310831list_a: partia2670972154091845814t_unit > ( list_a > list_a ) > set_list_a > list_a ).
thf(sy_c_FiniteProduct_Ofinprod_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001tf__a,type,
finpro4329226410377213737unit_a: partia2670972154091845814t_unit > ( a > list_a ) > set_a > list_a ).
thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
finpro5422611492341532390st_a_a: partia2175431115845679010xt_a_b > ( ( list_a > a ) > a ) > set_list_a_a > a ).
thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
finpro4838020199848830884list_a: partia2175431115845679010xt_a_b > ( ( nat > list_a ) > a ) > set_nat_list_a > a ).
thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_It__Nat__Onat_Mtf__a_J,type,
finpro5839458686994656414_nat_a: partia2175431115845679010xt_a_b > ( ( nat > a ) > a ) > set_nat_a > a ).
thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
finpro4938371440467910406st_a_a: partia2175431115845679010xt_a_b > ( ( set_list_a > a ) > a ) > set_set_list_a_a > a ).
thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001t__List__Olist_Itf__a_J,type,
finpro6052973074229812797list_a: partia2175431115845679010xt_a_b > ( list_a > a ) > set_list_a > a ).
thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
finpro1280035270526425175_b_nat: partia2175431115845679010xt_a_b > ( nat > a ) > set_nat > a ).
thf(sy_c_FiniteProduct_Ofinprod_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_001tf__a,type,
finpro205304725090349623_a_b_a: partia2175431115845679010xt_a_b > ( a > a ) > set_a > a ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
finite2458174228029419510st_a_a: set_list_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
finite7630042315537210004list_a: set_nat_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Nat__Onat_Mtf__a_J,type,
finite_finite_nat_a: set_nat_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
finite6385009043124570134st_a_a: set_set_list_a_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
finite1660835950917165235list_a: set_list_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
finite5282473924520328461list_a: set_set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
inj_on_list_a_list_a: ( list_a > list_a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_Itf__a_J_001tf__a,type,
inj_on_list_a_a: ( list_a > a ) > set_list_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(sy_c_Group_OUnits_001t__List__Olist_It__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_001t__Set__Oset_Itf__a_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_Itf__a_J_Mt__Product____Type__Ounit_J,type,
units_2471184348132832486t_unit: partia6043505979758434576t_unit > set_set_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_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_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_001t__Set__Oset_Itf__a_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_Itf__a_J_Mt__Product____Type__Ounit_J,type,
mult_s7930653359683758801t_unit: partia6043505979758434576t_unit > set_a > set_a > set_a ).
thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
mult_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Group_Omonoid_Oone_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
one_li8234411390022467901t_unit: partia2956882679547061052t_unit > list_list_a ).
thf(sy_c_Group_Omonoid_Oone_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
one_li8328186300101108157t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Group_Omonoid_Oone_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
one_se1127990129394575805t_unit: partia7496981018696276118t_unit > set_list_a ).
thf(sy_c_Group_Omonoid_Oone_001t__Set__Oset_Itf__a_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_Itf__a_J_Mt__Product____Type__Ounit_J,type,
one_se211549098623999037t_unit: partia6043505979758434576t_unit > set_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_Osubmonoid_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
submon977613251886402007t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Group_Osubmonoid_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
submon8907322713594755401xt_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
minus_921748639838131438st_a_a: set_list_a_a > set_list_a_a > set_list_a_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
minus_4169782841487898290list_a: set_nat_list_a > set_nat_list_a > set_nat_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
minus_490503922182417452_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
minus_5613498140476352782st_a_a: set_set_list_a_a > set_set_list_a_a > set_set_list_a_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
minus_5335179877275218001list_a: set_list_list_a > set_list_list_a > set_list_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_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_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_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__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Ideal_Ocgenideal_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
cgenid9131348535277946915t_unit: partia2670972154091845814t_unit > list_a > set_list_a ).
thf(sy_c_Ideal_Ocgenideal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
cgenid547466209912283029xt_a_b: partia2175431115845679010xt_a_b > a > set_a ).
thf(sy_c_Ideal_Ogenideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
genide3243992037924705879t_unit: partia2670972154091845814t_unit > set_list_a > set_list_a ).
thf(sy_c_Ideal_Ogenideal_001tf__a_001tf__b,type,
genideal_a_b: partia2175431115845679010xt_a_b > set_a > set_a ).
thf(sy_c_Ideal_Omaximalideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
maxima6585700282301356660t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Ideal_Omaximalideal_001tf__a_001tf__b,type,
maximalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ideal_Oprimeideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
primei6309817859076077608t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Ideal_Oprimeideal_001tf__a_001tf__b,type,
primeideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ideal_Oprincipalideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
princi8786919440553033881t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_If_001t__List__Olist_Itf__a_J,type,
if_list_a: $o > list_a > list_a > list_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lagrange__Interpolation_Oring_Olagrange__basis__polynomial_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
lagran6985349428869127715t_unit: partia2670972154091845814t_unit > set_list_a > list_a > list_list_a ).
thf(sy_c_Lagrange__Interpolation_Oring_Olagrange__basis__polynomial_001tf__a_001tf__b,type,
lagran2649660974587678107al_a_b: partia2175431115845679010xt_a_b > set_a > a > list_a ).
thf(sy_c_Lagrange__Interpolation_Oring_Olagrange__basis__polynomial__aux_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
lagran3534788790333317459t_unit: partia2670972154091845814t_unit > set_list_a > list_list_a ).
thf(sy_c_Lagrange__Interpolation_Oring_Olagrange__basis__polynomial__aux_001tf__a_001tf__b,type,
lagran9092808442999052491ux_a_b: partia2175431115845679010xt_a_b > set_a > list_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_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_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_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > 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__List__Olist_Itf__a_J_Mtf__a_J_J,type,
bot_bot_set_list_a_a: set_list_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
bot_bo3806784159821827511list_a: set_nat_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
bot_bot_set_nat_a: set_nat_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
bot_bo8301825967528238409st_a_a: set_set_list_a_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
bot_bo1875519244922727510list_a: set_list_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
bot_bo3186585308812441520list_a: set_set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__List__Olist_Itf__a_J_Mtf__a_J_M_Eo_J,type,
ord_le5538412863658560464_a_a_o: ( ( list_a > a ) > $o ) > ( ( list_a > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_M_Eo_J,type,
ord_le4184171100712167858st_a_o: ( ( nat > list_a ) > $o ) > ( ( nat > list_a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
ord_less_eq_nat_a_o: ( ( nat > a ) > $o ) > ( ( nat > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J_M_Eo_J,type,
ord_le6553425858663066544_a_a_o: ( ( set_list_a > a ) > $o ) > ( ( set_list_a > a ) > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
ord_less_eq_list_a_o: ( list_a > $o ) > ( list_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $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__List__Olist_Itf__a_J_Mtf__a_J_J,type,
ord_le6942402695062981877st_a_a: set_list_a_a > set_list_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
ord_le2145805922479659755list_a: set_nat_list_a > set_nat_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
ord_le871467723717165285_nat_a: set_nat_a > set_nat_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J_J,type,
ord_le4799719167512954133st_a_a: set_set_list_a_a > set_set_list_a_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
ord_le8488217952732425610list_a: set_list_list_a > set_list_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
ord_le8877086941679407844list_a: set_set_list_a > set_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_001tf__a_001tf__b,type,
polyno5814909790663948098es_a_b: partia2175431115845679010xt_a_b > list_a > list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
polyno6951661231331188332t_unit: partia2670972154091845814t_unit > list_list_a > list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001tf__a_001tf__b,type,
polyno4133073214067823460ot_a_b: partia2175431115845679010xt_a_b > list_a > a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Olong__divides_001tf__a_001tf__b,type,
polyno2806191415236617128es_a_b: partia2175431115845679010xt_a_b > list_a > list_a > produc9164743771328383783list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Opdiv_001tf__a_001tf__b,type,
polynomial_pdiv_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Opmod_001tf__a_001tf__b,type,
polynomial_pmod_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Osplitted_001tf__a_001tf__b,type,
polyno8329700637149614481ed_a_b: partia2175431115845679010xt_a_b > list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Orupture_001tf__a_001tf__b,type,
polyno5459750281392823787re_a_b: partia2175431115845679010xt_a_b > set_a > list_a > partia7496981018696276118t_unit ).
thf(sy_c_Polynomials_Oring_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_001tf__a_001tf__b,type,
eval_a_b: partia2175431115845679010xt_a_b > list_a > a > a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
poly_o8716471131768098070t_unit: partia2670972154091845814t_unit > list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001tf__a_001tf__b,type,
poly_of_const_a_b: partia2175431115845679010xt_a_b > a > list_a ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
univ_p7953238456130426574t_unit: partia2670972154091845814t_unit > set_list_a > partia2956882679547061052t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001tf__a_001tf__b,type,
univ_poly_a_b: partia2175431115845679010xt_a_b > set_a > partia2670972154091845814t_unit ).
thf(sy_c_Polynomials_Ovar_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
var_li8453953174693405341t_unit: partia2670972154091845814t_unit > list_list_a ).
thf(sy_c_Polynomials_Ovar_001tf__a_001tf__b,type,
var_a_b: partia2175431115845679010xt_a_b > list_a ).
thf(sy_c_QuotRing_OFactRing_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
factRi3329376332477095402t_unit: partia2670972154091845814t_unit > set_list_a > partia7496981018696276118t_unit ).
thf(sy_c_QuotRing_OFactRing_001tf__a_001tf__b,type,
factRing_a_b: partia2175431115845679010xt_a_b > set_a > partia6043505979758434576t_unit ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
is_rin2993610189962786360t_unit: partia2670972154091845814t_unit > partia7496981018696276118t_unit > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
is_rin4843644836746533432t_unit: partia7496981018696276118t_unit > partia2670972154091845814t_unit > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
is_rin5597148638330396976it_a_b: partia7496981018696276118t_unit > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
is_rin6001486760346555702it_a_b: partia6043505979758434576t_unit > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001tf__a_001tf__b_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
is_rin9099215527551818550t_unit: partia2175431115845679010xt_a_b > partia6043505979758434576t_unit > $o ).
thf(sy_c_QuotRing_Oring__iso_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_i8122894263081988538it_a_b: partia7496981018696276118t_unit > partia2175431115845679010xt_a_b > set_set_list_a_a ).
thf(sy_c_Ring_Oa__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_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_Itf__a_J_001t__Product____Type__Ounit,type,
domain4236798911309298543t_unit: partia6043505979758434576t_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_Itf__a_J_J_001t__Product____Type__Ounit,type,
field_26233345952514695t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
field_6045675692312731021t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
field_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_l6212528067271185461t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_s2910681146719230829t_unit: partia7496981018696276118t_unit > set_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
zero_s2174465271003423091t_unit: partia6043505979758434576t_unit > set_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h5031276006722532742t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > set_li3422455791611400469list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h8078271382950527358it_a_b: partia2956882679547061052t_unit > partia2175431115845679010xt_a_b > set_list_list_a_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h8002040739877300486t_unit: partia2670972154091845814t_unit > partia2956882679547061052t_unit > set_li6773872926390105121list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h7399960747407462284t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > set_list_a_list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h6188449271506562988t_unit: partia2670972154091845814t_unit > partia7496981018696276118t_unit > set_li1071299071675007611list_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h2895973938487309444it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > set_list_a_a ).
thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h8038483918290310060t_unit: partia7496981018696276118t_unit > partia2670972154091845814t_unit > set_se5067313844698916539list_a ).
thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h6076331213207892940t_unit: partia7496981018696276118t_unit > partia7496981018696276118t_unit > set_se1917860372504128155list_a ).
thf(sy_c_Ring_Oring__hom_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h8906680420194085028it_a_b: partia7496981018696276118t_unit > partia2175431115845679010xt_a_b > set_set_list_a_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h6858658657455840382t_unit: partia2175431115845679010xt_a_b > partia2956882679547061052t_unit > set_a_list_list_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h405018892823518980t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > set_a_list_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h6109298854714515236t_unit: partia2175431115845679010xt_a_b > partia7496981018696276118t_unit > set_a_set_list_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_hom_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h8282015026914974507t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > ( list_a > list_a ) > $o ).
thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h5296475915237130059t_unit: partia2670972154091845814t_unit > partia7496981018696276118t_unit > ( list_a > set_list_a ) > $o ).
thf(sy_c_Ring_Oring__hom__cring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h1547129875642963619it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > ( list_a > a ) > $o ).
thf(sy_c_Ring_Oring__hom__cring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h7146510562020877131t_unit: partia7496981018696276118t_unit > partia2670972154091845814t_unit > ( set_list_a > list_a ) > $o ).
thf(sy_c_Ring_Oring__hom__cring_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_h1755269146545522539t_unit: partia7496981018696276118t_unit > partia7496981018696276118t_unit > ( set_list_a > set_list_a ) > $o ).
thf(sy_c_Ring_Oring__hom__cring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_h1101743994381864643it_a_b: partia7496981018696276118t_unit > partia2175431115845679010xt_a_b > ( set_list_a > a ) > $o ).
thf(sy_c_Ring_Oring__hom__cring_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h8279546866833948963t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > ( a > list_a ) > $o ).
thf(sy_c_Ring_Oring__hom__cring_001tf__a_001tf__b_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h7527734465757070659t_unit: partia2175431115845679010xt_a_b > partia7496981018696276118t_unit > ( a > set_list_a ) > $o ).
thf(sy_c_Ring_Oring__hom__cring_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_h661254511236296859_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > ( a > a ) > $o ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
semiri2265994252334843677t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
semiri2871908745932252451t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Osemiring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
semiri4000464634269493571t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_e7478897652244013592t_unit: partia2670972154091845814t_unit > ( list_a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001tf__a_001tf__b,type,
ring_e8745995371659049232in_a_b: partia2175431115845679010xt_a_b > ( a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001tf__a_001tf__b,type,
ring_f5272581269873410839in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001tf__a_001tf__b,type,
ring_n4045954140777738665in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001tf__a_001tf__b,type,
ring_n3639167112692572309ng_a_b: partia2175431115845679010xt_a_b > $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_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Set_OCollect_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
collect_list_a_a: ( ( list_a > a ) > $o ) > set_list_a_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
collect_nat_list_a: ( ( nat > list_a ) > $o ) > set_nat_list_a ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mtf__a_J,type,
collect_nat_a: ( ( nat > a ) > $o ) > set_nat_a ).
thf(sy_c_Set_OCollect_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
collect_set_list_a_a: ( ( set_list_a > a ) > $o ) > set_set_list_a_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
collect_set_list_a: ( set_list_a > $o ) > set_set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001_062_It__List__Olist_Itf__a_J_Mtf__a_J_001t__List__Olist_Itf__a_J,type,
image_8715568566693251358list_a: ( ( list_a > a ) > list_a ) > set_list_a_a > set_list_a ).
thf(sy_c_Set_Oimage_001_062_It__List__Olist_Itf__a_J_Mtf__a_J_001tf__a,type,
image_list_a_a_a: ( ( list_a > a ) > a ) > set_list_a_a > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
image_2569751527683375326list_a: ( ( nat > list_a ) > list_a ) > set_nat_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_001tf__a,type,
image_nat_list_a_a: ( ( nat > list_a ) > a ) > set_nat_list_a > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001t__List__Olist_Itf__a_J,type,
image_nat_a_list_a: ( ( nat > a ) > list_a ) > set_nat_a > set_list_a ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mtf__a_J_001tf__a,type,
image_nat_a_a: ( ( nat > a ) > a ) > set_nat_a > set_a ).
thf(sy_c_Set_Oimage_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J_001t__List__Olist_Itf__a_J,type,
image_2095547256931763454list_a: ( ( set_list_a > a ) > list_a ) > set_set_list_a_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__List__Olist_Itf__a_J_J_001tf__a,type,
image_list_list_a_a: ( list_list_a > a ) > set_list_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_8260866953997875467list_a: ( list_a > list_list_a ) > set_list_a > set_list_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Nat__Onat,type,
image_list_a_nat: ( list_a > nat ) > set_list_a > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mtf__a_J,type,
image_nat_nat_a: ( nat > nat > a ) > set_nat > set_nat_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_Itf__a_J,type,
image_nat_list_a: ( nat > list_a ) > set_nat > set_list_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
image_7934165218391885221list_a: ( set_list_a > list_a ) > set_set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001tf__a,type,
image_set_list_a_a: ( set_list_a > a ) > set_set_list_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
image_a_list_a_a: ( a > list_a > a ) > set_a > set_list_a_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
image_a_nat_list_a: ( a > nat > list_a ) > set_a > set_nat_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001_062_It__Nat__Onat_Mtf__a_J,type,
image_a_nat_a: ( a > nat > a ) > set_a > set_nat_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_a_list_list_a: ( a > list_list_a ) > set_a > set_list_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_Itf__a_J,type,
image_a_list_a: ( a > list_a ) > set_a > set_list_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
insert_list_a_a: ( list_a > a ) > set_list_a_a > set_list_a_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
insert_nat_list_a: ( nat > list_a ) > set_nat_list_a > set_nat_list_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mtf__a_J,type,
insert_nat_a: ( nat > a ) > set_nat_a > set_nat_a ).
thf(sy_c_Set_Oinsert_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
insert_set_list_a_a: ( set_list_a > a ) > set_set_list_a_a > set_set_list_a_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
insert_list_list_a: list_list_a > set_list_list_a > set_list_list_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
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_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_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_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subfie4546268998243038636t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
bound_list_list_a: list_list_a > nat > ( nat > list_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bound_set_list_a: set_list_a > nat > ( nat > set_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_lis8963924889346801084t_unit: partia2956882679547061052t_unit > set_nat_list_list_a ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_list_a ).
thf(sy_c_UnivPoly_Oup_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_set529185716248919906t_unit: partia7496981018696276118t_unit > set_nat_set_list_a ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member7168557129179038582list_a: ( list_list_a > list_a ) > set_li3422455791611400469list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_list_list_a_a: ( list_list_a > a ) > set_list_list_a_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member6714375691612171394list_a: ( list_a > list_list_a ) > set_li6773872926390105121list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member4263473470251683292list_a: ( list_a > set_list_a ) > set_li1071299071675007611list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8650753269014980122list_a: ( nat > list_list_a ) > set_nat_list_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
member_nat_list_a: ( nat > list_a ) > set_nat_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member491565700723299188list_a: ( nat > set_list_a ) > set_nat_set_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member5910328476188217884list_a: ( set_list_a > list_a ) > set_se5067313844698916539list_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5068272912271824380list_a: ( set_list_a > set_list_a ) > set_se1917860372504128155list_a > $o ).
thf(sy_c_member_001_062_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_set_list_a_a: ( set_list_a > a ) > set_set_list_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member_a_list_list_a: ( a > list_list_a ) > set_a_list_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member_a_set_list_a: ( a > set_list_a ) > set_a_set_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_S,type,
s: set_a ).
thf(sy_v_p____,type,
p: list_a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1278)
thf(fact_0_b,axiom,
member_list_a @ p @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% b
thf(fact_1_poly__of__const__in__carrier,axiom,
! [S: a] :
( ( member_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_2_c,axiom,
member_a @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) @ ( units_a_ring_ext_a_b @ r ) ).
% c
thf(fact_3_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_4_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_5_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_6_p__def,axiom,
( p
= ( lagran9092808442999052491ux_a_b @ r @ s ) ) ).
% p_def
thf(fact_7_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_8_x_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.m_assoc
thf(fact_9_x_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% x.m_comm
thf(fact_10_x_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% x.m_lcomm
thf(fact_11_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_12__092_060open_062_092_060And_062y_O_Ay_A_092_060in_062_AS_A_092_060Longrightarrow_062_Ax_A_092_060ominus_062_Ay_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
! [Y: a] :
( ( member_a @ Y @ s )
=> ( member_a @ ( a_minus_a_b @ r @ x @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% \<open>\<And>y. y \<in> S \<Longrightarrow> x \<ominus> y \<in> carrier R\<close>
thf(fact_13_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_14_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_15_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_16_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_17_assms_I3_J,axiom,
member_a @ x @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ s ) ).
% assms(3)
thf(fact_18_ring_Olagrange__basis__polynomial_Ocong,axiom,
lagran2649660974587678107al_a_b = lagran2649660974587678107al_a_b ).
% ring.lagrange_basis_polynomial.cong
thf(fact_19_ring_Olagrange__basis__polynomial_Ocong,axiom,
lagran6985349428869127715t_unit = lagran6985349428869127715t_unit ).
% ring.lagrange_basis_polynomial.cong
thf(fact_20_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_21_x_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.onepideal
thf(fact_22_x_Oring_Ohom__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ X @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.hom_closed
thf(fact_23_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_24_x_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.m_closed
thf(fact_25_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_26_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_27_x_Ocarrier__is__subcring,axiom,
subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.carrier_is_subcring
thf(fact_28_x_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.semiring_axioms
thf(fact_29_x_Oring__hom__restrict,axiom,
! [F: list_a > a,S2: partia2175431115845679010xt_a_b,G: list_a > a] :
( ( member_list_a_a @ F @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) )
=> ( ! [R: list_a] :
( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_list_a_a @ G @ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) ) ) ) ).
% x.ring_hom_restrict
thf(fact_30_x_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= bot_bot_set_list_a ) ).
% x.carrier_not_empty
thf(fact_31_lagrange__aux__poly,axiom,
! [S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% lagrange_aux_poly
thf(fact_32_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_33_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_34_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_35_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_36_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_37_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran3534788790333317459t_unit = lagran3534788790333317459t_unit ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_38_x_Omonoid__cancelI,axiom,
( ! [A: list_a,B: list_a,C: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) )
=> ( ! [A: list_a,B: list_a,C: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monoid_cancelI
thf(fact_39_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_40_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_41_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_42_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_43_lagrange__basis__polynomial__def,axiom,
! [S2: set_a,X: a] :
( ( lagran2649660974587678107al_a_b @ r @ S2 @ X )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( lagran9092808442999052491ux_a_b @ r @ S2 ) @ ( poly_of_const_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ X ) @ S2 ) ) ) ) ) ).
% lagrange_basis_polynomial_def
thf(fact_44__092_060open_062_092_060And_062s_O_As_A_092_060in_062_AS_A_092_060Longrightarrow_062_Alocal_Oeval_A_Ilagrange__basis__polynomial_AS_Ax_J_As_A_061_A_092_060zero_062_092_060close_062,axiom,
! [S: a] :
( ( member_a @ S @ s )
=> ( ( eval_a_b @ r @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) @ S )
= ( zero_a_b @ r ) ) ) ).
% \<open>\<And>s. s \<in> S \<Longrightarrow> local.eval (lagrange_basis_polynomial S x) s = \<zero>\<close>
thf(fact_45_a,axiom,
( ( eval_a_b @ r @ p @ x )
= ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ).
% a
thf(fact_46__092_060open_062local_Oeval_A_Ilagrange__basis__polynomial_AS_Ax_J_Ax_A_061_A_092_060one_062_092_060close_062,axiom,
( ( eval_a_b @ r @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) @ x )
= ( one_a_ring_ext_a_b @ r ) ) ).
% \<open>local.eval (lagrange_basis_polynomial S x) x = \<one>\<close>
thf(fact_47_mem__Collect__eq,axiom,
! [A3: list_a > a,P2: ( list_a > a ) > $o] :
( ( member_list_a_a @ A3 @ ( collect_list_a_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A3: set_list_a > a,P2: ( set_list_a > a ) > $o] :
( ( member_set_list_a_a @ A3 @ ( collect_set_list_a_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A3: nat > list_a,P2: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ A3 @ ( collect_nat_list_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_50_mem__Collect__eq,axiom,
! [A3: nat > a,P2: ( nat > a ) > $o] :
( ( member_nat_a @ A3 @ ( collect_nat_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A3: a,P2: a > $o] :
( ( member_a @ A3 @ ( collect_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A3: nat,P2: nat > $o] :
( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A3: list_a,P2: list_a > $o] :
( ( member_list_a @ A3 @ ( collect_list_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A3: list_list_a,P2: list_list_a > $o] :
( ( member_list_list_a @ A3 @ ( collect_list_list_a @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A2: set_list_a_a] :
( ( collect_list_a_a
@ ^ [X2: list_a > a] : ( member_list_a_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_set_list_a_a] :
( ( collect_set_list_a_a
@ ^ [X2: set_list_a > a] : ( member_set_list_a_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_nat_list_a] :
( ( collect_nat_list_a
@ ^ [X2: nat > list_a] : ( member_nat_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X2: list_list_a] : ( member_list_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_Collect__cong,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_64_Collect__cong,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_65_Collect__cong,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ! [X3: list_a] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_list_a @ P2 )
= ( collect_list_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_66_Collect__cong,axiom,
! [P2: list_list_a > $o,Q2: list_list_a > $o] :
( ! [X3: list_list_a] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect_list_list_a @ P2 )
= ( collect_list_list_a @ Q2 ) ) ) ).
% Collect_cong
thf(fact_67_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_68_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_69_finite__Diff,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_70_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_71_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_72_finite__Diff2,axiom,
! [B2: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
= ( finite_finite_list_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_73_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_74_Diff__empty,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Diff_empty
thf(fact_75_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_76_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_77_finprod__one__eqI,axiom,
! [A2: set_list_a_a,F: ( list_a > a ) > a] :
( ! [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A2 )
=> ( ( F @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro5422611492341532390st_a_a @ r @ F @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_78_finprod__one__eqI,axiom,
! [A2: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ! [X3: set_list_a > a] :
( ( member_set_list_a_a @ X3 @ A2 )
=> ( ( F @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro4938371440467910406st_a_a @ r @ F @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_79_finprod__one__eqI,axiom,
! [A2: set_nat_list_a,F: ( nat > list_a ) > a] :
( ! [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A2 )
=> ( ( F @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro4838020199848830884list_a @ r @ F @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_80_finprod__one__eqI,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A2 )
=> ( ( F @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro5839458686994656414_nat_a @ r @ F @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_81_finprod__one__eqI,axiom,
! [A2: set_a,F: a > a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( F @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_one_eqI
thf(fact_82_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_83_subset__antisym,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_84_subsetI,axiom,
! [A2: set_list_a_a,B2: set_list_a_a] :
( ! [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A2 )
=> ( member_list_a_a @ X3 @ B2 ) )
=> ( ord_le6942402695062981877st_a_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_85_subsetI,axiom,
! [A2: set_set_list_a_a,B2: set_set_list_a_a] :
( ! [X3: set_list_a > a] :
( ( member_set_list_a_a @ X3 @ A2 )
=> ( member_set_list_a_a @ X3 @ B2 ) )
=> ( ord_le4799719167512954133st_a_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_86_subsetI,axiom,
! [A2: set_nat_list_a,B2: set_nat_list_a] :
( ! [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A2 )
=> ( member_nat_list_a @ X3 @ B2 ) )
=> ( ord_le2145805922479659755list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_87_subsetI,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A2 )
=> ( member_nat_a @ X3 @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_88_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_89_subsetI,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( member_list_a @ X3 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_90_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_91_empty__Collect__eq,axiom,
! [P2: list_list_a > $o] :
( ( bot_bo1875519244922727510list_a
= ( collect_list_list_a @ P2 ) )
= ( ! [X2: list_list_a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_92_empty__Collect__eq,axiom,
! [P2: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P2 ) )
= ( ! [X2: list_a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_93_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_94_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_95_Collect__empty__eq,axiom,
! [P2: list_list_a > $o] :
( ( ( collect_list_list_a @ P2 )
= bot_bo1875519244922727510list_a )
= ( ! [X2: list_list_a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_96_Collect__empty__eq,axiom,
! [P2: list_a > $o] :
( ( ( collect_list_a @ P2 )
= bot_bot_set_list_a )
= ( ! [X2: list_a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_97_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_98_all__not__in__conv,axiom,
! [A2: set_list_a_a] :
( ( ! [X2: list_a > a] :
~ ( member_list_a_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_list_a_a ) ) ).
% all_not_in_conv
thf(fact_99_all__not__in__conv,axiom,
! [A2: set_set_list_a_a] :
( ( ! [X2: set_list_a > a] :
~ ( member_set_list_a_a @ X2 @ A2 ) )
= ( A2 = bot_bo8301825967528238409st_a_a ) ) ).
% all_not_in_conv
thf(fact_100_all__not__in__conv,axiom,
! [A2: set_nat_list_a] :
( ( ! [X2: nat > list_a] :
~ ( member_nat_list_a @ X2 @ A2 ) )
= ( A2 = bot_bo3806784159821827511list_a ) ) ).
% all_not_in_conv
thf(fact_101_all__not__in__conv,axiom,
! [A2: set_nat_a] :
( ( ! [X2: nat > a] :
~ ( member_nat_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_102_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X2: list_a] :
~ ( member_list_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_103_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_104_empty__iff,axiom,
! [C2: list_a > a] :
~ ( member_list_a_a @ C2 @ bot_bot_set_list_a_a ) ).
% empty_iff
thf(fact_105_empty__iff,axiom,
! [C2: set_list_a > a] :
~ ( member_set_list_a_a @ C2 @ bot_bo8301825967528238409st_a_a ) ).
% empty_iff
thf(fact_106_empty__iff,axiom,
! [C2: nat > list_a] :
~ ( member_nat_list_a @ C2 @ bot_bo3806784159821827511list_a ) ).
% empty_iff
thf(fact_107_empty__iff,axiom,
! [C2: nat > a] :
~ ( member_nat_a @ C2 @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_108_empty__iff,axiom,
! [C2: list_a] :
~ ( member_list_a @ C2 @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_109_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_110_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_111_Diff__idemp,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ B2 )
= ( minus_646659088055828811list_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_112_Diff__iff,axiom,
! [C2: list_a > a,A2: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C2 @ ( minus_921748639838131438st_a_a @ A2 @ B2 ) )
= ( ( member_list_a_a @ C2 @ A2 )
& ~ ( member_list_a_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_113_Diff__iff,axiom,
! [C2: set_list_a > a,A2: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( minus_5613498140476352782st_a_a @ A2 @ B2 ) )
= ( ( member_set_list_a_a @ C2 @ A2 )
& ~ ( member_set_list_a_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_114_Diff__iff,axiom,
! [C2: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( minus_4169782841487898290list_a @ A2 @ B2 ) )
= ( ( member_nat_list_a @ C2 @ A2 )
& ~ ( member_nat_list_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_115_Diff__iff,axiom,
! [C2: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C2 @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
= ( ( member_nat_a @ C2 @ A2 )
& ~ ( member_nat_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_116_Diff__iff,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_117_Diff__iff,axiom,
! [C2: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
= ( ( member_list_a @ C2 @ A2 )
& ~ ( member_list_a @ C2 @ B2 ) ) ) ).
