TPTP Problem File: SLH0902^1.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_00313_012510__17419784_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1582 ( 379 unt; 304 typ; 0 def)
% Number of atoms : 3939 (1298 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 16642 ( 377 ~; 47 |; 191 &;13928 @)
% ( 0 <=>;2099 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 42 ( 41 usr)
% Number of type conns : 939 ( 939 >; 0 *; 0 +; 0 <<)
% Number of symbols : 264 ( 263 usr; 13 con; 0-4 aty)
% Number of variables : 3256 ( 55 ^;3099 !; 102 ?;3256 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:38:41.094
%------------------------------------------------------------------------------
% Could-be-implicit typings (41)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia5333488208502193986t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia4960592913263135132t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia4556295656693239580t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2956882679547061052t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia7496981018696276118t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__Set__Oset_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia3925755165846298134t_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__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_li5608457238520824219list_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_It__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J_J,type,
set_set_nat_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_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_set_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
set_list_set_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_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_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
set_nat_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_J,type,
set_set_nat_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
list_set_list_a: $tType ).
thf(ty_n_t__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_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
set_list_set_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
set_a_list_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_Itf__a_J_Mtf__a_J_J,type,
set_list_a_a: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__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__List__Olist_It__Set__Oset_Itf__a_J_J,type,
list_set_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_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (263)
thf(sy_c_AbelCoset_Oa__l__coset_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
a_l_co3970804650394549132t_unit: partia2956882679547061052t_unit > list_list_a > set_list_list_a > set_list_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_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J,type,
partia5038748322285217333t_unit: partia5333488208502193986t_unit > set_list_list_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
partia2464479390973590831t_unit: partia2956882679547061052t_unit > set_list_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_Mt__Product____Type__Ounit_J_J,type,
partia7265347635606999311t_unit: partia4556295656693239580t_unit > set_list_set_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__Set__Oset_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
partia3893404292425143049t_unit: partia3925755165846298134t_unit > set_list_set_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_Coset_Oorder_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
order_3240872107759947550t_unit: partia2670972154091845814t_unit > nat ).
thf(sy_c_Coset_Oorder_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
order_a_ring_ext_a_b: partia2175431115845679010xt_a_b > nat ).
thf(sy_c_Divisibility_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_Ofinite__dimension_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
embedd1345800358437254783t_unit: partia2670972154091845814t_unit > set_list_a > set_list_a > $o ).
thf(sy_c_Embedded__Algebras_Oring_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_Osubalgebra_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
embedd1768981623711841426t_unit: set_list_a > set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Embedded__Algebras_Osubalgebra_001tf__a_001tf__b,type,
embedd9027525575939734154ra_a_b: set_a > set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_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_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_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Group_OUnits_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
units_4903515905731149798t_unit: partia2956882679547061052t_unit > set_list_list_a ).
thf(sy_c_Group_OUnits_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
units_2932844235741507942t_unit: partia2670972154091845814t_unit > set_list_a ).
thf(sy_c_Group_OUnits_001t__Set__Oset_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_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_Opow_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_001t__Nat__Onat,type,
pow_li1142815632869257134it_nat: partia2670972154091845814t_unit > list_a > nat > list_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
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__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_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__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_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
uminus8475698304107272002list_a: set_nat_list_a > set_nat_list_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
uminus5121639726568051900_nat_a: set_nat_a > set_nat_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
uminus4049073354455507169list_a: set_list_list_a > set_list_list_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
uminus7925729386456332763list_a: set_list_a > set_list_a ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Ideal_Ocgenideal_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J,type,
cgenid24865672677839267t_unit: partia2956882679547061052t_unit > list_list_a > set_list_list_a ).
thf(sy_c_Ideal_Ocgenideal_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
cgenid9131348535277946915t_unit: partia2670972154091845814t_unit > list_a > set_list_a ).
thf(sy_c_Ideal_Ocgenideal_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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
genide2671672708880404049t_unit: partia2956882679547061052t_unit > set_list_list_a > set_list_list_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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
maxima7552488817642790894t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Ideal_Omaximalideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
maxima6585700282301356660t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Ideal_Omaximalideal_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
maxima3875439991530298004t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Ideal_Omaximalideal_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
maxima2253313296322093082t_unit: set_set_a > partia6043505979758434576t_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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
primei2288432046033540002t_unit: set_list_list_a > partia2956882679547061052t_unit > $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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
princi2534607884127416211t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Ideal_Oprincipalideal_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
princi8786919440553033881t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Lagrange__Interpolation_Oring_Ointerpolate_001tf__a_001tf__b,type,
lagran1063865941317790773te_a_b: partia2175431115845679010xt_a_b > set_a > ( a > a ) > list_a ).
thf(sy_c_Lagrange__Interpolation_Oring_Olagrange__basis__polynomial_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
lagran8662613185911980061t_unit: partia2956882679547061052t_unit > set_list_list_a > list_list_a > list_list_list_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_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
lagran8541024212194239043t_unit: partia7496981018696276118t_unit > set_set_list_a > set_list_a > list_set_list_a ).
thf(sy_c_Lagrange__Interpolation_Oring_Olagrange__basis__polynomial_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
lagran5736318333021047625t_unit: partia6043505979758434576t_unit > set_set_a > set_a > list_set_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_001tf__a_001tf__b,type,
lagran9092808442999052491ux_a_b: partia2175431115845679010xt_a_b > set_a > list_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_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_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_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__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_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__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_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_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_Oalg__mult_001tf__a_001tf__b,type,
polyno4422430861927485590lt_a_b: partia2175431115845679010xt_a_b > list_a > a > nat ).
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_Opoly__of__const_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
poly_o8716471131768098070t_unit: partia2670972154091845814t_unit > list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001tf__a_001tf__b,type,
poly_of_const_a_b: partia2175431115845679010xt_a_b > a > list_a ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
univ_p2250591967980070728t_unit: partia2956882679547061052t_unit > set_list_list_a > partia5333488208502193986t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
univ_p7953238456130426574t_unit: partia2670972154091845814t_unit > set_list_a > partia2956882679547061052t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
univ_p863672496597069550t_unit: partia7496981018696276118t_unit > set_set_list_a > partia4556295656693239580t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
univ_p6720748963476187508t_unit: partia6043505979758434576t_unit > set_set_a > partia3925755165846298134t_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_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
thf(sy_c_QuotRing_OFactRing_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
factRi7259693425559269476t_unit: partia2956882679547061052t_unit > set_list_list_a > partia4960592913263135132t_unit ).
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_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__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_i6186174840089424918t_unit: partia2956882679547061052t_unit > partia2956882679547061052t_unit > set_li5608457238520824219list_a ).
thf(sy_c_QuotRing_Oring__iso_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_i4611353245267337884t_unit: partia2956882679547061052t_unit > partia2670972154091845814t_unit > set_li3422455791611400469list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_i5684343068699926420it_a_b: partia2956882679547061052t_unit > partia2175431115845679010xt_a_b > set_list_list_a_a ).
thf(sy_c_QuotRing_Oring__iso_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_i7582117978422105628t_unit: partia2670972154091845814t_unit > partia2956882679547061052t_unit > set_li6773872926390105121list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i7414513579304222626t_unit: partia2670972154091845814t_unit > partia2670972154091845814t_unit > set_list_a_list_a ).
thf(sy_c_QuotRing_Oring__iso_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
ring_i7048835797181109658it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > set_list_a_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_i4464730343205239444t_unit: partia2175431115845679010xt_a_b > partia2956882679547061052t_unit > set_a_list_list_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_i4557880751517319194t_unit: partia2175431115845679010xt_a_b > partia2670972154091845814t_unit > set_a_list_a ).
thf(sy_c_QuotRing_Oring__iso_001tf__a_001tf__b_001tf__a_001tf__b,type,
ring_iso_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_inv_8944721093294617173t_unit: partia2670972154091845814t_unit > list_a > list_a ).
thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_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_Oadd__pow_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001t__Int__Oint,type,
add_po2638046716968164713it_int: partia2670972154091845814t_unit > int > list_a > list_a ).
thf(sy_c_Ring_Odomain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
domain7810152921033798211t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
domain6553523120543210313t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Set__Oset_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_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
field_1540243473349940225t_unit: partia4960592913263135132t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
field_26233345952514695t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Ofield_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_Oadd_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
add_li174743652000525320t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_s2910681146719230829t_unit: partia7496981018696276118t_unit > set_list_a ).
thf(sy_c_Ring_Oring_Ozero_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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_h8129544334414776832t_unit: partia2956882679547061052t_unit > partia2956882679547061052t_unit > set_li5608457238520824219list_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_001tf__a_001tf__b,type,
ring_h2895973938487309444it_a_b: partia2670972154091845814t_unit > partia2175431115845679010xt_a_b > 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_001tf__a_001tf__b,type,
ring_hom_a_b_a_b: partia2175431115845679010xt_a_b > partia2175431115845679010xt_a_b > set_a_a ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_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_001tf__a_001tf__b,type,
semiring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001tf__a_001tf__b,type,
ring_e8745995371659049232in_a_b: partia2175431115845679010xt_a_b > ( a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_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_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r7790391342995787508t_unit: partia6043505979758434576t_unit > set_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5437400583859147359t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
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_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_001tf__a,type,
collect_a: ( a > $o ) > set_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_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_Itf__a_J,type,
image_4177985004758493951list_a: ( list_list_a > list_a ) > set_list_list_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_001_062_It__Nat__Onat_Mtf__a_J,type,
image_list_a_nat_a: ( list_a > nat > a ) > set_list_a > set_nat_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
image_8260866953997875467list_a: ( list_a > list_list_a ) > set_list_a > set_list_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
image_list_a_list_a: ( list_a > list_a ) > set_list_a > set_list_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001tf__a,type,
image_list_a_a: ( list_a > a ) > set_list_a > set_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
image_3157922941172939973list_a: ( set_nat_list_a > set_nat_list_a ) > set_set_nat_list_a > set_set_nat_list_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
image_6965494298868581957_nat_a: ( set_nat_a > set_nat_a ) > set_set_nat_a > set_set_nat_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
image_452714708507473285list_a: ( set_list_list_a > set_list_list_a ) > set_set_list_list_a > set_set_list_list_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
image_5749939591322298757list_a: ( set_list_a > set_list_a ) > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_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_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_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_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__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_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_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $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_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
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_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_It__List__Olist_Itf__a_J_J_J,type,
member8231385768148312316list_a: ( list_list_a > list_list_a ) > set_li5608457238520824219list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member7168557129179038582list_a: ( list_list_a > list_a ) > set_li3422455791611400469list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_list_list_a_a: ( list_list_a > a ) > set_list_list_a_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member6714375691612171394list_a: ( list_a > list_list_a ) > set_li6773872926390105121list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_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_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_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_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_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5524387281408368019list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
member_list_set_a: list_set_a > set_list_set_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_I_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J_J,type,
member8163584906436000884list_a: set_nat_list_a > set_set_nat_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
member_set_nat_a: set_nat_a > set_set_nat_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member334759470184282131list_a: set_list_list_a > set_set_list_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_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_f,type,
f: a > a ).
% Relevant facts (1277)
thf(fact_0_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_1_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_2_b,axiom,
! [X: a] :
( ( member_a @ X @ s )
=> ( member_a @ ( f @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% b
thf(fact_3_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_4_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_5_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_6_ring_Ointerpolate_Ocong,axiom,
lagran1063865941317790773te_a_b = lagran1063865941317790773te_a_b ).
% ring.interpolate.cong
thf(fact_7_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_8_r_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% r.onepideal
thf(fact_9_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_10_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_11_r_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 ) ) ).
% r.carrier_is_subcring
thf(fact_12_r_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% r.semiring_axioms
thf(fact_13_r_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= bot_bot_set_list_a ) ).
% r.carrier_not_empty
thf(fact_14_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_15_r_Oadd_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A )
= ( add_li7652885771158616974t_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 ) ) ) ) ) ).
% r.add.l_cancel
thf(fact_16_r_Oadd_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C )
= ( add_li7652885771158616974t_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 ) ) ) ) ) ).
% r.add.r_cancel
thf(fact_17_r_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% r.add.m_lcomm
thf(fact_18_r_Oadd_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% r.add.m_comm
thf(fact_19_r_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% r.add.m_assoc
thf(fact_20_r_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% r.add.right_cancel
thf(fact_21_r_Oadd_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.add.m_closed
thf(fact_22_assms_I3_J,axiom,
ord_less_eq_set_a @ ( image_a_a @ f @ s ) @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(3)
thf(fact_23_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_24_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_25_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_26_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_27_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_28_r_Oadd_Oint__pow__mult__distrib,axiom,
! [X: list_a,Y: list_a,I: int] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ Y ) ) ) ) ) ) ).
% r.add.int_pow_mult_distrib
thf(fact_29_r_Oadd_Oint__pow__distrib,axiom,
! [X: list_a,Y: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ Y ) ) ) ) ) ).
% r.add.int_pow_distrib
thf(fact_30_r_Ocgenideal__is__principalideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% r.cgenideal_is_principalideal
thf(fact_31_subalgebra__in__carrier,axiom,
! [K: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_32_carrier__is__subalgebra,axiom,
! [K: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_33_r_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% r.minus_unique
thf(fact_34_r_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.add.r_inv_ex
thf(fact_35_r_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.add.one_unique
thf(fact_36_r_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.add.l_inv_ex
thf(fact_37_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_38_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_39_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_40_subsetI,axiom,
! [A2: set_list_list_a,B2: set_list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A2 )
=> ( member_list_list_a @ X2 @ B2 ) )
=> ( ord_le8488217952732425610list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_41_subsetI,axiom,
! [A2: set_nat_list_a,B2: set_nat_list_a] :
( ! [X2: nat > list_a] :
( ( member_nat_list_a @ X2 @ A2 )
=> ( member_nat_list_a @ X2 @ B2 ) )
=> ( ord_le2145805922479659755list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_42_subsetI,axiom,
! [A2: set_nat_a,B2: set_nat_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_nat_a @ X2 @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_43_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_44_subsetI,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_45_empty__Collect__eq,axiom,
! [P: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P ) )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_46_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_47_Collect__empty__eq,axiom,
! [P: list_a > $o] :
( ( ( collect_list_a @ P )
= bot_bot_set_list_a )
= ( ! [X3: list_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_48_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_49_all__not__in__conv,axiom,
! [A2: set_list_list_a] :
( ( ! [X3: list_list_a] :
~ ( member_list_list_a @ X3 @ A2 ) )
= ( A2 = bot_bo1875519244922727510list_a ) ) ).
% all_not_in_conv
thf(fact_50_all__not__in__conv,axiom,
! [A2: set_nat_list_a] :
( ( ! [X3: nat > list_a] :
~ ( member_nat_list_a @ X3 @ A2 ) )
= ( A2 = bot_bo3806784159821827511list_a ) ) ).
% all_not_in_conv
thf(fact_51_all__not__in__conv,axiom,
! [A2: set_nat_a] :
( ( ! [X3: nat > a] :
~ ( member_nat_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_52_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X3: list_a] :
~ ( member_list_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_53_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_54_empty__iff,axiom,
! [C: list_list_a] :
~ ( member_list_list_a @ C @ bot_bo1875519244922727510list_a ) ).
% empty_iff
thf(fact_55_empty__iff,axiom,
! [C: nat > list_a] :
~ ( member_nat_list_a @ C @ bot_bo3806784159821827511list_a ) ).
% empty_iff
thf(fact_56_empty__iff,axiom,
! [C: nat > a] :
~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_57_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_58_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_59_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: nat > list_a,P: ( nat > list_a ) > $o] :
( ( member_nat_list_a @ A @ ( collect_nat_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X3: list_list_a] : ( member_list_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_nat_list_a] :
( ( collect_nat_list_a
@ ^ [X3: nat > list_a] : ( member_nat_list_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_nat_a] :
( ( collect_nat_a
@ ^ [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_image__eqI,axiom,
! [B: a,F: a > a,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_70_image__eqI,axiom,
! [B: a,F: list_a > a,X: list_a,A2: set_list_a] :
( ( B
= ( F @ X ) )
=> ( ( member_list_a @ X @ A2 )
=> ( member_a @ B @ ( image_list_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_71_image__eqI,axiom,
! [B: list_a,F: a > list_a,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_list_a @ B @ ( image_a_list_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_72_image__eqI,axiom,
! [B: list_a,F: list_a > list_a,X: list_a,A2: set_list_a] :
( ( B
= ( F @ X ) )
=> ( ( member_list_a @ X @ A2 )
=> ( member_list_a @ B @ ( image_list_a_list_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_73_image__eqI,axiom,
! [B: list_list_a,F: a > list_list_a,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_list_list_a @ B @ ( image_a_list_list_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_74_image__eqI,axiom,
! [B: nat > a,F: a > nat > a,X: a,A2: set_a] :
( ( B
= ( F @ X ) )
=> ( ( member_a @ X @ A2 )
=> ( member_nat_a @ B @ ( image_a_nat_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_75_image__eqI,axiom,
! [B: a,F: list_list_a > a,X: list_list_a,A2: set_list_list_a] :
( ( B
= ( F @ X ) )
=> ( ( member_list_list_a @ X @ A2 )
=> ( member_a @ B @ ( image_list_list_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_76_image__eqI,axiom,
! [B: a,F: ( nat > a ) > a,X: nat > a,A2: set_nat_a] :
( ( B
= ( F @ X ) )
=> ( ( member_nat_a @ X @ A2 )
=> ( member_a @ B @ ( image_nat_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_77_image__eqI,axiom,
! [B: list_list_a,F: list_a > list_list_a,X: list_a,A2: set_list_a] :
( ( B
= ( F @ X ) )
=> ( ( member_list_a @ X @ A2 )
=> ( member_list_list_a @ B @ ( image_8260866953997875467list_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_78_image__eqI,axiom,
! [B: nat > a,F: list_a > nat > a,X: list_a,A2: set_list_a] :
( ( B
= ( F @ X ) )
=> ( ( member_list_a @ X @ A2 )
=> ( member_nat_a @ B @ ( image_list_a_nat_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_79_r_Ocgenideal__self,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I ) ) ) ).
% r.cgenideal_self
thf(fact_80_r_Oadd_Oinv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% r.add.inv_comm
thf(fact_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_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_93_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_94_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_95_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_96_r_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 ) ) ) ).
% r.zero_closed
thf(fact_97_r_Oadd_Oint__pow__closed,axiom,
! [X: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.add.int_pow_closed
thf(fact_98_r_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% r.add.int_pow_one
thf(fact_99_r_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% r.add.l_cancel_one
thf(fact_100_r_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% r.add.l_cancel_one'
thf(fact_101_r_Oadd_Or__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% r.add.r_cancel_one
thf(fact_102_r_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% r.add.r_cancel_one'
thf(fact_103_r_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% r.l_zero
thf(fact_104_r_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% r.r_zero
thf(fact_105_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: a,F: a > a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_106_rev__image__eqI,axiom,
! [X: list_a,A2: set_list_a,B: a,F: list_a > a] :
( ( member_list_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_list_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_107_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: list_a,F: a > list_a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_list_a @ B @ ( image_a_list_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_108_rev__image__eqI,axiom,
! [X: list_a,A2: set_list_a,B: list_a,F: list_a > list_a] :
( ( member_list_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_list_a @ B @ ( image_list_a_list_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_109_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: list_list_a,F: a > list_list_a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_list_list_a @ B @ ( image_a_list_list_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_110_rev__image__eqI,axiom,
! [X: a,A2: set_a,B: nat > a,F: a > nat > a] :
( ( member_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_a @ B @ ( image_a_nat_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_111_rev__image__eqI,axiom,
! [X: list_list_a,A2: set_list_list_a,B: a,F: list_list_a > a] :
( ( member_list_list_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_list_list_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_112_rev__image__eqI,axiom,
! [X: nat > a,A2: set_nat_a,B: a,F: ( nat > a ) > a] :
( ( member_nat_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_a @ B @ ( image_nat_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_113_rev__image__eqI,axiom,
! [X: list_a,A2: set_list_a,B: list_list_a,F: list_a > list_list_a] :
( ( member_list_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_list_list_a @ B @ ( image_8260866953997875467list_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_114_rev__image__eqI,axiom,
! [X: list_a,A2: set_list_a,B: nat > a,F: list_a > nat > a] :
( ( member_list_a @ X @ A2 )
=> ( ( B
= ( F @ X ) )
=> ( member_nat_a @ B @ ( image_list_a_nat_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_115_ball__imageD,axiom,
! [F: a > a,A2: set_a,P: a > $o] :
( ! [X2: a] :
( ( member_a @ X2 @ ( image_a_a @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_116_ball__imageD,axiom,
! [F: a > list_a,A2: set_a,P: list_a > $o] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( image_a_list_a @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_117_ball__imageD,axiom,
! [F: list_a > list_list_a,A2: set_list_a,P: list_list_a > $o] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( image_8260866953997875467list_a @ F @ A2 ) )
=> ( P @ X2 ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_118_image__cong,axiom,
! [M: set_list_a,N: set_list_a,F: list_a > list_list_a,G: list_a > list_list_a] :
( ( M = N )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_8260866953997875467list_a @ F @ M )
= ( image_8260866953997875467list_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_119_image__cong,axiom,
! [M: set_a,N: set_a,F: a > a,G: a > a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_a @ F @ M )
= ( image_a_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_120_image__cong,axiom,
! [M: set_a,N: set_a,F: a > list_a,G: a > list_a] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_a_list_a @ F @ M )
= ( image_a_list_a @ G @ N ) ) ) ) ).
% image_cong
thf(fact_121_bex__imageD,axiom,
! [F: a > a,A2: set_a,P: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( image_a_a @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_122_bex__imageD,axiom,
! [F: a > list_a,A2: set_a,P: list_a > $o] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( image_a_list_a @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_123_bex__imageD,axiom,
! [F: list_a > list_list_a,A2: set_list_a,P: list_list_a > $o] :
( ? [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( image_8260866953997875467list_a @ F @ A2 ) )
& ( P @ X4 ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_124_image__iff,axiom,
! [Z: list_a,F: a > list_a,A2: set_a] :
( ( member_list_a @ Z @ ( image_a_list_a @ F @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_125_image__iff,axiom,
! [Z: a,F: a > a,A2: set_a] :
( ( member_a @ Z @ ( image_a_a @ F @ A2 ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_126_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 ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
& ( Z
= ( F @ X3 ) ) ) ) ) ).
% image_iff
thf(fact_127_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_128_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_129_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_130_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_131_imageI,axiom,
! [X: a,A2: set_a,F: a > list_list_a] :
( ( member_a @ X @ A2 )
=> ( member_list_list_a @ ( F @ X ) @ ( image_a_list_list_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_132_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_133_imageI,axiom,
! [X: list_list_a,A2: set_list_list_a,F: list_list_a > a] :
( ( member_list_list_a @ X @ A2 )
=> ( member_a @ ( F @ X ) @ ( image_list_list_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_134_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_135_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_136_imageI,axiom,
! [X: list_a,A2: set_list_a,F: list_a > nat > a] :
( ( member_list_a @ X @ A2 )
=> ( member_nat_a @ ( F @ X ) @ ( image_list_a_nat_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_137_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_138_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_139_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_140_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_141_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_142_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 )
= ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( member_list_list_a @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_143_image__subset__iff,axiom,
! [F: a > a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B2 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_144_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 )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_list_a @ ( F @ X3 ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_145_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 ) )
=> ~ ! [C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
=> ( B2
!= ( image_8260866953997875467list_a @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_146_subset__imageE,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( B2
!= ( image_a_a @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_147_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 ) )
=> ~ ! [C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
=> ( B2
!= ( image_list_a_a @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_148_subset__imageE,axiom,
! [B2: set_list_a,F: a > list_a,A2: set_a] :
( ( ord_le8861187494160871172list_a @ B2 @ ( image_a_list_a @ F @ A2 ) )
=> ~ ! [C2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( B2
!= ( image_a_list_a @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_149_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 ) )
=> ~ ! [C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
=> ( B2
!= ( image_list_a_list_a @ F @ C2 ) ) ) ) ).