% Diff_iff
thf(fact_118_DiffI,axiom,
! [C2: list_a > a,A2: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C2 @ A2 )
=> ( ~ ( member_list_a_a @ C2 @ B2 )
=> ( member_list_a_a @ C2 @ ( minus_921748639838131438st_a_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_119_DiffI,axiom,
! [C2: set_list_a > a,A2: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ A2 )
=> ( ~ ( member_set_list_a_a @ C2 @ B2 )
=> ( member_set_list_a_a @ C2 @ ( minus_5613498140476352782st_a_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_120_DiffI,axiom,
! [C2: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ A2 )
=> ( ~ ( member_nat_list_a @ C2 @ B2 )
=> ( member_nat_list_a @ C2 @ ( minus_4169782841487898290list_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_121_DiffI,axiom,
! [C2: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C2 @ A2 )
=> ( ~ ( member_nat_a @ C2 @ B2 )
=> ( member_nat_a @ C2 @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_122_DiffI,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B2 )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_123_DiffI,axiom,
! [C2: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C2 @ A2 )
=> ( ~ ( member_list_a @ C2 @ B2 )
=> ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_124_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_125_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_126_finprod__zero__iff,axiom,
! [A2: set_list_a_a,F: ( list_a > a ) > a] :
( ( finite2458174228029419510st_a_a @ A2 )
=> ( ! [A: list_a > a] :
( ( member_list_a_a @ A @ A2 )
=> ( member_a @ ( F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro5422611492341532390st_a_a @ r @ F @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: list_a > a] :
( ( member_list_a_a @ X2 @ A2 )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_127_finprod__zero__iff,axiom,
! [A2: set_set_list_a_a,F: ( set_list_a > a ) > a] :
( ( finite6385009043124570134st_a_a @ A2 )
=> ( ! [A: set_list_a > a] :
( ( member_set_list_a_a @ A @ A2 )
=> ( member_a @ ( F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro4938371440467910406st_a_a @ r @ F @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A2 )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_128_finprod__zero__iff,axiom,
! [A2: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( finite7630042315537210004list_a @ A2 )
=> ( ! [A: nat > list_a] :
( ( member_nat_list_a @ A @ A2 )
=> ( member_a @ ( F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro4838020199848830884list_a @ r @ F @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A2 )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_129_finprod__zero__iff,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a] :
( ( finite_finite_nat_a @ A2 )
=> ( ! [A: nat > a] :
( ( member_nat_a @ A @ A2 )
=> ( member_a @ ( F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro5839458686994656414_nat_a @ r @ F @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_130_finprod__zero__iff,axiom,
! [A2: set_list_a,F: list_a > a] :
( ( finite_finite_list_a @ A2 )
=> ( ! [A: list_a] :
( ( member_list_a @ A @ A2 )
=> ( member_a @ ( F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro6052973074229812797list_a @ r @ F @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_131_finprod__zero__iff,axiom,
! [A2: set_nat,F: nat > a] :
( ( finite_finite_nat @ A2 )
=> ( ! [A: nat] :
( ( member_nat @ A @ A2 )
=> ( member_a @ ( F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro1280035270526425175_b_nat @ r @ F @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_132_finprod__zero__iff,axiom,
! [A2: set_a,F: a > a] :
( ( finite_finite_a @ A2 )
=> ( ! [A: a] :
( ( member_a @ A @ A2 )
=> ( member_a @ ( F @ A ) @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( finpro205304725090349623_a_b_a @ r @ F @ A2 )
= ( zero_a_b @ r ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( ( F @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ) ) ).
% finprod_zero_iff
thf(fact_133_finite__Collect__conjI,axiom,
! [P2: list_list_a > $o,Q2: list_list_a > $o] :
( ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P2 ) )
| ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q2 ) ) )
=> ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_134_finite__Collect__conjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P2 ) )
| ( finite_finite_a @ ( collect_a @ Q2 ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_135_finite__Collect__conjI,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
| ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) )
=> ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_136_finite__Collect__conjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
| ( finite_finite_nat @ ( collect_nat @ Q2 ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_137_finite__Collect__disjI,axiom,
! [P2: list_list_a > $o,Q2: list_list_a > $o] :
( ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite1660835950917165235list_a @ ( collect_list_list_a @ P2 ) )
& ( finite1660835950917165235list_a @ ( collect_list_list_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_138_finite__Collect__disjI,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P2 ) )
& ( finite_finite_a @ ( collect_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_139_finite__Collect__disjI,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( finite_finite_list_a
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
& ( finite_finite_list_a @ ( collect_list_a @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_140_finite__Collect__disjI,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
& ( finite_finite_nat @ ( collect_nat @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_141_finite__number__of__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823460ot_a_b @ r @ P ) ) ) ) ).
% finite_number_of_roots
thf(fact_142_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_143_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_144_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_145_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_146_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_147_Diff__cancel,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ A2 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_148_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_149_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_150_finite__Collect__subsets,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite1152437895449049373et_nat
@ ( collect_set_nat
@ ^ [B3: set_nat] : ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_151_finite__Collect__subsets,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_set_a
@ ( collect_set_a
@ ^ [B3: set_a] : ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_152_finite__Collect__subsets,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite5282473924520328461list_a
@ ( collect_set_list_a
@ ^ [B3: set_list_a] : ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_153_lagrange__aux__eval,axiom,
! [S2: set_a,X: a] :
( ( finite_finite_a @ S2 )
=> ( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( lagran9092808442999052491ux_a_b @ r @ S2 ) @ X )
= ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ X ) @ S2 ) ) ) ) ) ).
% lagrange_aux_eval
thf(fact_154_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_155_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_156_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_157_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_158_finprod__one,axiom,
! [A2: set_a] :
( ( finpro205304725090349623_a_b_a @ r
@ ^ [I: a] : ( one_a_ring_ext_a_b @ r )
@ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_one
thf(fact_159_finprod__empty,axiom,
! [F: list_a > a] :
( ( finpro6052973074229812797list_a @ r @ F @ bot_bot_set_list_a )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_empty
thf(fact_160_finprod__empty,axiom,
! [F: a > a] :
( ( finpro205304725090349623_a_b_a @ r @ F @ bot_bot_set_a )
= ( one_a_ring_ext_a_b @ r ) ) ).
% finprod_empty
thf(fact_161_finprod__infinite,axiom,
! [A2: set_list_a,F: list_a > a] :
( ~ ( finite_finite_list_a @ A2 )
=> ( ( finpro6052973074229812797list_a @ r @ F @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_162_finprod__infinite,axiom,
! [A2: set_nat,F: nat > a] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finpro1280035270526425175_b_nat @ r @ F @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_163_finprod__infinite,axiom,
! [A2: set_a,F: a > a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finpro205304725090349623_a_b_a @ r @ F @ A2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ).
% finprod_infinite
thf(fact_164_r__right__minus__eq,axiom,
! [A3: a,B4: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A3 @ B4 )
= ( zero_a_b @ r ) )
= ( A3 = B4 ) ) ) ) ).
% r_right_minus_eq
thf(fact_165_x_Oring_Ohomh,axiom,
( member_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( ring_h2895973938487309444it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r ) ) ).
% x.ring.homh
thf(fact_166_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% empty_def
thf(fact_167_empty__def,axiom,
( bot_bo1875519244922727510list_a
= ( collect_list_list_a
@ ^ [X2: list_list_a] : $false ) ) ).
% empty_def
thf(fact_168_empty__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a
@ ^ [X2: list_a] : $false ) ) ).
% empty_def
thf(fact_169_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_170_set__diff__eq,axiom,
( minus_921748639838131438st_a_a
= ( ^ [A4: set_list_a_a,B3: set_list_a_a] :
( collect_list_a_a
@ ^ [X2: list_a > a] :
( ( member_list_a_a @ X2 @ A4 )
& ~ ( member_list_a_a @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_171_set__diff__eq,axiom,
( minus_5613498140476352782st_a_a
= ( ^ [A4: set_set_list_a_a,B3: set_set_list_a_a] :
( collect_set_list_a_a
@ ^ [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A4 )
& ~ ( member_set_list_a_a @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_172_set__diff__eq,axiom,
( minus_4169782841487898290list_a
= ( ^ [A4: set_nat_list_a,B3: set_nat_list_a] :
( collect_nat_list_a
@ ^ [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A4 )
& ~ ( member_nat_list_a @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_173_set__diff__eq,axiom,
( minus_490503922182417452_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ A4 )
& ~ ( member_nat_a @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_174_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A4: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A4 )
& ~ ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_175_set__diff__eq,axiom,
( minus_5335179877275218001list_a
= ( ^ [A4: set_list_list_a,B3: set_list_list_a] :
( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A4 )
& ~ ( member_list_list_a @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_176_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A4: set_a,B3: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A4 )
& ~ ( member_a @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_177_set__diff__eq,axiom,
( minus_646659088055828811list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ A4 )
& ~ ( member_list_a @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_178_Collect__subset,axiom,
! [A2: set_list_a_a,P2: ( list_a > a ) > $o] :
( ord_le6942402695062981877st_a_a
@ ( collect_list_a_a
@ ^ [X2: list_a > a] :
( ( member_list_a_a @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_179_Collect__subset,axiom,
! [A2: set_set_list_a_a,P2: ( set_list_a > a ) > $o] :
( ord_le4799719167512954133st_a_a
@ ( collect_set_list_a_a
@ ^ [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_180_Collect__subset,axiom,
! [A2: set_nat_list_a,P2: ( nat > list_a ) > $o] :
( ord_le2145805922479659755list_a
@ ( collect_nat_list_a
@ ^ [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_181_Collect__subset,axiom,
! [A2: set_nat_a,P2: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_182_Collect__subset,axiom,
! [A2: set_nat,P2: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_183_Collect__subset,axiom,
! [A2: set_list_list_a,P2: list_list_a > $o] :
( ord_le8488217952732425610list_a
@ ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_184_Collect__subset,axiom,
! [A2: set_a,P2: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_185_Collect__subset,axiom,
! [A2: set_list_a,P2: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( P2 @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_186_not__finite__existsD,axiom,
! [P2: list_list_a > $o] :
( ~ ( finite1660835950917165235list_a @ ( collect_list_list_a @ P2 ) )
=> ? [X_1: list_list_a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_187_not__finite__existsD,axiom,
! [P2: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P2 ) )
=> ? [X_1: a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_188_not__finite__existsD,axiom,
! [P2: list_a > $o] :
( ~ ( finite_finite_list_a @ ( collect_list_a @ P2 ) )
=> ? [X_1: list_a] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_189_not__finite__existsD,axiom,
! [P2: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P2 ) )
=> ? [X_1: nat] : ( P2 @ X_1 ) ) ).
% not_finite_existsD
thf(fact_190_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_a,R3: a > a > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A2 )
& ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_191_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_nat,R3: a > nat > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A2 )
& ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_192_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_a,R3: nat > a > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_193_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_nat,R3: nat > nat > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_194_pigeonhole__infinite__rel,axiom,
! [A2: set_a,B2: set_list_a,R3: a > list_a > $o] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_finite_list_a @ B2 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B2 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ B2 )
& ~ ( finite_finite_a
@ ( collect_a
@ ^ [A5: a] :
( ( member_a @ A5 @ A2 )
& ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_195_pigeonhole__infinite__rel,axiom,
! [A2: set_list_a,B2: set_a,R3: list_a > a > $o] :
( ~ ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A5: list_a] :
( ( member_list_a @ A5 @ A2 )
& ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_196_pigeonhole__infinite__rel,axiom,
! [A2: set_list_a,B2: set_nat,R3: list_a > nat > $o] :
( ~ ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ? [Xa: nat] :
( ( member_nat @ Xa @ B2 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite_finite_list_a
@ ( collect_list_a
@ ^ [A5: list_a] :
( ( member_list_a @ A5 @ A2 )
& ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_197_pigeonhole__infinite__rel,axiom,
! [A2: set_nat,B2: set_list_a,R3: nat > list_a > $o] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_finite_list_a @ B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ B2 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ B2 )
& ~ ( finite_finite_nat
@ ( collect_nat
@ ^ [A5: nat] :
( ( member_nat @ A5 @ A2 )
& ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_198_pigeonhole__infinite__rel,axiom,
! [A2: set_nat_a,B2: set_a,R3: ( nat > a ) > a > $o] :
( ~ ( finite_finite_nat_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ~ ( finite_finite_nat_a
@ ( collect_nat_a
@ ^ [A5: nat > a] :
( ( member_nat_a @ A5 @ A2 )
& ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_199_pigeonhole__infinite__rel,axiom,
! [A2: set_list_list_a,B2: set_a,R3: list_list_a > a > $o] :
( ~ ( finite1660835950917165235list_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A2 )
=> ? [Xa: a] :
( ( member_a @ Xa @ B2 )
& ( R3 @ X3 @ Xa ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ~ ( finite1660835950917165235list_a
@ ( collect_list_list_a
@ ^ [A5: list_list_a] :
( ( member_list_list_a @ A5 @ A2 )
& ( R3 @ A5 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_200_Collect__mono__iff,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) )
= ( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_201_Collect__mono__iff,axiom,
! [P2: list_list_a > $o,Q2: list_list_a > $o] :
( ( ord_le8488217952732425610list_a @ ( collect_list_list_a @ P2 ) @ ( collect_list_list_a @ Q2 ) )
= ( ! [X2: list_list_a] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_202_Collect__mono__iff,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) )
= ( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_203_Collect__mono__iff,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) )
= ( ! [X2: list_a] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_204_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z2: set_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A4 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_205_set__eq__subset,axiom,
( ( ^ [Y2: set_list_a,Z2: set_list_a] : ( Y2 = Z2 ) )
= ( ^ [A4: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_206_subset__trans,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_207_subset__trans,axiom,
! [A2: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C3 )
=> ( ord_le8861187494160871172list_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_208_Collect__mono,axiom,
! [P2: nat > $o,Q2: nat > $o] :
( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q2 ) ) ) ).
% Collect_mono
thf(fact_209_Collect__mono,axiom,
! [P2: list_list_a > $o,Q2: list_list_a > $o] :
( ! [X3: list_list_a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le8488217952732425610list_a @ ( collect_list_list_a @ P2 ) @ ( collect_list_list_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_210_Collect__mono,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_211_Collect__mono,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ! [X3: list_a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_212_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_213_subset__refl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_214_subset__iff,axiom,
( ord_le6942402695062981877st_a_a
= ( ^ [A4: set_list_a_a,B3: set_list_a_a] :
! [T: list_a > a] :
( ( member_list_a_a @ T @ A4 )
=> ( member_list_a_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_215_subset__iff,axiom,
( ord_le4799719167512954133st_a_a
= ( ^ [A4: set_set_list_a_a,B3: set_set_list_a_a] :
! [T: set_list_a > a] :
( ( member_set_list_a_a @ T @ A4 )
=> ( member_set_list_a_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_216_subset__iff,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A4: set_nat_list_a,B3: set_nat_list_a] :
! [T: nat > list_a] :
( ( member_nat_list_a @ T @ A4 )
=> ( member_nat_list_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_217_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
! [T: nat > a] :
( ( member_nat_a @ T @ A4 )
=> ( member_nat_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_218_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_219_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A4 )
=> ( member_list_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_220_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_221_equalityD2,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_222_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_223_equalityD1,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_224_subset__eq,axiom,
( ord_le6942402695062981877st_a_a
= ( ^ [A4: set_list_a_a,B3: set_list_a_a] :
! [X2: list_a > a] :
( ( member_list_a_a @ X2 @ A4 )
=> ( member_list_a_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_225_subset__eq,axiom,
( ord_le4799719167512954133st_a_a
= ( ^ [A4: set_set_list_a_a,B3: set_set_list_a_a] :
! [X2: set_list_a > a] :
( ( member_set_list_a_a @ X2 @ A4 )
=> ( member_set_list_a_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_226_subset__eq,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A4: set_nat_list_a,B3: set_nat_list_a] :
! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A4 )
=> ( member_nat_list_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_227_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
! [X2: nat > a] :
( ( member_nat_a @ X2 @ A4 )
=> ( member_nat_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_228_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A4 )
=> ( member_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_229_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
! [X2: list_a] :
( ( member_list_a @ X2 @ A4 )
=> ( member_list_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_230_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_231_equalityE,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ~ ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_232_subsetD,axiom,
! [A2: set_list_a_a,B2: set_list_a_a,C2: list_a > a] :
( ( ord_le6942402695062981877st_a_a @ A2 @ B2 )
=> ( ( member_list_a_a @ C2 @ A2 )
=> ( member_list_a_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_233_subsetD,axiom,
! [A2: set_set_list_a_a,B2: set_set_list_a_a,C2: set_list_a > a] :
( ( ord_le4799719167512954133st_a_a @ A2 @ B2 )
=> ( ( member_set_list_a_a @ C2 @ A2 )
=> ( member_set_list_a_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_234_subsetD,axiom,
! [A2: set_nat_list_a,B2: set_nat_list_a,C2: nat > list_a] :
( ( ord_le2145805922479659755list_a @ A2 @ B2 )
=> ( ( member_nat_list_a @ C2 @ A2 )
=> ( member_nat_list_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_235_subsetD,axiom,
! [A2: set_nat_a,B2: set_nat_a,C2: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( member_nat_a @ C2 @ A2 )
=> ( member_nat_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_236_subsetD,axiom,
! [A2: set_a,B2: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_237_subsetD,axiom,
! [A2: set_list_a,B2: set_list_a,C2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( member_list_a @ C2 @ A2 )
=> ( member_list_a @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_238_in__mono,axiom,
! [A2: set_list_a_a,B2: set_list_a_a,X: list_a > a] :
( ( ord_le6942402695062981877st_a_a @ A2 @ B2 )
=> ( ( member_list_a_a @ X @ A2 )
=> ( member_list_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_239_in__mono,axiom,
! [A2: set_set_list_a_a,B2: set_set_list_a_a,X: set_list_a > a] :
( ( ord_le4799719167512954133st_a_a @ A2 @ B2 )
=> ( ( member_set_list_a_a @ X @ A2 )
=> ( member_set_list_a_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_240_in__mono,axiom,
! [A2: set_nat_list_a,B2: set_nat_list_a,X: nat > list_a] :
( ( ord_le2145805922479659755list_a @ A2 @ B2 )
=> ( ( member_nat_list_a @ X @ A2 )
=> ( member_nat_list_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_241_in__mono,axiom,
! [A2: set_nat_a,B2: set_nat_a,X: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_nat_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_242_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_243_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_244_ex__in__conv,axiom,
! [A2: set_list_a_a] :
( ( ? [X2: list_a > a] : ( member_list_a_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_list_a_a ) ) ).
% ex_in_conv
thf(fact_245_ex__in__conv,axiom,
! [A2: set_set_list_a_a] :
( ( ? [X2: set_list_a > a] : ( member_set_list_a_a @ X2 @ A2 ) )
= ( A2 != bot_bo8301825967528238409st_a_a ) ) ).
% ex_in_conv
thf(fact_246_ex__in__conv,axiom,
! [A2: set_nat_list_a] :
( ( ? [X2: nat > list_a] : ( member_nat_list_a @ X2 @ A2 ) )
= ( A2 != bot_bo3806784159821827511list_a ) ) ).
% ex_in_conv
thf(fact_247_ex__in__conv,axiom,
! [A2: set_nat_a] :
( ( ? [X2: nat > a] : ( member_nat_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat_a ) ) ).
% ex_in_conv
thf(fact_248_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_249_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_250_equals0I,axiom,
! [A2: set_list_a_a] :
( ! [Y3: list_a > a] :
~ ( member_list_a_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_list_a_a ) ) ).
% equals0I
thf(fact_251_equals0I,axiom,
! [A2: set_set_list_a_a] :
( ! [Y3: set_list_a > a] :
~ ( member_set_list_a_a @ Y3 @ A2 )
=> ( A2 = bot_bo8301825967528238409st_a_a ) ) ).
% equals0I
thf(fact_252_equals0I,axiom,
! [A2: set_nat_list_a] :
( ! [Y3: nat > list_a] :
~ ( member_nat_list_a @ Y3 @ A2 )
=> ( A2 = bot_bo3806784159821827511list_a ) ) ).
% equals0I
thf(fact_253_equals0I,axiom,
! [A2: set_nat_a] :
( ! [Y3: nat > a] :
~ ( member_nat_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_nat_a ) ) ).
% equals0I
thf(fact_254_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y3: list_a] :
~ ( member_list_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_255_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_256_equals0D,axiom,
! [A2: set_list_a_a,A3: list_a > a] :
( ( A2 = bot_bot_set_list_a_a )
=> ~ ( member_list_a_a @ A3 @ A2 ) ) ).
% equals0D
thf(fact_257_equals0D,axiom,
! [A2: set_set_list_a_a,A3: set_list_a > a] :
( ( A2 = bot_bo8301825967528238409st_a_a )
=> ~ ( member_set_list_a_a @ A3 @ A2 ) ) ).
% equals0D
thf(fact_258_equals0D,axiom,
! [A2: set_nat_list_a,A3: nat > list_a] :
( ( A2 = bot_bo3806784159821827511list_a )
=> ~ ( member_nat_list_a @ A3 @ A2 ) ) ).
% equals0D
thf(fact_259_equals0D,axiom,
! [A2: set_nat_a,A3: nat > a] :
( ( A2 = bot_bot_set_nat_a )
=> ~ ( member_nat_a @ A3 @ A2 ) ) ).
% equals0D
thf(fact_260_equals0D,axiom,
! [A2: set_list_a,A3: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A3 @ A2 ) ) ).
% equals0D
thf(fact_261_equals0D,axiom,
! [A2: set_a,A3: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A3 @ A2 ) ) ).
% equals0D
thf(fact_262_emptyE,axiom,
! [A3: list_a > a] :
~ ( member_list_a_a @ A3 @ bot_bot_set_list_a_a ) ).
% emptyE
thf(fact_263_emptyE,axiom,
! [A3: set_list_a > a] :
~ ( member_set_list_a_a @ A3 @ bot_bo8301825967528238409st_a_a ) ).
% emptyE
thf(fact_264_emptyE,axiom,
! [A3: nat > list_a] :
~ ( member_nat_list_a @ A3 @ bot_bo3806784159821827511list_a ) ).
% emptyE
thf(fact_265_emptyE,axiom,
! [A3: nat > a] :
~ ( member_nat_a @ A3 @ bot_bot_set_nat_a ) ).
% emptyE
thf(fact_266_emptyE,axiom,
! [A3: list_a] :
~ ( member_list_a @ A3 @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_267_emptyE,axiom,
! [A3: a] :
~ ( member_a @ A3 @ bot_bot_set_a ) ).
% emptyE
thf(fact_268_DiffD2,axiom,
! [C2: list_a > a,A2: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C2 @ ( minus_921748639838131438st_a_a @ A2 @ B2 ) )
=> ~ ( member_list_a_a @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_269_DiffD2,axiom,
! [C2: set_list_a > a,A2: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( minus_5613498140476352782st_a_a @ A2 @ B2 ) )
=> ~ ( member_set_list_a_a @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_270_DiffD2,axiom,
! [C2: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( minus_4169782841487898290list_a @ A2 @ B2 ) )
=> ~ ( member_nat_list_a @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_271_DiffD2,axiom,
! [C2: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C2 @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ~ ( member_nat_a @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_272_DiffD2,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_273_DiffD2,axiom,
! [C2: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ~ ( member_list_a @ C2 @ B2 ) ) ).
% DiffD2
thf(fact_274_DiffD1,axiom,
! [C2: list_a > a,A2: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C2 @ ( minus_921748639838131438st_a_a @ A2 @ B2 ) )
=> ( member_list_a_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_275_DiffD1,axiom,
! [C2: set_list_a > a,A2: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( minus_5613498140476352782st_a_a @ A2 @ B2 ) )
=> ( member_set_list_a_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_276_DiffD1,axiom,
! [C2: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( minus_4169782841487898290list_a @ A2 @ B2 ) )
=> ( member_nat_list_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_277_DiffD1,axiom,
! [C2: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C2 @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ( member_nat_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_278_DiffD1,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_279_DiffD1,axiom,
! [C2: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ( member_list_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_280_DiffE,axiom,
! [C2: list_a > a,A2: set_list_a_a,B2: set_list_a_a] :
( ( member_list_a_a @ C2 @ ( minus_921748639838131438st_a_a @ A2 @ B2 ) )
=> ~ ( ( member_list_a_a @ C2 @ A2 )
=> ( member_list_a_a @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_281_DiffE,axiom,
! [C2: set_list_a > a,A2: set_set_list_a_a,B2: set_set_list_a_a] :
( ( member_set_list_a_a @ C2 @ ( minus_5613498140476352782st_a_a @ A2 @ B2 ) )
=> ~ ( ( member_set_list_a_a @ C2 @ A2 )
=> ( member_set_list_a_a @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_282_DiffE,axiom,
! [C2: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C2 @ ( minus_4169782841487898290list_a @ A2 @ B2 ) )
=> ~ ( ( member_nat_list_a @ C2 @ A2 )
=> ( member_nat_list_a @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_283_DiffE,axiom,
! [C2: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C2 @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ~ ( ( member_nat_a @ C2 @ A2 )
=> ( member_nat_a @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_284_DiffE,axiom,
! [C2: a,A2: set_a,B2: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_285_DiffE,axiom,
! [C2: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C2 @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ~ ( ( member_list_a @ C2 @ A2 )
=> ( member_list_a @ C2 @ B2 ) ) ) ).
% DiffE
thf(fact_286_finite__has__minimal2,axiom,
! [A2: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A3 @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_287_finite__has__minimal2,axiom,
! [A2: set_set_list_a,A3: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A3 @ A2 )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ( ord_le8861187494160871172list_a @ X3 @ A3 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_288_finite__has__minimal2,axiom,
! [A2: set_nat,A3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A3 @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_289_finite__has__maximal2,axiom,
! [A2: set_set_a,A3: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A3 @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A3 @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_290_finite__has__maximal2,axiom,
! [A2: set_set_list_a,A3: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A3 @ A2 )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ( ord_le8861187494160871172list_a @ A3 @ X3 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_291_finite__has__maximal2,axiom,
! [A2: set_nat,A3: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A3 @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A3 @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_292_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_293_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_294_rev__finite__subset,axiom,
! [B2: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_295_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_296_infinite__super,axiom,
! [S2: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_297_infinite__super,axiom,
! [S2: set_list_a,T2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S2 @ T2 )
=> ( ~ ( finite_finite_list_a @ S2 )
=> ~ ( finite_finite_list_a @ T2 ) ) ) ).
% infinite_super
thf(fact_298_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_299_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_300_finite__subset,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( finite_finite_list_a @ B2 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% finite_subset
thf(fact_301_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_302_infinite__imp__nonempty,axiom,
! [S2: set_list_a] :
( ~ ( finite_finite_list_a @ S2 )
=> ( S2 != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_303_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_304_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_305_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_306_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_307_double__diff,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_308_double__diff,axiom,
! [A2: set_list_a,B2: set_list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C3 )
=> ( ( minus_646659088055828811list_a @ B2 @ ( minus_646659088055828811list_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_309_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_310_Diff__subset,axiom,
! [A2: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_311_Diff__mono,axiom,
! [A2: set_a,C3: set_a,D: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ D @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_312_Diff__mono,axiom,
! [A2: set_list_a,C3: set_list_a,D: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C3 )
=> ( ( ord_le8861187494160871172list_a @ D @ B2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ ( minus_646659088055828811list_a @ C3 @ D ) ) ) ) ).
% Diff_mono
thf(fact_313_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_314_Diff__infinite__finite,axiom,
! [T2: set_a,S2: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_315_Diff__infinite__finite,axiom,
! [T2: set_list_a,S2: set_list_a] :
( ( finite_finite_list_a @ T2 )
=> ( ~ ( finite_finite_list_a @ S2 )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_316_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_317_finite__has__minimal,axiom,
! [A2: set_set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( A2 != bot_bo3186585308812441520list_a )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_318_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_319_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_320_finite__has__maximal,axiom,
! [A2: set_set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( A2 != bot_bo3186585308812441520list_a )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_321_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_322_primeness__condition,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeness_condition
thf(fact_323__092_060open_062local_Oeval_A_Ilagrange__basis__polynomial_AS_Ax_J_Ax_A_061_Afinprod_AR_A_Ia__minus_AR_Ax_J_AS_A_092_060otimes_062_Alocal_Oeval_A_Ipoly__of__const_A_Iinv_Afinprod_AR_A_Ia__minus_AR_Ax_J_AS_J_J_Ax_092_060close_062,axiom,
( ( eval_a_b @ r @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) @ x )
= ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) @ ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ) @ x ) ) ) ).
% \<open>local.eval (lagrange_basis_polynomial S x) x = finprod R (a_minus R x) S \<otimes> local.eval (poly_of_const (inv finprod R (a_minus R x) S)) x\<close>
thf(fact_324__092_060open_062finprod_AR_A_Ia__minus_AR_Ax_J_AS_A_092_060otimes_062_Alocal_Oeval_A_Ipoly__of__const_A_Iinv_Afinprod_AR_A_Ia__minus_AR_Ax_J_AS_J_J_Ax_A_061_A_092_060one_062_092_060close_062,axiom,
( ( mult_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) @ ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ) @ x ) )
= ( one_a_ring_ext_a_b @ r ) ) ).
% \<open>finprod R (a_minus R x) S \<otimes> local.eval (poly_of_const (inv finprod R (a_minus R x) S)) x = \<one>\<close>
thf(fact_325_ring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( P
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_326_x_Oring__hom__cring__axioms,axiom,
( ring_h1547129875642963619it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ).
% x.ring_hom_cring_axioms
thf(fact_327_x_Oring_Ois__abelian__group__hom,axiom,
( abelia8217020544048703197it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ).
% x.ring.is_abelian_group_hom
thf(fact_328_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_329_x_Oup__smult__closed,axiom,
! [A3: list_a,P: nat > list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [I: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ ( P @ I ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.up_smult_closed
thf(fact_330_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_331_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_332_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_333_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_334_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_335_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_336_m__rcancel,axiom,
! [A3: a,B4: a,C2: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B4 @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ A3 ) )
= ( B4 = C2 ) ) ) ) ) ) ).
% m_rcancel
thf(fact_337_m__lcancel,axiom,
! [A3: a,B4: a,C2: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A3 @ B4 )
= ( mult_a_ring_ext_a_b @ r @ A3 @ C2 ) )
= ( B4 = C2 ) ) ) ) ) ) ).