% subset_imageE
thf(fact_150_image__subsetI,axiom,
! [A2: set_a,F: a > a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_151_image__subsetI,axiom,
! [A2: set_list_a,F: list_a > a,B2: set_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_152_image__subsetI,axiom,
! [A2: set_a,F: a > list_a,B2: set_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_list_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_153_image__subsetI,axiom,
! [A2: set_a,F: a > list_list_a,B2: set_list_list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_list_list_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_le8488217952732425610list_a @ ( image_a_list_list_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_154_image__subsetI,axiom,
! [A2: set_a,F: a > nat > a,B2: set_nat_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_nat_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ ( image_a_nat_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_155_image__subsetI,axiom,
! [A2: set_list_list_a,F: list_list_a > a,B2: set_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_list_list_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_156_image__subsetI,axiom,
! [A2: set_nat_a,F: ( nat > a ) > a,B2: set_a] :
( ! [X2: nat > a] :
( ( member_nat_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_less_eq_set_a @ ( image_nat_a_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_157_image__subsetI,axiom,
! [A2: set_list_a,F: list_a > list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_158_image__subsetI,axiom,
! [A2: set_list_a,F: list_a > list_list_a,B2: set_list_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_list_list_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_le8488217952732425610list_a @ ( image_8260866953997875467list_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_159_image__subsetI,axiom,
! [A2: set_list_a,F: list_a > nat > a,B2: set_nat_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_nat_a @ ( F @ X2 ) @ B2 ) )
=> ( ord_le871467723717165285_nat_a @ ( image_list_a_nat_a @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_160_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_161_image__mono,axiom,
! [A2: set_a,B2: set_a,F: a > list_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A2 ) @ ( image_a_list_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_162_image__mono,axiom,
! [A2: set_list_a,B2: set_list_a,F: list_a > list_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ord_le8488217952732425610list_a @ ( image_8260866953997875467list_a @ F @ A2 ) @ ( image_8260866953997875467list_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_163_image__mono,axiom,
! [A2: set_list_a,B2: set_list_a,F: list_a > a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A2 ) @ ( image_list_a_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_164_image__mono,axiom,
! [A2: set_list_a,B2: set_list_a,F: list_a > list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ord_le8861187494160871172list_a @ ( image_list_a_list_a @ F @ A2 ) @ ( image_list_a_list_a @ F @ B2 ) ) ) ).
% image_mono
thf(fact_165_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_166_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_167_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_168_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z2: set_a] : ( Y3 = Z2 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_169_set__eq__subset,axiom,
( ( ^ [Y3: set_list_a,Z2: set_list_a] : ( Y3 = Z2 ) )
= ( ^ [A3: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_170_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_171_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_172_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_173_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X2: list_a] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_174_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_175_subset__refl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_176_subset__iff,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A3: set_list_list_a,B3: set_list_list_a] :
! [T: list_list_a] :
( ( member_list_list_a @ T @ A3 )
=> ( member_list_list_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_177_subset__iff,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A3: set_nat_list_a,B3: set_nat_list_a] :
! [T: nat > list_a] :
( ( member_nat_list_a @ T @ A3 )
=> ( member_nat_list_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_178_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
! [T: nat > a] :
( ( member_nat_a @ T @ A3 )
=> ( member_nat_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_179_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_180_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A3 )
=> ( member_list_a @ T @ B3 ) ) ) ) ).
% subset_iff
thf(fact_181_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_182_equalityD2,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_183_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_184_equalityD1,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_185_subset__eq,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A3: set_list_list_a,B3: set_list_list_a] :
! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A3 )
=> ( member_list_list_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_186_subset__eq,axiom,
( ord_le2145805922479659755list_a
= ( ^ [A3: set_nat_list_a,B3: set_nat_list_a] :
! [X3: nat > list_a] :
( ( member_nat_list_a @ X3 @ A3 )
=> ( member_nat_list_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_187_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A3: set_nat_a,B3: set_nat_a] :
! [X3: nat > a] :
( ( member_nat_a @ X3 @ A3 )
=> ( member_nat_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_188_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_189_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B3: set_list_a] :
! [X3: list_a] :
( ( member_list_a @ X3 @ A3 )
=> ( member_list_a @ X3 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_190_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_191_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_192_subsetD,axiom,
! [A2: set_list_list_a,B2: set_list_list_a,C: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ B2 )
=> ( ( member_list_list_a @ C @ A2 )
=> ( member_list_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_193_subsetD,axiom,
! [A2: set_nat_list_a,B2: set_nat_list_a,C: nat > list_a] :
( ( ord_le2145805922479659755list_a @ A2 @ B2 )
=> ( ( member_nat_list_a @ C @ A2 )
=> ( member_nat_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_194_subsetD,axiom,
! [A2: set_nat_a,B2: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ B2 )
=> ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_195_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_196_subsetD,axiom,
! [A2: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_197_in__mono,axiom,
! [A2: set_list_list_a,B2: set_list_list_a,X: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ B2 )
=> ( ( member_list_list_a @ X @ A2 )
=> ( member_list_list_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_198_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_199_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_200_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_201_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_202_ex__in__conv,axiom,
! [A2: set_list_list_a] :
( ( ? [X3: list_list_a] : ( member_list_list_a @ X3 @ A2 ) )
= ( A2 != bot_bo1875519244922727510list_a ) ) ).
% ex_in_conv
thf(fact_203_ex__in__conv,axiom,
! [A2: set_nat_list_a] :
( ( ? [X3: nat > list_a] : ( member_nat_list_a @ X3 @ A2 ) )
= ( A2 != bot_bo3806784159821827511list_a ) ) ).
% ex_in_conv
thf(fact_204_ex__in__conv,axiom,
! [A2: set_nat_a] :
( ( ? [X3: nat > a] : ( member_nat_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat_a ) ) ).
% ex_in_conv
thf(fact_205_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X3: list_a] : ( member_list_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_206_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_207_equals0I,axiom,
! [A2: set_list_list_a] :
( ! [Y4: list_list_a] :
~ ( member_list_list_a @ Y4 @ A2 )
=> ( A2 = bot_bo1875519244922727510list_a ) ) ).
% equals0I
thf(fact_208_equals0I,axiom,
! [A2: set_nat_list_a] :
( ! [Y4: nat > list_a] :
~ ( member_nat_list_a @ Y4 @ A2 )
=> ( A2 = bot_bo3806784159821827511list_a ) ) ).
% equals0I
thf(fact_209_equals0I,axiom,
! [A2: set_nat_a] :
( ! [Y4: nat > a] :
~ ( member_nat_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_nat_a ) ) ).
% equals0I
thf(fact_210_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y4: list_a] :
~ ( member_list_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_211_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_212_equals0D,axiom,
! [A2: set_list_list_a,A: list_list_a] :
( ( A2 = bot_bo1875519244922727510list_a )
=> ~ ( member_list_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_213_equals0D,axiom,
! [A2: set_nat_list_a,A: nat > list_a] :
( ( A2 = bot_bo3806784159821827511list_a )
=> ~ ( member_nat_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_214_equals0D,axiom,
! [A2: set_nat_a,A: nat > a] :
( ( A2 = bot_bot_set_nat_a )
=> ~ ( member_nat_a @ A @ A2 ) ) ).
% equals0D
thf(fact_215_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_216_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_217_emptyE,axiom,
! [A: list_list_a] :
~ ( member_list_list_a @ A @ bot_bo1875519244922727510list_a ) ).
% emptyE
thf(fact_218_emptyE,axiom,
! [A: nat > list_a] :
~ ( member_nat_list_a @ A @ bot_bo3806784159821827511list_a ) ).
% emptyE
thf(fact_219_emptyE,axiom,
! [A: nat > a] :
~ ( member_nat_a @ A @ bot_bot_set_nat_a ) ).
% emptyE
thf(fact_220_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_221_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_222_r_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 ) ) ).
% r.a_lcos_mult_one
thf(fact_223_r_Oa__lcos__m__assoc,axiom,
! [M: set_list_a,G: list_a,H: list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ M ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ H ) @ M ) ) ) ) ) ).
% r.a_lcos_m_assoc
thf(fact_224_r_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 ) ) ).
% r.zeropideal
thf(fact_225_r_Ocarrier__is__subalgebra,axiom,
! [K: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% r.carrier_is_subalgebra
thf(fact_226_r_Osubalgebra__in__carrier,axiom,
! [K: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.subalgebra_in_carrier
thf(fact_227_r_Oadd_Oint__pow__mult,axiom,
! [X: list_a,I: int,J: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( plus_plus_int @ I @ J ) @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ J @ X ) ) ) ) ).
% r.add.int_pow_mult
thf(fact_228_r_Oa__l__coset__subset__G,axiom,
! [H2: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ H2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.a_l_coset_subset_G
thf(fact_229_finite__imageI,axiom,
! [F2: set_a,H: a > a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_a @ ( image_a_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_230_finite__imageI,axiom,
! [F2: set_a,H: a > list_a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_list_a @ ( image_a_list_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_231_finite__imageI,axiom,
! [F2: set_list_a,H: list_a > list_list_a] :
( ( finite_finite_list_a @ F2 )
=> ( finite1660835950917165235list_a @ ( image_8260866953997875467list_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_232_finite__imageI,axiom,
! [F2: set_list_a,H: list_a > a] :
( ( finite_finite_list_a @ F2 )
=> ( finite_finite_a @ ( image_list_a_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_233_finite__imageI,axiom,
! [F2: set_list_a,H: list_a > list_a] :
( ( finite_finite_list_a @ F2 )
=> ( finite_finite_list_a @ ( image_list_a_list_a @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_234_r_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 ) ) ) ).
% r.genideal_self
thf(fact_235_r_Osubset__Idl__subset,axiom,
! [I2: set_list_a,H2: set_list_a] :
( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ I2 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 ) ) ) ) ).
% r.subset_Idl_subset
thf(fact_236_r_Oadd_Oone__in__subset,axiom,
! [H2: set_list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( H2 != bot_bot_set_list_a )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ H2 )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 ) @ H2 ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ H2 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ H2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Xa ) @ H2 ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ H2 ) ) ) ) ) ).
% r.add.one_in_subset
thf(fact_237_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_238_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_239_insert__iff,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_240_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_241_insert__iff,axiom,
! [A: list_list_a,B: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ ( insert_list_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_list_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_242_insert__iff,axiom,
! [A: nat > list_a,B: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A @ ( insert_nat_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_243_insert__iff,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
= ( ( A = B )
| ( member_nat_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_244_insertCI,axiom,
! [A: list_a,B2: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A @ B2 )
=> ( A = B ) )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_245_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_246_insertCI,axiom,
! [A: list_list_a,B2: set_list_list_a,B: list_list_a] :
( ( ~ ( member_list_list_a @ A @ B2 )
=> ( A = B ) )
=> ( member_list_list_a @ A @ ( insert_list_list_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_247_insertCI,axiom,
! [A: nat > list_a,B2: set_nat_list_a,B: nat > list_a] :
( ( ~ ( member_nat_list_a @ A @ B2 )
=> ( A = B ) )
=> ( member_nat_list_a @ A @ ( insert_nat_list_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_248_insertCI,axiom,
! [A: nat > a,B2: set_nat_a,B: nat > a] :
( ( ~ ( member_nat_a @ A @ B2 )
=> ( A = B ) )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_249_r_Or__neg2,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y ) )
= Y ) ) ) ).
% r.r_neg2
thf(fact_250_r_Or__neg1,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= Y ) ) ) ).
% r.r_neg1
thf(fact_251_r_Ominus__add,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ) ) ).
% r.minus_add
thf(fact_252_r_Oadd_Oinv__solve__right_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( 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 ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C ) )
= A )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ) ) ).
% r.add.inv_solve_right'
thf(fact_253_r_Oadd_Oinv__solve__right,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( 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
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C ) ) )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ) ) ).
% r.add.inv_solve_right
thf(fact_254_r_Oadd_Oinv__solve__left_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( 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 ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) @ C )
= A )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A ) ) ) ) ) ) ).
% r.add.inv_solve_left'
thf(fact_255_r_Oadd_Oinv__solve__left,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( 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
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) @ C ) )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A ) ) ) ) ) ) ).
% r.add.inv_solve_left
thf(fact_256_r_Oadd_Oinv__mult__group,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) ) ) ) ) ).
% r.add.inv_mult_group
thf(fact_257_r_Oa__transpose__inv,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= Z )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% r.a_transpose_inv
thf(fact_258_r_Oadd_Oint__pow__inv,axiom,
! [X: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X ) ) ) ) ).
% r.add.int_pow_inv
thf(fact_259_r_Osum__zero__eq__neg,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( X
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ) ) ).
% r.sum_zero_eq_neg
thf(fact_260_r_Or__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.r_neg
thf(fact_261_r_Ominus__equality,axiom,
! [Y: list_a,X: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ).
% r.minus_equality
thf(fact_262_r_Ol__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.l_neg
thf(fact_263_r_Ogenideal__self_H,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% r.genideal_self'
thf(fact_264_r_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 ) ) ).
% r.genideal_zero
thf(fact_265_insert__subset,axiom,
! [X: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( insert_list_list_a @ X @ A2 ) @ B2 )
= ( ( member_list_list_a @ X @ B2 )
& ( ord_le8488217952732425610list_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_266_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_267_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_268_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_269_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_270_singletonI,axiom,
! [A: list_list_a] : ( member_list_list_a @ A @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ).
% singletonI
thf(fact_271_singletonI,axiom,
! [A: nat > list_a] : ( member_nat_list_a @ A @ ( insert_nat_list_a @ A @ bot_bo3806784159821827511list_a ) ) ).
% singletonI
thf(fact_272_singletonI,axiom,
! [A: nat > a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) ).
% singletonI
thf(fact_273_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_274_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_275_image__insert,axiom,
! [F: list_a > list_list_a,A: list_a,B2: set_list_a] :
( ( image_8260866953997875467list_a @ F @ ( insert_list_a @ A @ B2 ) )
= ( insert_list_list_a @ ( F @ A ) @ ( image_8260866953997875467list_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_276_image__insert,axiom,
! [F: list_a > list_a,A: list_a,B2: set_list_a] :
( ( image_list_a_list_a @ F @ ( insert_list_a @ A @ B2 ) )
= ( insert_list_a @ ( F @ A ) @ ( image_list_a_list_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_277_image__insert,axiom,
! [F: list_a > a,A: list_a,B2: set_list_a] :
( ( image_list_a_a @ F @ ( insert_list_a @ A @ B2 ) )
= ( insert_a @ ( F @ A ) @ ( image_list_a_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_278_image__insert,axiom,
! [F: a > list_a,A: a,B2: set_a] :
( ( image_a_list_a @ F @ ( insert_a @ A @ B2 ) )
= ( insert_list_a @ ( F @ A ) @ ( image_a_list_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_279_image__insert,axiom,
! [F: a > a,A: a,B2: set_a] :
( ( image_a_a @ F @ ( insert_a @ A @ B2 ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a @ F @ B2 ) ) ) ).
% image_insert
thf(fact_280_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_281_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_282_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_283_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_284_insert__image,axiom,
! [X: list_list_a,A2: set_list_list_a,F: list_list_a > a] :
( ( member_list_list_a @ X @ A2 )
=> ( ( insert_a @ ( F @ X ) @ ( image_list_list_a_a @ F @ A2 ) )
= ( image_list_list_a_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_285_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_286_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_287_insert__image,axiom,
! [X: list_list_a,A2: set_list_list_a,F: list_list_a > list_a] :
( ( member_list_list_a @ X @ A2 )
=> ( ( insert_list_a @ ( F @ X ) @ ( image_4177985004758493951list_a @ F @ A2 ) )
= ( image_4177985004758493951list_a @ F @ A2 ) ) ) ).
% insert_image
thf(fact_288_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_289_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_290_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_291_finite__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) )
= ( finite_finite_list_a @ A2 ) ) ).
% finite_insert
thf(fact_292_r_OIdl__subset__ideal_H,axiom,
! [A: list_a,B: list_a] :
( ( 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 ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ) ) ).
% r.Idl_subset_ideal'
thf(fact_293_r_Ocgenideal__eq__genideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I )
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% r.cgenideal_eq_genideal
thf(fact_294_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_295_singleton__insert__inj__eq,axiom,
! [B: list_a,A: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_296_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_297_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A2: set_list_a,B: list_a] :
( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_298_r_Ominus__minus,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= X ) ) ).
% r.minus_minus
thf(fact_299_r_Oadd_Oinv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.add.inv_closed
thf(fact_300_r_Ominus__zero,axiom,
( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% r.minus_zero
thf(fact_301_r_Oadd_Oinv__eq__1__iff,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( X
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.add.inv_eq_1_iff
thf(fact_302_mk__disjoint__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ? [B4: set_list_a] :
( ( A2
= ( insert_list_a @ A @ B4 ) )
& ~ ( member_list_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_303_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B4: set_a] :
( ( A2
= ( insert_a @ A @ B4 ) )
& ~ ( member_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_304_mk__disjoint__insert,axiom,
! [A: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ A2 )
=> ? [B4: set_list_list_a] :
( ( A2
= ( insert_list_list_a @ A @ B4 ) )
& ~ ( member_list_list_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_305_mk__disjoint__insert,axiom,
! [A: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A @ A2 )
=> ? [B4: set_nat_list_a] :
( ( A2
= ( insert_nat_list_a @ A @ B4 ) )
& ~ ( member_nat_list_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_306_mk__disjoint__insert,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ? [B4: set_nat_a] :
( ( A2
= ( insert_nat_a @ A @ B4 ) )
& ~ ( member_nat_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_307_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_308_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_309_insert__eq__iff,axiom,
! [A: list_a,A2: set_list_a,B: list_a,B2: set_list_a] :
( ~ ( member_list_a @ A @ A2 )
=> ( ~ ( member_list_a @ B @ B2 )
=> ( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_list_a] :
( ( A2
= ( insert_list_a @ B @ C4 ) )
& ~ ( member_list_a @ B @ C4 )
& ( B2
= ( insert_list_a @ A @ C4 ) )
& ~ ( member_list_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_310_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B2
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_311_insert__eq__iff,axiom,
! [A: list_list_a,A2: set_list_list_a,B: list_list_a,B2: set_list_list_a] :
( ~ ( member_list_list_a @ A @ A2 )
=> ( ~ ( member_list_list_a @ B @ B2 )
=> ( ( ( insert_list_list_a @ A @ A2 )
= ( insert_list_list_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_list_list_a] :
( ( A2
= ( insert_list_list_a @ B @ C4 ) )
& ~ ( member_list_list_a @ B @ C4 )
& ( B2
= ( insert_list_list_a @ A @ C4 ) )
& ~ ( member_list_list_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_312_insert__eq__iff,axiom,
! [A: nat > list_a,A2: set_nat_list_a,B: nat > list_a,B2: set_nat_list_a] :
( ~ ( member_nat_list_a @ A @ A2 )
=> ( ~ ( member_nat_list_a @ B @ B2 )
=> ( ( ( insert_nat_list_a @ A @ A2 )
= ( insert_nat_list_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_nat_list_a] :
( ( A2
= ( insert_nat_list_a @ B @ C4 ) )
& ~ ( member_nat_list_a @ B @ C4 )
& ( B2
= ( insert_nat_list_a @ A @ C4 ) )
& ~ ( member_nat_list_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_313_insert__eq__iff,axiom,
! [A: nat > a,A2: set_nat_a,B: nat > a,B2: set_nat_a] :
( ~ ( member_nat_a @ A @ A2 )
=> ( ~ ( member_nat_a @ B @ B2 )
=> ( ( ( insert_nat_a @ A @ A2 )
= ( insert_nat_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C4: set_nat_a] :
( ( A2
= ( insert_nat_a @ B @ C4 ) )
& ~ ( member_nat_a @ B @ C4 )
& ( B2
= ( insert_nat_a @ A @ C4 ) )
& ~ ( member_nat_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_314_insert__absorb,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_315_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_316_insert__absorb,axiom,
! [A: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ A2 )
=> ( ( insert_list_list_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_317_insert__absorb,axiom,
! [A: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A @ A2 )
=> ( ( insert_nat_list_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_318_insert__absorb,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ( ( insert_nat_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_319_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_320_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_321_insert__ident,axiom,
! [X: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ~ ( member_list_list_a @ X @ A2 )
=> ( ~ ( member_list_list_a @ X @ B2 )
=> ( ( ( insert_list_list_a @ X @ A2 )
= ( insert_list_list_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_322_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_323_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_324_Set_Oset__insert,axiom,
! [X: list_a,A2: set_list_a] :
( ( member_list_a @ X @ A2 )
=> ~ ! [B4: set_list_a] :
( ( A2
= ( insert_list_a @ X @ B4 ) )
=> ( member_list_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_325_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B4: set_a] :
( ( A2
= ( insert_a @ X @ B4 ) )
=> ( member_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_326_Set_Oset__insert,axiom,
! [X: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ X @ A2 )
=> ~ ! [B4: set_list_list_a] :
( ( A2
= ( insert_list_list_a @ X @ B4 ) )
=> ( member_list_list_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_327_Set_Oset__insert,axiom,
! [X: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ X @ A2 )
=> ~ ! [B4: set_nat_list_a] :
( ( A2
= ( insert_nat_list_a @ X @ B4 ) )
=> ( member_nat_list_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_328_Set_Oset__insert,axiom,
! [X: nat > a,A2: set_nat_a] :
( ( member_nat_a @ X @ A2 )
=> ~ ! [B4: set_nat_a] :
( ( A2
= ( insert_nat_a @ X @ B4 ) )
=> ( member_nat_a @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_329_insertI2,axiom,
! [A: list_a,B2: set_list_a,B: list_a] :
( ( member_list_a @ A @ B2 )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_330_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_331_insertI2,axiom,
! [A: list_list_a,B2: set_list_list_a,B: list_list_a] :
( ( member_list_list_a @ A @ B2 )
=> ( member_list_list_a @ A @ ( insert_list_list_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_332_insertI2,axiom,
! [A: nat > list_a,B2: set_nat_list_a,B: nat > list_a] :
( ( member_nat_list_a @ A @ B2 )
=> ( member_nat_list_a @ A @ ( insert_nat_list_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_333_insertI2,axiom,
! [A: nat > a,B2: set_nat_a,B: nat > a] :
( ( member_nat_a @ A @ B2 )
=> ( member_nat_a @ A @ ( insert_nat_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_334_insertI1,axiom,
! [A: list_a,B2: set_list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ B2 ) ) ).
% insertI1
thf(fact_335_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_336_insertI1,axiom,
! [A: list_list_a,B2: set_list_list_a] : ( member_list_list_a @ A @ ( insert_list_list_a @ A @ B2 ) ) ).
% insertI1
thf(fact_337_insertI1,axiom,
! [A: nat > list_a,B2: set_nat_list_a] : ( member_nat_list_a @ A @ ( insert_nat_list_a @ A @ B2 ) ) ).
% insertI1
thf(fact_338_insertI1,axiom,
! [A: nat > a,B2: set_nat_a] : ( member_nat_a @ A @ ( insert_nat_a @ A @ B2 ) ) ).