% m_lcancel
thf(fact_338_integral__iff,axiom,
! [A3: a,B4: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A3 @ B4 )
= ( zero_a_b @ r ) )
= ( ( A3
= ( zero_a_b @ r ) )
| ( B4
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_339_local_Ointegral,axiom,
! [A3: a,B4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ B4 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A3
= ( zero_a_b @ r ) )
| ( B4
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_340_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_341_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_342_unit__factor,axiom,
! [A3: a,B4: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B4 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_343_prod__unit__r,axiom,
! [A3: a,B4: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B4 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B4 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_344_prod__unit__l,axiom,
! [A3: a,B4: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A3 @ B4 ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B4 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_345_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_346_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_347_insert__iff,axiom,
! [A3: list_a,B4: list_a,A2: set_list_a] :
( ( member_list_a @ A3 @ ( insert_list_a @ B4 @ A2 ) )
= ( ( A3 = B4 )
| ( member_list_a @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_348_insert__iff,axiom,
! [A3: list_a > a,B4: list_a > a,A2: set_list_a_a] :
( ( member_list_a_a @ A3 @ ( insert_list_a_a @ B4 @ A2 ) )
= ( ( A3 = B4 )
| ( member_list_a_a @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_349_insert__iff,axiom,
! [A3: set_list_a > a,B4: set_list_a > a,A2: set_set_list_a_a] :
( ( member_set_list_a_a @ A3 @ ( insert_set_list_a_a @ B4 @ A2 ) )
= ( ( A3 = B4 )
| ( member_set_list_a_a @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_350_insert__iff,axiom,
! [A3: nat > list_a,B4: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A3 @ ( insert_nat_list_a @ B4 @ A2 ) )
= ( ( A3 = B4 )
| ( member_nat_list_a @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_351_insert__iff,axiom,
! [A3: nat > a,B4: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A3 @ ( insert_nat_a @ B4 @ A2 ) )
= ( ( A3 = B4 )
| ( member_nat_a @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_352_insert__iff,axiom,
! [A3: a,B4: a,A2: set_a] :
( ( member_a @ A3 @ ( insert_a @ B4 @ A2 ) )
= ( ( A3 = B4 )
| ( member_a @ A3 @ A2 ) ) ) ).
% insert_iff
thf(fact_353_insertCI,axiom,
! [A3: list_a,B2: set_list_a,B4: list_a] :
( ( ~ ( member_list_a @ A3 @ B2 )
=> ( A3 = B4 ) )
=> ( member_list_a @ A3 @ ( insert_list_a @ B4 @ B2 ) ) ) ).
% insertCI
thf(fact_354_insertCI,axiom,
! [A3: list_a > a,B2: set_list_a_a,B4: list_a > a] :
( ( ~ ( member_list_a_a @ A3 @ B2 )
=> ( A3 = B4 ) )
=> ( member_list_a_a @ A3 @ ( insert_list_a_a @ B4 @ B2 ) ) ) ).
% insertCI
thf(fact_355_insertCI,axiom,
! [A3: set_list_a > a,B2: set_set_list_a_a,B4: set_list_a > a] :
( ( ~ ( member_set_list_a_a @ A3 @ B2 )
=> ( A3 = B4 ) )
=> ( member_set_list_a_a @ A3 @ ( insert_set_list_a_a @ B4 @ B2 ) ) ) ).
% insertCI
thf(fact_356_insertCI,axiom,
! [A3: nat > list_a,B2: set_nat_list_a,B4: nat > list_a] :
( ( ~ ( member_nat_list_a @ A3 @ B2 )
=> ( A3 = B4 ) )
=> ( member_nat_list_a @ A3 @ ( insert_nat_list_a @ B4 @ B2 ) ) ) ).
% insertCI
thf(fact_357_insertCI,axiom,
! [A3: nat > a,B2: set_nat_a,B4: nat > a] :
( ( ~ ( member_nat_a @ A3 @ B2 )
=> ( A3 = B4 ) )
=> ( member_nat_a @ A3 @ ( insert_nat_a @ B4 @ B2 ) ) ) ).
% insertCI
thf(fact_358_insertCI,axiom,
! [A3: a,B2: set_a,B4: a] :
( ( ~ ( member_a @ A3 @ B2 )
=> ( A3 = B4 ) )
=> ( member_a @ A3 @ ( insert_a @ B4 @ B2 ) ) ) ).
% insertCI
thf(fact_359_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_360_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_361_ring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).
% ring_primeE(3)
thf(fact_362_ring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI
thf(fact_363_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_364_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_365_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_366_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_367_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_368_ring__irreducibleE_I5_J,axiom,
! [R2: a,A3: a,B4: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A3 @ B4 ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B4 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_369_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_370_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_371_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_372_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_373_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_374_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_375_local_Ofield__Units,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 ) ) ) ).
% local.field_Units
thf(fact_376_insert__subset,axiom,
! [X: list_a > a,A2: set_list_a_a,B2: set_list_a_a] :
( ( ord_le6942402695062981877st_a_a @ ( insert_list_a_a @ X @ A2 ) @ B2 )
= ( ( member_list_a_a @ X @ B2 )
& ( ord_le6942402695062981877st_a_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_377_insert__subset,axiom,
! [X: set_list_a > a,A2: set_set_list_a_a,B2: set_set_list_a_a] :
( ( ord_le4799719167512954133st_a_a @ ( insert_set_list_a_a @ X @ A2 ) @ B2 )
= ( ( member_set_list_a_a @ X @ B2 )
& ( ord_le4799719167512954133st_a_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_378_insert__subset,axiom,
! [X: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( ord_le2145805922479659755list_a @ ( insert_nat_list_a @ X @ A2 ) @ B2 )
= ( ( member_nat_list_a @ X @ B2 )
& ( ord_le2145805922479659755list_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_379_insert__subset,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( ( member_nat_a @ X @ B2 )
& ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_380_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_381_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_382_singletonI,axiom,
! [A3: list_a > a] : ( member_list_a_a @ A3 @ ( insert_list_a_a @ A3 @ bot_bot_set_list_a_a ) ) ).
% singletonI
thf(fact_383_singletonI,axiom,
! [A3: set_list_a > a] : ( member_set_list_a_a @ A3 @ ( insert_set_list_a_a @ A3 @ bot_bo8301825967528238409st_a_a ) ) ).
% singletonI
thf(fact_384_singletonI,axiom,
! [A3: nat > list_a] : ( member_nat_list_a @ A3 @ ( insert_nat_list_a @ A3 @ bot_bo3806784159821827511list_a ) ) ).
% singletonI
thf(fact_385_singletonI,axiom,
! [A3: nat > a] : ( member_nat_a @ A3 @ ( insert_nat_a @ A3 @ bot_bot_set_nat_a ) ) ).
% singletonI
thf(fact_386_singletonI,axiom,
! [A3: list_a] : ( member_list_a @ A3 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_387_singletonI,axiom,
! [A3: a] : ( member_a @ A3 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_388_finite__insert,axiom,
! [A3: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A3 @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_389_finite__insert,axiom,
! [A3: list_a,A2: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A3 @ A2 ) )
= ( finite_finite_list_a @ A2 ) ) ).
% finite_insert
thf(fact_390_finite__insert,axiom,
! [A3: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A3 @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_391_Diff__insert0,axiom,
! [X: list_a > a,A2: set_list_a_a,B2: set_list_a_a] :
( ~ ( member_list_a_a @ X @ A2 )
=> ( ( minus_921748639838131438st_a_a @ A2 @ ( insert_list_a_a @ X @ B2 ) )
= ( minus_921748639838131438st_a_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_392_Diff__insert0,axiom,
! [X: set_list_a > a,A2: set_set_list_a_a,B2: set_set_list_a_a] :
( ~ ( member_set_list_a_a @ X @ A2 )
=> ( ( minus_5613498140476352782st_a_a @ A2 @ ( insert_set_list_a_a @ X @ B2 ) )
= ( minus_5613498140476352782st_a_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_393_Diff__insert0,axiom,
! [X: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ~ ( member_nat_list_a @ X @ A2 )
=> ( ( minus_4169782841487898290list_a @ A2 @ ( insert_nat_list_a @ X @ B2 ) )
= ( minus_4169782841487898290list_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_394_Diff__insert0,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
= ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_395_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_396_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_397_insert__Diff1,axiom,
! [X: list_a > a,B2: set_list_a_a,A2: set_list_a_a] :
( ( member_list_a_a @ X @ B2 )
=> ( ( minus_921748639838131438st_a_a @ ( insert_list_a_a @ X @ A2 ) @ B2 )
= ( minus_921748639838131438st_a_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_398_insert__Diff1,axiom,
! [X: set_list_a > a,B2: set_set_list_a_a,A2: set_set_list_a_a] :
( ( member_set_list_a_a @ X @ B2 )
=> ( ( minus_5613498140476352782st_a_a @ ( insert_set_list_a_a @ X @ A2 ) @ B2 )
= ( minus_5613498140476352782st_a_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_399_insert__Diff1,axiom,
! [X: nat > list_a,B2: set_nat_list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ X @ B2 )
=> ( ( minus_4169782841487898290list_a @ ( insert_nat_list_a @ X @ A2 ) @ B2 )
= ( minus_4169782841487898290list_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_400_insert__Diff1,axiom,
! [X: nat > a,B2: set_nat_a,A2: set_nat_a] :
( ( member_nat_a @ X @ B2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_401_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_402_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_403_singleton__conv2,axiom,
! [A3: nat] :
( ( collect_nat
@ ( ^ [Y2: nat,Z2: nat] : ( Y2 = Z2 )
@ A3 ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_404_singleton__conv2,axiom,
! [A3: list_list_a] :
( ( collect_list_list_a
@ ( ^ [Y2: list_list_a,Z2: list_list_a] : ( Y2 = Z2 )
@ A3 ) )
= ( insert_list_list_a @ A3 @ bot_bo1875519244922727510list_a ) ) ).
% singleton_conv2
thf(fact_405_singleton__conv2,axiom,
! [A3: list_a] :
( ( collect_list_a
@ ( ^ [Y2: list_a,Z2: list_a] : ( Y2 = Z2 )
@ A3 ) )
= ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ).
% singleton_conv2
thf(fact_406_singleton__conv2,axiom,
! [A3: a] :
( ( collect_a
@ ( ^ [Y2: a,Z2: a] : ( Y2 = Z2 )
@ A3 ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_407_singleton__conv,axiom,
! [A3: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A3 ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_408_singleton__conv,axiom,
! [A3: list_list_a] :
( ( collect_list_list_a
@ ^ [X2: list_list_a] : ( X2 = A3 ) )
= ( insert_list_list_a @ A3 @ bot_bo1875519244922727510list_a ) ) ).
% singleton_conv
thf(fact_409_singleton__conv,axiom,
! [A3: list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( X2 = A3 ) )
= ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ).
% singleton_conv
thf(fact_410_singleton__conv,axiom,
! [A3: a] :
( ( collect_a
@ ^ [X2: a] : ( X2 = A3 ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_411_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 ) )
=> ( ! [A: a,B: a] :
( ( 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_r999134135267193926le_a_b @ r @ R2 ) ) ) ) ).
% ring_irreducibleI
thf(fact_412_singleton__insert__inj__eq,axiom,
! [B4: a,A3: a,A2: set_a] :
( ( ( insert_a @ B4 @ bot_bot_set_a )
= ( insert_a @ A3 @ A2 ) )
= ( ( A3 = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_413_singleton__insert__inj__eq,axiom,
! [B4: list_a,A3: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B4 @ bot_bot_set_list_a )
= ( insert_list_a @ A3 @ A2 ) )
= ( ( A3 = B4 )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B4 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_414_singleton__insert__inj__eq_H,axiom,
! [A3: a,A2: set_a,B4: a] :
( ( ( insert_a @ A3 @ A2 )
= ( insert_a @ B4 @ bot_bot_set_a ) )
= ( ( A3 = B4 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_415_singleton__insert__inj__eq_H,axiom,
! [A3: list_a,A2: set_list_a,B4: list_a] :
( ( ( insert_list_a @ A3 @ A2 )
= ( insert_list_a @ B4 @ bot_bot_set_list_a ) )
= ( ( A3 = B4 )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B4 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_416_insert__Diff__single,axiom,
! [A3: a,A2: set_a] :
( ( insert_a @ A3 @ ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= ( insert_a @ A3 @ A2 ) ) ).
% insert_Diff_single
thf(fact_417_insert__Diff__single,axiom,
! [A3: list_a,A2: set_list_a] :
( ( insert_list_a @ A3 @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A3 @ A2 ) ) ).
% insert_Diff_single
thf(fact_418_finite__Diff__insert,axiom,
! [A2: set_nat,A3: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A3 @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_419_finite__Diff__insert,axiom,
! [A2: set_a,A3: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_420_finite__Diff__insert,axiom,
! [A2: set_list_a,A3: list_a,B2: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A3 @ B2 ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_421_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_422_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_423_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_424_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_425_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_426_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_427_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_428_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_429_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_430_x_Oring_Ohom__mult,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.ring.hom_mult
thf(fact_431_mk__disjoint__insert,axiom,
! [A3: list_a,A2: set_list_a] :
( ( member_list_a @ A3 @ A2 )
=> ? [B5: set_list_a] :
( ( A2
= ( insert_list_a @ A3 @ B5 ) )
& ~ ( member_list_a @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_432_mk__disjoint__insert,axiom,
! [A3: list_a > a,A2: set_list_a_a] :
( ( member_list_a_a @ A3 @ A2 )
=> ? [B5: set_list_a_a] :
( ( A2
= ( insert_list_a_a @ A3 @ B5 ) )
& ~ ( member_list_a_a @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_433_mk__disjoint__insert,axiom,
! [A3: set_list_a > a,A2: set_set_list_a_a] :
( ( member_set_list_a_a @ A3 @ A2 )
=> ? [B5: set_set_list_a_a] :
( ( A2
= ( insert_set_list_a_a @ A3 @ B5 ) )
& ~ ( member_set_list_a_a @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_434_mk__disjoint__insert,axiom,
! [A3: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A3 @ A2 )
=> ? [B5: set_nat_list_a] :
( ( A2
= ( insert_nat_list_a @ A3 @ B5 ) )
& ~ ( member_nat_list_a @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_435_mk__disjoint__insert,axiom,
! [A3: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A3 @ A2 )
=> ? [B5: set_nat_a] :
( ( A2
= ( insert_nat_a @ A3 @ B5 ) )
& ~ ( member_nat_a @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_436_mk__disjoint__insert,axiom,
! [A3: a,A2: set_a] :
( ( member_a @ A3 @ A2 )
=> ? [B5: set_a] :
( ( A2
= ( insert_a @ A3 @ B5 ) )
& ~ ( member_a @ A3 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_437_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_438_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_439_insert__eq__iff,axiom,
! [A3: list_a,A2: set_list_a,B4: list_a,B2: set_list_a] :
( ~ ( member_list_a @ A3 @ A2 )
=> ( ~ ( member_list_a @ B4 @ B2 )
=> ( ( ( insert_list_a @ A3 @ A2 )
= ( insert_list_a @ B4 @ B2 ) )
= ( ( ( A3 = B4 )
=> ( A2 = B2 ) )
& ( ( A3 != B4 )
=> ? [C4: set_list_a] :
( ( A2
= ( insert_list_a @ B4 @ C4 ) )
& ~ ( member_list_a @ B4 @ C4 )
& ( B2
= ( insert_list_a @ A3 @ C4 ) )
& ~ ( member_list_a @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_440_insert__eq__iff,axiom,
! [A3: list_a > a,A2: set_list_a_a,B4: list_a > a,B2: set_list_a_a] :
( ~ ( member_list_a_a @ A3 @ A2 )
=> ( ~ ( member_list_a_a @ B4 @ B2 )
=> ( ( ( insert_list_a_a @ A3 @ A2 )
= ( insert_list_a_a @ B4 @ B2 ) )
= ( ( ( A3 = B4 )
=> ( A2 = B2 ) )
& ( ( A3 != B4 )
=> ? [C4: set_list_a_a] :
( ( A2
= ( insert_list_a_a @ B4 @ C4 ) )
& ~ ( member_list_a_a @ B4 @ C4 )
& ( B2
= ( insert_list_a_a @ A3 @ C4 ) )
& ~ ( member_list_a_a @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_441_insert__eq__iff,axiom,
! [A3: set_list_a > a,A2: set_set_list_a_a,B4: set_list_a > a,B2: set_set_list_a_a] :
( ~ ( member_set_list_a_a @ A3 @ A2 )
=> ( ~ ( member_set_list_a_a @ B4 @ B2 )
=> ( ( ( insert_set_list_a_a @ A3 @ A2 )
= ( insert_set_list_a_a @ B4 @ B2 ) )
= ( ( ( A3 = B4 )
=> ( A2 = B2 ) )
& ( ( A3 != B4 )
=> ? [C4: set_set_list_a_a] :
( ( A2
= ( insert_set_list_a_a @ B4 @ C4 ) )
& ~ ( member_set_list_a_a @ B4 @ C4 )
& ( B2
= ( insert_set_list_a_a @ A3 @ C4 ) )
& ~ ( member_set_list_a_a @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_442_insert__eq__iff,axiom,
! [A3: nat > list_a,A2: set_nat_list_a,B4: nat > list_a,B2: set_nat_list_a] :
( ~ ( member_nat_list_a @ A3 @ A2 )
=> ( ~ ( member_nat_list_a @ B4 @ B2 )
=> ( ( ( insert_nat_list_a @ A3 @ A2 )
= ( insert_nat_list_a @ B4 @ B2 ) )
= ( ( ( A3 = B4 )
=> ( A2 = B2 ) )
& ( ( A3 != B4 )
=> ? [C4: set_nat_list_a] :
( ( A2
= ( insert_nat_list_a @ B4 @ C4 ) )
& ~ ( member_nat_list_a @ B4 @ C4 )
& ( B2
= ( insert_nat_list_a @ A3 @ C4 ) )
& ~ ( member_nat_list_a @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_443_insert__eq__iff,axiom,
! [A3: nat > a,A2: set_nat_a,B4: nat > a,B2: set_nat_a] :
( ~ ( member_nat_a @ A3 @ A2 )
=> ( ~ ( member_nat_a @ B4 @ B2 )
=> ( ( ( insert_nat_a @ A3 @ A2 )
= ( insert_nat_a @ B4 @ B2 ) )
= ( ( ( A3 = B4 )
=> ( A2 = B2 ) )
& ( ( A3 != B4 )
=> ? [C4: set_nat_a] :
( ( A2
= ( insert_nat_a @ B4 @ C4 ) )
& ~ ( member_nat_a @ B4 @ C4 )
& ( B2
= ( insert_nat_a @ A3 @ C4 ) )
& ~ ( member_nat_a @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_444_insert__eq__iff,axiom,
! [A3: a,A2: set_a,B4: a,B2: set_a] :
( ~ ( member_a @ A3 @ A2 )
=> ( ~ ( member_a @ B4 @ B2 )
=> ( ( ( insert_a @ A3 @ A2 )
= ( insert_a @ B4 @ B2 ) )
= ( ( ( A3 = B4 )
=> ( A2 = B2 ) )
& ( ( A3 != B4 )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B4 @ C4 ) )
& ~ ( member_a @ B4 @ C4 )
& ( B2
= ( insert_a @ A3 @ C4 ) )
& ~ ( member_a @ A3 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_445_insert__absorb,axiom,
! [A3: list_a,A2: set_list_a] :
( ( member_list_a @ A3 @ A2 )
=> ( ( insert_list_a @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_446_insert__absorb,axiom,
! [A3: list_a > a,A2: set_list_a_a] :
( ( member_list_a_a @ A3 @ A2 )
=> ( ( insert_list_a_a @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_447_insert__absorb,axiom,
! [A3: set_list_a > a,A2: set_set_list_a_a] :
( ( member_set_list_a_a @ A3 @ A2 )
=> ( ( insert_set_list_a_a @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_448_insert__absorb,axiom,
! [A3: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A3 @ A2 )
=> ( ( insert_nat_list_a @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_449_insert__absorb,axiom,
! [A3: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A3 @ A2 )
=> ( ( insert_nat_a @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_450_insert__absorb,axiom,
! [A3: a,A2: set_a] :
( ( member_a @ A3 @ A2 )
=> ( ( insert_a @ A3 @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_451_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_452_insert__ident,axiom,
! [X: list_a > a,A2: set_list_a_a,B2: set_list_a_a] :
( ~ ( member_list_a_a @ X @ A2 )
=> ( ~ ( member_list_a_a @ X @ B2 )
=> ( ( ( insert_list_a_a @ X @ A2 )
= ( insert_list_a_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_453_insert__ident,axiom,
! [X: set_list_a > a,A2: set_set_list_a_a,B2: set_set_list_a_a] :
( ~ ( member_set_list_a_a @ X @ A2 )
=> ( ~ ( member_set_list_a_a @ X @ B2 )
=> ( ( ( insert_set_list_a_a @ X @ A2 )
= ( insert_set_list_a_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_454_insert__ident,axiom,
! [X: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ~ ( member_nat_list_a @ X @ A2 )
=> ( ~ ( member_nat_list_a @ X @ B2 )
=> ( ( ( insert_nat_list_a @ X @ A2 )
= ( insert_nat_list_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_455_insert__ident,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ~ ( member_nat_a @ X @ B2 )
=> ( ( ( insert_nat_a @ X @ A2 )
= ( insert_nat_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_456_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_457_Set_Oset__insert,axiom,
! [X: list_a,A2: set_list_a] :
( ( member_list_a @ X @ A2 )
=> ~ ! [B5: set_list_a] :
( ( A2
= ( insert_list_a @ X @ B5 ) )
=> ( member_list_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_458_Set_Oset__insert,axiom,
! [X: list_a > a,A2: set_list_a_a] :
( ( member_list_a_a @ X @ A2 )
=> ~ ! [B5: set_list_a_a] :
( ( A2
= ( insert_list_a_a @ X @ B5 ) )
=> ( member_list_a_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_459_Set_Oset__insert,axiom,
! [X: set_list_a > a,A2: set_set_list_a_a] :
( ( member_set_list_a_a @ X @ A2 )
=> ~ ! [B5: set_set_list_a_a] :
( ( A2
= ( insert_set_list_a_a @ X @ B5 ) )
=> ( member_set_list_a_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_460_Set_Oset__insert,axiom,
! [X: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ X @ A2 )
=> ~ ! [B5: set_nat_list_a] :
( ( A2
= ( insert_nat_list_a @ X @ B5 ) )
=> ( member_nat_list_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_461_Set_Oset__insert,axiom,
! [X: nat > a,A2: set_nat_a] :
( ( member_nat_a @ X @ A2 )
=> ~ ! [B5: set_nat_a] :
( ( A2
= ( insert_nat_a @ X @ B5 ) )
=> ( member_nat_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_462_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B5: set_a] :
( ( A2
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_463_insertI2,axiom,
! [A3: list_a,B2: set_list_a,B4: list_a] :
( ( member_list_a @ A3 @ B2 )
=> ( member_list_a @ A3 @ ( insert_list_a @ B4 @ B2 ) ) ) ).
% insertI2
thf(fact_464_insertI2,axiom,
! [A3: list_a > a,B2: set_list_a_a,B4: list_a > a] :
( ( member_list_a_a @ A3 @ B2 )
=> ( member_list_a_a @ A3 @ ( insert_list_a_a @ B4 @ B2 ) ) ) ).
% insertI2
thf(fact_465_insertI2,axiom,
! [A3: set_list_a > a,B2: set_set_list_a_a,B4: set_list_a > a] :
( ( member_set_list_a_a @ A3 @ B2 )
=> ( member_set_list_a_a @ A3 @ ( insert_set_list_a_a @ B4 @ B2 ) ) ) ).
% insertI2
thf(fact_466_insertI2,axiom,
! [A3: nat > list_a,B2: set_nat_list_a,B4: nat > list_a] :
( ( member_nat_list_a @ A3 @ B2 )
=> ( member_nat_list_a @ A3 @ ( insert_nat_list_a @ B4 @ B2 ) ) ) ).
% insertI2
thf(fact_467_insertI2,axiom,
! [A3: nat > a,B2: set_nat_a,B4: nat > a] :
( ( member_nat_a @ A3 @ B2 )
=> ( member_nat_a @ A3 @ ( insert_nat_a @ B4 @ B2 ) ) ) ).
% insertI2
thf(fact_468_insertI2,axiom,
! [A3: a,B2: set_a,B4: a] :
( ( member_a @ A3 @ B2 )
=> ( member_a @ A3 @ ( insert_a @ B4 @ B2 ) ) ) ).
% insertI2
thf(fact_469_insertI1,axiom,
! [A3: list_a,B2: set_list_a] : ( member_list_a @ A3 @ ( insert_list_a @ A3 @ B2 ) ) ).
% insertI1
thf(fact_470_insertI1,axiom,
! [A3: list_a > a,B2: set_list_a_a] : ( member_list_a_a @ A3 @ ( insert_list_a_a @ A3 @ B2 ) ) ).
% insertI1
thf(fact_471_insertI1,axiom,
! [A3: set_list_a > a,B2: set_set_list_a_a] : ( member_set_list_a_a @ A3 @ ( insert_set_list_a_a @ A3 @ B2 ) ) ).
% insertI1
thf(fact_472_insertI1,axiom,
! [A3: nat > list_a,B2: set_nat_list_a] : ( member_nat_list_a @ A3 @ ( insert_nat_list_a @ A3 @ B2 ) ) ).
% insertI1
thf(fact_473_insertI1,axiom,
! [A3: nat > a,B2: set_nat_a] : ( member_nat_a @ A3 @ ( insert_nat_a @ A3 @ B2 ) ) ).
% insertI1
thf(fact_474_insertI1,axiom,
! [A3: a,B2: set_a] : ( member_a @ A3 @ ( insert_a @ A3 @ B2 ) ) ).
% insertI1
thf(fact_475_insertE,axiom,
! [A3: list_a,B4: list_a,A2: set_list_a] :
( ( member_list_a @ A3 @ ( insert_list_a @ B4 @ A2 ) )
=> ( ( A3 != B4 )
=> ( member_list_a @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_476_insertE,axiom,
! [A3: list_a > a,B4: list_a > a,A2: set_list_a_a] :
( ( member_list_a_a @ A3 @ ( insert_list_a_a @ B4 @ A2 ) )
=> ( ( A3 != B4 )
=> ( member_list_a_a @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_477_insertE,axiom,
! [A3: set_list_a > a,B4: set_list_a > a,A2: set_set_list_a_a] :
( ( member_set_list_a_a @ A3 @ ( insert_set_list_a_a @ B4 @ A2 ) )
=> ( ( A3 != B4 )
=> ( member_set_list_a_a @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_478_insertE,axiom,
! [A3: nat > list_a,B4: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A3 @ ( insert_nat_list_a @ B4 @ A2 ) )
=> ( ( A3 != B4 )
=> ( member_nat_list_a @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_479_insertE,axiom,
! [A3: nat > a,B4: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A3 @ ( insert_nat_a @ B4 @ A2 ) )
=> ( ( A3 != B4 )
=> ( member_nat_a @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_480_insertE,axiom,
! [A3: a,B4: a,A2: set_a] :
( ( member_a @ A3 @ ( insert_a @ B4 @ A2 ) )
=> ( ( A3 != B4 )
=> ( member_a @ A3 @ A2 ) ) ) ).
% insertE
thf(fact_481_insert__compr,axiom,
( insert_list_a_a
= ( ^ [A5: list_a > a,B3: set_list_a_a] :
( collect_list_a_a
@ ^ [X2: list_a > a] :
( ( X2 = A5 )
| ( member_list_a_a @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_482_insert__compr,axiom,
( insert_set_list_a_a
= ( ^ [A5: set_list_a > a,B3: set_set_list_a_a] :
( collect_set_list_a_a
@ ^ [X2: set_list_a > a] :
( ( X2 = A5 )
| ( member_set_list_a_a @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_483_insert__compr,axiom,
( insert_nat_list_a
= ( ^ [A5: nat > list_a,B3: set_nat_list_a] :
( collect_nat_list_a
@ ^ [X2: nat > list_a] :
( ( X2 = A5 )
| ( member_nat_list_a @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_484_insert__compr,axiom,
( insert_nat_a
= ( ^ [A5: nat > a,B3: set_nat_a] :
( collect_nat_a
@ ^ [X2: nat > a] :
( ( X2 = A5 )
| ( member_nat_a @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_485_insert__compr,axiom,
( insert_a
= ( ^ [A5: a,B3: set_a] :
( collect_a
@ ^ [X2: a] :
( ( X2 = A5 )
| ( member_a @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_486_insert__compr,axiom,
( insert_nat
= ( ^ [A5: nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( X2 = A5 )
| ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_487_insert__compr,axiom,
( insert_list_a
= ( ^ [A5: list_a,B3: set_list_a] :
( collect_list_a
@ ^ [X2: list_a] :
( ( X2 = A5 )
| ( member_list_a @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_488_insert__compr,axiom,
( insert_list_list_a
= ( ^ [A5: list_list_a,B3: set_list_list_a] :
( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( X2 = A5 )
| ( member_list_list_a @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_489_insert__Collect,axiom,
! [A3: a,P2: a > $o] :
( ( insert_a @ A3 @ ( collect_a @ P2 ) )
= ( collect_a
@ ^ [U2: a] :
( ( U2 != A3 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_490_insert__Collect,axiom,
! [A3: nat,P2: nat > $o] :
( ( insert_nat @ A3 @ ( collect_nat @ P2 ) )
= ( collect_nat
@ ^ [U2: nat] :
( ( U2 != A3 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_491_insert__Collect,axiom,
! [A3: list_a,P2: list_a > $o] :
( ( insert_list_a @ A3 @ ( collect_list_a @ P2 ) )
= ( collect_list_a
@ ^ [U2: list_a] :
( ( U2 != A3 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_492_insert__Collect,axiom,
! [A3: list_list_a,P2: list_list_a > $o] :
( ( insert_list_list_a @ A3 @ ( collect_list_list_a @ P2 ) )
= ( collect_list_list_a
@ ^ [U2: list_list_a] :
( ( U2 != A3 )
=> ( P2 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_493_insert__mono,axiom,
! [C3: set_a,D: set_a,A3: a] :
( ( ord_less_eq_set_a @ C3 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A3 @ C3 ) @ ( insert_a @ A3 @ D ) ) ) ).
% insert_mono
thf(fact_494_insert__mono,axiom,
! [C3: set_list_a,D: set_list_a,A3: list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ D )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ A3 @ C3 ) @ ( insert_list_a @ A3 @ D ) ) ) ).
% insert_mono
thf(fact_495_subset__insert,axiom,
! [X: list_a > a,A2: set_list_a_a,B2: set_list_a_a] :
( ~ ( member_list_a_a @ X @ A2 )
=> ( ( ord_le6942402695062981877st_a_a @ A2 @ ( insert_list_a_a @ X @ B2 ) )
= ( ord_le6942402695062981877st_a_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_496_subset__insert,axiom,
! [X: set_list_a > a,A2: set_set_list_a_a,B2: set_set_list_a_a] :
( ~ ( member_set_list_a_a @ X @ A2 )
=> ( ( ord_le4799719167512954133st_a_a @ A2 @ ( insert_set_list_a_a @ X @ B2 ) )
= ( ord_le4799719167512954133st_a_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_497_subset__insert,axiom,
! [X: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ~ ( member_nat_list_a @ X @ A2 )
=> ( ( ord_le2145805922479659755list_a @ A2 @ ( insert_nat_list_a @ X @ B2 ) )
= ( ord_le2145805922479659755list_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_498_subset__insert,axiom,
! [X: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
= ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_499_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_500_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_501_subset__insertI,axiom,
! [B2: set_a,A3: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_502_subset__insertI,axiom,
! [B2: set_list_a,A3: list_a] : ( ord_le8861187494160871172list_a @ B2 @ ( insert_list_a @ A3 @ B2 ) ) ).