% insertI1
thf(fact_339_insertE,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_list_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_340_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_341_insertE,axiom,
! [A: list_list_a,B: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ ( insert_list_list_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_list_list_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_342_insertE,axiom,
! [A: nat > list_a,B: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A @ ( insert_nat_list_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_list_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_343_insertE,axiom,
! [A: nat > a,B: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ ( insert_nat_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_344_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_345_finite_OinsertI,axiom,
! [A2: set_list_a,A: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_346_insert__mono,axiom,
! [C3: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_347_insert__mono,axiom,
! [C3: set_list_a,D: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ D )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ A @ C3 ) @ ( insert_list_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_348_subset__insert,axiom,
! [X: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ~ ( member_list_list_a @ X @ A2 )
=> ( ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ X @ B2 ) )
= ( ord_le8488217952732425610list_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_349_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_350_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_351_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_352_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_353_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_354_subset__insertI,axiom,
! [B2: set_list_a,A: list_a] : ( ord_le8861187494160871172list_a @ B2 @ ( insert_list_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_355_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_356_subset__insertI2,axiom,
! [A2: set_list_a,B2: set_list_a,B: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_357_singletonD,axiom,
! [B: list_list_a,A: list_list_a] :
( ( member_list_list_a @ B @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_358_singletonD,axiom,
! [B: nat > list_a,A: nat > list_a] :
( ( member_nat_list_a @ B @ ( insert_nat_list_a @ A @ bot_bo3806784159821827511list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_359_singletonD,axiom,
! [B: nat > a,A: nat > a] :
( ( member_nat_a @ B @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_360_singletonD,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_361_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_362_singleton__iff,axiom,
! [B: list_list_a,A: list_list_a] :
( ( member_list_list_a @ B @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_363_singleton__iff,axiom,
! [B: nat > list_a,A: nat > list_a] :
( ( member_nat_list_a @ B @ ( insert_nat_list_a @ A @ bot_bo3806784159821827511list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_364_singleton__iff,axiom,
! [B: nat > a,A: nat > a] :
( ( member_nat_a @ B @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_365_singleton__iff,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_366_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_367_doubleton__eq__iff,axiom,
! [A: list_a,B: list_a,C: list_a,D2: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D2 @ bot_bot_set_list_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_368_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D2 ) )
| ( ( A = D2 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_369_insert__not__empty,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ A2 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_370_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_371_singleton__inject,axiom,
! [A: list_a,B: list_a] :
( ( ( insert_list_a @ A @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_372_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_373_finite_Ocases,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( A != bot_bot_set_list_a )
=> ~ ! [A4: set_list_a] :
( ? [A5: list_a] :
( A
= ( insert_list_a @ A5 @ A4 ) )
=> ~ ( finite_finite_list_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_374_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A4: set_a] :
( ? [A5: a] :
( A
= ( insert_a @ A5 @ A4 ) )
=> ~ ( finite_finite_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_375_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A6: set_list_a] :
( ( A6 = bot_bot_set_list_a )
| ? [A3: set_list_a,B5: list_a] :
( ( A6
= ( insert_list_a @ B5 @ A3 ) )
& ( finite_finite_list_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_376_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A6: set_a] :
( ( A6 = bot_bot_set_a )
| ? [A3: set_a,B5: a] :
( ( A6
= ( insert_a @ B5 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_377_finite__induct,axiom,
! [F2: set_list_list_a,P: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ F2 )
=> ( ( P @ bot_bo1875519244922727510list_a )
=> ( ! [X2: list_list_a,F3: set_list_list_a] :
( ( finite1660835950917165235list_a @ F3 )
=> ( ~ ( member_list_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_list_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_378_finite__induct,axiom,
! [F2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( P @ bot_bo3806784159821827511list_a )
=> ( ! [X2: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ~ ( member_nat_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_list_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_379_finite__induct,axiom,
! [F2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_380_finite__induct,axiom,
! [F2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_381_finite__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% finite_induct
thf(fact_382_finite__ne__induct,axiom,
! [F2: set_list_list_a,P: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ F2 )
=> ( ( F2 != bot_bo1875519244922727510list_a )
=> ( ! [X2: list_list_a] : ( P @ ( insert_list_list_a @ X2 @ bot_bo1875519244922727510list_a ) )
=> ( ! [X2: list_list_a,F3: set_list_list_a] :
( ( finite1660835950917165235list_a @ F3 )
=> ( ( F3 != bot_bo1875519244922727510list_a )
=> ( ~ ( member_list_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_list_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_383_finite__ne__induct,axiom,
! [F2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( F2 != bot_bo3806784159821827511list_a )
=> ( ! [X2: nat > list_a] : ( P @ ( insert_nat_list_a @ X2 @ bot_bo3806784159821827511list_a ) )
=> ( ! [X2: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ( F3 != bot_bo3806784159821827511list_a )
=> ( ~ ( member_nat_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_list_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_384_finite__ne__induct,axiom,
! [F2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( F2 != bot_bot_set_nat_a )
=> ( ! [X2: nat > a] : ( P @ ( insert_nat_a @ X2 @ bot_bot_set_nat_a ) )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( F3 != bot_bot_set_nat_a )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_385_finite__ne__induct,axiom,
! [F2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( F2 != bot_bot_set_list_a )
=> ( ! [X2: list_a] : ( P @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
=> ( ! [X2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_386_finite__ne__induct,axiom,
! [F2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X2: a] : ( P @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_387_infinite__finite__induct,axiom,
! [P: set_list_list_a > $o,A2: set_list_list_a] :
( ! [A4: set_list_list_a] :
( ~ ( finite1660835950917165235list_a @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bo1875519244922727510list_a )
=> ( ! [X2: list_list_a,F3: set_list_list_a] :
( ( finite1660835950917165235list_a @ F3 )
=> ( ~ ( member_list_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_list_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_388_infinite__finite__induct,axiom,
! [P: set_nat_list_a > $o,A2: set_nat_list_a] :
( ! [A4: set_nat_list_a] :
( ~ ( finite7630042315537210004list_a @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bo3806784159821827511list_a )
=> ( ! [X2: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ~ ( member_nat_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_list_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_389_infinite__finite__induct,axiom,
! [P: set_nat_a > $o,A2: set_nat_a] :
( ! [A4: set_nat_a] :
( ~ ( finite_finite_nat_a @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [X2: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ~ ( member_nat_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_390_infinite__finite__induct,axiom,
! [P: set_list_a > $o,A2: set_list_a] :
( ! [A4: set_list_a] :
( ~ ( finite_finite_list_a @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [X2: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ~ ( member_list_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_391_infinite__finite__induct,axiom,
! [P: set_a > $o,A2: set_a] :
( ! [A4: set_a] :
( ~ ( finite_finite_a @ A4 )
=> ( P @ A4 ) )
=> ( ( P @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_392_finite__subset__induct,axiom,
! [F2: set_list_list_a,A2: set_list_list_a,P: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ F2 )
=> ( ( ord_le8488217952732425610list_a @ F2 @ A2 )
=> ( ( P @ bot_bo1875519244922727510list_a )
=> ( ! [A5: list_list_a,F3: set_list_list_a] :
( ( finite1660835950917165235list_a @ F3 )
=> ( ( member_list_list_a @ A5 @ A2 )
=> ( ~ ( member_list_list_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_list_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_393_finite__subset__induct,axiom,
! [F2: set_nat_list_a,A2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( ord_le2145805922479659755list_a @ F2 @ A2 )
=> ( ( P @ bot_bo3806784159821827511list_a )
=> ( ! [A5: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ( member_nat_list_a @ A5 @ A2 )
=> ( ~ ( member_nat_list_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_list_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_394_finite__subset__induct,axiom,
! [F2: set_nat_a,A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( ord_le871467723717165285_nat_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A5: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A5 @ A2 )
=> ( ~ ( member_nat_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_395_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A5 @ A2 )
=> ( ~ ( member_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_396_finite__subset__induct,axiom,
! [F2: set_list_a,A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A5: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a @ A5 @ A2 )
=> ( ~ ( member_list_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ A5 @ F3 ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_397_finite__subset__induct_H,axiom,
! [F2: set_list_list_a,A2: set_list_list_a,P: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ F2 )
=> ( ( ord_le8488217952732425610list_a @ F2 @ A2 )
=> ( ( P @ bot_bo1875519244922727510list_a )
=> ( ! [A5: list_list_a,F3: set_list_list_a] :
( ( finite1660835950917165235list_a @ F3 )
=> ( ( member_list_list_a @ A5 @ A2 )
=> ( ( ord_le8488217952732425610list_a @ F3 @ A2 )
=> ( ~ ( member_list_list_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_list_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_398_finite__subset__induct_H,axiom,
! [F2: set_nat_list_a,A2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ F2 )
=> ( ( ord_le2145805922479659755list_a @ F2 @ A2 )
=> ( ( P @ bot_bo3806784159821827511list_a )
=> ( ! [A5: nat > list_a,F3: set_nat_list_a] :
( ( finite7630042315537210004list_a @ F3 )
=> ( ( member_nat_list_a @ A5 @ A2 )
=> ( ( ord_le2145805922479659755list_a @ F3 @ A2 )
=> ( ~ ( member_nat_list_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_list_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_399_finite__subset__induct_H,axiom,
! [F2: set_nat_a,A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ F2 )
=> ( ( ord_le871467723717165285_nat_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A5: nat > a,F3: set_nat_a] :
( ( finite_finite_nat_a @ F3 )
=> ( ( member_nat_a @ A5 @ A2 )
=> ( ( ord_le871467723717165285_nat_a @ F3 @ A2 )
=> ( ~ ( member_nat_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_nat_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_400_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A5: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_401_finite__subset__induct_H,axiom,
! [F2: set_list_a,A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ F2 )
=> ( ( ord_le8861187494160871172list_a @ F2 @ A2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A5: list_a,F3: set_list_a] :
( ( finite_finite_list_a @ F3 )
=> ( ( member_list_a @ A5 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A2 )
=> ( ~ ( member_list_a @ A5 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert_list_a @ A5 @ F3 ) ) ) ) ) ) )
=> ( P @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_402_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_403_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_404_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_405_subset__singleton__iff,axiom,
! [X5: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ X5 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( ( X5 = bot_bot_set_list_a )
| ( X5
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_406_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_407_finite__has__maximal2,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A @ A2 )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
& ( ord_le8861187494160871172list_a @ A @ X2 )
& ! [Xa2: set_list_a] :
( ( member_set_list_a @ Xa2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_408_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_409_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_410_finite__has__minimal2,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A @ A2 )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
& ( ord_le8861187494160871172list_a @ X2 @ A )
& ! [Xa2: set_list_a] :
( ( member_set_list_a @ Xa2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_411_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_412_all__subset__image,axiom,
! [F: list_a > list_list_a,A2: set_list_a,P: set_list_list_a > $o] :
( ( ! [B3: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ B3 @ ( image_8260866953997875467list_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( P @ ( image_8260866953997875467list_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_413_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P @ ( image_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_414_all__subset__image,axiom,
! [F: list_a > a,A2: set_list_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_list_a_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( P @ ( image_list_a_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_415_all__subset__image,axiom,
! [F: a > list_a,A2: set_a,P: set_list_a > $o] :
( ( ! [B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ ( image_a_list_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( P @ ( image_a_list_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_416_all__subset__image,axiom,
! [F: list_a > list_a,A2: set_list_a,P: set_list_a > $o] :
( ( ! [B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ ( image_list_a_list_a @ F @ A2 ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( P @ ( image_list_a_list_a @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_417_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_418_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_419_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_420_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_421_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_422_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_423_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_424_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_425_infinite__imp__nonempty,axiom,
! [S2: set_list_a] :
( ~ ( finite_finite_list_a @ S2 )
=> ( S2 != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_426_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_427_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_428_finite__has__maximal,axiom,
! [A2: set_set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( A2 != bot_bo3186585308812441520list_a )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
& ! [Xa2: set_list_a] :
( ( member_set_list_a @ Xa2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_429_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_430_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_431_finite__has__minimal,axiom,
! [A2: set_set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( A2 != bot_bo3186585308812441520list_a )
=> ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ A2 )
& ! [Xa2: set_list_a] :
( ( member_set_list_a @ Xa2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_432_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ord_less_eq_nat @ Xa2 @ X2 )
=> ( X2 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_433_finite__surj,axiom,
! [A2: set_list_a,B2: set_list_list_a,F: list_a > list_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( ord_le8488217952732425610list_a @ B2 @ ( image_8260866953997875467list_a @ F @ A2 ) )
=> ( finite1660835950917165235list_a @ B2 ) ) ) ).
% finite_surj
thf(fact_434_finite__surj,axiom,
! [A2: set_a,B2: set_a,F: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_435_finite__surj,axiom,
! [A2: set_list_a,B2: set_a,F: list_a > a] :
( ( finite_finite_list_a @ A2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_list_a_a @ F @ A2 ) )
=> ( finite_finite_a @ B2 ) ) ) ).
% finite_surj
thf(fact_436_finite__surj,axiom,
! [A2: set_a,B2: set_list_a,F: a > list_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ ( image_a_list_a @ F @ A2 ) )
=> ( finite_finite_list_a @ B2 ) ) ) ).
% finite_surj
thf(fact_437_finite__surj,axiom,
! [A2: set_list_a,B2: set_list_a,F: list_a > list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ ( image_list_a_list_a @ F @ A2 ) )
=> ( finite_finite_list_a @ B2 ) ) ) ).
% finite_surj
thf(fact_438_finite__subset__image,axiom,
! [B2: set_list_list_a,F: list_a > list_list_a,A2: set_list_a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( ord_le8488217952732425610list_a @ B2 @ ( image_8260866953997875467list_a @ F @ A2 ) )
=> ? [C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
& ( finite_finite_list_a @ C2 )
& ( B2
= ( image_8260866953997875467list_a @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_439_finite__subset__image,axiom,
! [B2: set_a,F: a > a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_a_a @ F @ A2 ) )
=> ? [C2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
& ( finite_finite_a @ C2 )
& ( B2
= ( image_a_a @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_440_finite__subset__image,axiom,
! [B2: set_a,F: list_a > a,A2: set_list_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ ( image_list_a_a @ F @ A2 ) )
=> ? [C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
& ( finite_finite_list_a @ C2 )
& ( B2
= ( image_list_a_a @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_441_finite__subset__image,axiom,
! [B2: set_list_a,F: a > list_a,A2: set_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ ( image_a_list_a @ F @ A2 ) )
=> ? [C2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
& ( finite_finite_a @ C2 )
& ( B2
= ( image_a_list_a @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_442_finite__subset__image,axiom,
! [B2: set_list_a,F: list_a > list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ ( image_list_a_list_a @ F @ A2 ) )
=> ? [C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A2 )
& ( finite_finite_list_a @ C2 )
& ( B2
= ( image_list_a_list_a @ F @ C2 ) ) ) ) ) ).
% finite_subset_image
thf(fact_443_ex__finite__subset__image,axiom,
! [F: list_a > list_list_a,A2: set_list_a,P: set_list_list_a > $o] :
( ( ? [B3: set_list_list_a] :
( ( finite1660835950917165235list_a @ B3 )
& ( ord_le8488217952732425610list_a @ B3 @ ( image_8260866953997875467list_a @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_list_a] :
( ( finite_finite_list_a @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A2 )
& ( P @ ( image_8260866953997875467list_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_444_ex__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 )
& ( P @ ( image_a_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_445_ex__finite__subset__image,axiom,
! [F: list_a > a,A2: set_list_a,P: set_a > $o] :
( ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_list_a_a @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_list_a] :
( ( finite_finite_list_a @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A2 )
& ( P @ ( image_list_a_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_446_ex__finite__subset__image,axiom,
! [F: a > list_a,A2: set_a,P: set_list_a > $o] :
( ( ? [B3: set_list_a] :
( ( finite_finite_list_a @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ ( image_a_list_a @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 )
& ( P @ ( image_a_list_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_447_ex__finite__subset__image,axiom,
! [F: list_a > list_a,A2: set_list_a,P: set_list_a > $o] :
( ( ? [B3: set_list_a] :
( ( finite_finite_list_a @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ ( image_list_a_list_a @ F @ A2 ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_list_a] :
( ( finite_finite_list_a @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A2 )
& ( P @ ( image_list_a_list_a @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_448_all__finite__subset__image,axiom,
! [F: list_a > list_list_a,A2: set_list_a,P: set_list_list_a > $o] :
( ( ! [B3: set_list_list_a] :
( ( ( finite1660835950917165235list_a @ B3 )
& ( ord_le8488217952732425610list_a @ B3 @ ( image_8260866953997875467list_a @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_list_a] :
( ( ( finite_finite_list_a @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A2 ) )
=> ( P @ ( image_8260866953997875467list_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_449_all__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 ) )
=> ( P @ ( image_a_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_450_all__finite__subset__image,axiom,
! [F: list_a > a,A2: set_list_a,P: set_a > $o] :
( ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ ( image_list_a_a @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_list_a] :
( ( ( finite_finite_list_a @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A2 ) )
=> ( P @ ( image_list_a_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_451_all__finite__subset__image,axiom,
! [F: a > list_a,A2: set_a,P: set_list_a > $o] :
( ( ! [B3: set_list_a] :
( ( ( finite_finite_list_a @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ ( image_a_list_a @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A2 ) )
=> ( P @ ( image_a_list_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_452_all__finite__subset__image,axiom,
! [F: list_a > list_a,A2: set_list_a,P: set_list_a > $o] :
( ( ! [B3: set_list_a] :
( ( ( finite_finite_list_a @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ ( image_list_a_list_a @ F @ A2 ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_list_a] :
( ( ( finite_finite_list_a @ B3 )
& ( ord_le8861187494160871172list_a @ B3 @ A2 ) )
=> ( P @ ( image_list_a_list_a @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_453_r_Ominus__eq,axiom,
! [X: list_a,Y: list_a] :
( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ).
% r.minus_eq
thf(fact_454_r_Oadd_Oint__pow__diff,axiom,
! [X: list_a,N2: int,M2: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( minus_minus_int @ N2 @ M2 ) @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N2 @ X ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M2 @ X ) ) ) ) ) ).
% r.add.int_pow_diff
thf(fact_455_r_Oline__extension__in__carrier,axiom,
! [K: set_list_a,A: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% r.line_extension_in_carrier
thf(fact_456_r_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 ) ) ) ) ).
% r.genideal_one
thf(fact_457_principalideal_Ogenerate,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( princi8786919440553033881t_unit @ I2 @ R )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
& ( I2
= ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_458_principalideal_Ogenerate,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_459_principalideal_Ogenerate,axiom,
! [I2: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( princi2534607884127416211t_unit @ I2 @ R )
=> ? [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
& ( I2
= ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ X2 @ bot_bo1875519244922727510list_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_460_r_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 ) ) ) ) ) ).
% r.one_zeroI
thf(fact_461_r_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 ) ) ) ).
% r.one_zeroD
thf(fact_462_r_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 ) ) ) ) ) ).
% r.carrier_one_zero
thf(fact_463_r_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 ) ) ) ) ) ).
% r.carrier_one_not_zero
thf(fact_464_a__coset__hom_I1_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,I2: set_list_a,A: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
=> ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( image_list_a_list_a @ H @ ( a_l_co7008843373686234386t_unit @ R @ A @ I2 ) )
= ( a_l_co7008843373686234386t_unit @ S2 @ ( H @ A ) @ ( image_list_a_list_a @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_465_a__coset__hom_I1_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,I2: set_list_a,A: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
=> ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( image_list_a_a @ H @ ( a_l_co7008843373686234386t_unit @ R @ A @ I2 ) )
= ( a_l_coset_a_b @ S2 @ ( H @ A ) @ ( image_list_a_a @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_466_a__coset__hom_I1_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,I2: set_a,A: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( image_a_list_a @ H @ ( a_l_coset_a_b @ R @ A @ I2 ) )
= ( a_l_co7008843373686234386t_unit @ S2 @ ( H @ A ) @ ( image_a_list_a @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_467_a__coset__hom_I1_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,I2: set_a,A: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( image_a_a @ H @ ( a_l_coset_a_b @ R @ A @ I2 ) )
= ( a_l_coset_a_b @ S2 @ ( H @ A ) @ ( image_a_a @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_468_a__coset__hom_I1_J,axiom,
! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,I2: set_list_list_a,A: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S2 ) )
=> ( ( ord_le8488217952732425610list_a @ I2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( image_4177985004758493951list_a @ H @ ( a_l_co3970804650394549132t_unit @ R @ A @ I2 ) )
= ( a_l_co7008843373686234386t_unit @ S2 @ ( H @ A ) @ ( image_4177985004758493951list_a @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_469_a__coset__hom_I1_J,axiom,
! [H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,I2: set_list_list_a,A: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) )
=> ( ( ord_le8488217952732425610list_a @ I2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( image_list_list_a_a @ H @ ( a_l_co3970804650394549132t_unit @ R @ A @ I2 ) )
= ( a_l_coset_a_b @ S2 @ ( H @ A ) @ ( image_list_list_a_a @ H @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_470_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_471_subset__Idl__subset,axiom,
! [I2: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_472_a__l__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_l_coset_subset_G
thf(fact_473_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_474_Idl__subset__ideal_H,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_475_r_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 ) ) ) ).
% r.one_closed
thf(fact_476_r_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 ) ) ) ) ) ) ).
% r.minus_closed
thf(fact_477_r_Or__right__minus__eq,axiom,
! [A: list_a,B: list_a] :
( ( 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 ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A = B ) ) ) ) ).
% r.r_right_minus_eq
thf(fact_478_ring__iso__memE_I4_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S2 ) )
=> ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ).
% ring_iso_memE(4)
thf(fact_479_ring__iso__memE_I4_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S2 ) )
=> ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ).
% ring_iso_memE(4)
thf(fact_480_ring__iso__memE_I4_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S2 ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ).
% ring_iso_memE(4)
thf(fact_481_ring__iso__memE_I4_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S2 ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ).
% ring_iso_memE(4)
thf(fact_482_principalideal_Ois__principalideal,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( princi8786919440553033881t_unit @ I2 @ R )
=> ( princi8786919440553033881t_unit @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_483_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_484_ring__iso__memE_I1_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_485_ring__iso__memE_I1_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_486_ring__iso__memE_I1_J,axiom,
! [H: list_a > list_list_a,R: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a] :
( ( member6714375691612171394list_a @ H @ ( ring_i7582117978422105628t_unit @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_487_ring__iso__memE_I1_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_488_ring__iso__memE_I1_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_489_ring__iso__memE_I1_J,axiom,
! [H: a > list_list_a,R: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a] :
( ( member_a_list_list_a @ H @ ( ring_i4464730343205239444t_unit @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_490_ring__iso__memE_I1_J,axiom,
! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,X: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_i4611353245267337884t_unit @ R @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_491_ring__iso__memE_I1_J,axiom,
! [H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_i5684343068699926420it_a_b @ R @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_492_ring__iso__memE_I1_J,axiom,
! [H: list_list_a > list_list_a,R: partia2956882679547061052t_unit,S2: partia2956882679547061052t_unit,X: list_list_a] :
( ( member8231385768148312316list_a @ H @ ( ring_i6186174840089424918t_unit @ R @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_493_ring__iso__memE_I3_J,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_i7414513579304222626t_unit @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_494_ring__iso__memE_I3_J,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109658it_a_b @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_495_ring__iso__memE_I3_J,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_i4557880751517319194t_unit @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_496_ring__iso__memE_I3_J,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_iso_a_b_a_b @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_497_ring__iso__memE_I3_J,axiom,
! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_i4611353245267337884t_unit @ R @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_498_ring__iso__memE_I3_J,axiom,
! [H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_i5684343068699926420it_a_b @ R @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_499_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_500_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_501_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_502_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_503_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_504_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
!= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_505_semiring_Ocarrier__one__not__zero,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( partia2464479390973590831t_unit @ R )
!= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) )
= ( ( one_li8234411390022467901t_unit @ R )
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.carrier_one_not_zero
thf(fact_506_semiring_Ocarrier__one__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_507_semiring_Ocarrier__one__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_508_semiring_Ocarrier__one__zero,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( partia2464479390973590831t_unit @ R )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) )
= ( ( one_li8234411390022467901t_unit @ R )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.carrier_one_zero
thf(fact_509_semiring_Oone__zeroI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_510_semiring_Oone__zeroI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_511_semiring_Oone__zeroI,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( partia2464479390973590831t_unit @ R )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) )
=> ( ( one_li8234411390022467901t_unit @ R )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.one_zeroI
thf(fact_512_semiring_Oone__zeroD,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( ( one_li8328186300101108157t_unit @ R )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( partia5361259788508890537t_unit @ R )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_513_semiring_Oone__zeroD,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
= ( zero_a_b @ R ) )
=> ( ( partia707051561876973205xt_a_b @ R )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_514_semiring_Oone__zeroD,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( ( one_li8234411390022467901t_unit @ R )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( partia2464479390973590831t_unit @ R )
= ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% semiring.one_zeroD
thf(fact_515_r_Oadd_Oint__pow__neg,axiom,
! [X: list_a,I: int] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( uminus_uminus_int @ I ) @ X )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X ) ) ) ) ).
% r.add.int_pow_neg
thf(fact_516_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_517_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_518_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_519_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_520_local_Ominus__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_521_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_522_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_523_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_524_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_525_a__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_526_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_527_a__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_528_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_529_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_530_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_531_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_532_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_533_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_534_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_535_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_536_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_537_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_538_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_539_Diff__empty,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Diff_empty
thf(fact_540_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_541_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_542_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_543_Diff__cancel,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ A2 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_544_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_545_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_546_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_547_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_548_insert__Diff1,axiom,
! [X: list_list_a,B2: set_list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ X @ B2 )
=> ( ( minus_5335179877275218001list_a @ ( insert_list_list_a @ X @ A2 ) @ B2 )
= ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_549_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_550_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_551_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_552_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_553_Diff__insert0,axiom,
! [X: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ~ ( member_list_list_a @ X @ A2 )
=> ( ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ X @ B2 ) )
= ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_554_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_555_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_556_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_557_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_558_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_559_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_560_a__lcos__m__assoc,axiom,
! [M: set_a,G: a,H: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H @ M ) )
= ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H ) @ M ) ) ) ) ) ).