% subset_insertI
thf(fact_503_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B4: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B4 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_504_subset__insertI2,axiom,
! [A2: set_list_a,B2: set_list_a,B4: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B4 @ B2 ) ) ) ).
% subset_insertI2
thf(fact_505_singletonD,axiom,
! [B4: list_a > a,A3: list_a > a] :
( ( member_list_a_a @ B4 @ ( insert_list_a_a @ A3 @ bot_bot_set_list_a_a ) )
=> ( B4 = A3 ) ) ).
% singletonD
thf(fact_506_singletonD,axiom,
! [B4: set_list_a > a,A3: set_list_a > a] :
( ( member_set_list_a_a @ B4 @ ( insert_set_list_a_a @ A3 @ bot_bo8301825967528238409st_a_a ) )
=> ( B4 = A3 ) ) ).
% singletonD
thf(fact_507_singletonD,axiom,
! [B4: nat > list_a,A3: nat > list_a] :
( ( member_nat_list_a @ B4 @ ( insert_nat_list_a @ A3 @ bot_bo3806784159821827511list_a ) )
=> ( B4 = A3 ) ) ).
% singletonD
thf(fact_508_singletonD,axiom,
! [B4: nat > a,A3: nat > a] :
( ( member_nat_a @ B4 @ ( insert_nat_a @ A3 @ bot_bot_set_nat_a ) )
=> ( B4 = A3 ) ) ).
% singletonD
thf(fact_509_singletonD,axiom,
! [B4: list_a,A3: list_a] :
( ( member_list_a @ B4 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) )
=> ( B4 = A3 ) ) ).
% singletonD
thf(fact_510_singletonD,axiom,
! [B4: a,A3: a] :
( ( member_a @ B4 @ ( insert_a @ A3 @ bot_bot_set_a ) )
=> ( B4 = A3 ) ) ).
% singletonD
thf(fact_511_singleton__iff,axiom,
! [B4: list_a > a,A3: list_a > a] :
( ( member_list_a_a @ B4 @ ( insert_list_a_a @ A3 @ bot_bot_set_list_a_a ) )
= ( B4 = A3 ) ) ).
% singleton_iff
thf(fact_512_singleton__iff,axiom,
! [B4: set_list_a > a,A3: set_list_a > a] :
( ( member_set_list_a_a @ B4 @ ( insert_set_list_a_a @ A3 @ bot_bo8301825967528238409st_a_a ) )
= ( B4 = A3 ) ) ).
% singleton_iff
thf(fact_513_singleton__iff,axiom,
! [B4: nat > list_a,A3: nat > list_a] :
( ( member_nat_list_a @ B4 @ ( insert_nat_list_a @ A3 @ bot_bo3806784159821827511list_a ) )
= ( B4 = A3 ) ) ).
% singleton_iff
thf(fact_514_singleton__iff,axiom,
! [B4: nat > a,A3: nat > a] :
( ( member_nat_a @ B4 @ ( insert_nat_a @ A3 @ bot_bot_set_nat_a ) )
= ( B4 = A3 ) ) ).
% singleton_iff
thf(fact_515_singleton__iff,axiom,
! [B4: list_a,A3: list_a] :
( ( member_list_a @ B4 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) )
= ( B4 = A3 ) ) ).
% singleton_iff
thf(fact_516_singleton__iff,axiom,
! [B4: a,A3: a] :
( ( member_a @ B4 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( B4 = A3 ) ) ).
% singleton_iff
thf(fact_517_doubleton__eq__iff,axiom,
! [A3: list_a,B4: list_a,C2: list_a,D2: list_a] :
( ( ( insert_list_a @ A3 @ ( insert_list_a @ B4 @ bot_bot_set_list_a ) )
= ( insert_list_a @ C2 @ ( insert_list_a @ D2 @ bot_bot_set_list_a ) ) )
= ( ( ( A3 = C2 )
& ( B4 = D2 ) )
| ( ( A3 = D2 )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_518_doubleton__eq__iff,axiom,
! [A3: a,B4: a,C2: a,D2: a] :
( ( ( insert_a @ A3 @ ( insert_a @ B4 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A3 = C2 )
& ( B4 = D2 ) )
| ( ( A3 = D2 )
& ( B4 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_519_insert__not__empty,axiom,
! [A3: list_a,A2: set_list_a] :
( ( insert_list_a @ A3 @ A2 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_520_insert__not__empty,axiom,
! [A3: a,A2: set_a] :
( ( insert_a @ A3 @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_521_singleton__inject,axiom,
! [A3: list_a,B4: list_a] :
( ( ( insert_list_a @ A3 @ bot_bot_set_list_a )
= ( insert_list_a @ B4 @ bot_bot_set_list_a ) )
=> ( A3 = B4 ) ) ).
% singleton_inject
thf(fact_522_singleton__inject,axiom,
! [A3: a,B4: a] :
( ( ( insert_a @ A3 @ bot_bot_set_a )
= ( insert_a @ B4 @ bot_bot_set_a ) )
=> ( A3 = B4 ) ) ).
% singleton_inject
thf(fact_523_finite_OinsertI,axiom,
! [A2: set_a,A3: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A3 @ A2 ) ) ) ).
% finite.insertI
thf(fact_524_finite_OinsertI,axiom,
! [A2: set_list_a,A3: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( insert_list_a @ A3 @ A2 ) ) ) ).
% finite.insertI
thf(fact_525_finite_OinsertI,axiom,
! [A2: set_nat,A3: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A3 @ A2 ) ) ) ).
% finite.insertI
thf(fact_526_insert__Diff__if,axiom,
! [X: list_a > a,B2: set_list_a_a,A2: set_list_a_a] :
( ( ( member_list_a_a @ X @ B2 )
=> ( ( minus_921748639838131438st_a_a @ ( insert_list_a_a @ X @ A2 ) @ B2 )
= ( minus_921748639838131438st_a_a @ A2 @ B2 ) ) )
& ( ~ ( member_list_a_a @ X @ B2 )
=> ( ( minus_921748639838131438st_a_a @ ( insert_list_a_a @ X @ A2 ) @ B2 )
= ( insert_list_a_a @ X @ ( minus_921748639838131438st_a_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_527_insert__Diff__if,axiom,
! [X: set_list_a > a,B2: set_set_list_a_a,A2: set_set_list_a_a] :
( ( ( member_set_list_a_a @ X @ B2 )
=> ( ( minus_5613498140476352782st_a_a @ ( insert_set_list_a_a @ X @ A2 ) @ B2 )
= ( minus_5613498140476352782st_a_a @ A2 @ B2 ) ) )
& ( ~ ( member_set_list_a_a @ X @ B2 )
=> ( ( minus_5613498140476352782st_a_a @ ( insert_set_list_a_a @ X @ A2 ) @ B2 )
= ( insert_set_list_a_a @ X @ ( minus_5613498140476352782st_a_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_528_insert__Diff__if,axiom,
! [X: nat > list_a,B2: set_nat_list_a,A2: set_nat_list_a] :
( ( ( member_nat_list_a @ X @ B2 )
=> ( ( minus_4169782841487898290list_a @ ( insert_nat_list_a @ X @ A2 ) @ B2 )
= ( minus_4169782841487898290list_a @ A2 @ B2 ) ) )
& ( ~ ( member_nat_list_a @ X @ B2 )
=> ( ( minus_4169782841487898290list_a @ ( insert_nat_list_a @ X @ A2 ) @ B2 )
= ( insert_nat_list_a @ X @ ( minus_4169782841487898290list_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_529_insert__Diff__if,axiom,
! [X: nat > a,B2: set_nat_a,A2: set_nat_a] :
( ( ( member_nat_a @ X @ B2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) )
& ( ~ ( member_nat_a @ X @ B2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ B2 )
= ( insert_nat_a @ X @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_530_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_531_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_532_less__eq__set__def,axiom,
( ord_le6942402695062981877st_a_a
= ( ^ [A4: set_list_a_a,B3: set_list_a_a] :
( ord_le5538412863658560464_a_a_o
@ ^ [X2: list_a > a] : ( member_list_a_a @ X2 @ A4 )
@ ^ [X2: list_a > a] : ( member_list_a_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_533_less__eq__set__def,axiom,
( ord_le4799719167512954133st_a_a
= ( ^ [A4: set_set_list_a_a,B3: set_set_list_a_a] :
( ord_le6553425858663066544_a_a_o
@ ^ [X2: set_list_a > a] : ( member_set_list_a_a @ X2 @ A4 )
@ ^ [X2: set_list_a > a] : ( member_set_list_a_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_534_less__eq__set__def,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A4: set_nat_list_a,B3: set_nat_list_a] :
( ord_le4184171100712167858st_a_o
@ ^ [X2: nat > list_a] : ( member_nat_list_a @ X2 @ A4 )
@ ^ [X2: nat > list_a] : ( member_nat_list_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_535_less__eq__set__def,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A4: set_nat_a,B3: set_nat_a] :
( ord_less_eq_nat_a_o
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ A4 )
@ ^ [X2: nat > a] : ( member_nat_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_536_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A4 )
@ ^ [X2: a] : ( member_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_537_less__eq__set__def,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B3: set_list_a] :
( ord_less_eq_list_a_o
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A4 )
@ ^ [X2: list_a] : ( member_list_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_538_Collect__conv__if2,axiom,
! [P2: nat > $o,A3: nat] :
( ( ( P2 @ A3 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A3 = X2 )
& ( P2 @ X2 ) ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A3 = X2 )
& ( P2 @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_539_Collect__conv__if2,axiom,
! [P2: list_list_a > $o,A3: list_list_a] :
( ( ( P2 @ A3 )
=> ( ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( A3 = X2 )
& ( P2 @ X2 ) ) )
= ( insert_list_list_a @ A3 @ bot_bo1875519244922727510list_a ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( A3 = X2 )
& ( P2 @ X2 ) ) )
= bot_bo1875519244922727510list_a ) ) ) ).
% Collect_conv_if2
thf(fact_540_Collect__conv__if2,axiom,
! [P2: list_a > $o,A3: list_a] :
( ( ( P2 @ A3 )
=> ( ( collect_list_a
@ ^ [X2: list_a] :
( ( A3 = X2 )
& ( P2 @ X2 ) ) )
= ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collect_list_a
@ ^ [X2: list_a] :
( ( A3 = X2 )
& ( P2 @ X2 ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if2
thf(fact_541_Collect__conv__if2,axiom,
! [P2: a > $o,A3: a] :
( ( ( P2 @ A3 )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A3 = X2 )
& ( P2 @ X2 ) ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A3 = X2 )
& ( P2 @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_542_Collect__conv__if,axiom,
! [P2: nat > $o,A3: nat] :
( ( ( P2 @ A3 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A3 )
& ( P2 @ X2 ) ) )
= ( insert_nat @ A3 @ bot_bot_set_nat ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A3 )
& ( P2 @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_543_Collect__conv__if,axiom,
! [P2: list_list_a > $o,A3: list_list_a] :
( ( ( P2 @ A3 )
=> ( ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( X2 = A3 )
& ( P2 @ X2 ) ) )
= ( insert_list_list_a @ A3 @ bot_bo1875519244922727510list_a ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collect_list_list_a
@ ^ [X2: list_list_a] :
( ( X2 = A3 )
& ( P2 @ X2 ) ) )
= bot_bo1875519244922727510list_a ) ) ) ).
% Collect_conv_if
thf(fact_544_Collect__conv__if,axiom,
! [P2: list_a > $o,A3: list_a] :
( ( ( P2 @ A3 )
=> ( ( collect_list_a
@ ^ [X2: list_a] :
( ( X2 = A3 )
& ( P2 @ X2 ) ) )
= ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collect_list_a
@ ^ [X2: list_a] :
( ( X2 = A3 )
& ( P2 @ X2 ) ) )
= bot_bot_set_list_a ) ) ) ).
% Collect_conv_if
thf(fact_545_Collect__conv__if,axiom,
! [P2: a > $o,A3: a] :
( ( ( P2 @ A3 )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A3 )
& ( P2 @ X2 ) ) )
= ( insert_a @ A3 @ bot_bot_set_a ) ) )
& ( ~ ( P2 @ A3 )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A3 )
& ( P2 @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_546_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_547_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_548_subset__singleton__iff,axiom,
! [X5: set_a,A3: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A3 @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_549_subset__singleton__iff,axiom,
! [X5: set_list_a,A3: list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) )
= ( ( X5 = bot_bot_set_list_a )
| ( X5
= ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_550_finite_Ocases,axiom,
! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ( A3 != bot_bot_set_nat )
=> ~ ! [A6: set_nat] :
( ? [A: nat] :
( A3
= ( insert_nat @ A @ A6 ) )
=> ~ ( finite_finite_nat @ A6 ) ) ) ) ).
% finite.cases
thf(fact_551_finite_Ocases,axiom,
! [A3: set_list_a] :
( ( finite_finite_list_a @ A3 )
=> ( ( A3 != bot_bot_set_list_a )
=> ~ ! [A6: set_list_a] :
( ? [A: list_a] :
( A3
= ( insert_list_a @ A @ A6 ) )
=> ~ ( finite_finite_list_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_552_finite_Ocases,axiom,
! [A3: set_a] :
( ( finite_finite_a @ A3 )
=> ( ( A3 != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A: a] :
( A3
= ( insert_a @ A @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_553_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A5: set_nat] :
( ( A5 = bot_bot_set_nat )
| ? [A4: set_nat,B6: nat] :
( ( A5
= ( insert_nat @ B6 @ A4 ) )
& ( finite_finite_nat @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_554_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A5: set_list_a] :
( ( A5 = bot_bot_set_list_a )
| ? [A4: set_list_a,B6: list_a] :
( ( A5
= ( insert_list_a @ B6 @ A4 ) )
& ( finite_finite_list_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_555_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A5: set_a] :
( ( A5 = bot_bot_set_a )
| ? [A4: set_a,B6: a] :
( ( A5
= ( insert_a @ B6 @ A4 ) )
& ( finite_finite_a @ A4 ) ) ) ) ) ).
% finite.simps
thf(fact_556_finite__induct,axiom,
! [F2: set_list_a_a,P2: set_list_a_a > $o] :
( ( finite2458174228029419510st_a_a @ F2 )
=> ( ( P2 @ bot_bot_set_list_a_a )
=> ( ! [X3: list_a > a,F3: set_list_a_a] :
( ( finite2458174228029419510st_a_a @ F3 )
=> ( ~ ( member_list_a_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_557_finite__induct,axiom,
! [F2: set_set_list_a_a,P2: set_set_list_a_a > $o] :
( ( finite6385009043124570134st_a_a @ F2 )
=> ( ( P2 @ bot_bo8301825967528238409st_a_a )
=> ( ! [X3: set_list_a > a,F3: set_set_list_a_a] :
( ( finite6385009043124570134st_a_a @ F3 )
=> ( ~ ( member_set_list_a_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_list_a_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_558_finite__induct,axiom,
! [F2: set_nat_list_a,P2: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( P2 @ bot_bo3806784159821827511list_a )
=> ( ! [X3: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ~ ( member_nat_list_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_list_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_559_finite__induct,axiom,
! [F2: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( P2 @ bot_bot_set_nat_a )
=> ( ! [X3: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_560_finite__induct,axiom,
! [F2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_561_finite__induct,axiom,
! [F2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [X3: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_562_finite__induct,axiom,
! [F2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_563_finite__ne__induct,axiom,
! [F2: set_list_a_a,P2: set_list_a_a > $o] :
( ( finite2458174228029419510st_a_a @ F2 )
=> ( ( F2 != bot_bot_set_list_a_a )
=> ( ! [X3: list_a > a] : ( P2 @ ( insert_list_a_a @ X3 @ bot_bot_set_list_a_a ) )
=> ( ! [X3: list_a > a,F3: set_list_a_a] :
( ( finite2458174228029419510st_a_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a_a )
=> ( ~ ( member_list_a_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a_a @ X3 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_564_finite__ne__induct,axiom,
! [F2: set_set_list_a_a,P2: set_set_list_a_a > $o] :
( ( finite6385009043124570134st_a_a @ F2 )
=> ( ( F2 != bot_bo8301825967528238409st_a_a )
=> ( ! [X3: set_list_a > a] : ( P2 @ ( insert_set_list_a_a @ X3 @ bot_bo8301825967528238409st_a_a ) )
=> ( ! [X3: set_list_a > a,F3: set_set_list_a_a] :
( ( finite6385009043124570134st_a_a @ F3 )
=> ( ( F3 != bot_bo8301825967528238409st_a_a )
=> ( ~ ( member_set_list_a_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_list_a_a @ X3 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_565_finite__ne__induct,axiom,
! [F2: set_nat_list_a,P2: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( F2 != bot_bo3806784159821827511list_a )
=> ( ! [X3: nat > list_a] : ( P2 @ ( insert_nat_list_a @ X3 @ bot_bo3806784159821827511list_a ) )
=> ( ! [X3: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ( F3 != bot_bo3806784159821827511list_a )
=> ( ~ ( member_nat_list_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_list_a @ X3 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_566_finite__ne__induct,axiom,
! [F2: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( F2 != bot_bot_set_nat_a )
=> ( ! [X3: nat > a] : ( P2 @ ( insert_nat_a @ X3 @ bot_bot_set_nat_a ) )
=> ( ! [X3: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( F3 != bot_bot_set_nat_a )
=> ( ~ ( member_nat_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_a @ X3 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_567_finite__ne__induct,axiom,
! [F2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X3: nat] : ( P2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X3 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_568_finite__ne__induct,axiom,
! [F2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( F2 != bot_bot_set_list_a )
=> ( ! [X3: list_a] : ( P2 @ ( insert_list_a @ X3 @ bot_bot_set_list_a ) )
=> ( ! [X3: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ X3 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_569_finite__ne__induct,axiom,
! [F2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X3: a] : ( P2 @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X3 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_570_infinite__finite__induct,axiom,
! [P2: set_list_a_a > $o,A2: set_list_a_a] :
( ! [A6: set_list_a_a] :
( ~ ( finite2458174228029419510st_a_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_list_a_a )
=> ( ! [X3: list_a > a,F3: set_list_a_a] :
( ( finite2458174228029419510st_a_a @ F3 )
=> ( ~ ( member_list_a_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_571_infinite__finite__induct,axiom,
! [P2: set_set_list_a_a > $o,A2: set_set_list_a_a] :
( ! [A6: set_set_list_a_a] :
( ~ ( finite6385009043124570134st_a_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bo8301825967528238409st_a_a )
=> ( ! [X3: set_list_a > a,F3: set_set_list_a_a] :
( ( finite6385009043124570134st_a_a @ F3 )
=> ( ~ ( member_set_list_a_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_list_a_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_572_infinite__finite__induct,axiom,
! [P2: set_nat_list_a > $o,A2: set_nat_list_a] :
( ! [A6: set_nat_list_a] :
( ~ ( finite7630042315537210004list_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bo3806784159821827511list_a )
=> ( ! [X3: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ~ ( member_nat_list_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_list_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_573_infinite__finite__induct,axiom,
! [P2: set_nat_a > $o,A2: set_nat_a] :
( ! [A6: set_nat_a] :
( ~ ( finite_finite_nat_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_nat_a )
=> ( ! [X3: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_574_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A2: set_nat] :
( ! [A6: set_nat] :
( ~ ( finite_finite_nat @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X3: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X3 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_575_infinite__finite__induct,axiom,
! [P2: set_list_a > $o,A2: set_list_a] :
( ! [A6: set_list_a] :
( ~ ( finite_finite_list_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [X3: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_576_infinite__finite__induct,axiom,
! [P2: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X3: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X3 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X3 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_577_subset__Diff__insert,axiom,
! [A2: set_list_a_a,B2: set_list_a_a,X: list_a > a,C3: set_list_a_a] :
( ( ord_le6942402695062981877st_a_a @ A2 @ ( minus_921748639838131438st_a_a @ B2 @ ( insert_list_a_a @ X @ C3 ) ) )
= ( ( ord_le6942402695062981877st_a_a @ A2 @ ( minus_921748639838131438st_a_a @ B2 @ C3 ) )
& ~ ( member_list_a_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_578_subset__Diff__insert,axiom,
! [A2: set_set_list_a_a,B2: set_set_list_a_a,X: set_list_a > a,C3: set_set_list_a_a] :
( ( ord_le4799719167512954133st_a_a @ A2 @ ( minus_5613498140476352782st_a_a @ B2 @ ( insert_set_list_a_a @ X @ C3 ) ) )
= ( ( ord_le4799719167512954133st_a_a @ A2 @ ( minus_5613498140476352782st_a_a @ B2 @ C3 ) )
& ~ ( member_set_list_a_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_579_subset__Diff__insert,axiom,
! [A2: set_nat_list_a,B2: set_nat_list_a,X: nat > list_a,C3: set_nat_list_a] :
( ( ord_le2145805922479659755list_a @ A2 @ ( minus_4169782841487898290list_a @ B2 @ ( insert_nat_list_a @ X @ C3 ) ) )
= ( ( ord_le2145805922479659755list_a @ A2 @ ( minus_4169782841487898290list_a @ B2 @ C3 ) )
& ~ ( member_nat_list_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_580_subset__Diff__insert,axiom,
! [A2: set_nat_a,B2: set_nat_a,X: nat > a,C3: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( minus_490503922182417452_nat_a @ B2 @ ( insert_nat_a @ X @ C3 ) ) )
= ( ( ord_le871467723717165285_nat_a @ A2 @ ( minus_490503922182417452_nat_a @ B2 @ C3 ) )
& ~ ( member_nat_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_581_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C3 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_582_subset__Diff__insert,axiom,
! [A2: set_list_a,B2: set_list_a,X: list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B2 @ ( insert_list_a @ X @ C3 ) ) )
= ( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B2 @ C3 ) )
& ~ ( member_list_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_583_Diff__insert,axiom,
! [A2: set_a,A3: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_584_Diff__insert,axiom,
! [A2: set_list_a,A3: list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A3 @ B2 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_585_insert__Diff,axiom,
! [A3: list_a > a,A2: set_list_a_a] :
( ( member_list_a_a @ A3 @ A2 )
=> ( ( insert_list_a_a @ A3 @ ( minus_921748639838131438st_a_a @ A2 @ ( insert_list_a_a @ A3 @ bot_bot_set_list_a_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_586_insert__Diff,axiom,
! [A3: set_list_a > a,A2: set_set_list_a_a] :
( ( member_set_list_a_a @ A3 @ A2 )
=> ( ( insert_set_list_a_a @ A3 @ ( minus_5613498140476352782st_a_a @ A2 @ ( insert_set_list_a_a @ A3 @ bot_bo8301825967528238409st_a_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_587_insert__Diff,axiom,
! [A3: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A3 @ A2 )
=> ( ( insert_nat_list_a @ A3 @ ( minus_4169782841487898290list_a @ A2 @ ( insert_nat_list_a @ A3 @ bot_bo3806784159821827511list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_588_insert__Diff,axiom,
! [A3: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A3 @ A2 )
=> ( ( insert_nat_a @ A3 @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ A3 @ bot_bot_set_nat_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_589_insert__Diff,axiom,
! [A3: a,A2: set_a] :
( ( member_a @ A3 @ A2 )
=> ( ( insert_a @ A3 @ ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_590_insert__Diff,axiom,
! [A3: list_a,A2: set_list_a] :
( ( member_list_a @ A3 @ A2 )
=> ( ( insert_list_a @ A3 @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_591_Diff__insert2,axiom,
! [A2: set_a,A3: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_592_Diff__insert2,axiom,
! [A2: set_list_a,A3: list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A3 @ B2 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_593_Diff__insert__absorb,axiom,
! [X: list_a > a,A2: set_list_a_a] :
( ~ ( member_list_a_a @ X @ A2 )
=> ( ( minus_921748639838131438st_a_a @ ( insert_list_a_a @ X @ A2 ) @ ( insert_list_a_a @ X @ bot_bot_set_list_a_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_594_Diff__insert__absorb,axiom,
! [X: set_list_a > a,A2: set_set_list_a_a] :
( ~ ( member_set_list_a_a @ X @ A2 )
=> ( ( minus_5613498140476352782st_a_a @ ( insert_set_list_a_a @ X @ A2 ) @ ( insert_set_list_a_a @ X @ bot_bo8301825967528238409st_a_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_595_Diff__insert__absorb,axiom,
! [X: nat > list_a,A2: set_nat_list_a] :
( ~ ( member_nat_list_a @ X @ A2 )
=> ( ( minus_4169782841487898290list_a @ ( insert_nat_list_a @ X @ A2 ) @ ( insert_nat_list_a @ X @ bot_bo3806784159821827511list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_596_Diff__insert__absorb,axiom,
! [X: nat > a,A2: set_nat_a] :
( ~ ( member_nat_a @ X @ A2 )
=> ( ( minus_490503922182417452_nat_a @ ( insert_nat_a @ X @ A2 ) @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_597_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_598_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_599_finite__subset__induct,axiom,
! [F2: set_list_a_a,A2: set_list_a_a,P2: set_list_a_a > $o] :
( ( finite2458174228029419510st_a_a @ F2 )
=> ( ( ord_le6942402695062981877st_a_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_list_a_a )
=> ( ! [A: list_a > a,F3: set_list_a_a] :
( ( finite2458174228029419510st_a_a @ F3 )
=> ( ( member_list_a_a @ A @ A2 )
=> ( ~ ( member_list_a_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a_a @ A @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_600_finite__subset__induct,axiom,
! [F2: set_set_list_a_a,A2: set_set_list_a_a,P2: set_set_list_a_a > $o] :
( ( finite6385009043124570134st_a_a @ F2 )
=> ( ( ord_le4799719167512954133st_a_a @ F2 @ A2 )
=> ( ( P2 @ bot_bo8301825967528238409st_a_a )
=> ( ! [A: set_list_a > a,F3: set_set_list_a_a] :
( ( finite6385009043124570134st_a_a @ F3 )
=> ( ( member_set_list_a_a @ A @ A2 )
=> ( ~ ( member_set_list_a_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_list_a_a @ A @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_601_finite__subset__induct,axiom,
! [F2: set_nat_list_a,A2: set_nat_list_a,P2: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( ord_le2145805922479659755list_a @ F2 @ A2 )
=> ( ( P2 @ bot_bo3806784159821827511list_a )
=> ( ! [A: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ( member_nat_list_a @ A @ A2 )
=> ( ~ ( member_nat_list_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_list_a @ A @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_602_finite__subset__induct,axiom,
! [F2: set_nat_a,A2: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( ord_le871467723717165285_nat_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_nat_a )
=> ( ! [A: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A @ A2 )
=> ( ~ ( member_nat_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_a @ A @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_603_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_604_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ A @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_605_finite__subset__induct,axiom,
! [F2: set_list_a,A2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [A: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a @ A @ A2 )
=> ( ~ ( member_list_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ A @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_606_finite__subset__induct_H,axiom,
! [F2: set_list_a_a,A2: set_list_a_a,P2: set_list_a_a > $o] :
( ( finite2458174228029419510st_a_a @ F2 )
=> ( ( ord_le6942402695062981877st_a_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_list_a_a )
=> ( ! [A: list_a > a,F3: set_list_a_a] :
( ( finite2458174228029419510st_a_a @ F3 )
=> ( ( member_list_a_a @ A @ A2 )
=> ( ( ord_le6942402695062981877st_a_a @ F3 @ A2 )
=> ( ~ ( member_list_a_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a_a @ A @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_607_finite__subset__induct_H,axiom,
! [F2: set_set_list_a_a,A2: set_set_list_a_a,P2: set_set_list_a_a > $o] :
( ( finite6385009043124570134st_a_a @ F2 )
=> ( ( ord_le4799719167512954133st_a_a @ F2 @ A2 )
=> ( ( P2 @ bot_bo8301825967528238409st_a_a )
=> ( ! [A: set_list_a > a,F3: set_set_list_a_a] :
( ( finite6385009043124570134st_a_a @ F3 )
=> ( ( member_set_list_a_a @ A @ A2 )
=> ( ( ord_le4799719167512954133st_a_a @ F3 @ A2 )
=> ( ~ ( member_set_list_a_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_list_a_a @ A @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_608_finite__subset__induct_H,axiom,
! [F2: set_nat_list_a,A2: set_nat_list_a,P2: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( ord_le2145805922479659755list_a @ F2 @ A2 )
=> ( ( P2 @ bot_bo3806784159821827511list_a )
=> ( ! [A: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ( member_nat_list_a @ A @ A2 )
=> ( ( ord_le2145805922479659755list_a @ F3 @ A2 )
=> ( ~ ( member_nat_list_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_list_a @ A @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_609_finite__subset__induct_H,axiom,
! [F2: set_nat_a,A2: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( ord_le871467723717165285_nat_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_nat_a )
=> ( ! [A: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ F3 @ A2 )
=> ( ~ ( member_nat_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat_a @ A @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_610_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_611_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ A @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_612_finite__subset__induct_H,axiom,
! [F2: set_list_a,A2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [A: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a @ A @ A2 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A2 )
=> ( ~ ( member_list_a @ A @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_list_a @ A @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_613_subset__insert__iff,axiom,
! [A2: set_list_a_a,X: list_a > a,B2: set_list_a_a] :
( ( ord_le6942402695062981877st_a_a @ A2 @ ( insert_list_a_a @ X @ B2 ) )
= ( ( ( member_list_a_a @ X @ A2 )
=> ( ord_le6942402695062981877st_a_a @ ( minus_921748639838131438st_a_a @ A2 @ ( insert_list_a_a @ X @ bot_bot_set_list_a_a ) ) @ B2 ) )
& ( ~ ( member_list_a_a @ X @ A2 )
=> ( ord_le6942402695062981877st_a_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_614_subset__insert__iff,axiom,
! [A2: set_set_list_a_a,X: set_list_a > a,B2: set_set_list_a_a] :
( ( ord_le4799719167512954133st_a_a @ A2 @ ( insert_set_list_a_a @ X @ B2 ) )
= ( ( ( member_set_list_a_a @ X @ A2 )
=> ( ord_le4799719167512954133st_a_a @ ( minus_5613498140476352782st_a_a @ A2 @ ( insert_set_list_a_a @ X @ bot_bo8301825967528238409st_a_a ) ) @ B2 ) )
& ( ~ ( member_set_list_a_a @ X @ A2 )
=> ( ord_le4799719167512954133st_a_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_615_subset__insert__iff,axiom,
! [A2: set_nat_list_a,X: nat > list_a,B2: set_nat_list_a] :
( ( ord_le2145805922479659755list_a @ A2 @ ( insert_nat_list_a @ X @ B2 ) )
= ( ( ( member_nat_list_a @ X @ A2 )
=> ( ord_le2145805922479659755list_a @ ( minus_4169782841487898290list_a @ A2 @ ( insert_nat_list_a @ X @ bot_bo3806784159821827511list_a ) ) @ B2 ) )
& ( ~ ( member_nat_list_a @ X @ A2 )
=> ( ord_le2145805922479659755list_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_616_subset__insert__iff,axiom,
! [A2: set_nat_a,X: nat > a,B2: set_nat_a] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( insert_nat_a @ X @ B2 ) )
= ( ( ( member_nat_a @ X @ A2 )
=> ( ord_le871467723717165285_nat_a @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) ) @ B2 ) )
& ( ~ ( member_nat_a @ X @ A2 )
=> ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_617_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_618_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_619_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_620_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_621_infinite__remove,axiom,
! [S2: set_nat,A3: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_622_infinite__remove,axiom,
! [S2: set_a,A3: a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_623_infinite__remove,axiom,
! [S2: set_list_a,A3: list_a] :
( ~ ( finite_finite_list_a @ S2 )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S2 @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_624_infinite__coinduct,axiom,
! [X5: set_nat > $o,A2: set_nat] :
( ( X5 @ A2 )
=> ( ! [A6: set_nat] :
( ( X5 @ A6 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A6 )
& ( ( X5 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_625_infinite__coinduct,axiom,
! [X5: set_a > $o,A2: set_a] :
( ( X5 @ A2 )
=> ( ! [A6: set_a] :
( ( X5 @ A6 )
=> ? [X4: a] :
( ( member_a @ X4 @ A6 )
& ( ( X5 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_626_infinite__coinduct,axiom,
! [X5: set_list_a > $o,A2: set_list_a] :
( ( X5 @ A2 )
=> ( ! [A6: set_list_a] :
( ( X5 @ A6 )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
& ( ( X5 @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_627_finite__empty__induct,axiom,
! [A2: set_list_a_a,P2: set_list_a_a > $o] :
( ( finite2458174228029419510st_a_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A: list_a > a,A6: set_list_a_a] :
( ( finite2458174228029419510st_a_a @ A6 )
=> ( ( member_list_a_a @ A @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_921748639838131438st_a_a @ A6 @ ( insert_list_a_a @ A @ bot_bot_set_list_a_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_list_a_a ) ) ) ) ).