% a_lcos_m_assoc
thf(fact_561_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_562_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_563_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_564_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_565_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_566_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_567_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_568_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_569_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_570_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_571_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_572_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_573_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_574_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_575_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_576_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_577_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_578_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_579_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_580_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_581_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_582_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_583_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_584_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_585_insert__Diff__single,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_586_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_587_finite__Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B2: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B2 ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_588_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_589_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_590_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_591_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_592_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_593_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_594_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_595_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_596_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_597_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_598_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_599_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_600_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_601_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_602_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_603_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_604_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_605_group__cancel_Oneg1,axiom,
! [A2: int,K2: int,A: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_606_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_607_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_608_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_609_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_610_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_611_Diff__subset,axiom,
! [A2: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_612_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_613_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_614_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_615_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_616_insert__Diff__if,axiom,
! [X: list_list_a,B2: set_list_list_a,A2: set_list_list_a] :
( ( ( member_list_list_a @ X @ B2 )
=> ( ( minus_5335179877275218001list_a @ ( insert_list_list_a @ X @ A2 ) @ B2 )
= ( minus_5335179877275218001list_a @ A2 @ B2 ) ) )
& ( ~ ( member_list_list_a @ X @ B2 )
=> ( ( minus_5335179877275218001list_a @ ( insert_list_list_a @ X @ A2 ) @ B2 )
= ( insert_list_list_a @ X @ ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_617_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_618_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_619_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_620_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_621_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A6: int,B5: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B5 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_622_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A6: int,B5: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B5 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_623_group__cancel_Osub2,axiom,
! [B2: int,K2: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K2 @ B ) )
=> ( ( minus_minus_int @ A @ B2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_624_image__diff__subset,axiom,
! [F: list_a > list_list_a,A2: set_list_a,B2: set_list_a] : ( ord_le8488217952732425610list_a @ ( minus_5335179877275218001list_a @ ( image_8260866953997875467list_a @ F @ A2 ) @ ( image_8260866953997875467list_a @ F @ B2 ) ) @ ( image_8260866953997875467list_a @ F @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_625_image__diff__subset,axiom,
! [F: a > a,A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B2 ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_626_image__diff__subset,axiom,
! [F: list_a > a,A2: set_list_a,B2: set_list_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_list_a_a @ F @ A2 ) @ ( image_list_a_a @ F @ B2 ) ) @ ( image_list_a_a @ F @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_627_image__diff__subset,axiom,
! [F: a > list_a,A2: set_a,B2: set_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ ( image_a_list_a @ F @ A2 ) @ ( image_a_list_a @ F @ B2 ) ) @ ( image_a_list_a @ F @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_628_image__diff__subset,axiom,
! [F: list_a > list_a,A2: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ ( image_list_a_list_a @ F @ A2 ) @ ( image_list_a_list_a @ F @ B2 ) ) @ ( image_list_a_list_a @ F @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% image_diff_subset
thf(fact_629_subset__Diff__insert,axiom,
! [A2: set_list_list_a,B2: set_list_list_a,X: list_list_a,C3: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( minus_5335179877275218001list_a @ B2 @ ( insert_list_list_a @ X @ C3 ) ) )
= ( ( ord_le8488217952732425610list_a @ A2 @ ( minus_5335179877275218001list_a @ B2 @ C3 ) )
& ~ ( member_list_list_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_630_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_631_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_632_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_633_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_634_in__image__insert__iff,axiom,
! [B2: set_set_list_list_a,X: list_list_a,A2: set_list_list_a] :
( ! [C2: set_list_list_a] :
( ( member334759470184282131list_a @ C2 @ B2 )
=> ~ ( member_list_list_a @ X @ C2 ) )
=> ( ( member334759470184282131list_a @ A2 @ ( image_452714708507473285list_a @ ( insert_list_list_a @ X ) @ B2 ) )
= ( ( member_list_list_a @ X @ A2 )
& ( member334759470184282131list_a @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ X @ bot_bo1875519244922727510list_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_635_in__image__insert__iff,axiom,
! [B2: set_set_nat_list_a,X: nat > list_a,A2: set_nat_list_a] :
( ! [C2: set_nat_list_a] :
( ( member8163584906436000884list_a @ C2 @ B2 )
=> ~ ( member_nat_list_a @ X @ C2 ) )
=> ( ( member8163584906436000884list_a @ A2 @ ( image_3157922941172939973list_a @ ( insert_nat_list_a @ X ) @ B2 ) )
= ( ( member_nat_list_a @ X @ A2 )
& ( member8163584906436000884list_a @ ( minus_4169782841487898290list_a @ A2 @ ( insert_nat_list_a @ X @ bot_bo3806784159821827511list_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_636_in__image__insert__iff,axiom,
! [B2: set_set_nat_a,X: nat > a,A2: set_nat_a] :
( ! [C2: set_nat_a] :
( ( member_set_nat_a @ C2 @ B2 )
=> ~ ( member_nat_a @ X @ C2 ) )
=> ( ( member_set_nat_a @ A2 @ ( image_6965494298868581957_nat_a @ ( insert_nat_a @ X ) @ B2 ) )
= ( ( member_nat_a @ X @ A2 )
& ( member_set_nat_a @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ X @ bot_bot_set_nat_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_637_in__image__insert__iff,axiom,
! [B2: set_set_a,X: a,A2: set_a] :
( ! [C2: set_a] :
( ( member_set_a @ C2 @ B2 )
=> ~ ( member_a @ X @ C2 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X ) @ B2 ) )
= ( ( member_a @ X @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_638_in__image__insert__iff,axiom,
! [B2: set_set_list_a,X: list_a,A2: set_list_a] :
( ! [C2: set_list_a] :
( ( member_set_list_a @ C2 @ B2 )
=> ~ ( member_list_a @ X @ C2 ) )
=> ( ( member_set_list_a @ A2 @ ( image_5749939591322298757list_a @ ( insert_list_a @ X ) @ B2 ) )
= ( ( member_list_a @ X @ A2 )
& ( member_set_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) @ B2 ) ) ) ) ).
% in_image_insert_iff
thf(fact_639_Diff__insert__absorb,axiom,
! [X: list_list_a,A2: set_list_list_a] :
( ~ ( member_list_list_a @ X @ A2 )
=> ( ( minus_5335179877275218001list_a @ ( insert_list_list_a @ X @ A2 ) @ ( insert_list_list_a @ X @ bot_bo1875519244922727510list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_640_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_641_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_642_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_643_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_644_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_645_Diff__insert2,axiom,
! [A2: set_list_a,A: list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_646_insert__Diff,axiom,
! [A: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ A2 )
=> ( ( insert_list_list_a @ A @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_647_insert__Diff,axiom,
! [A: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ A @ A2 )
=> ( ( insert_nat_list_a @ A @ ( minus_4169782841487898290list_a @ A2 @ ( insert_nat_list_a @ A @ bot_bo3806784159821827511list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_648_insert__Diff,axiom,
! [A: nat > a,A2: set_nat_a] :
( ( member_nat_a @ A @ A2 )
=> ( ( insert_nat_a @ A @ ( minus_490503922182417452_nat_a @ A2 @ ( insert_nat_a @ A @ bot_bot_set_nat_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_649_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_650_insert__Diff,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_651_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_652_Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_653_subset__insert__iff,axiom,
! [A2: set_list_list_a,X: list_list_a,B2: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ X @ B2 ) )
= ( ( ( member_list_list_a @ X @ A2 )
=> ( ord_le8488217952732425610list_a @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ X @ bot_bo1875519244922727510list_a ) ) @ B2 ) )
& ( ~ ( member_list_list_a @ X @ A2 )
=> ( ord_le8488217952732425610list_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_654_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_655_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_656_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_657_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_658_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_659_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_660_infinite__remove,axiom,
! [S2: set_a,A: a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_661_infinite__remove,axiom,
! [S2: set_list_a,A: list_a] :
( ~ ( finite_finite_list_a @ S2 )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_662_infinite__coinduct,axiom,
! [X5: set_a > $o,A2: set_a] :
( ( X5 @ A2 )
=> ( ! [A4: set_a] :
( ( X5 @ A4 )
=> ? [X4: a] :
( ( member_a @ X4 @ A4 )
& ( ( X5 @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_663_infinite__coinduct,axiom,
! [X5: set_list_a > $o,A2: set_list_a] :
( ( X5 @ A2 )
=> ( ! [A4: set_list_a] :
( ( X5 @ A4 )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ A4 )
& ( ( X5 @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_664_finite__empty__induct,axiom,
! [A2: set_list_list_a,P: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: list_list_a,A4: set_list_list_a] :
( ( finite1660835950917165235list_a @ A4 )
=> ( ( member_list_list_a @ A5 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_5335179877275218001list_a @ A4 @ ( insert_list_list_a @ A5 @ bot_bo1875519244922727510list_a ) ) ) ) ) )
=> ( P @ bot_bo1875519244922727510list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_665_finite__empty__induct,axiom,
! [A2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: nat > list_a,A4: set_nat_list_a] :
( ( finite7630042315537210004list_a @ A4 )
=> ( ( member_nat_list_a @ A5 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_4169782841487898290list_a @ A4 @ ( insert_nat_list_a @ A5 @ bot_bo3806784159821827511list_a ) ) ) ) ) )
=> ( P @ bot_bo3806784159821827511list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_666_finite__empty__induct,axiom,
! [A2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: nat > a,A4: set_nat_a] :
( ( finite_finite_nat_a @ A4 )
=> ( ( member_nat_a @ A5 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_490503922182417452_nat_a @ A4 @ ( insert_nat_a @ A5 @ bot_bot_set_nat_a ) ) ) ) ) )
=> ( P @ bot_bot_set_nat_a ) ) ) ) ).
% finite_empty_induct
thf(fact_667_finite__empty__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: a,A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A5 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ) )
=> ( P @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_668_finite__empty__induct,axiom,
! [A2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ A2 )
=> ( ( P @ A2 )
=> ( ! [A5: list_a,A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ( member_list_a @ A5 @ A4 )
=> ( ( P @ A4 )
=> ( P @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ A5 @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_669_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_670_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_671_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_672_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_673_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_674_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_675_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_676_group__cancel_Oadd1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_677_group__cancel_Oadd1,axiom,
! [A2: nat,K2: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_678_group__cancel_Oadd2,axiom,
! [B2: int,K2: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K2 @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_679_group__cancel_Oadd2,axiom,
! [B2: nat,K2: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_680_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_681_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_682_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_683_group__add__class_Oadd_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% group_add_class.add.right_cancel
thf(fact_684_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A6: int,B5: int] : ( plus_plus_int @ B5 @ A6 ) ) ) ).
% add.commute
thf(fact_685_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A6: nat,B5: nat] : ( plus_plus_nat @ B5 @ A6 ) ) ) ).