% finite_empty_induct
thf(fact_628_finite__empty__induct,axiom,
! [A2: set_set_list_a_a,P2: set_set_list_a_a > $o] :
( ( finite6385009043124570134st_a_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A: set_list_a > a,A6: set_set_list_a_a] :
( ( finite6385009043124570134st_a_a @ A6 )
=> ( ( member_set_list_a_a @ A @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_5613498140476352782st_a_a @ A6 @ ( insert_set_list_a_a @ A @ bot_bo8301825967528238409st_a_a ) ) ) ) ) )
=> ( P2 @ bot_bo8301825967528238409st_a_a ) ) ) ) ).
% finite_empty_induct
thf(fact_629_finite__empty__induct,axiom,
! [A2: set_nat_list_a,P2: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A: nat > list_a,A6: set_nat_list_a] :
( ( finite7630042315537210004list_a @ A6 )
=> ( ( member_nat_list_a @ A @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_4169782841487898290list_a @ A6 @ ( insert_nat_list_a @ A @ bot_bo3806784159821827511list_a ) ) ) ) ) )
=> ( P2 @ bot_bo3806784159821827511list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_630_finite__empty__induct,axiom,
! [A2: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A: nat > a,A6: set_nat_a] :
( ( finite_finite_nat_a @ A6 )
=> ( ( member_nat_a @ A @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_490503922182417452_nat_a @ A6 @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat_a ) ) ) ) ).
% finite_empty_induct
thf(fact_631_finite__empty__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A: nat,A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( member_nat @ A @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_632_finite__empty__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_633_finite__empty__induct,axiom,
! [A2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A: list_a,A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ( member_list_a @ A @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_634_remove__induct,axiom,
! [P2: set_list_a_a > $o,B2: set_list_a_a] :
( ( P2 @ bot_bot_set_list_a_a )
=> ( ( ~ ( finite2458174228029419510st_a_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A6: set_list_a_a] :
( ( finite2458174228029419510st_a_a @ A6 )
=> ( ( A6 != bot_bot_set_list_a_a )
=> ( ( ord_le6942402695062981877st_a_a @ A6 @ B2 )
=> ( ! [X4: list_a > a] :
( ( member_list_a_a @ X4 @ A6 )
=> ( P2 @ ( minus_921748639838131438st_a_a @ A6 @ ( insert_list_a_a @ X4 @ bot_bot_set_list_a_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_635_remove__induct,axiom,
! [P2: set_set_list_a_a > $o,B2: set_set_list_a_a] :
( ( P2 @ bot_bo8301825967528238409st_a_a )
=> ( ( ~ ( finite6385009043124570134st_a_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A6: set_set_list_a_a] :
( ( finite6385009043124570134st_a_a @ A6 )
=> ( ( A6 != bot_bo8301825967528238409st_a_a )
=> ( ( ord_le4799719167512954133st_a_a @ A6 @ B2 )
=> ( ! [X4: set_list_a > a] :
( ( member_set_list_a_a @ X4 @ A6 )
=> ( P2 @ ( minus_5613498140476352782st_a_a @ A6 @ ( insert_set_list_a_a @ X4 @ bot_bo8301825967528238409st_a_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_636_remove__induct,axiom,
! [P2: set_nat_list_a > $o,B2: set_nat_list_a] :
( ( P2 @ bot_bo3806784159821827511list_a )
=> ( ( ~ ( finite7630042315537210004list_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A6: set_nat_list_a] :
( ( finite7630042315537210004list_a @ A6 )
=> ( ( A6 != bot_bo3806784159821827511list_a )
=> ( ( ord_le2145805922479659755list_a @ A6 @ B2 )
=> ( ! [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A6 )
=> ( P2 @ ( minus_4169782841487898290list_a @ A6 @ ( insert_nat_list_a @ X4 @ bot_bo3806784159821827511list_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_637_remove__induct,axiom,
! [P2: set_nat_a > $o,B2: set_nat_a] :
( ( P2 @ bot_bot_set_nat_a )
=> ( ( ~ ( finite_finite_nat_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A6: set_nat_a] :
( ( finite_finite_nat_a @ A6 )
=> ( ( A6 != bot_bot_set_nat_a )
=> ( ( ord_le871467723717165285_nat_a @ A6 @ B2 )
=> ( ! [X4: nat > a] :
( ( member_nat_a @ X4 @ A6 )
=> ( P2 @ ( minus_490503922182417452_nat_a @ A6 @ ( insert_nat_a @ X4 @ bot_bot_set_nat_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_638_remove__induct,axiom,
! [P2: set_nat > $o,B2: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( P2 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_639_remove__induct,axiom,
! [P2: set_a > $o,B2: set_a] :
( ( P2 @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( P2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_640_remove__induct,axiom,
! [P2: set_list_a > $o,B2: set_list_a] :
( ( P2 @ bot_bot_set_list_a )
=> ( ( ~ ( finite_finite_list_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ( A6 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A6 @ B2 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
=> ( P2 @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_641_finite__remove__induct,axiom,
! [B2: set_list_a_a,P2: set_list_a_a > $o] :
( ( finite2458174228029419510st_a_a @ B2 )
=> ( ( P2 @ bot_bot_set_list_a_a )
=> ( ! [A6: set_list_a_a] :
( ( finite2458174228029419510st_a_a @ A6 )
=> ( ( A6 != bot_bot_set_list_a_a )
=> ( ( ord_le6942402695062981877st_a_a @ A6 @ B2 )
=> ( ! [X4: list_a > a] :
( ( member_list_a_a @ X4 @ A6 )
=> ( P2 @ ( minus_921748639838131438st_a_a @ A6 @ ( insert_list_a_a @ X4 @ bot_bot_set_list_a_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_642_finite__remove__induct,axiom,
! [B2: set_set_list_a_a,P2: set_set_list_a_a > $o] :
( ( finite6385009043124570134st_a_a @ B2 )
=> ( ( P2 @ bot_bo8301825967528238409st_a_a )
=> ( ! [A6: set_set_list_a_a] :
( ( finite6385009043124570134st_a_a @ A6 )
=> ( ( A6 != bot_bo8301825967528238409st_a_a )
=> ( ( ord_le4799719167512954133st_a_a @ A6 @ B2 )
=> ( ! [X4: set_list_a > a] :
( ( member_set_list_a_a @ X4 @ A6 )
=> ( P2 @ ( minus_5613498140476352782st_a_a @ A6 @ ( insert_set_list_a_a @ X4 @ bot_bo8301825967528238409st_a_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_643_finite__remove__induct,axiom,
! [B2: set_nat_list_a,P2: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ B2 )
=> ( ( P2 @ bot_bo3806784159821827511list_a )
=> ( ! [A6: set_nat_list_a] :
( ( finite7630042315537210004list_a @ A6 )
=> ( ( A6 != bot_bo3806784159821827511list_a )
=> ( ( ord_le2145805922479659755list_a @ A6 @ B2 )
=> ( ! [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A6 )
=> ( P2 @ ( minus_4169782841487898290list_a @ A6 @ ( insert_nat_list_a @ X4 @ bot_bo3806784159821827511list_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_644_finite__remove__induct,axiom,
! [B2: set_nat_a,P2: set_nat_a > $o] :
( ( finite_finite_nat_a @ B2 )
=> ( ( P2 @ bot_bot_set_nat_a )
=> ( ! [A6: set_nat_a] :
( ( finite_finite_nat_a @ A6 )
=> ( ( A6 != bot_bot_set_nat_a )
=> ( ( ord_le871467723717165285_nat_a @ A6 @ B2 )
=> ( ! [X4: nat > a] :
( ( member_nat_a @ X4 @ A6 )
=> ( P2 @ ( minus_490503922182417452_nat_a @ A6 @ ( insert_nat_a @ X4 @ bot_bot_set_nat_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_645_finite__remove__induct,axiom,
! [B2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ( A6 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A6 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A6 )
=> ( P2 @ ( minus_minus_set_nat @ A6 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_646_finite__remove__induct,axiom,
! [B2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A6 )
=> ( P2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_647_finite__remove__induct,axiom,
! [B2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ B2 )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ( A6 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A6 @ B2 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A6 )
=> ( P2 @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_648_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_649_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_650_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_651_Ring_Ofield__Units,axiom,
! [R3: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R3 )
=> ( ( units_2471184348132832486t_unit @ R3 )
= ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R3 ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R3 ) @ bot_bot_set_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_652_Ring_Ofield__Units,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R3 )
=> ( ( units_2932844235741507942t_unit @ R3 )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R3 ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ bot_bot_set_list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_653_Ring_Ofield__Units,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R3 )
=> ( ( units_a_ring_ext_a_b @ R3 )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R3 ) @ ( insert_a @ ( zero_a_b @ R3 ) @ bot_bot_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_654_Ring_Ofield__Units,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R3 )
=> ( ( units_5837875185506529638t_unit @ R3 )
= ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R3 ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ bot_bo3186585308812441520list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_655_Ring_Ofield__Units,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R3 )
=> ( ( units_4903515905731149798t_unit @ R3 )
= ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R3 ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_656_semiring_Oone__zeroD,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( ( one_li8328186300101108157t_unit @ R3 )
= ( zero_l4142658623432671053t_unit @ R3 ) )
=> ( ( partia5361259788508890537t_unit @ R3 )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ bot_bot_set_list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_657_semiring_Oone__zeroD,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( ( ( one_a_ring_ext_a_b @ R3 )
= ( zero_a_b @ R3 ) )
=> ( ( partia707051561876973205xt_a_b @ R3 )
= ( insert_a @ ( zero_a_b @ R3 ) @ bot_bot_set_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_658_semiring_Oone__zeroD,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( ( one_se1127990129394575805t_unit @ R3 )
= ( zero_s2910681146719230829t_unit @ R3 ) )
=> ( ( partia141011252114345353t_unit @ R3 )
= ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ bot_bo3186585308812441520list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_659_semiring_Oone__zeroD,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( ( one_li8234411390022467901t_unit @ R3 )
= ( zero_l347298301471573063t_unit @ R3 ) )
=> ( ( partia2464479390973590831t_unit @ R3 )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_660_semiring_Oone__zeroI,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( ( partia5361259788508890537t_unit @ R3 )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ R3 )
= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_661_semiring_Oone__zeroI,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( ( ( partia707051561876973205xt_a_b @ R3 )
= ( insert_a @ ( zero_a_b @ R3 ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ R3 )
= ( zero_a_b @ R3 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_662_semiring_Oone__zeroI,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( ( partia141011252114345353t_unit @ R3 )
= ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ bot_bo3186585308812441520list_a ) )
=> ( ( one_se1127990129394575805t_unit @ R3 )
= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_663_semiring_Oone__zeroI,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( ( partia2464479390973590831t_unit @ R3 )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ bot_bo1875519244922727510list_a ) )
=> ( ( one_li8234411390022467901t_unit @ R3 )
= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ).
% semiring.one_zeroI
thf(fact_664_semiring_Ocarrier__one__zero,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( ( partia5361259788508890537t_unit @ R3 )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R3 )
= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_665_semiring_Ocarrier__one__zero,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( ( ( partia707051561876973205xt_a_b @ R3 )
= ( insert_a @ ( zero_a_b @ R3 ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R3 )
= ( zero_a_b @ R3 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_666_semiring_Ocarrier__one__zero,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( ( partia141011252114345353t_unit @ R3 )
= ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ bot_bo3186585308812441520list_a ) )
= ( ( one_se1127990129394575805t_unit @ R3 )
= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_667_semiring_Ocarrier__one__zero,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( ( partia2464479390973590831t_unit @ R3 )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ bot_bo1875519244922727510list_a ) )
= ( ( one_li8234411390022467901t_unit @ R3 )
= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_668_semiring_Ocarrier__one__not__zero,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( ( partia5361259788508890537t_unit @ R3 )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R3 )
!= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_669_semiring_Ocarrier__one__not__zero,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( ( ( partia707051561876973205xt_a_b @ R3 )
!= ( insert_a @ ( zero_a_b @ R3 ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R3 )
!= ( zero_a_b @ R3 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_670_semiring_Ocarrier__one__not__zero,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( ( partia141011252114345353t_unit @ R3 )
!= ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ bot_bo3186585308812441520list_a ) )
= ( ( one_se1127990129394575805t_unit @ R3 )
!= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_671_semiring_Ocarrier__one__not__zero,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( ( partia2464479390973590831t_unit @ R3 )
!= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ bot_bo1875519244922727510list_a ) )
= ( ( one_li8234411390022467901t_unit @ R3 )
!= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_672_x_Oup__minus__closed,axiom,
! [P: nat > list_a,Q: nat > list_a] :
( ( member_nat_list_a @ P @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_nat_list_a @ Q @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [I: nat] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( P @ I ) @ ( Q @ I ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.up_minus_closed
thf(fact_673_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_674_x_OUnits__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_closed
thf(fact_675_x_OUnits__inv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.Units_inv_comm
thf(fact_676_x_Ocring__fieldI2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.cring_fieldI2
thf(fact_677_x_OUnits__l__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_l_inv_ex
thf(fact_678_x_OUnits__r__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X3 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_r_inv_ex
thf(fact_679_x_Oprod__unit__l,axiom,
! [A3: list_a,B4: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B4 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ B4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_l
thf(fact_680_x_Oprod__unit__r,axiom,
! [A3: list_a,B4: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B4 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.prod_unit_r
thf(fact_681_x_Ounit__factor,axiom,
! [A3: list_a,B4: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B4 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.unit_factor
thf(fact_682_x_Oinv__unique,axiom,
! [Y: list_a,X: list_a,Y4: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y4 ) ) ) ) ) ) ).
% x.inv_unique
thf(fact_683_x_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.one_unique
thf(fact_684_x_Ofield__intro2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.field_intro2
thf(fact_685_x_Ocring__fieldI,axiom,
( ( ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cring_fieldI
thf(fact_686_x_Ocarrier__one__not__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.carrier_one_not_zero
thf(fact_687_x_Ocarrier__one__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.carrier_one_zero
thf(fact_688_x_Oone__zeroD,axiom,
( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% x.one_zeroD
thf(fact_689_x_Oone__zeroI,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.one_zeroI
thf(fact_690_x_Oring_Ozero__closed,axiom,
member_a @ ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ).
% x.ring.zero_closed
thf(fact_691_x_Ozeropideal,axiom,
princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.zeropideal
thf(fact_692_x_OUnits__one__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.Units_one_closed
thf(fact_693_x_Ofinite__ring__finite__units,axiom,
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_list_a @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.finite_ring_finite_units
thf(fact_694_x_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.one_closed
thf(fact_695_x_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.zero_closed
thf(fact_696_x_OUnits__m__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.Units_m_closed
thf(fact_697_x_OUnits__l__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% x.Units_l_cancel
thf(fact_698_x_Ol__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.l_one
thf(fact_699_x_Or__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% x.r_one
thf(fact_700_x_Ol__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.l_null
thf(fact_701_x_Or__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.r_null
thf(fact_702_x_Or__right__minus__eq,axiom,
! [A3: list_a,B4: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B4 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A3 = B4 ) ) ) ) ).
% x.r_right_minus_eq
thf(fact_703_x_Ominus__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.minus_closed
thf(fact_704_x_Oring_Ohom__zero,axiom,
( ( eval_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
= ( zero_a_b @ r ) ) ).
% x.ring.hom_zero
thf(fact_705_x_Oring_Ohom__one,axiom,
( ( eval_a_b @ r @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ x )
= ( one_a_ring_ext_a_b @ r ) ) ).
% x.ring.hom_one
thf(fact_706_x_Ohom__sub,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ x )
= ( a_minus_a_b @ r @ ( eval_a_b @ r @ X @ x ) @ ( eval_a_b @ r @ Y @ x ) ) ) ) ) ).
% x.hom_sub
thf(fact_707_ring__hom__closed,axiom,
! [H2: a > a,R3: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_708_ring__hom__closed,axiom,
! [H2: list_a > a,R3: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_709_ring__hom__closed,axiom,
! [H2: a > list_a,R3: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_710_ring__hom__closed,axiom,
! [H2: list_a > list_a,R3: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_711_ring__hom__closed,axiom,
! [H2: a > set_list_a,R3: partia2175431115845679010xt_a_b,S2: partia7496981018696276118t_unit,X: a] :
( ( member_a_set_list_a @ H2 @ ( ring_h6109298854714515236t_unit @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_set_list_a @ ( H2 @ X ) @ ( partia141011252114345353t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_712_ring__hom__closed,axiom,
! [H2: a > list_list_a,R3: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a] :
( ( member_a_list_list_a @ H2 @ ( ring_h6858658657455840382t_unit @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_list_a @ ( H2 @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_713_ring__hom__closed,axiom,
! [H2: set_list_a > a,R3: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,X: set_list_a] :
( ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R3 @ S2 ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_714_ring__hom__closed,axiom,
! [H2: list_list_a > a,R3: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_h8078271382950527358it_a_b @ R3 @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_715_ring__hom__closed,axiom,
! [H2: list_a > set_list_a,R3: partia2670972154091845814t_unit,S2: partia7496981018696276118t_unit,X: list_a] :
( ( member4263473470251683292list_a @ H2 @ ( ring_h6188449271506562988t_unit @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_set_list_a @ ( H2 @ X ) @ ( partia141011252114345353t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_716_ring__hom__closed,axiom,
! [H2: list_a > list_list_a,R3: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a] :
( ( member6714375691612171394list_a @ H2 @ ( ring_h8002040739877300486t_unit @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_list_a @ ( H2 @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_717_ring__hom__one,axiom,
! [H2: a > a,R3: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R3 @ S2 ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ).
% ring_hom_one
thf(fact_718_ring__hom__one,axiom,
! [H2: a > list_a,R3: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ).
% ring_hom_one
thf(fact_719_ring__hom__one,axiom,
! [H2: a > set_list_a,R3: partia2175431115845679010xt_a_b,S2: partia7496981018696276118t_unit] :
( ( member_a_set_list_a @ H2 @ ( ring_h6109298854714515236t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_se1127990129394575805t_unit @ S2 ) ) ) ).
% ring_hom_one
thf(fact_720_ring__hom__one,axiom,
! [H2: list_a > a,R3: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R3 @ S2 ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ).
% ring_hom_one
thf(fact_721_ring__hom__one,axiom,
! [H2: list_a > list_a,R3: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ).
% ring_hom_one
thf(fact_722_ring__hom__one,axiom,
! [H2: list_a > set_list_a,R3: partia2670972154091845814t_unit,S2: partia7496981018696276118t_unit] :
( ( member4263473470251683292list_a @ H2 @ ( ring_h6188449271506562988t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_se1127990129394575805t_unit @ S2 ) ) ) ).
% ring_hom_one
thf(fact_723_ring__hom__one,axiom,
! [H2: set_list_a > a,R3: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b] :
( ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R3 @ S2 ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ).
% ring_hom_one
thf(fact_724_ring__hom__one,axiom,
! [H2: set_list_a > list_a,R3: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit] :
( ( member5910328476188217884list_a @ H2 @ ( ring_h8038483918290310060t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ).
% ring_hom_one
thf(fact_725_ring__hom__one,axiom,
! [H2: set_list_a > set_list_a,R3: partia7496981018696276118t_unit,S2: partia7496981018696276118t_unit] :
( ( member5068272912271824380list_a @ H2 @ ( ring_h6076331213207892940t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_se1127990129394575805t_unit @ S2 ) ) ) ).
% ring_hom_one
thf(fact_726_ring__hom__cring_Ohom__zero,axiom,
! [R3: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,H2: a > a] :
( ( ring_h661254511236296859_b_a_b @ R3 @ S2 @ H2 )
=> ( ( H2 @ ( zero_a_b @ R3 ) )
= ( zero_a_b @ S2 ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_727_ring__hom__cring_Ohom__zero,axiom,
! [R3: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,H2: a > list_a] :
( ( ring_h8279546866833948963t_unit @ R3 @ S2 @ H2 )
=> ( ( H2 @ ( zero_a_b @ R3 ) )
= ( zero_l4142658623432671053t_unit @ S2 ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_728_ring__hom__cring_Ohom__zero,axiom,
! [R3: partia2175431115845679010xt_a_b,S2: partia7496981018696276118t_unit,H2: a > set_list_a] :
( ( ring_h7527734465757070659t_unit @ R3 @ S2 @ H2 )
=> ( ( H2 @ ( zero_a_b @ R3 ) )
= ( zero_s2910681146719230829t_unit @ S2 ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_729_ring__hom__cring_Ohom__zero,axiom,
! [R3: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,H2: list_a > list_a] :
( ( ring_h8282015026914974507t_unit @ R3 @ S2 @ H2 )
=> ( ( H2 @ ( zero_l4142658623432671053t_unit @ R3 ) )
= ( zero_l4142658623432671053t_unit @ S2 ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_730_ring__hom__cring_Ohom__zero,axiom,
! [R3: partia2670972154091845814t_unit,S2: partia7496981018696276118t_unit,H2: list_a > set_list_a] :
( ( ring_h5296475915237130059t_unit @ R3 @ S2 @ H2 )
=> ( ( H2 @ ( zero_l4142658623432671053t_unit @ R3 ) )
= ( zero_s2910681146719230829t_unit @ S2 ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_731_ring__hom__cring_Ohom__zero,axiom,
! [R3: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,H2: set_list_a > a] :
( ( ring_h1101743994381864643it_a_b @ R3 @ S2 @ H2 )
=> ( ( H2 @ ( zero_s2910681146719230829t_unit @ R3 ) )
= ( zero_a_b @ S2 ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_732_ring__hom__cring_Ohom__zero,axiom,
! [R3: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit,H2: set_list_a > list_a] :
( ( ring_h7146510562020877131t_unit @ R3 @ S2 @ H2 )
=> ( ( H2 @ ( zero_s2910681146719230829t_unit @ R3 ) )
= ( zero_l4142658623432671053t_unit @ S2 ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_733_ring__hom__cring_Ohom__zero,axiom,
! [R3: partia7496981018696276118t_unit,S2: partia7496981018696276118t_unit,H2: set_list_a > set_list_a] :
( ( ring_h1755269146545522539t_unit @ R3 @ S2 @ H2 )
=> ( ( H2 @ ( zero_s2910681146719230829t_unit @ R3 ) )
= ( zero_s2910681146719230829t_unit @ S2 ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_734_ring__hom__cring_Ohom__zero,axiom,
! [R3: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,H2: list_a > a] :
( ( ring_h1547129875642963619it_a_b @ R3 @ S2 @ H2 )
=> ( ( H2 @ ( zero_l4142658623432671053t_unit @ R3 ) )
= ( zero_a_b @ S2 ) ) ) ).
% ring_hom_cring.hom_zero
thf(fact_735_ring__hom__cring_Ohomh,axiom,
! [R3: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,H2: set_list_a > a] :
( ( ring_h1101743994381864643it_a_b @ R3 @ S2 @ H2 )
=> ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R3 @ S2 ) ) ) ).
% ring_hom_cring.homh
thf(fact_736_ring__hom__cring_Ohomh,axiom,
! [R3: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,H2: list_a > a] :
( ( ring_h1547129875642963619it_a_b @ R3 @ S2 @ H2 )
=> ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R3 @ S2 ) ) ) ).
% ring_hom_cring.homh
thf(fact_737_ring__hom__mult,axiom,
! [H2: a > a,R3: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_738_ring__hom__mult,axiom,
! [H2: list_a > a,R3: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R3 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_739_ring__hom__mult,axiom,
! [H2: a > list_a,R3: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_740_ring__hom__mult,axiom,
! [H2: list_a > list_a,R3: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R3 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_741_ring__hom__mult,axiom,
! [H2: a > list_list_a,R3: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a,Y: a] :
( ( member_a_list_list_a @ H2 @ ( ring_h6858658657455840382t_unit @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) )
= ( mult_l4853965630390486993t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_742_ring__hom__mult,axiom,
! [H2: set_list_a > a,R3: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,X: set_list_a,Y: set_list_a] :
( ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R3 @ S2 ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( H2 @ ( mult_s7802724872828879953t_unit @ R3 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_743_ring__hom__mult,axiom,
! [H2: list_list_a > a,R3: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_h8078271382950527358it_a_b @ R3 @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R3 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_744_ring__hom__mult,axiom,
! [H2: list_a > list_list_a,R3: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a,Y: list_a] :
( ( member6714375691612171394list_a @ H2 @ ( ring_h8002040739877300486t_unit @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R3 @ X @ Y ) )
= ( mult_l4853965630390486993t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_745_ring__hom__mult,axiom,
! [H2: set_list_a > list_a,R3: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit,X: set_list_a,Y: set_list_a] :
( ( member5910328476188217884list_a @ H2 @ ( ring_h8038483918290310060t_unit @ R3 @ S2 ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( H2 @ ( mult_s7802724872828879953t_unit @ R3 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_746_ring__hom__mult,axiom,
! [H2: list_list_a > list_a,R3: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_h5031276006722532742t_unit @ R3 @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R3 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_mult
thf(fact_747_Ring_Oone__not__zero,axiom,
! [R3: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R3 )
=> ( ( one_se211549098623999037t_unit @ R3 )
!= ( zero_s2174465271003423091t_unit @ R3 ) ) ) ).
% Ring.one_not_zero
thf(fact_748_Ring_Oone__not__zero,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R3 )
=> ( ( one_a_ring_ext_a_b @ R3 )
!= ( zero_a_b @ R3 ) ) ) ).
% Ring.one_not_zero
thf(fact_749_Ring_Oone__not__zero,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R3 )
=> ( ( one_li8328186300101108157t_unit @ R3 )
!= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ).
% Ring.one_not_zero
thf(fact_750_Ring_Oone__not__zero,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R3 )
=> ( ( one_se1127990129394575805t_unit @ R3 )
!= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ).
% Ring.one_not_zero
thf(fact_751_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_752_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( member_a @ ( zero_a_b @ R3 ) @ ( partia707051561876973205xt_a_b @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_753_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ ( partia141011252114345353t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_754_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_755_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( mult_l7073676228092353617t_unit @ R3 @ ( mult_l7073676228092353617t_unit @ R3 @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ R3 @ X @ ( mult_l7073676228092353617t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_756_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ R3 @ X @ ( mult_a_ring_ext_a_b @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_757_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Z @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( mult_s7802724872828879953t_unit @ R3 @ ( mult_s7802724872828879953t_unit @ R3 @ X @ Y ) @ Z )
= ( mult_s7802724872828879953t_unit @ R3 @ X @ ( mult_s7802724872828879953t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_758_semiring_Osemiring__simprules_I8_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( mult_l4853965630390486993t_unit @ R3 @ ( mult_l4853965630390486993t_unit @ R3 @ X @ Y ) @ Z )
= ( mult_l4853965630390486993t_unit @ R3 @ X @ ( mult_l4853965630390486993t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_759_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R3 @ X @ Y ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_760_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_761_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia7496981018696276118t_unit,X: set_list_a,Y: set_list_a] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R3 ) )
=> ( member_set_list_a @ ( mult_s7802724872828879953t_unit @ R3 @ X @ Y ) @ ( partia141011252114345353t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_762_semiring_Osemiring__simprules_I3_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R3 @ X @ Y ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_763_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_764_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R3 )
=> ( member_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( partia707051561876973205xt_a_b @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_765_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( partia141011252114345353t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_766_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_767_ring__hom__cring_Ohom__closed,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: a > a,R3: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ) ).
% ring_hom_cring.hom_closed
thf(fact_768_ring__hom__cring_Ohom__closed,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_a > a,R3: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ) ).
% ring_hom_cring.hom_closed
thf(fact_769_ring__hom__cring_Ohom__closed,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: a > list_a,R3: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ) ).
% ring_hom_cring.hom_closed
thf(fact_770_ring__hom__cring_Ohom__closed,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_a > list_a,R3: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ) ).
% ring_hom_cring.hom_closed
thf(fact_771_ring__hom__cring_Ohom__closed,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: a > set_list_a,R3: partia2175431115845679010xt_a_b,S2: partia7496981018696276118t_unit,X: a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_a_set_list_a @ H2 @ ( ring_h6109298854714515236t_unit @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_set_list_a @ ( H2 @ X ) @ ( partia141011252114345353t_unit @ S2 ) ) ) ) ) ).
% ring_hom_cring.hom_closed
thf(fact_772_ring__hom__cring_Ohom__closed,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: a > list_list_a,R3: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_a_list_list_a @ H2 @ ( ring_h6858658657455840382t_unit @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( member_list_list_a @ ( H2 @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ) ).
% ring_hom_cring.hom_closed
thf(fact_773_ring__hom__cring_Ohom__closed,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: set_list_a > a,R3: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,X: set_list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R3 @ S2 ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ) ).
% ring_hom_cring.hom_closed
thf(fact_774_ring__hom__cring_Ohom__closed,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_list_a > a,R3: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_list_list_a_a @ H2 @ ( ring_h8078271382950527358it_a_b @ R3 @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ) ).
% ring_hom_cring.hom_closed
thf(fact_775_ring__hom__cring_Ohom__closed,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_a > set_list_a,R3: partia2670972154091845814t_unit,S2: partia7496981018696276118t_unit,X: list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member4263473470251683292list_a @ H2 @ ( ring_h6188449271506562988t_unit @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_set_list_a @ ( H2 @ X ) @ ( partia141011252114345353t_unit @ S2 ) ) ) ) ) ).
% ring_hom_cring.hom_closed
thf(fact_776_ring__hom__cring_Ohom__closed,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_a > list_list_a,R3: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member6714375691612171394list_a @ H2 @ ( ring_h8002040739877300486t_unit @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_list_a @ ( H2 @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ) ).
% ring_hom_cring.hom_closed
thf(fact_777_ring__hom__cring_Ohom__one,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: a > a,R3: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R3 @ S2 ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ) ).
% ring_hom_cring.hom_one
thf(fact_778_ring__hom__cring_Ohom__one,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: a > list_a,R3: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ) ).
% ring_hom_cring.hom_one
thf(fact_779_ring__hom__cring_Ohom__one,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: a > set_list_a,R3: partia2175431115845679010xt_a_b,S2: partia7496981018696276118t_unit] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_a_set_list_a @ H2 @ ( ring_h6109298854714515236t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_a_ring_ext_a_b @ R3 ) )
= ( one_se1127990129394575805t_unit @ S2 ) ) ) ) ).
% ring_hom_cring.hom_one
thf(fact_780_ring__hom__cring_Ohom__one,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_a > a,R3: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R3 @ S2 ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ) ).
% ring_hom_cring.hom_one
thf(fact_781_ring__hom__cring_Ohom__one,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_a > list_a,R3: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ) ).
% ring_hom_cring.hom_one
thf(fact_782_ring__hom__cring_Ohom__one,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_a > set_list_a,R3: partia2670972154091845814t_unit,S2: partia7496981018696276118t_unit] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member4263473470251683292list_a @ H2 @ ( ring_h6188449271506562988t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_se1127990129394575805t_unit @ S2 ) ) ) ) ).
% ring_hom_cring.hom_one
thf(fact_783_ring__hom__cring_Ohom__one,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: set_list_a > a,R3: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R3 @ S2 ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ) ).
% ring_hom_cring.hom_one
thf(fact_784_ring__hom__cring_Ohom__one,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: set_list_a > list_a,R3: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member5910328476188217884list_a @ H2 @ ( ring_h8038483918290310060t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ) ).
% ring_hom_cring.hom_one
thf(fact_785_ring__hom__cring_Ohom__one,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: set_list_a > set_list_a,R3: partia7496981018696276118t_unit,S2: partia7496981018696276118t_unit] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member5068272912271824380list_a @ H2 @ ( ring_h6076331213207892940t_unit @ R3 @ S2 ) )
=> ( ( H2 @ ( one_se1127990129394575805t_unit @ R3 ) )
= ( one_se1127990129394575805t_unit @ S2 ) ) ) ) ).
% ring_hom_cring.hom_one
thf(fact_786_Ring_Ointegral,axiom,
! [R3: partia6043505979758434576t_unit,A3: set_a,B4: set_a] :
( ( field_6045675692312731021t_unit @ R3 )
=> ( ( ( mult_s7930653359683758801t_unit @ R3 @ A3 @ B4 )
= ( zero_s2174465271003423091t_unit @ R3 ) )
=> ( ( member_set_a @ A3 @ ( partia5907974310037520643t_unit @ R3 ) )
=> ( ( member_set_a @ B4 @ ( partia5907974310037520643t_unit @ R3 ) )
=> ( ( A3
= ( zero_s2174465271003423091t_unit @ R3 ) )
| ( B4
= ( zero_s2174465271003423091t_unit @ R3 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_787_Ring_Ointegral,axiom,
! [R3: partia2670972154091845814t_unit,A3: list_a,B4: list_a] :
( ( field_6388047844668329575t_unit @ R3 )
=> ( ( ( mult_l7073676228092353617t_unit @ R3 @ A3 @ B4 )
= ( zero_l4142658623432671053t_unit @ R3 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( A3
= ( zero_l4142658623432671053t_unit @ R3 ) )
| ( B4
= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_788_Ring_Ointegral,axiom,
! [R3: partia2175431115845679010xt_a_b,A3: a,B4: a] :
( ( field_a_b @ R3 )
=> ( ( ( mult_a_ring_ext_a_b @ R3 @ A3 @ B4 )
= ( zero_a_b @ R3 ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( A3
= ( zero_a_b @ R3 ) )
| ( B4
= ( zero_a_b @ R3 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_789_Ring_Ointegral,axiom,
! [R3: partia7496981018696276118t_unit,A3: set_list_a,B4: set_list_a] :
( ( field_26233345952514695t_unit @ R3 )
=> ( ( ( mult_s7802724872828879953t_unit @ R3 @ A3 @ B4 )
= ( zero_s2910681146719230829t_unit @ R3 ) )
=> ( ( member_set_list_a @ A3 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ B4 @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( A3
= ( zero_s2910681146719230829t_unit @ R3 ) )
| ( B4
= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_790_Ring_Ointegral,axiom,
! [R3: partia2956882679547061052t_unit,A3: list_list_a,B4: list_list_a] :
( ( field_1861437471013600865t_unit @ R3 )
=> ( ( ( mult_l4853965630390486993t_unit @ R3 @ A3 @ B4 )
= ( zero_l347298301471573063t_unit @ R3 ) )
=> ( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ B4 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( A3
= ( zero_l347298301471573063t_unit @ R3 ) )
| ( B4
= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_791_semiring_Or__null,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( mult_l7073676228092353617t_unit @ R3 @ X @ ( zero_l4142658623432671053t_unit @ R3 ) )
= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_792_semiring_Or__null,axiom,
! [R3: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ X @ ( zero_a_b @ R3 ) )
= ( zero_a_b @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_793_semiring_Or__null,axiom,
! [R3: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( mult_s7802724872828879953t_unit @ R3 @ X @ ( zero_s2910681146719230829t_unit @ R3 ) )
= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_794_semiring_Or__null,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( mult_l4853965630390486993t_unit @ R3 @ X @ ( zero_l347298301471573063t_unit @ R3 ) )
= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ).
% semiring.r_null
thf(fact_795_semiring_Ol__null,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( mult_l7073676228092353617t_unit @ R3 @ ( zero_l4142658623432671053t_unit @ R3 ) @ X )
= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_796_semiring_Ol__null,axiom,
! [R3: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( zero_a_b @ R3 ) @ X )
= ( zero_a_b @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_797_semiring_Ol__null,axiom,
! [R3: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( mult_s7802724872828879953t_unit @ R3 @ ( zero_s2910681146719230829t_unit @ R3 ) @ X )
= ( zero_s2910681146719230829t_unit @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_798_semiring_Ol__null,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( mult_l4853965630390486993t_unit @ R3 @ ( zero_l347298301471573063t_unit @ R3 ) @ X )
= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ).
% semiring.l_null
thf(fact_799_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( mult_l7073676228092353617t_unit @ R3 @ ( one_li8328186300101108157t_unit @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_800_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R3 )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( mult_a_ring_ext_a_b @ R3 @ ( one_a_ring_ext_a_b @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_801_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia7496981018696276118t_unit,X: set_list_a] :
( ( semiri4000464634269493571t_unit @ R3 )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( mult_s7802724872828879953t_unit @ R3 @ ( one_se1127990129394575805t_unit @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_802_semiring_Osemiring__simprules_I9_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( mult_l4853965630390486993t_unit @ R3 @ ( one_li8234411390022467901t_unit @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_803_ring__hom__cring_Ohom__mult,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: a > a,R3: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_a_a @ H2 @ ( ring_hom_a_b_a_b @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ) ).
% ring_hom_cring.hom_mult
thf(fact_804_ring__hom__cring_Ohom__mult,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_a > a,R3: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_list_a_a @ H2 @ ( ring_h2895973938487309444it_a_b @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R3 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ) ).
% ring_hom_cring.hom_mult
thf(fact_805_ring__hom__cring_Ohom__mult,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: a > list_a,R3: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a,Y: a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_a_list_a @ H2 @ ( ring_h405018892823518980t_unit @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ) ).
% ring_hom_cring.hom_mult
thf(fact_806_ring__hom__cring_Ohom__mult,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_a > list_a,R3: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_list_a_list_a @ H2 @ ( ring_h7399960747407462284t_unit @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R3 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ) ).
% ring_hom_cring.hom_mult
thf(fact_807_ring__hom__cring_Ohom__mult,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: a > list_list_a,R3: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a,Y: a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_a_list_list_a @ H2 @ ( ring_h6858658657455840382t_unit @ R3 @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R3 @ X @ Y ) )
= ( mult_l4853965630390486993t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ) ).
% ring_hom_cring.hom_mult
thf(fact_808_ring__hom__cring_Ohom__mult,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: set_list_a > a,R3: partia7496981018696276118t_unit,S2: partia2175431115845679010xt_a_b,X: set_list_a,Y: set_list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_set_list_a_a @ H2 @ ( ring_h8906680420194085028it_a_b @ R3 @ S2 ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( H2 @ ( mult_s7802724872828879953t_unit @ R3 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ) ).
% ring_hom_cring.hom_mult
thf(fact_809_ring__hom__cring_Ohom__mult,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_list_a > a,R3: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member_list_list_a_a @ H2 @ ( ring_h8078271382950527358it_a_b @ R3 @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R3 @ X @ Y ) )
= ( mult_a_ring_ext_a_b @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ) ).
% ring_hom_cring.hom_mult
thf(fact_810_ring__hom__cring_Ohom__mult,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_a > list_list_a,R3: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a,Y: list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member6714375691612171394list_a @ H2 @ ( ring_h8002040739877300486t_unit @ R3 @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R3 @ X @ Y ) )
= ( mult_l4853965630390486993t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ) ).
% ring_hom_cring.hom_mult
thf(fact_811_ring__hom__cring_Ohom__mult,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: set_list_a > list_a,R3: partia7496981018696276118t_unit,S2: partia2670972154091845814t_unit,X: set_list_a,Y: set_list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member5910328476188217884list_a @ H2 @ ( ring_h8038483918290310060t_unit @ R3 @ S2 ) )
=> ( ( member_set_list_a @ X @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( member_set_list_a @ Y @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ( H2 @ ( mult_s7802724872828879953t_unit @ R3 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ) ).
% ring_hom_cring.hom_mult
thf(fact_812_ring__hom__cring_Ohom__mult,axiom,
! [Ra: partia2670972154091845814t_unit,Sa: partia2175431115845679010xt_a_b,Ha: list_a > a,H2: list_list_a > list_a,R3: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( ring_h1547129875642963619it_a_b @ Ra @ Sa @ Ha )
=> ( ( member7168557129179038582list_a @ H2 @ ( ring_h5031276006722532742t_unit @ R3 @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R3 @ X @ Y ) )
= ( mult_l7073676228092353617t_unit @ S2 @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ) ).
% ring_hom_cring.hom_mult
thf(fact_813_x_Obound__upD,axiom,
! [F: nat > list_a] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [N: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F ) ) ).
% x.bound_upD
thf(fact_814_x_Ozeromaximalideal__eq__field,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeromaximalideal_eq_field
thf(fact_815_x_Ozeromaximalideal__fieldI,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeromaximalideal_fieldI
thf(fact_816_x_OsubdomainI,axiom,
! [H: set_list_a] :
( ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( H1
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( H22
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subdomainI
thf(fact_817_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_818_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_819_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_820_carrier__is__subfield,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subfield
thf(fact_821_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_822_subring__props_I6_J,axiom,
! [K: set_a,H12: a,H23: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H12 @ K )
=> ( ( member_a @ H23 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H23 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_823_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_824_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_825_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_826_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_827_genideal__self,axiom,
! [S2: set_a] :
( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S2 @ ( genideal_a_b @ r @ S2 ) ) ) ).
% genideal_self
thf(fact_828_genideal__self_H,axiom,
! [I3: a] :
( ( member_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I3 @ ( genideal_a_b @ r @ ( insert_a @ I3 @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_829_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_830_Idl__subset__ideal_H,axiom,
! [A3: a,B4: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A3 @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B4 @ bot_bot_set_a ) ) )
= ( member_a @ A3 @ ( genideal_a_b @ r @ ( insert_a @ B4 @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_831_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_832_subfield__m__inv__simprule,axiom,
! [K: set_a,K2: a,A3: 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 @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ A3 ) @ K )
=> ( member_a @ A3 @ K ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_833_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_834_pprimeE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_835_mem__upI,axiom,
! [F: nat > list_a,R3: partia2670972154091845814t_unit] :
( ! [N: nat] : ( member_list_a @ ( F @ N ) @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ? [N2: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ N2 @ F )
=> ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R3 ) ) ) ) ).
% mem_upI
thf(fact_836_mem__upI,axiom,
! [F: nat > a,R3: partia2175431115845679010xt_a_b] :
( ! [N: nat] : ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R3 ) )
=> ( ? [N2: nat] : ( bound_a @ ( zero_a_b @ R3 ) @ N2 @ F )
=> ( member_nat_a @ F @ ( up_a_b @ R3 ) ) ) ) ).
% mem_upI
thf(fact_837_mem__upI,axiom,
! [F: nat > set_list_a,R3: partia7496981018696276118t_unit] :
( ! [N: nat] : ( member_set_list_a @ ( F @ N ) @ ( partia141011252114345353t_unit @ R3 ) )
=> ( ? [N2: nat] : ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ N2 @ F )
=> ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R3 ) ) ) ) ).
% mem_upI
thf(fact_838_mem__upI,axiom,
! [F: nat > list_list_a,R3: partia2956882679547061052t_unit] :
( ! [N: nat] : ( member_list_list_a @ ( F @ N ) @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ? [N2: nat] : ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ N2 @ F )
=> ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R3 ) ) ) ) ).
% mem_upI
thf(fact_839_x_Ogenideal__one,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.genideal_one
thf(fact_840_long__division__closed_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_841_x_Omaximalideal__prime,axiom,
! [I2: set_list_a] :
( ( maxima6585700282301356660t_unit @ I2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( primei6309817859076077608t_unit @ I2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.maximalideal_prime
thf(fact_842_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N: nat] : ( bound_a @ ( zero_a_b @ r ) @ N @ F ) ) ).
% bound_upD
thf(fact_843_up__minus__closed,axiom,
! [P: nat > a,Q: nat > a] :
( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( ( member_nat_a @ Q @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I: nat] : ( a_minus_a_b @ r @ ( P @ I ) @ ( Q @ I ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_minus_closed
thf(fact_844_up__smult__closed,axiom,
! [A3: a,P: nat > a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_nat_a @ P @ ( up_a_b @ r ) )
=> ( member_nat_a
@ ^ [I: nat] : ( mult_a_ring_ext_a_b @ r @ A3 @ ( P @ I ) )
@ ( up_a_b @ r ) ) ) ) ).
% up_smult_closed
thf(fact_845_x_Ogenideal__self,axiom,
! [S2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ S2 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) ) ) ).
% x.genideal_self
thf(fact_846_x_Osubset__Idl__subset,axiom,
! [I2: set_list_a,H: set_list_a] :
( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H @ I2 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 ) ) ) ) ).
% x.subset_Idl_subset
thf(fact_847_x_OIdl__subset__ideal_H,axiom,
! [A3: list_a,B4: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ A3 @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B4 @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A3 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B4 @ bot_bot_set_list_a ) ) ) ) ) ) ).
% x.Idl_subset_ideal'
thf(fact_848_x_Ogenideal__self_H,axiom,
! [I3: list_a] :
( ( member_list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I3 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I3 @ bot_bot_set_list_a ) ) ) ) ).
% x.genideal_self'
thf(fact_849_x_Ogenideal__zero,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% x.genideal_zero
thf(fact_850_bound__below,axiom,
! [Z: list_a,M2: nat,F: nat > list_a,N3: nat] :
( ( bound_list_a @ Z @ M2 @ F )
=> ( ( ( F @ N3 )
!= Z )
=> ( ord_less_eq_nat @ N3 @ M2 ) ) ) ).
% bound_below
thf(fact_851_bound__below,axiom,
! [Z: a,M2: nat,F: nat > a,N3: nat] :
( ( bound_a @ Z @ M2 @ F )
=> ( ( ( F @ N3 )
!= Z )
=> ( ord_less_eq_nat @ N3 @ M2 ) ) ) ).
% bound_below
thf(fact_852_mem__upD,axiom,
! [F: nat > list_a,R3: partia2670972154091845814t_unit,N3: nat] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R3 ) )
=> ( member_list_a @ ( F @ N3 ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ).
% mem_upD
thf(fact_853_mem__upD,axiom,
! [F: nat > a,R3: partia2175431115845679010xt_a_b,N3: nat] :
( ( member_nat_a @ F @ ( up_a_b @ R3 ) )
=> ( member_a @ ( F @ N3 ) @ ( partia707051561876973205xt_a_b @ R3 ) ) ) ).
% mem_upD
thf(fact_854_mem__upD,axiom,
! [F: nat > set_list_a,R3: partia7496981018696276118t_unit,N3: nat] :
( ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R3 ) )
=> ( member_set_list_a @ ( F @ N3 ) @ ( partia141011252114345353t_unit @ R3 ) ) ) ).
% mem_upD
thf(fact_855_mem__upD,axiom,
! [F: nat > list_list_a,R3: partia2956882679547061052t_unit,N3: nat] :
( ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R3 ) )
=> ( member_list_list_a @ ( F @ N3 ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ).
% mem_upD
thf(fact_856_x_Oring__primeI,axiom,
! [P: list_a] :
( ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.ring_primeI
thf(fact_857_x_Oup__one__closed,axiom,
( member_nat_list_a
@ ^ [N4: nat] : ( if_list_a @ ( N4 = zero_zero_nat ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.up_one_closed
thf(fact_858_x_Osubfield__m__inv__simprule,axiom,
! [K: set_list_a,K2: list_a,A3: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A3 ) @ K )
=> ( member_list_a @ A3 @ K ) ) ) ) ) ).
% x.subfield_m_inv_simprule
thf(fact_859_x_Oa__lcos__mult__one,axiom,
! [M: set_list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M )
= M ) ) ).
% x.a_lcos_mult_one
thf(fact_860_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_861_x_Oirreducible__prod__lI,axiom,
! [B4: list_a,A3: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B4 )
=> ( ( member_list_a @ A3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B4 ) ) ) ) ) ) ).
% x.irreducible_prod_lI
thf(fact_862_maximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ r )
=> ( primeideal_a_b @ I2 @ r ) ) ).
% maximalideal_prime
thf(fact_863_x_Osubring__props_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).
% x.subring_props(2)
thf(fact_864_x_Osubring__props_I6_J,axiom,
! [K: set_list_a,H12: list_a,H23: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H12 @ K )
=> ( ( member_list_a @ H23 @ K )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H23 ) @ K ) ) ) ) ).
% x.subring_props(6)
thf(fact_865_x_Osubring__props_I4_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( K != bot_bot_set_list_a ) ) ).
% x.subring_props(4)
thf(fact_866_x_Osubring__props_I3_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).
% x.subring_props(3)
thf(fact_867_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_868_up__one__closed,axiom,
( member_nat_a
@ ^ [N4: nat] : ( if_a @ ( N4 = zero_zero_nat ) @ ( one_a_ring_ext_a_b @ r ) @ ( zero_a_b @ r ) )
@ ( up_a_b @ r ) ) ).
% up_one_closed
thf(fact_869_x_Osubring__props_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subring_props(1)
thf(fact_870_x_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.a_l_coset_subset_G
thf(fact_871_x_Oirreducible__prod__rI,axiom,
! [A3: list_a,B4: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 )
=> ( ( member_list_a @ B4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ B4 ) ) ) ) ) ) ).
% x.irreducible_prod_rI
thf(fact_872_euclidean__domain__axioms,axiom,
( ring_e8745995371659049232in_a_b @ r
@ ^ [Uu: a] : zero_zero_nat ) ).
% euclidean_domain_axioms
thf(fact_873_primeideal__iff__prime,axiom,
! [P: a] :
( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeideal_iff_prime
thf(fact_874_x_Oline__extension__smult__closed,axiom,
! [K: set_list_a,E: set_list_a,A3: list_a,K2: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K3: list_a,V: list_a] :
( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ V @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ V ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A3 @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A3 @ E ) ) ) ) ) ) ) ) ).
% x.line_extension_smult_closed
thf(fact_875_x_Oring_Oimg__is__subfield_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( subfield_a_b
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ r ) ) ) ).
% x.ring.img_is_subfield(2)
thf(fact_876_x_Osubfield__m__inv_I3_J,axiom,
! [K: set_list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ K2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subfield_m_inv(3)
thf(fact_877_x_Osubfield__m__inv_I2_J,axiom,
! [K: set_list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subfield_m_inv(2)
thf(fact_878_cgenideal__self,axiom,
! [I3: a] :
( ( member_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I3 @ ( cgenid547466209912283029xt_a_b @ r @ I3 ) ) ) ).
% cgenideal_self
thf(fact_879_image__eqI,axiom,
! [B4: a,F: a > a,X: a,A2: set_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B4 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_880_image__eqI,axiom,
! [B4: a,F: list_a > a,X: list_a,A2: set_list_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_list_a @ X @ A2 )
=> ( member_a @ B4 @ ( image_list_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_881_image__eqI,axiom,
! [B4: list_a,F: a > list_a,X: a,A2: set_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_list_a @ B4 @ ( image_a_list_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_882_image__eqI,axiom,
! [B4: list_a,F: list_a > list_a,X: list_a,A2: set_list_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_list_a @ X @ A2 )
=> ( member_list_a @ B4 @ ( image_list_a_list_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_883_image__eqI,axiom,
! [B4: a,F: set_list_a > a,X: set_list_a,A2: set_set_list_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_set_list_a @ X @ A2 )
=> ( member_a @ B4 @ ( image_set_list_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_884_image__eqI,axiom,
! [B4: a,F: ( nat > a ) > a,X: nat > a,A2: set_nat_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_a @ B4 @ ( image_nat_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_885_image__eqI,axiom,
! [B4: nat > a,F: a > nat > a,X: a,A2: set_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat_a @ B4 @ ( image_a_nat_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_886_image__eqI,axiom,
! [B4: list_list_a,F: list_a > list_list_a,X: list_a,A2: set_list_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_list_a @ X @ A2 )
=> ( member_list_list_a @ B4 @ ( image_8260866953997875467list_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_887_image__eqI,axiom,
! [B4: a,F: ( list_a > a ) > a,X: list_a > a,A2: set_list_a_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_list_a_a @ X @ A2 )
=> ( member_a @ B4 @ ( image_list_a_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_888_image__eqI,axiom,
! [B4: a,F: ( nat > list_a ) > a,X: nat > list_a,A2: set_nat_list_a] :
( ( B4
= ( F @ X ) )
=> ( ( member_nat_list_a @ X @ A2 )
=> ( member_a @ B4 @ ( image_nat_list_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_889_ideal__eq__carrier__iff,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A3 ) )
= ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_890_cgenideal__is__principalideal,axiom,
! [I3: a] :
( ( member_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I3 ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_891_x_Oinv__eq__imp__eq,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) )
=> ( X = Y ) ) ) ) ).
% x.inv_eq_imp_eq
thf(fact_892_image__ident,axiom,
! [Y5: set_list_a] :
( ( image_list_a_list_a
@ ^ [X2: list_a] : X2
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_893_x_Oline__extension__in__carrier,axiom,
! [K: set_list_a,A3: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A3 @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.line_extension_in_carrier
thf(fact_894_irreducible__imp__maximalideal,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r ) ) ) ).
% irreducible_imp_maximalideal
thf(fact_895_x_Oinv__eq__one__eq,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( X
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.inv_eq_one_eq
thf(fact_896_cgenideal__eq__genideal,axiom,
! [I3: a] :
( ( member_a @ I3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I3 )
= ( genideal_a_b @ r @ ( insert_a @ I3 @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_897_x_Ocomm__inv__char,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ).
% x.comm_inv_char
thf(fact_898_x_Oinv__char,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ) ).
% x.inv_char
thf(fact_899_x_Oinv__unique_H,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y
= ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) ) ) ) ) ) ).
% x.inv_unique'
thf(fact_900_le__zero__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_901_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_902_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_903_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_904_image__is__empty,axiom,
! [F: list_a > list_list_a,A2: set_list_a] :
( ( ( image_8260866953997875467list_a @ F @ A2 )
= bot_bo1875519244922727510list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_905_image__is__empty,axiom,
! [F: list_a > list_a,A2: set_list_a] :
( ( ( image_list_a_list_a @ F @ A2 )
= bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_906_image__is__empty,axiom,
! [F: a > list_a,A2: set_a] :
( ( ( image_a_list_a @ F @ A2 )
= bot_bot_set_list_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_907_image__is__empty,axiom,
! [F: set_list_a > a,A2: set_set_list_a] :
( ( ( image_set_list_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bo3186585308812441520list_a ) ) ).
% image_is_empty
thf(fact_908_image__is__empty,axiom,
! [F: list_a > a,A2: set_list_a] :
( ( ( image_list_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_909_image__is__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_910_empty__is__image,axiom,
! [F: list_a > list_list_a,A2: set_list_a] :
( ( bot_bo1875519244922727510list_a
= ( image_8260866953997875467list_a @ F @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_911_empty__is__image,axiom,
! [F: list_a > list_a,A2: set_list_a] :
( ( bot_bot_set_list_a
= ( image_list_a_list_a @ F @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_912_empty__is__image,axiom,
! [F: a > list_a,A2: set_a] :
( ( bot_bot_set_list_a
= ( image_a_list_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_913_empty__is__image,axiom,
! [F: set_list_a > a,A2: set_set_list_a] :
( ( bot_bot_set_a
= ( image_set_list_a_a @ F @ A2 ) )
= ( A2 = bot_bo3186585308812441520list_a ) ) ).
% empty_is_image
thf(fact_914_empty__is__image,axiom,
! [F: list_a > a,A2: set_list_a] :
( ( bot_bot_set_a
= ( image_list_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_915_empty__is__image,axiom,
! [F: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_916_image__empty,axiom,
! [F: set_list_a > a] :
( ( image_set_list_a_a @ F @ bot_bo3186585308812441520list_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_917_image__empty,axiom,
! [F: list_a > list_list_a] :
( ( image_8260866953997875467list_a @ F @ bot_bot_set_list_a )
= bot_bo1875519244922727510list_a ) ).
% image_empty
thf(fact_918_image__empty,axiom,
! [F: list_a > list_a] :
( ( image_list_a_list_a @ F @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% image_empty
thf(fact_919_image__empty,axiom,
! [F: list_a > a] :
( ( image_list_a_a @ F @ bot_bot_set_list_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_920_image__empty,axiom,
! [F: a > list_a] :
( ( image_a_list_a @ F @ bot_bot_set_a )
= bot_bot_set_list_a ) ).
% image_empty
thf(fact_921_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_922_finite__imageI,axiom,
! [F2: set_a,H2: a > a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_a @ ( image_a_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_923_finite__imageI,axiom,
! [F2: set_a,H2: a > nat] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_nat @ ( image_a_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_924_finite__imageI,axiom,
! [F2: set_nat,H2: nat > a] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_a @ ( image_nat_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_925_finite__imageI,axiom,
! [F2: set_nat,H2: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_926_finite__imageI,axiom,
! [F2: set_a,H2: a > list_a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_list_a @ ( image_a_list_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_927_finite__imageI,axiom,
! [F2: set_list_a,H2: list_a > a] :
( ( finite_finite_list_a @ F2 )
=> ( finite_finite_a @ ( image_list_a_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_928_finite__imageI,axiom,
! [F2: set_list_a,H2: list_a > nat] :
( ( finite_finite_list_a @ F2 )
=> ( finite_finite_nat @ ( image_list_a_nat @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_929_finite__imageI,axiom,
! [F2: set_nat,H2: nat > list_a] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_list_a @ ( image_nat_list_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_930_finite__imageI,axiom,
! [F2: set_set_list_a,H2: set_list_a > a] :
( ( finite5282473924520328461list_a @ F2 )
=> ( finite_finite_a @ ( image_set_list_a_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_931_finite__imageI,axiom,
! [F2: set_list_a,H2: list_a > list_a] :
( ( finite_finite_list_a @ F2 )
=> ( finite_finite_list_a @ ( image_list_a_list_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_932_image__insert,axiom,
! [F: set_list_a > a,A3: set_list_a,B2: set_set_list_a] :
( ( image_set_list_a_a @ F @ ( insert_set_list_a @ A3 @ B2 ) )
= ( insert_a @ ( F @ A3 ) @ ( image_set_list_a_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_933_image__insert,axiom,
! [F: a > a,A3: a,B2: set_a] :
( ( image_a_a @ F @ ( insert_a @ A3 @ B2 ) )
= ( insert_a @ ( F @ A3 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_934_image__insert,axiom,
! [F: a > list_a,A3: a,B2: set_a] :
( ( image_a_list_a @ F @ ( insert_a @ A3 @ B2 ) )
= ( insert_list_a @ ( F @ A3 ) @ ( image_a_list_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_935_image__insert,axiom,
! [F: list_a > list_list_a,A3: list_a,B2: set_list_a] :
( ( image_8260866953997875467list_a @ F @ ( insert_list_a @ A3 @ B2 ) )
= ( insert_list_list_a @ ( F @ A3 ) @ ( image_8260866953997875467list_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_936_image__insert,axiom,
! [F: list_a > a,A3: list_a,B2: set_list_a] :
( ( image_list_a_a @ F @ ( insert_list_a @ A3 @ B2 ) )
= ( insert_a @ ( F @ A3 ) @ ( image_list_a_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_937_image__insert,axiom,
! [F: list_a > list_a,A3: list_a,B2: set_list_a] :
( ( image_list_a_list_a @ F @ ( insert_list_a @ A3 @ B2 ) )
= ( insert_list_a @ ( F @ A3 ) @ ( image_list_a_list_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_938_insert__image,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) )
= ( image_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_939_insert__image,axiom,
! [X: list_a,A2: set_list_a,F: list_a > a] :
( ( member_list_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_list_a_a @ F @ A2 ) )
= ( image_list_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_940_insert__image,axiom,
! [X: a,A2: set_a,F: a > list_a] :
( ( member_a @ X @ A2 )
=> ( ( insert_list_a @ ( F @ X ) @ ( image_a_list_a @ F @ A2 ) )
= ( image_a_list_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_941_insert__image,axiom,
! [X: set_list_a,A2: set_set_list_a,F: set_list_a > a] :
( ( member_set_list_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_set_list_a_a @ F @ A2 ) )
= ( image_set_list_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_942_insert__image,axiom,
! [X: list_a,A2: set_list_a,F: list_a > list_a] :
( ( member_list_a @ X @ A2 )
=> ( ( insert_list_a @ ( F @ X ) @ ( image_list_a_list_a @ F @ A2 ) )
= ( image_list_a_list_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_943_insert__image,axiom,
! [X: nat > a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( member_nat_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_nat_a_a @ F @ A2 ) )
= ( image_nat_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_944_insert__image,axiom,
! [X: list_a,A2: set_list_a,F: list_a > list_list_a] :
( ( member_list_a @ X @ A2 )
=> ( ( insert_list_list_a @ ( F @ X ) @ ( image_8260866953997875467list_a @ F @ A2 ) )
= ( image_8260866953997875467list_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_945_insert__image,axiom,
! [X: list_a > a,A2: set_list_a_a,F: ( list_a > a ) > a] :
( ( member_list_a_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_list_a_a_a @ F @ A2 ) )
= ( image_list_a_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_946_insert__image,axiom,
! [X: nat > list_a,A2: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( member_nat_list_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_nat_list_a_a @ F @ A2 ) )
= ( image_nat_list_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_947_insert__image,axiom,
! [X: nat > a,A2: set_nat_a,F: ( nat > a ) > list_a] :
( ( member_nat_a @ X @ A2 )
=> ( ( insert_list_a @ ( F @ X ) @ ( image_nat_a_list_a @ F @ A2 ) )
= ( image_nat_a_list_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_948_x_Osubfield__m__inv_I1_J,axiom,
! [K: set_list_a,K2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ).
% x.subfield_m_inv(1)
thf(fact_949_x_OUnits__inv__Units,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_inv_Units
thf(fact_950_x_OUnits__inv__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= X ) ) ).
% x.Units_inv_inv
thf(fact_951_x_Oinv__one,axiom,
( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.inv_one
thf(fact_952_x_OUnits__inv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_inv_closed
thf(fact_953_x_OUnits__l__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_l_inv
thf(fact_954_x_OUnits__r__inv,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.Units_r_inv
thf(fact_955_imageE,axiom,
! [B4: a,F: a > a,A2: set_a] :
( ( member_a @ B4 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_956_imageE,axiom,
! [B4: list_a,F: a > list_a,A2: set_a] :
( ( member_list_a @ B4 @ ( image_a_list_a @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_957_imageE,axiom,
! [B4: a,F: list_a > a,A2: set_list_a] :
( ( member_a @ B4 @ ( image_list_a_a @ F @ A2 ) )
=> ~ ! [X3: list_a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_list_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_958_imageE,axiom,
! [B4: list_a,F: list_a > list_a,A2: set_list_a] :
( ( member_list_a @ B4 @ ( image_list_a_list_a @ F @ A2 ) )
=> ~ ! [X3: list_a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_list_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_959_imageE,axiom,
! [B4: nat > a,F: a > nat > a,A2: set_a] :
( ( member_nat_a @ B4 @ ( image_a_nat_a @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_960_imageE,axiom,
! [B4: a,F: set_list_a > a,A2: set_set_list_a] :
( ( member_a @ B4 @ ( image_set_list_a_a @ F @ A2 ) )
=> ~ ! [X3: set_list_a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_set_list_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_961_imageE,axiom,
! [B4: a,F: ( nat > a ) > a,A2: set_nat_a] :
( ( member_a @ B4 @ ( image_nat_a_a @ F @ A2 ) )
=> ~ ! [X3: nat > a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_nat_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_962_imageE,axiom,
! [B4: list_list_a,F: list_a > list_list_a,A2: set_list_a] :
( ( member_list_list_a @ B4 @ ( image_8260866953997875467list_a @ F @ A2 ) )
=> ~ ! [X3: list_a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_list_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_963_imageE,axiom,
! [B4: list_a > a,F: a > list_a > a,A2: set_a] :
( ( member_list_a_a @ B4 @ ( image_a_list_a_a @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_964_imageE,axiom,
! [B4: nat > list_a,F: a > nat > list_a,A2: set_a] :
( ( member_nat_list_a @ B4 @ ( image_a_nat_list_a @ F @ A2 ) )
=> ~ ! [X3: a] :
( ( B4
= ( F @ X3 ) )
=> ~ ( member_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_965_image__image,axiom,
! [F: a > a,G: list_a > a,A2: set_list_a] :
( ( image_a_a @ F @ ( image_list_a_a @ G @ A2 ) )
= ( image_list_a_a
@ ^ [X2: list_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_966_image__image,axiom,
! [F: list_a > a,G: a > list_a,A2: set_a] :
( ( image_list_a_a @ F @ ( image_a_list_a @ G @ A2 ) )
= ( image_a_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_967_image__image,axiom,
! [F: a > list_a,G: a > a,A2: set_a] :
( ( image_a_list_a @ F @ ( image_a_a @ G @ A2 ) )
= ( image_a_list_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_968_image__image,axiom,
! [F: a > a,G: set_list_a > a,A2: set_set_list_a] :
( ( image_a_a @ F @ ( image_set_list_a_a @ G @ A2 ) )
= ( image_set_list_a_a
@ ^ [X2: set_list_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_969_image__image,axiom,
! [F: list_a > a,G: list_a > list_a,A2: set_list_a] :
( ( image_list_a_a @ F @ ( image_list_a_list_a @ G @ A2 ) )
= ( image_list_a_a
@ ^ [X2: list_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_970_image__image,axiom,
! [F: a > list_a,G: list_a > a,A2: set_list_a] :
( ( image_a_list_a @ F @ ( image_list_a_a @ G @ A2 ) )
= ( image_list_a_list_a
@ ^ [X2: list_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_971_image__image,axiom,
! [F: list_a > list_a,G: a > list_a,A2: set_a] :
( ( image_list_a_list_a @ F @ ( image_a_list_a @ G @ A2 ) )
= ( image_a_list_a
@ ^ [X2: a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_972_image__image,axiom,
! [F: a > list_list_a,G: list_a > a,A2: set_list_a] :
( ( image_a_list_list_a @ F @ ( image_list_a_a @ G @ A2 ) )
= ( image_8260866953997875467list_a
@ ^ [X2: list_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_973_image__image,axiom,
! [F: list_list_a > a,G: list_a > list_list_a,A2: set_list_a] :
( ( image_list_list_a_a @ F @ ( image_8260866953997875467list_a @ G @ A2 ) )
= ( image_list_a_a
@ ^ [X2: list_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_974_image__image,axiom,
! [F: list_a > a,G: set_list_a > list_a,A2: set_set_list_a] :
( ( image_list_a_a @ F @ ( image_7934165218391885221list_a @ G @ A2 ) )
= ( image_set_list_a_a
@ ^ [X2: set_list_a] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_975_Compr__image__eq,axiom,
! [F: a > a,A2: set_a,P2: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_976_Compr__image__eq,axiom,
! [F: nat > a,A2: set_nat,P2: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_nat_a @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_nat_a @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_977_Compr__image__eq,axiom,
! [F: a > nat,A2: set_a,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_a_nat @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_a_nat @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_978_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_979_Compr__image__eq,axiom,
! [F: list_a > a,A2: set_list_a,P2: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ ( image_list_a_a @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_list_a_a @ F
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_980_Compr__image__eq,axiom,
! [F: list_a > nat,A2: set_list_a,P2: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_list_a_nat @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_list_a_nat @ F
@ ( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_981_Compr__image__eq,axiom,
! [F: a > list_a,A2: set_a,P2: list_a > $o] :
( ( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ ( image_a_list_a @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_a_list_a @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_982_Compr__image__eq,axiom,
! [F: nat > list_a,A2: set_nat,P2: list_a > $o] :
( ( collect_list_a
@ ^ [X2: list_a] :
( ( member_list_a @ X2 @ ( image_nat_list_a @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_nat_list_a @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_983_Compr__image__eq,axiom,
! [F: a > nat > a,A2: set_a,P2: ( nat > a ) > $o] :
( ( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ ( image_a_nat_a @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_a_nat_a @ F
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_984_Compr__image__eq,axiom,
! [F: nat > nat > a,A2: set_nat,P2: ( nat > a ) > $o] :
( ( collect_nat_a
@ ^ [X2: nat > a] :
( ( member_nat_a @ X2 @ ( image_nat_nat_a @ F @ A2 ) )
& ( P2 @ X2 ) ) )
= ( image_nat_nat_a @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P2 @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_985_imageI,axiom,
! [X: a,A2: set_a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_986_imageI,axiom,
! [X: list_a,A2: set_list_a,F: list_a > a] :
( ( member_list_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_list_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_987_imageI,axiom,
! [X: a,A2: set_a,F: a > list_a] :
( ( member_a @ X @ A2 )
=> ( member_list_a @ ( F @ X ) @ ( image_a_list_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_988_imageI,axiom,
! [X: list_a,A2: set_list_a,F: list_a > list_a] :
( ( member_list_a @ X @ A2 )
=> ( member_list_a @ ( F @ X ) @ ( image_list_a_list_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_989_imageI,axiom,
! [X: set_list_a,A2: set_set_list_a,F: set_list_a > a] :
( ( member_set_list_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_set_list_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_990_imageI,axiom,
! [X: nat > a,A2: set_nat_a,F: ( nat > a ) > a] :
( ( member_nat_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_nat_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_991_imageI,axiom,
! [X: a,A2: set_a,F: a > nat > a] :
( ( member_a @ X @ A2 )
=> ( member_nat_a @ ( F @ X ) @ ( image_a_nat_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_992_imageI,axiom,
! [X: list_a,A2: set_list_a,F: list_a > list_list_a] :
( ( member_list_a @ X @ A2 )
=> ( member_list_list_a @ ( F @ X ) @ ( image_8260866953997875467list_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_993_imageI,axiom,
! [X: list_a > a,A2: set_list_a_a,F: ( list_a > a ) > a] :
( ( member_list_a_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_list_a_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_994_imageI,axiom,
! [X: nat > list_a,A2: set_nat_list_a,F: ( nat > list_a ) > a] :
( ( member_nat_list_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_nat_list_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_995_image__iff,axiom,
! [Z: list_a,F: a > list_a,A2: set_a] :
( ( member_list_a @ Z @ ( image_a_list_a @ F @ A2 ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_996_image__iff,axiom,
! [Z: list_list_a,F: list_a > list_list_a,A2: set_list_a] :
( ( member_list_list_a @ Z @ ( image_8260866953997875467list_a @ F @ A2 ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_997_image__iff,axiom,
! [Z: list_a,F: list_a > list_a,A2: set_list_a] :
( ( member_list_a @ Z @ ( image_list_a_list_a @ F @ A2 ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_998_image__iff,axiom,
! [Z: a,F: list_a > a,A2: set_list_a] :
( ( member_a @ Z @ ( image_list_a_a @ F @ A2 ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_999_image__iff,axiom,
! [Z: a,F: set_list_a > a,A2: set_set_list_a] :
( ( member_a @ Z @ ( image_set_list_a_a @ F @ A2 ) )
= ( ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
& ( Z
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_1000_bex__imageD,axiom,
! [F: list_a > a,A2: set_list_a,P2: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_list_a_a @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1001_bex__imageD,axiom,
! [F: a > list_a,A2: set_a,P2: list_a > $o] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( image_a_list_a @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1002_bex__imageD,axiom,
! [F: set_list_a > a,A2: set_set_list_a,P2: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_set_list_a_a @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1003_bex__imageD,axiom,
! [F: list_a > list_list_a,A2: set_list_a,P2: list_list_a > $o] :
( ? [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( image_8260866953997875467list_a @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1004_bex__imageD,axiom,
! [F: list_a > list_a,A2: set_list_a,P2: list_a > $o] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( image_list_a_list_a @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_1005_image__cong,axiom,
! [M: set_list_a,N5: set_list_a,F: list_a > a,G: list_a > a] :
( ( M = N5 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_list_a_a @ F @ M )
= ( image_list_a_a @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_1006_image__cong,axiom,
! [M: set_set_list_a,N5: set_set_list_a,F: set_list_a > a,G: set_list_a > a] :
( ( M = N5 )
=> ( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_set_list_a_a @ F @ M )
= ( image_set_list_a_a @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_1007_image__cong,axiom,
! [M: set_list_a,N5: set_list_a,F: list_a > list_list_a,G: list_a > list_list_a] :
( ( M = N5 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_8260866953997875467list_a @ F @ M )
= ( image_8260866953997875467list_a @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_1008_image__cong,axiom,
! [M: set_list_a,N5: set_list_a,F: list_a > list_a,G: list_a > list_a] :
( ( M = N5 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_list_a_list_a @ F @ M )
= ( image_list_a_list_a @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_1009_image__cong,axiom,
! [M: set_a,N5: set_a,F: a > list_a,G: a > list_a] :
( ( M = N5 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ N5 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_a_list_a @ F @ M )
= ( image_a_list_a @ G @ N5 ) ) ) ) ).
% image_cong
thf(fact_1010_ball__imageD,axiom,
! [F: list_a > a,A2: set_list_a,P2: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_list_a_a @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1011_ball__imageD,axiom,
! [F: a > list_a,A2: set_a,P2: list_a > $o] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( image_a_list_a @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1012_ball__imageD,axiom,
! [F: set_list_a > a,A2: set_set_list_a,P2: a > $o] :
( ! [X3: a] :
( ( member_a @ X3 @ ( image_set_list_a_a @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: set_list_a] :
( ( member_set_list_a @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1013_ball__imageD,axiom,
! [F: list_a > list_list_a,A2: set_list_a,P2: list_list_a > $o] :
( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( image_8260866953997875467list_a @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1014_ball__imageD,axiom,
! [F: list_a > list_a,A2: set_list_a,P2: list_a > $o] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( image_list_a_list_a @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_1015_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_a @ B4 @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1016_rev__image__eqI,axiom,
! [X: list_a,A2: set_list_a,B4: a,F: list_a > a] :
( ( member_list_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_a @ B4 @ ( image_list_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1017_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: list_a,F: a > list_a] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_list_a @ B4 @ ( image_a_list_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1018_rev__image__eqI,axiom,
! [X: list_a,A2: set_list_a,B4: list_a,F: list_a > list_a] :
( ( member_list_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_list_a @ B4 @ ( image_list_a_list_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1019_rev__image__eqI,axiom,
! [X: set_list_a,A2: set_set_list_a,B4: a,F: set_list_a > a] :
( ( member_set_list_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_a @ B4 @ ( image_set_list_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1020_rev__image__eqI,axiom,
! [X: nat > a,A2: set_nat_a,B4: a,F: ( nat > a ) > a] :
( ( member_nat_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_a @ B4 @ ( image_nat_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1021_rev__image__eqI,axiom,
! [X: a,A2: set_a,B4: nat > a,F: a > nat > a] :
( ( member_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_nat_a @ B4 @ ( image_a_nat_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1022_rev__image__eqI,axiom,
! [X: list_a,A2: set_list_a,B4: list_list_a,F: list_a > list_list_a] :
( ( member_list_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_list_list_a @ B4 @ ( image_8260866953997875467list_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1023_rev__image__eqI,axiom,
! [X: list_a > a,A2: set_list_a_a,B4: a,F: ( list_a > a ) > a] :
( ( member_list_a_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_a @ B4 @ ( image_list_a_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1024_rev__image__eqI,axiom,
! [X: nat > list_a,A2: set_nat_list_a,B4: a,F: ( nat > list_a ) > a] :
( ( member_nat_list_a @ X @ A2 )
=> ( ( B4
= ( F @ X ) )
=> ( member_a @ B4 @ ( image_nat_list_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_1025_all__subset__image,axiom,
! [F: list_a > list_list_a,A2: set_list_a,P2: set_list_list_a > $o] :
( ( ! [B3: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ B3 @ ( image_8260866953997875467list_a @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( P2 @ ( image_8260866953997875467list_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1026_all__subset__image,axiom,
! [F: set_list_a > a,A2: set_set_list_a,P2: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_set_list_a_a @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ B3 @ A2 )
=> ( P2 @ ( image_set_list_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1027_all__subset__image,axiom,
! [F: a > a,A2: set_a,P2: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P2 @ ( image_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1028_all__subset__image,axiom,
! [F: list_a > a,A2: set_list_a,P2: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_list_a_a @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( P2 @ ( image_list_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1029_all__subset__image,axiom,
! [F: a > list_a,A2: set_a,P2: set_list_a > $o] :
( ( ! [B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ ( image_a_list_a @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P2 @ ( image_a_list_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1030_all__subset__image,axiom,
! [F: list_a > list_a,A2: set_list_a,P2: set_list_a > $o] :
( ( ! [B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ ( image_list_a_list_a @ F @ A2 ) )
=> ( P2 @ B3 ) ) )
= ( ! [B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( P2 @ ( image_list_a_list_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_1031_subset__image__iff,axiom,
! [B2: set_list_list_a,F: list_a > list_list_a,A2: set_list_a] :
( ( ord_le8488217952732425610list_a @ B2 @ ( image_8260866953997875467list_a @ F @ A2 ) )
= ( ? [AA: set_list_a] :
( ( ord_le8861187494160871172list_a @ AA @ A2 )
& ( B2
= ( image_8260866953997875467list_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1032_subset__image__iff,axiom,
! [B2: set_a,F: set_list_a > a,A2: set_set_list_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_set_list_a_a @ F @ A2 ) )
= ( ? [AA: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ AA @ A2 )
& ( B2
= ( image_set_list_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1033_subset__image__iff,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1034_subset__image__iff,axiom,
! [B2: set_a,F: list_a > a,A2: set_list_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_list_a_a @ F @ A2 ) )
= ( ? [AA: set_list_a] :
( ( ord_le8861187494160871172list_a @ AA @ A2 )
& ( B2
= ( image_list_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1035_subset__image__iff,axiom,
! [B2: set_list_a,F: a > list_a,A2: set_a] :
( ( ord_le8861187494160871172list_a @ B2 @ ( image_a_list_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B2
= ( image_a_list_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1036_subset__image__iff,axiom,
! [B2: set_list_a,F: list_a > list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ ( image_list_a_list_a @ F @ A2 ) )
= ( ? [AA: set_list_a] :
( ( ord_le8861187494160871172list_a @ AA @ A2 )
& ( B2
= ( image_list_a_list_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_1037_image__subset__iff,axiom,
! [F: list_a > list_list_a,A2: set_list_a,B2: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( image_8260866953997875467list_a @ F @ A2 ) @ B2 )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_list_list_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1038_image__subset__iff,axiom,
! [F: list_a > a,A2: set_list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A2 ) @ B2 )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1039_image__subset__iff,axiom,
! [F: set_list_a > a,A2: set_set_list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_set_list_a_a @ F @ A2 ) @ B2 )
= ( ! [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1040_image__subset__iff,axiom,
! [F: a > list_a,A2: set_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A2 ) @ B2 )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_list_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1041_image__subset__iff,axiom,
! [F: list_a > list_a,A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A2 ) @ B2 )
= ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ ( F @ X2 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_1042_subset__imageE,axiom,
! [B2: set_list_list_a,F: list_a > list_list_a,A2: set_list_a] :
( ( ord_le8488217952732425610list_a @ B2 @ ( image_8260866953997875467list_a @ F @ A2 ) )
=> ~ ! [C5: set_list_a] :
( ( ord_le8861187494160871172list_a @ C5 @ A2 )
=> ( B2
!= ( image_8260866953997875467list_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1043_subset__imageE,axiom,
! [B2: set_a,F: set_list_a > a,A2: set_set_list_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_set_list_a_a @ F @ A2 ) )
=> ~ ! [C5: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ C5 @ A2 )
=> ( B2
!= ( image_set_list_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1044_subset__imageE,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
=> ( B2
!= ( image_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1045_subset__imageE,axiom,
! [B2: set_a,F: list_a > a,A2: set_list_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_list_a_a @ F @ A2 ) )
=> ~ ! [C5: set_list_a] :
( ( ord_le8861187494160871172list_a @ C5 @ A2 )
=> ( B2
!= ( image_list_a_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1046_subset__imageE,axiom,
! [B2: set_list_a,F: a > list_a,A2: set_a] :
( ( ord_le8861187494160871172list_a @ B2 @ ( image_a_list_a @ F @ A2 ) )
=> ~ ! [C5: set_a] :
( ( ord_less_eq_set_a @ C5 @ A2 )
=> ( B2
!= ( image_a_list_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1047_subset__imageE,axiom,
! [B2: set_list_a,F: list_a > list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ ( image_list_a_list_a @ F @ A2 ) )
=> ~ ! [C5: set_list_a] :
( ( ord_le8861187494160871172list_a @ C5 @ A2 )
=> ( B2
!= ( image_list_a_list_a @ F @ C5 ) ) ) ) ).
% subset_imageE
thf(fact_1048_image__subsetI,axiom,
! [A2: set_list_a_a,F: ( list_a > a ) > list_a,B2: set_list_a] :
( ! [X3: list_a > a] :
( ( member_list_a_a @ X3 @ A2 )
=> ( member_list_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le8861187494160871172list_a @ ( image_8715568566693251358list_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1049_image__subsetI,axiom,
! [A2: set_set_list_a_a,F: ( set_list_a > a ) > list_a,B2: set_list_a] :
( ! [X3: set_list_a > a] :
( ( member_set_list_a_a @ X3 @ A2 )
=> ( member_list_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le8861187494160871172list_a @ ( image_2095547256931763454list_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1050_image__subsetI,axiom,
! [A2: set_nat_list_a,F: ( nat > list_a ) > list_a,B2: set_list_a] :
( ! [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A2 )
=> ( member_list_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le8861187494160871172list_a @ ( image_2569751527683375326list_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1051_image__subsetI,axiom,
! [A2: set_nat_a,F: ( nat > a ) > list_a,B2: set_list_a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A2 )
=> ( member_list_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le8861187494160871172list_a @ ( image_nat_a_list_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1052_image__subsetI,axiom,
! [A2: set_a,F: a > list_a,B2: set_list_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_list_a @ ( F @ X3 ) @ B2 ) )
=> ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_1053_x_Oring_Oline__extension__hom,axiom,
! [K: set_list_a,A3: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd971793762689825387on_a_b @ r
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ ( eval_a_b @ r @ A3 @ x )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ E ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A3 @ E ) ) ) ) ) ) ).
% x.ring.line_extension_hom
thf(fact_1054_field__iff__prime,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A3 ) ) )
= ( ring_ring_prime_a_b @ r @ A3 ) ) ) ).
% field_iff_prime
thf(fact_1055_x_Oring_Onon__trivial__field__hom__imp__inj,axiom,
( ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.ring.non_trivial_field_hom_imp_inj
thf(fact_1056_x_Oring_Otrivial__hom__iff,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.trivial_hom_iff
thf(fact_1057_x_OsubfieldI,axiom,
! [K: set_list_a] :
( ( subcri7763218559781929323t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( units_2932844235741507942t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subfieldI
thf(fact_1058_x_Ospace__subgroup__props_I6_J,axiom,
! [K: set_list_a,N3: nat,E: set_list_a,K2: list_a,A3: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A3 ) @ E )
=> ( member_list_a @ A3 @ E ) ) ) ) ) ) ).
% x.space_subgroup_props(6)
thf(fact_1059_line__extension__in__carrier,axiom,
! [K: set_a,A3: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A3 @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_1060_x_Odimension__is__inj,axiom,
! [K: set_list_a,N3: nat,E: set_list_a,M2: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M2 @ K @ E )
=> ( N3 = M2 ) ) ) ) ).
% x.dimension_is_inj
thf(fact_1061_univ__poly__carrier__subfield__of__consts,axiom,
subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% univ_poly_carrier_subfield_of_consts
thf(fact_1062_univ__poly__subfield__of__consts,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( subfie1779122896746047282t_unit @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_subfield_of_consts
thf(fact_1063_x_Ospace__subgroup__props_I2_J,axiom,
! [K: set_list_a,N3: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ E ) ) ) ).
% x.space_subgroup_props(2)
thf(fact_1064_x_Ospace__subgroup__props_I5_J,axiom,
! [K: set_list_a,N3: nat,E: set_list_a,K2: list_a,V2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ V2 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ V2 ) @ E ) ) ) ) ) ).
% x.space_subgroup_props(5)
thf(fact_1065_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A3: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K3: a,V: a] :
( ( member_a @ K3 @ K )
=> ( ( member_a @ V @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ V ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A3 @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A3 @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_1066_x_Osubfield__iff_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( field_6388047844668329575t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subfield_iff(2)
thf(fact_1067_x_Ospace__subgroup__props_I1_J,axiom,
! [K: set_list_a,N3: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E )
=> ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.space_subgroup_props(1)
thf(fact_1068_x_Ozero__dim,axiom,
! [K: set_list_a] : ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ zero_zero_nat @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% x.zero_dim
thf(fact_1069_x_Osubfield__iff_I1_J,axiom,
! [K: set_list_a] :
( ( field_6388047844668329575t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : K
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subfield_iff(1)
thf(fact_1070_x_Odimension__zero,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ zero_zero_nat @ K @ E )
=> ( E
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ).
% x.dimension_zero
thf(fact_1071_x_Oring_Oimg__is__subfield_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K ) ) ) ).
% x.ring.img_is_subfield(1)
thf(fact_1072_x_Oring_Otrivial__ker__imp__inj,axiom,
( ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.trivial_ker_imp_inj
thf(fact_1073_x_Oring_Oinj__iff__trivial__ker,axiom,
( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% x.ring.inj_iff_trivial_ker
thf(fact_1074_FactRing__zeroideal_I2_J,axiom,
is_rin9099215527551818550t_unit @ r @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% FactRing_zeroideal(2)
thf(fact_1075_FactRing__zeroideal_I1_J,axiom,
is_rin6001486760346555702it_a_b @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) @ r ).
% FactRing_zeroideal(1)
thf(fact_1076_x_Om__inv__monoid__consistent,axiom,
! [X: list_a,H: set_list_a] :
( ( member_list_a @ X
@ ( units_2932844235741507942t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( submon977613251886402007t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( m_inv_2802811658206063947t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
@ X )
= ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) ) ) ) ).
% x.m_inv_monoid_consistent
thf(fact_1077_x_Oring_Oadditive__subgroup__a__kernel,axiom,
( additi4714453376129182166t_unit
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.additive_subgroup_a_kernel
thf(fact_1078_x_Oring_Othe__elem__wf_H,axiom,
! [X5: set_list_a] :
( ( member_set_list_a @ X5
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X5 )
= ( insert_a @ ( eval_a_b @ r @ X3 @ x ) @ bot_bot_set_a ) ) ) ) ).
% x.ring.the_elem_wf'
thf(fact_1079_inv__inj__on__Units,axiom,
inj_on_a_a @ ( m_inv_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% inv_inj_on_Units
thf(fact_1080_x_Ocgenideal__self,axiom,
! [I3: list_a] :
( ( member_list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I3 @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 ) ) ) ).
% x.cgenideal_self
thf(fact_1081_x_Oideal__eq__carrier__iff,axiom,
! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 ) )
= ( member_list_a @ A3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.ideal_eq_carrier_iff
thf(fact_1082_x_Oinv__inj__on__Units,axiom,
inj_on_list_a_list_a @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.inv_inj_on_Units
thf(fact_1083_x_Ocgenideal__is__principalideal,axiom,
! [I3: list_a] :
( ( member_list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.cgenideal_is_principalideal
thf(fact_1084_x_Ocgenideal__eq__genideal,axiom,
! [I3: list_a] :
( ( member_list_a @ I3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I3 )
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I3 @ bot_bot_set_list_a ) ) ) ) ).
% x.cgenideal_eq_genideal
thf(fact_1085_x_Oring_Othe__elem__wf,axiom,
! [X5: set_list_a] :
( ( member_set_list_a @ X5
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X5 )
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% x.ring.the_elem_wf
thf(fact_1086_x_OFactRing__zeroideal_I2_J,axiom,
is_rin2993610189962786360t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% x.FactRing_zeroideal(2)
thf(fact_1087_x_OFactRing__zeroideal_I1_J,axiom,
is_rin4843644836746533432t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.FactRing_zeroideal(1)
thf(fact_1088_x_Oring_OFactRing__iso,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia707051561876973205xt_a_b @ r ) )
=> ( is_rin5597148638330396976it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ r ) ) ).
% x.ring.FactRing_iso
thf(fact_1089_x_Oring_Othe__elem__surj,axiom,
( ( image_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.the_elem_surj
thf(fact_1090_x_Oring_Othe__elem__inj,axiom,
! [X5: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X5
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ( ( member_set_list_a @ Y5
@ ( partia141011252114345353t_unit
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) ) )
=> ( ( ( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X5 ) )
= ( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ Y5 ) ) )
=> ( X5 = Y5 ) ) ) ) ).
% x.ring.the_elem_inj
thf(fact_1091_x_Oring_Othe__elem__hom,axiom,
( member_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( ring_h8906680420194085028it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ r ) ) ).
% x.ring.the_elem_hom
thf(fact_1092_x_Oring_Oset__add__ker__hom_I2_J,axiom,
! [I2: set_list_a] :
( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
@ I2 ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ I2 ) ) ) ).
% x.ring.set_add_ker_hom(2)
thf(fact_1093_ring__irreducibleE_I2_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( irredu6211895646901577903xt_a_b @ r @ R2 ) ) ) ).
% ring_irreducibleE(2)
thf(fact_1094_zero__is__irreducible__iff__field,axiom,
( ( irredu6211895646901577903xt_a_b @ r @ ( zero_a_b @ r ) )
= ( field_a_b @ r ) ) ).
% zero_is_irreducible_iff_field
thf(fact_1095_irreducible__prod__rI,axiom,
! [A3: a,B4: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A3 )
=> ( ( member_a @ B4 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A3 @ B4 ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_1096_irreducible__prod__lI,axiom,
! [B4: a,A3: a] :
( ( irredu6211895646901577903xt_a_b @ r @ B4 )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A3 @ B4 ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_1097_pprime__iff__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_1098_x_Oadd__additive__subgroups,axiom,
! [H: set_list_a,K: set_list_a] :
( ( additi4714453376129182166t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( additi4714453376129182166t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( additi4714453376129182166t_unit @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ K ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add_additive_subgroups
thf(fact_1099_x_Osetadd__subset__G,axiom,
! [H: set_list_a,K: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.setadd_subset_G
thf(fact_1100_x_Oset__add__comm,axiom,
! [I2: set_list_a,J: set_list_a] :
( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ J @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ J )
= ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ J @ I2 ) ) ) ) ).
% x.set_add_comm
thf(fact_1101_x_Oset__add__closed,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.set_add_closed
thf(fact_1102_x_Oring_Oset__add__ker__hom_I1_J,axiom,
! [I2: set_list_a] :
( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) ) )
= ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ I2 ) ) ) ).
% x.ring.set_add_ker_hom(1)
thf(fact_1103_x_Oring_OFactRing__iso__set,axiom,
( ( ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( partia707051561876973205xt_a_b @ r ) )
=> ( member_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( ring_i8122894263081988538it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ r ) ) ) ).
% x.ring.FactRing_iso_set
thf(fact_1104_rupture__is__field__iff__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_1105_x_Odimension_Osimps,axiom,
! [A1: nat,A22: set_list_a,A32: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A1 @ A22 @ A32 )
= ( ? [K4: set_list_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K4 )
& ( A32
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
| ? [V3: list_a,E2: set_list_a,N4: nat,K4: set_list_a] :
( ( A1
= ( suc @ N4 ) )
& ( A22 = K4 )
& ( A32
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ V3 @ E2 ) )
& ( member_list_a @ V3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ~ ( member_list_a @ V3 @ E2 )
& ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N4 @ K4 @ E2 ) ) ) ) ).
% x.dimension.simps
thf(fact_1106_x_OSuc__dim,axiom,
! [V2: list_a,E: set_list_a,N3: nat,K: set_list_a] :
( ( member_list_a @ V2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( member_list_a @ V2 @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( suc @ N3 ) @ K @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ V2 @ E ) ) ) ) ) ).
% x.Suc_dim
thf(fact_1107_x_Odimension__backwards,axiom,
! [K: set_list_a,N3: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( suc @ N3 ) @ K @ E )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ? [E3: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E3 )
& ~ ( member_list_a @ X3 @ E3 )
& ( E
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ X3 @ E3 ) ) ) ) ) ) ).
% x.dimension_backwards
thf(fact_1108_x_Odimension_Ocases,axiom,
! [A1: nat,A22: set_list_a,A32: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ~ ! [V: list_a,E4: set_list_a,N: nat] :
( ( A1
= ( suc @ N ) )
=> ( ( A32
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A22 @ V @ E4 ) )
=> ( ( member_list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( member_list_a @ V @ E4 )
=> ~ ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ A22 @ E4 ) ) ) ) ) ) ) ).
% x.dimension.cases
thf(fact_1109_x_Oring_Orank__nullity__theorem,axiom,
! [K: set_list_a,N3: nat,E: set_list_a,M2: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M2 @ K
@ ( a_kern7116238624728830086it_a_b
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : E
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
@ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( minus_minus_nat @ N3 @ M2 )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ E ) ) ) ) ) ) ).
% x.ring.rank_nullity_theorem
thf(fact_1110_x_Oring_OFactRing__iso__set__aux,axiom,
( member_set_list_a_a
@ ^ [X6: set_list_a] :
( the_elem_a
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ X6 ) )
@ ( ring_i8122894263081988538it_a_b
@ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) ) )
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] :
( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ r ) ) ) ).
% x.ring.FactRing_iso_set_aux
thf(fact_1111_x_Oring_Oinj__hom__dimension,axiom,
! [K: set_list_a,E: set_list_a,N3: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E )
=> ( embedd2795209813406577254on_a_b @ r @ N3
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ K )
@ ( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ E ) ) ) ) ) ) ).
% x.ring.inj_hom_dimension
thf(fact_1112_dimension__is__inj,axiom,
! [K: set_a,N3: nat,E: set_a,M2: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ E )
=> ( N3 = M2 ) ) ) ) ).
% dimension_is_inj
thf(fact_1113_space__subgroup__props_I2_J,axiom,
! [K: set_a,N3: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E )
=> ( member_a @ ( zero_a_b @ r ) @ E ) ) ) ).
% space_subgroup_props(2)
thf(fact_1114_space__subgroup__props_I5_J,axiom,
! [K: set_a,N3: nat,E: set_a,K2: a,V2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ V2 @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ V2 ) @ E ) ) ) ) ) ).
% space_subgroup_props(5)
thf(fact_1115_subfield__iff_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( field_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) ) ) ).
% subfield_iff(2)
thf(fact_1116_space__subgroup__props_I1_J,axiom,
! [K: set_a,N3: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E )
=> ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% space_subgroup_props(1)
thf(fact_1117_Suc__dim,axiom,
! [V2: a,E: set_a,N3: nat,K: set_a] :
( ( member_a @ V2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V2 @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E )
=> ( embedd2795209813406577254on_a_b @ r @ ( suc @ N3 ) @ K @ ( embedd971793762689825387on_a_b @ r @ K @ V2 @ E ) ) ) ) ) ).
% Suc_dim
thf(fact_1118_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_1119_dimension__backwards,axiom,
! [K: set_a,N3: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ ( suc @ N3 ) @ K @ E )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ? [E3: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E3 )
& ~ ( member_a @ X3 @ E3 )
& ( E
= ( embedd971793762689825387on_a_b @ r @ K @ X3 @ E3 ) ) ) ) ) ) ).
% dimension_backwards
thf(fact_1120_subfield__iff_I1_J,axiom,
! [K: set_a] :
( ( field_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( subfield_a_b @ K @ r ) ) ) ).
% subfield_iff(1)
thf(fact_1121_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_1122_dimension_Osimps,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
= ( ? [K4: set_a] :
( ( A1 = zero_zero_nat )
& ( A22 = K4 )
& ( A32
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
| ? [V3: a,E2: set_a,N4: nat,K4: set_a] :
( ( A1
= ( suc @ N4 ) )
& ( A22 = K4 )
& ( A32
= ( embedd971793762689825387on_a_b @ r @ K4 @ V3 @ E2 ) )
& ( member_a @ V3 @ ( partia707051561876973205xt_a_b @ r ) )
& ~ ( member_a @ V3 @ E2 )
& ( embedd2795209813406577254on_a_b @ r @ N4 @ K4 @ E2 ) ) ) ) ).
% dimension.simps
thf(fact_1123_dimension_Ocases,axiom,
! [A1: nat,A22: set_a,A32: set_a] :
( ( embedd2795209813406577254on_a_b @ r @ A1 @ A22 @ A32 )
=> ( ( ( A1 = zero_zero_nat )
=> ( A32
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ~ ! [V: a,E4: set_a,N: nat] :
( ( A1
= ( suc @ N ) )
=> ( ( A32
= ( embedd971793762689825387on_a_b @ r @ A22 @ V @ E4 ) )
=> ( ( member_a @ V @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ~ ( member_a @ V @ E4 )
=> ~ ( embedd2795209813406577254on_a_b @ r @ N @ A22 @ E4 ) ) ) ) ) ) ) ).
% dimension.cases
thf(fact_1124_space__subgroup__props_I6_J,axiom,
! [K: set_a,N3: nat,E: set_a,K2: a,A3: a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ A3 ) @ E )
=> ( member_a @ A3 @ E ) ) ) ) ) ) ).
% space_subgroup_props(6)
thf(fact_1125_m__inv__monoid__consistent,axiom,
! [X: a,H: set_a] :
( ( member_a @ X
@ ( units_a_ring_ext_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) )
=> ( ( submon8907322713594755401xt_a_b @ H @ r )
=> ( ( m_inv_a_ring_ext_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r )
@ X )
= ( m_inv_a_ring_ext_a_b @ r @ X ) ) ) ) ).
% m_inv_monoid_consistent
thf(fact_1126_subfieldI,axiom,
! [K: set_a] :
( ( subcring_a_b @ K @ r )
=> ( ( ( units_a_ring_ext_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : K
@ r ) )
= ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( subfield_a_b @ K @ r ) ) ) ).
% subfieldI
thf(fact_1127_diff__is__0__eq_H,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( minus_minus_nat @ M2 @ N3 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1128_diff__is__0__eq,axiom,
! [M2: nat,N3: nat] :
( ( ( minus_minus_nat @ M2 @ N3 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% diff_is_0_eq
thf(fact_1129_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_1130_subdomainI,axiom,
! [H: set_a] :
( ( subcring_a_b @ H @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H1: a,H22: a] :
( ( 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_a_b @ H @ r ) ) ) ) ).
% subdomainI
thf(fact_1131_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1132_diff__0__eq__0,axiom,
! [N3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N3 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1133_diff__Suc__Suc,axiom,
! [M2: nat,N3: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N3 ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% diff_Suc_Suc
thf(fact_1134_Suc__diff__diff,axiom,
! [M2: nat,N3: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N3 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N3 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1135_diff__diff__cancel,axiom,
! [I3: nat,N3: nat] :
( ( ord_less_eq_nat @ I3 @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_1136_diff__commute,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K2 ) @ J2 ) ) ).
% diff_commute
thf(fact_1137_diffs0__imp__equal,axiom,
! [M2: nat,N3: nat] :
( ( ( minus_minus_nat @ M2 @ N3 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N3 @ M2 )
= zero_zero_nat )
=> ( M2 = N3 ) ) ) ).
% diffs0_imp_equal
thf(fact_1138_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1139_zero__induct__lemma,axiom,
! [P2: nat > $o,K2: nat,I3: nat] :
( ( P2 @ K2 )
=> ( ! [N: nat] :
( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) )
=> ( P2 @ ( minus_minus_nat @ K2 @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1140_diff__le__mono2,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1141_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B4: nat] :
( ( ord_less_eq_nat @ A3 @ C2 )
=> ( ( ord_less_eq_nat @ B4 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A3 ) @ ( minus_minus_nat @ C2 @ B4 ) )
= ( ord_less_eq_nat @ B4 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_1142_diff__le__self,axiom,
! [M2: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N3 ) @ M2 ) ).
% diff_le_self
thf(fact_1143_diff__le__mono,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_1144_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N3 @ K2 ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1145_le__diff__iff,axiom,
! [K2: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N3 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_1146_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N3 @ K2 ) )
= ( M2 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_1147_inj__on__diff__nat,axiom,
! [N5: set_nat,K2: nat] :
( ! [N: nat] :
( ( member_nat @ N @ N5 )
=> ( ord_less_eq_nat @ K2 @ N ) )
=> ( inj_on_nat_nat
@ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K2 )
@ N5 ) ) ).
% inj_on_diff_nat
thf(fact_1148_Suc__diff__le,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_eq_nat @ N3 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N3 )
= ( suc @ ( minus_minus_nat @ M2 @ N3 ) ) ) ) ).
% Suc_diff_le
thf(fact_1149_x_Oring_Oabelian__subgroup__a__kernel,axiom,
( abelia6695205329122750356t_unit
@ ( a_kern7116238624728830086it_a_b @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ r
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x ) )
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% x.ring.abelian_subgroup_a_kernel
thf(fact_1150_bounded__degree__dimension,axiom,
! [K: set_a,N3: nat] :
( ( subfield_a_b @ K @ r )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ K ) @ N3 @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K )
@ ( collect_list_a
@ ^ [Q3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( ord_less_eq_nat @ ( size_size_list_a @ Q3 ) @ N3 ) ) ) ) ) ).
% bounded_degree_dimension
thf(fact_1151_domain__iff__prime,axiom,
! [A3: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A3 ) ) )
= ( ring_ring_prime_a_b @ r @ A3 ) ) ) ).
% domain_iff_prime
thf(fact_1152_x_Oring_Oimg__is__ring,axiom,
( ring_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] :
( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ r ) ) ).
% x.ring.img_is_ring
thf(fact_1153_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_1154_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1155_x_Oring__axioms,axiom,
ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.ring_axioms
thf(fact_1156_x_Oeval__in__carrier__2,axiom,
! [X: list_list_a,Y: list_a] :
( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier_2
thf(fact_1157_x_Oeval__poly__of__const,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y )
= X ) ) ).
% x.eval_poly_of_const
thf(fact_1158_x_Opoly__of__const__in__carrier,axiom,
! [S: list_a] :
( ( member_list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.poly_of_const_in_carrier
thf(fact_1159_x_Ouniv__poly__subfield__of__consts,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subfie4546268998243038636t_unit @ ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).
% x.univ_poly_subfield_of_consts
thf(fact_1160_subdomain__is__domain,axiom,
! [H: set_a] :
( ( subdomain_a_b @ H @ r )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ).
% subdomain_is_domain
thf(fact_1161_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_1162_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_1163_subdomain__iff,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( subdomain_a_b @ H @ r )
= ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] : H
@ r ) ) ) ) ).
% subdomain_iff
thf(fact_1164_x_Osubdomain__is__domain,axiom,
! [H: set_list_a] :
( ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subdomain_is_domain
thf(fact_1165_x_Olagrange__basis__polynomial__def,axiom,
! [S2: set_list_a,X: list_a] :
( ( lagran6985349428869127715t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 @ X )
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( lagran3534788790333317459t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( finpro738134188688310831list_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ S2 ) ) ) ) ) ).
% x.lagrange_basis_polynomial_def
thf(fact_1166_x_Odomain__eq__zeroprimeideal,axiom,
( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.domain_eq_zeroprimeideal
thf(fact_1167_x_Ozeroprimeideal__domainI,axiom,
( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.zeroprimeideal_domainI
thf(fact_1168_x_Oring_Oinj__on__domain,axiom,
( ( inj_on_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( domain_a_b @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.ring.inj_on_domain
thf(fact_1169_x_Osubdomain__iff,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( domain6553523120543210313t_unit
@ ( partia9041243232023819264t_unit
@ ^ [Uu: set_list_a] : H
@ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.subdomain_iff
thf(fact_1170_x_Oring_Oimg__is__domain,axiom,
( ( domain_a_b @ r )
=> ( domain_a_b
@ ( partia8674076737563717228xt_a_b
@ ^ [Uu: set_a] :
( image_list_a_a
@ ^ [P3: list_a] : ( eval_a_b @ r @ P3 @ x )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ r ) ) ) ).
% x.ring.img_is_domain
thf(fact_1171_x_Olagrange__basis__polynomial__aux__def,axiom,
! [S2: set_list_a] :
( ( lagran3534788790333317459t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 )
= ( finpro3417560807142560175list_a @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
@ ^ [S3: list_a] : ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S3 ) )
@ S2 ) ) ).
% x.lagrange_basis_polynomial_aux_def
thf(fact_1172_poly__of__const__over__subfield,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K )
= ( collect_list_a
@ ^ [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% poly_of_const_over_subfield
thf(fact_1173_telescopic__base__aux,axiom,
! [K: set_a,F2: set_a,N3: nat,E: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( subfield_a_b @ F2 @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ F2 @ E )
=> ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E ) ) ) ) ) ).
% telescopic_base_aux
thf(fact_1174_x_Otelescopic__base__aux,axiom,
! [K: set_list_a,F2: set_list_a,N3: nat,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ F2 )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ one_one_nat @ F2 @ E )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E ) ) ) ) ) ).
% x.telescopic_base_aux
thf(fact_1175_x_Oeval__var,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% x.eval_var
thf(fact_1176_degree__one__imp__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_1177_degree__zero__imp__not__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_1178_poly__sub__degree__le,axiom,
! [X: list_a,N3: nat,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat ) @ N3 )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) @ one_one_nat ) @ N3 ) ) ) ) ) ).
% poly_sub_degree_le
thf(fact_1179_pirreducible__degree,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_1180_x_Opirreducible__degree,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% x.pirreducible_degree
thf(fact_1181_poly__of__const__over__carrier,axiom,
( ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) )
= ( collect_list_a
@ ^ [P3: list_a] :
( ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ).
% poly_of_const_over_carrier
thf(fact_1182_diff__Suc__1,axiom,
! [N3: nat] :
( ( minus_minus_nat @ ( suc @ N3 ) @ one_one_nat )
= N3 ) ).
% diff_Suc_1
thf(fact_1183_x_Opoly__of__const__over__subfield,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( image_8260866953997875467list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K )
= ( collect_list_list_a
@ ^ [P3: list_list_a] :
( ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
& ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P3 ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ) ).
% x.poly_of_const_over_subfield
thf(fact_1184_dimension__one,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( embedd2795209813406577254on_a_b @ r @ one_one_nat @ K @ K ) ) ).
% dimension_one
thf(fact_1185_x_Odimension__one,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ one_one_nat @ K @ K ) ) ).
% x.dimension_one
thf(fact_1186_univ__poly__is__euclidean,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ring_e7478897652244013592t_unit @ ( univ_poly_a_b @ r @ K )
@ ^ [P3: list_a] : ( minus_minus_nat @ ( size_size_list_a @ P3 ) @ one_one_nat ) ) ) ).
% univ_poly_is_euclidean
thf(fact_1187_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N3: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N3 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1188_pirreducible__imp__not__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ).
% pirreducible_imp_not_splitted
thf(fact_1189_degree__zero__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_zero_imp_splitted
thf(fact_1190_degree__one__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_one_imp_splitted
thf(fact_1191_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_1192_lagrange__basis__polynomial__aux__def,axiom,
! [S2: set_a] :
( ( lagran9092808442999052491ux_a_b @ r @ S2 )
= ( finpro4329226410377213737unit_a @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) )
@ ^ [S3: a] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( poly_of_const_a_b @ r @ S3 ) )
@ S2 ) ) ).
% lagrange_basis_polynomial_aux_def
thf(fact_1193__092_060open_062degree_A_Ilagrange__basis__polynomial_AS_Ax_J_A_092_060le_062_Adegree_Ap_A_L_Adegree_A_Ipoly__of__const_A_Iinv_Afinprod_AR_A_Ia__minus_AR_Ax_J_AS_J_J_092_060close_062,axiom,
ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ p ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ ( poly_of_const_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ) ) @ one_one_nat ) ) ).
% \<open>degree (lagrange_basis_polynomial S x) \<le> degree p + degree (poly_of_const (inv finprod R (a_minus R x) S))\<close>
thf(fact_1194_diff__diff__left,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K2 )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1195_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_1196_poly__mult__degree__le,axiom,
! [X: list_a,Y: list_a,N3: 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 ) @ N3 )
=> ( ( 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 @ N3 @ M2 ) ) ) ) ) ) ).
% poly_mult_degree_le
thf(fact_1197_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1198_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1199_Nat_Odiff__diff__right,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1200_diff__Suc__diff__eq1,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J2 @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K2 ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1201_diff__Suc__diff__eq2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K2 ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K2 @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1202_diff__add__inverse2,axiom,
! [M2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N3 ) @ N3 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1203_diff__add__inverse,axiom,
! [N3: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M2 ) @ N3 )
= M2 ) ).
% diff_add_inverse
thf(fact_1204_diff__cancel2,axiom,
! [M2: nat,K2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N3 @ K2 ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% diff_cancel2
thf(fact_1205_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N3 ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ).
% Nat.diff_cancel
thf(fact_1206_diff__add__0,axiom,
! [N3: nat,M2: nat] :
( ( minus_minus_nat @ N3 @ ( plus_plus_nat @ N3 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1207_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I3 )
= K2 )
= ( J2
= ( plus_plus_nat @ K2 @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1208_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1209_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K2 )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1210_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1211_le__diff__conv,axiom,
! [J2: nat,K2: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I3 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1212_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N4: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1213__092_060open_062degree_Ap_A_L_Adegree_A_Ipoly__of__const_A_Iinv_Afinprod_AR_A_Ia__minus_AR_Ax_J_AS_J_J_A_092_060le_062_Acard_AS_A_L_A0_092_060close_062,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ p ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ ( poly_of_const_a_b @ r @ ( m_inv_a_ring_ext_a_b @ r @ ( finpro205304725090349623_a_b_a @ r @ ( a_minus_a_b @ r @ x ) @ s ) ) ) ) @ one_one_nat ) ) @ ( plus_plus_nat @ ( finite_card_a @ s ) @ zero_zero_nat ) ).
% \<open>degree p + degree (poly_of_const (inv finprod R (a_minus R x) S)) \<le> card S + 0\<close>
thf(fact_1214_lagrange__aux__degree,axiom,
! [S2: set_a] :
( ( finite_finite_a @ S2 )
=> ( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S2 ) ) @ one_one_nat ) @ ( finite_card_a @ S2 ) ) ) ) ).
% lagrange_aux_degree
thf(fact_1215__092_060open_062degree_A_Ilagrange__basis__polynomial_AS_Ax_J_A_092_060le_062_Acard_AS_092_060close_062,axiom,
ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) ) @ one_one_nat ) @ ( finite_card_a @ s ) ).
% \<open>degree (lagrange_basis_polynomial S x) \<le> card S\<close>
thf(fact_1216_a__card__cosets__equal,axiom,
! [C2: set_a,H: set_a] :
( ( member_set_a @ C2 @ ( a_RCOSETS_a_b @ r @ H ) )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( finite_card_a @ C2 )
= ( finite_card_a @ H ) ) ) ) ) ).
% a_card_cosets_equal
thf(fact_1217_univ__poly__units_H,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
= ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( P != nil_a )
& ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ).
% univ_poly_units'
thf(fact_1218_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_1219_is__root__def,axiom,
! [P: list_a,X: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_1220_long__division__zero_I1_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_1221_pprimeE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_1222_card__Collect__le__nat,axiom,
! [N3: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_eq_nat @ I @ N3 ) ) )
= ( suc @ N3 ) ) ).
% card_Collect_le_nat
thf(fact_1223_exists__unique__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ X3 )
& ! [Y6: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y6 )
=> ( Y6 = X3 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_1224_x_Oa__card__cosets__equal,axiom,
! [C2: set_list_a,H: set_list_a] :
( ( member_set_list_a @ C2 @ ( a_RCOS6220190738183020281t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) )
=> ( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( finite_card_list_a @ C2 )
= ( finite_card_list_a @ H ) ) ) ) ) ).
% x.a_card_cosets_equal
thf(fact_1225_x_Oeval_Osimps_I1_J,axiom,
( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.eval.simps(1)
thf(fact_1226_x_Ois__root__def,axiom,
! [P: list_list_a,X: list_a] :
( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X )
= ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( P != nil_list_a ) ) ) ).
% x.is_root_def
thf(fact_1227_pmod__image__characterization,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ( image_list_a_list_a
@ ^ [Q3: list_a] : ( polynomial_pmod_a_b @ r @ Q3 @ P )
@ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
= ( collect_list_a
@ ^ [Q3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( ord_less_eq_nat @ ( size_size_list_a @ Q3 ) @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% pmod_image_characterization
thf(fact_1228_splitted__imp__trivial__factors,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q @ P )
=> ( ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ).
% splitted_imp_trivial_factors
thf(fact_1229_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_1230_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_1231_long__division__closed_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_1232_long__division__zero_I2_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_1233_pdivides__imp__root__sharing,axiom,
! [P: list_a,Q: list_a,A3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( eval_a_b @ r @ P @ A3 )
= ( zero_a_b @ r ) )
=> ( ( eval_a_b @ r @ Q @ A3 )
= ( zero_a_b @ r ) ) ) ) ) ) ).
% pdivides_imp_root_sharing
thf(fact_1234_pmod__zero__iff__pdivides,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q @ P ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_1235_pdivides__imp__splitted,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_1236_same__pmod__iff__pdivides,axiom,
! [K: set_a,A3: list_a,B4: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A3 @ Q )
= ( polynomial_pmod_a_b @ r @ B4 @ Q ) )
= ( polyno5814909790663948098es_a_b @ r @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ A3 @ B4 ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_1237_pprimeE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
| ( polyno5814909790663948098es_a_b @ r @ P @ R2 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_1238_pprimeI,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q4: list_a,R: list_a] :
( ( member_list_a @ Q4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q4 @ R ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q4 )
| ( polyno5814909790663948098es_a_b @ r @ P @ R ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).
% pprimeI
thf(fact_1239_trivial__factors__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [Q4: list_a] :
( ( member_list_a @ Q4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q4 )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q4 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Q4 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% trivial_factors_imp_splitted
thf(fact_1240_x_Odegree__oneE,axiom,
! [P: list_list_a,K: set_list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A: list_a] :
( ( member_list_a @ A @ K )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ! [B: list_a] :
( ( member_list_a @ B @ K )
=> ( P
!= ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) ) ) ) ) ) ) ).
% x.degree_oneE
thf(fact_1241_x_Odimension__direct__sum__space,axiom,
! [K: set_list_a,N3: nat,E: set_list_a,M2: nat,F2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M2 @ K @ F2 )
=> ( ( ( inf_inf_set_list_a @ E @ F2 )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( plus_plus_nat @ N3 @ M2 ) @ K @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ E @ F2 ) ) ) ) ) ) ).
% x.dimension_direct_sum_space
thf(fact_1242_x_Onormalize_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ~ ! [V: list_a,Va: list_list_a] :
( X
!= ( cons_list_a @ V @ Va ) ) ) ).
% x.normalize.cases
thf(fact_1243_x_Osubcring__inter,axiom,
! [I2: set_list_a,J: set_list_a] :
( ( subcri7763218559781929323t_unit @ I2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subcri7763218559781929323t_unit @ J @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subcri7763218559781929323t_unit @ ( inf_inf_set_list_a @ I2 @ J ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subcring_inter
thf(fact_1244_x_Odimension__sum__space,axiom,
! [K: set_list_a,N3: nat,E: set_list_a,M2: nat,F2: set_list_a,K2: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M2 @ K @ F2 )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ K @ ( inf_inf_set_list_a @ E @ F2 ) )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M2 ) @ K2 ) @ K @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ E @ F2 ) ) ) ) ) ) ).
% x.dimension_sum_space
thf(fact_1245_x_Ouniv__poly__units,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) )
= ( collect_list_list_a
@ ^ [Uu: list_list_a] :
? [K5: list_a] :
( ( Uu
= ( cons_list_a @ K5 @ nil_list_a ) )
& ( member_list_a @ K5 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ) ) ).
% x.univ_poly_units
thf(fact_1246_pmod__const_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ r @ P @ Q )
= nil_a ) ) ) ) ) ).
% pmod_const(1)
thf(fact_1247_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_1248_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_1249_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_1250_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_1251_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_1252_dimension__sum__space,axiom,
! [K: set_a,N3: nat,E: set_a,M2: nat,F2: set_a,K2: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ F2 )
=> ( ( embedd2795209813406577254on_a_b @ r @ K2 @ K @ ( inf_inf_set_a @ E @ F2 ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( minus_minus_nat @ ( plus_plus_nat @ N3 @ M2 ) @ K2 ) @ K @ ( set_add_a_b @ r @ E @ F2 ) ) ) ) ) ) ).
% dimension_sum_space
thf(fact_1253_boundD__carrier,axiom,
! [N3: nat,F: nat > a,M2: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N3 @ F )
=> ( ( ord_less_nat @ N3 @ M2 )
=> ( member_a @ ( F @ M2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_1254_dimension__direct__sum__space,axiom,
! [K: set_a,N3: nat,E: set_a,M2: nat,F2: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ( embedd2795209813406577254on_a_b @ r @ N3 @ K @ E )
=> ( ( embedd2795209813406577254on_a_b @ r @ M2 @ K @ F2 )
=> ( ( ( inf_inf_set_a @ E @ F2 )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( embedd2795209813406577254on_a_b @ r @ ( plus_plus_nat @ N3 @ M2 ) @ K @ ( set_add_a_b @ r @ E @ F2 ) ) ) ) ) ) ).
% dimension_direct_sum_space
thf(fact_1255_x_OboundD__carrier,axiom,
! [N3: nat,F: nat > list_a,M2: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N3 @ F )
=> ( ( ord_less_nat @ N3 @ M2 )
=> ( member_list_a @ ( F @ M2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.boundD_carrier
thf(fact_1256_degree__oneE,axiom,
! [P: list_a,K: set_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A: a] :
( ( member_a @ A @ K )
=> ( ( A
!= ( zero_a_b @ r ) )
=> ! [B: a] :
( ( member_a @ B @ K )
=> ( P
!= ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_1257_pmod__const_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ r @ P @ Q )
= P ) ) ) ) ) ).
% pmod_const(2)
thf(fact_1258_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1259_card__Collect__less__nat,axiom,
! [N3: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N3 ) ) )
= N3 ) ).
% card_Collect_less_nat
thf(fact_1260_pmod__degree,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pmod_degree
thf(fact_1261_rupture__one__not__zero,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) )
=> ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) )
!= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) ) ) ) ) ) ).
% rupture_one_not_zero
thf(fact_1262_zero__less__diff,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N3 @ M2 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ).
% zero_less_diff
thf(fact_1263_Suc__pred,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) )
= N3 ) ) ).
% Suc_pred
thf(fact_1264_Suc__diff__1,axiom,
! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( suc @ ( minus_minus_nat @ N3 @ one_one_nat ) )
= N3 ) ) ).
% Suc_diff_1
thf(fact_1265_diff__less,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N3 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1266_Suc__diff__Suc,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_nat @ N3 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N3 ) ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ) ).
% Suc_diff_Suc
thf(fact_1267_diff__less__Suc,axiom,
! [M2: nat,N3: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N3 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1268_less__diff__iff,axiom,
! [K2: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N3 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N3 @ K2 ) )
= ( ord_less_nat @ M2 @ N3 ) ) ) ) ).
% less_diff_iff
thf(fact_1269_diff__less__mono,axiom,
! [A3: nat,B4: nat,C2: nat] :
( ( ord_less_nat @ A3 @ B4 )
=> ( ( ord_less_eq_nat @ C2 @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C2 ) @ ( minus_minus_nat @ B4 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1270_less__diff__conv,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J2 @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K2 ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1271_add__diff__inverse__nat,axiom,
! [M2: nat,N3: nat] :
( ~ ( ord_less_nat @ M2 @ N3 )
=> ( ( plus_plus_nat @ N3 @ ( minus_minus_nat @ M2 @ N3 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1272_less__imp__diff__less,axiom,
! [J2: nat,K2: nat,N3: nat] :
( ( ord_less_nat @ J2 @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N3 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1273_diff__less__mono2,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1274_diff__Suc__less,axiom,
! [N3: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ord_less_nat @ ( minus_minus_nat @ N3 @ ( suc @ I3 ) ) @ N3 ) ) ).
% diff_Suc_less
thf(fact_1275_nat__diff__split,axiom,
! [P2: nat > $o,A3: nat,B4: nat] :
( ( P2 @ ( minus_minus_nat @ A3 @ B4 ) )
= ( ( ( ord_less_nat @ A3 @ B4 )
=> ( P2 @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A3
= ( plus_plus_nat @ B4 @ D3 ) )
=> ( P2 @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1276_nat__diff__split__asm,axiom,
! [P2: nat > $o,A3: nat,B4: nat] :
( ( P2 @ ( minus_minus_nat @ A3 @ B4 ) )
= ( ~ ( ( ( ord_less_nat @ A3 @ B4 )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A3
= ( plus_plus_nat @ B4 @ D3 ) )
& ~ ( P2 @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1277_less__diff__conv2,axiom,
! [K2: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I3 )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I3 @ K2 ) ) ) ) ).
% less_diff_conv2
% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
member_list_a @ ( lagran2649660974587678107al_a_b @ r @ s @ x ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
%------------------------------------------------------------------------------