% add.commute
thf(fact_686_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_687_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_688_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_689_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_690_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_691_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_692_remove__induct,axiom,
! [P: set_list_list_a > $o,B2: set_list_list_a] :
( ( P @ bot_bo1875519244922727510list_a )
=> ( ( ~ ( finite1660835950917165235list_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A4: set_list_list_a] :
( ( finite1660835950917165235list_a @ A4 )
=> ( ( A4 != bot_bo1875519244922727510list_a )
=> ( ( ord_le8488217952732425610list_a @ A4 @ B2 )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ A4 )
=> ( P @ ( minus_5335179877275218001list_a @ A4 @ ( insert_list_list_a @ X4 @ bot_bo1875519244922727510list_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_693_remove__induct,axiom,
! [P: set_nat_list_a > $o,B2: set_nat_list_a] :
( ( P @ bot_bo3806784159821827511list_a )
=> ( ( ~ ( finite7630042315537210004list_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A4: set_nat_list_a] :
( ( finite7630042315537210004list_a @ A4 )
=> ( ( A4 != bot_bo3806784159821827511list_a )
=> ( ( ord_le2145805922479659755list_a @ A4 @ B2 )
=> ( ! [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A4 )
=> ( P @ ( minus_4169782841487898290list_a @ A4 @ ( insert_nat_list_a @ X4 @ bot_bo3806784159821827511list_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_694_remove__induct,axiom,
! [P: set_nat_a > $o,B2: set_nat_a] :
( ( P @ bot_bot_set_nat_a )
=> ( ( ~ ( finite_finite_nat_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A4: set_nat_a] :
( ( finite_finite_nat_a @ A4 )
=> ( ( A4 != bot_bot_set_nat_a )
=> ( ( ord_le871467723717165285_nat_a @ A4 @ B2 )
=> ( ! [X4: nat > a] :
( ( member_nat_a @ X4 @ A4 )
=> ( P @ ( minus_490503922182417452_nat_a @ A4 @ ( insert_nat_a @ X4 @ bot_bot_set_nat_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_695_remove__induct,axiom,
! [P: set_a > $o,B2: set_a] :
( ( P @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_696_remove__induct,axiom,
! [P: set_list_a > $o,B2: set_list_a] :
( ( P @ bot_bot_set_list_a )
=> ( ( ~ ( finite_finite_list_a @ B2 )
=> ( P @ B2 ) )
=> ( ! [A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ( A4 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A4 @ B2 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A4 )
=> ( P @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% remove_induct
thf(fact_697_finite__remove__induct,axiom,
! [B2: set_list_list_a,P: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( P @ bot_bo1875519244922727510list_a )
=> ( ! [A4: set_list_list_a] :
( ( finite1660835950917165235list_a @ A4 )
=> ( ( A4 != bot_bo1875519244922727510list_a )
=> ( ( ord_le8488217952732425610list_a @ A4 @ B2 )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ A4 )
=> ( P @ ( minus_5335179877275218001list_a @ A4 @ ( insert_list_list_a @ X4 @ bot_bo1875519244922727510list_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_698_finite__remove__induct,axiom,
! [B2: set_nat_list_a,P: set_nat_list_a > $o] :
( ( finite7630042315537210004list_a @ B2 )
=> ( ( P @ bot_bo3806784159821827511list_a )
=> ( ! [A4: set_nat_list_a] :
( ( finite7630042315537210004list_a @ A4 )
=> ( ( A4 != bot_bo3806784159821827511list_a )
=> ( ( ord_le2145805922479659755list_a @ A4 @ B2 )
=> ( ! [X4: nat > list_a] :
( ( member_nat_list_a @ X4 @ A4 )
=> ( P @ ( minus_4169782841487898290list_a @ A4 @ ( insert_nat_list_a @ X4 @ bot_bo3806784159821827511list_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_699_finite__remove__induct,axiom,
! [B2: set_nat_a,P: set_nat_a > $o] :
( ( finite_finite_nat_a @ B2 )
=> ( ( P @ bot_bot_set_nat_a )
=> ( ! [A4: set_nat_a] :
( ( finite_finite_nat_a @ A4 )
=> ( ( A4 != bot_bot_set_nat_a )
=> ( ( ord_le871467723717165285_nat_a @ A4 @ B2 )
=> ( ! [X4: nat > a] :
( ( member_nat_a @ X4 @ A4 )
=> ( P @ ( minus_490503922182417452_nat_a @ A4 @ ( insert_nat_a @ X4 @ bot_bot_set_nat_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_700_finite__remove__induct,axiom,
! [B2: set_a,P: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P @ bot_bot_set_a )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( P @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_701_finite__remove__induct,axiom,
! [B2: set_list_a,P: set_list_a > $o] :
( ( finite_finite_list_a @ B2 )
=> ( ( P @ bot_bot_set_list_a )
=> ( ! [A4: set_list_a] :
( ( finite_finite_list_a @ A4 )
=> ( ( A4 != bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ A4 @ B2 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A4 )
=> ( P @ ( minus_646659088055828811list_a @ A4 @ ( insert_list_a @ X4 @ bot_bot_set_list_a ) ) ) )
=> ( P @ A4 ) ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_702_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_703_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_704_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_705_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_706_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_707_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_708_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_709_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_710_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_711_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_712_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_713_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_714_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_715_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_716_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C5: nat] :
( B
!= ( plus_plus_nat @ A @ C5 ) ) ) ).
% less_eqE
thf(fact_717_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_718_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_719_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B5: nat] :
? [C6: nat] :
( B5
= ( plus_plus_nat @ A6 @ C6 ) ) ) ) ).
% le_iff_add
thf(fact_720_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_721_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_722_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_723_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_724_group__cancel_Osub1,axiom,
! [A2: int,K2: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K2 @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_725_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_726_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_727_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_728_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_729_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_730_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_731_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_732_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_733_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_734_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_735_ring__hom__closed,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_736_ring__hom__closed,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_737_ring__hom__closed,axiom,
! [H: list_a > list_list_a,R: partia2670972154091845814t_unit,S2: partia2956882679547061052t_unit,X: list_a] :
( ( member6714375691612171394list_a @ H @ ( ring_h8002040739877300486t_unit @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_738_ring__hom__closed,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_739_ring__hom__closed,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_740_ring__hom__closed,axiom,
! [H: a > list_list_a,R: partia2175431115845679010xt_a_b,S2: partia2956882679547061052t_unit,X: a] :
( ( member_a_list_list_a @ H @ ( ring_h6858658657455840382t_unit @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_741_ring__hom__closed,axiom,
! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,X: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_a @ ( H @ X ) @ ( partia5361259788508890537t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_742_ring__hom__closed,axiom,
! [H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H @ X ) @ ( partia707051561876973205xt_a_b @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_743_ring__hom__closed,axiom,
! [H: list_list_a > list_list_a,R: partia2956882679547061052t_unit,S2: partia2956882679547061052t_unit,X: list_list_a] :
( ( member8231385768148312316list_a @ H @ ( ring_h8129544334414776832t_unit @ R @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( H @ X ) @ ( partia2464479390973590831t_unit @ S2 ) ) ) ) ).
% ring_hom_closed
thf(fact_744_ring__hom__one,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
=> ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ).
% ring_hom_one
thf(fact_745_ring__hom__one,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
=> ( ( H @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ).
% ring_hom_one
thf(fact_746_ring__hom__one,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S2 ) ) ) ).
% ring_hom_one
thf(fact_747_ring__hom__one,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
=> ( ( H @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S2 ) ) ) ).
% ring_hom_one
thf(fact_748_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_749_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_750_add__le__imp__le__diff,axiom,
! [I: int,K2: int,N2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N2 )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_751_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_752_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_753_add__le__add__imp__diff__le,axiom,
! [I: int,K2: int,N2: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N2 )
=> ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K2 ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N2 )
=> ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K2 ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_754_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N2: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_755_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_756_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_757_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_758_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_759_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_760_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_761_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_762_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_763_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_764_ring__hom__add,axiom,
! [H: list_a > list_a,R: partia2670972154091845814t_unit,S2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( member_list_a_list_a @ H @ ( ring_h7399960747407462284t_unit @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_765_ring__hom__add,axiom,
! [H: list_a > a,R: partia2670972154091845814t_unit,S2: partia2175431115845679010xt_a_b,X: list_a,Y: list_a] :
( ( member_list_a_a @ H @ ( ring_h2895973938487309444it_a_b @ R @ S2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) )
= ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_766_ring__hom__add,axiom,
! [H: a > list_a,R: partia2175431115845679010xt_a_b,S2: partia2670972154091845814t_unit,X: a,Y: a] :
( ( member_a_list_a @ H @ ( ring_h405018892823518980t_unit @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_767_ring__hom__add,axiom,
! [H: a > a,R: partia2175431115845679010xt_a_b,S2: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( member_a_a @ H @ ( ring_hom_a_b_a_b @ R @ S2 ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H @ ( add_a_b @ R @ X @ Y ) )
= ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_768_ring__hom__add,axiom,
! [H: list_list_a > list_a,R: partia2956882679547061052t_unit,S2: partia2670972154091845814t_unit,X: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H @ ( ring_h5031276006722532742t_unit @ R @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_769_ring__hom__add,axiom,
! [H: list_list_a > a,R: partia2956882679547061052t_unit,S2: partia2175431115845679010xt_a_b,X: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H @ ( ring_h8078271382950527358it_a_b @ R @ S2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H @ ( add_li174743652000525320t_unit @ R @ X @ Y ) )
= ( add_a_b @ S2 @ ( H @ X ) @ ( H @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_770_Ring_Oone__not__zero,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( one_se211549098623999037t_unit @ R )
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_771_Ring_Oone__not__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_772_Ring_Oone__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_773_Ring_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_774_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_775_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_776_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_777_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R @ Y @ ( add_li7652885771158616974t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_778_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
= ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_779_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ R @ Y @ ( add_li174743652000525320t_unit @ R @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_780_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ Y )
= ( add_li7652885771158616974t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_781_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ Y )
= ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_782_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ Y )
= ( add_li174743652000525320t_unit @ R @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_783_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R @ X @ ( add_li7652885771158616974t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_784_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a,Z: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
= ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_785_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R @ X @ ( add_li174743652000525320t_unit @ R @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_786_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_787_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a,Y: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_788_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_789_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R2: partia2670972154091845814t_unit,X3: list_a,Y5: list_a] : ( add_li7652885771158616974t_unit @ R2 @ X3 @ ( a_inv_8944721093294617173t_unit @ R2 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_790_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,X3: a,Y5: a] : ( add_a_b @ R2 @ X3 @ ( a_inv_a_b @ R2 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_791_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_792_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_793_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_794_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X @ ( zero_l4142658623432671053t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_795_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_796_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X @ ( zero_l347298301471573063t_unit @ R ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_797_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_798_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_799_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_800_a,axiom,
! [X: a] :
( ( member_a @ X @ s )
=> ( member_list_a @ ( lagran2649660974587678107al_a_b @ r @ ( minus_minus_set_a @ s @ ( insert_a @ X @ bot_bot_set_a ) ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% a
thf(fact_801_lagrange__poly,axiom,
! [S2: set_a,X: a] :
( ( finite_finite_a @ S2 )
=> ( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ S2 ) )
=> ( member_list_a @ ( lagran2649660974587678107al_a_b @ r @ S2 @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% lagrange_poly
thf(fact_802_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_803_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_804_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_805_ring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( P2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_806_ring__irreducibleE_I1_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( R3
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_807_cgenideal__eq__genideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I )
= ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_808_minus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus_a_b @ r @ X @ Y )
= ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% minus_eq
thf(fact_809_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_810_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_811_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_812_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_813_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_814_a__transpose__inv,axiom,
! [X: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_815_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_816_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_817_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_818_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_819_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_820_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_821_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_822_sum__zero__eq__neg,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( X
= ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_823_Compl__subset__Compl__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B2 ) )
= ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_824_Compl__subset__Compl__iff,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ A2 ) @ ( uminus7925729386456332763list_a @ B2 ) )
= ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_825_Compl__anti__mono,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B2 ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_826_Compl__anti__mono,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ord_le8861187494160871172list_a @ ( uminus7925729386456332763list_a @ B2 ) @ ( uminus7925729386456332763list_a @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_827_DiffI,axiom,
! [C: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ A2 )
=> ( ~ ( member_list_list_a @ C @ B2 )
=> ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_828_DiffI,axiom,
! [C: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ A2 )
=> ( ~ ( member_nat_list_a @ C @ B2 )
=> ( member_nat_list_a @ C @ ( minus_4169782841487898290list_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_829_DiffI,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ A2 )
=> ( ~ ( member_nat_a @ C @ B2 )
=> ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_830_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_831_DiffI,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_832_Diff__iff,axiom,
! [C: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A2 @ B2 ) )
= ( ( member_list_list_a @ C @ A2 )
& ~ ( member_list_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_833_Diff__iff,axiom,
! [C: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( minus_4169782841487898290list_a @ A2 @ B2 ) )
= ( ( member_nat_list_a @ C @ A2 )
& ~ ( member_nat_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_834_Diff__iff,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
= ( ( member_nat_a @ C @ A2 )
& ~ ( member_nat_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_835_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_836_Diff__iff,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
= ( ( member_list_a @ C @ A2 )
& ~ ( member_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_837_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_838_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_839_primeness__condition,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeness_condition
thf(fact_840_cgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_841_add_Oone__in__subset,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ H2 ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H2 )
=> ! [Xa: a] :
( ( member_a @ Xa @ H2 )
=> ( member_a @ ( add_a_b @ r @ X2 @ Xa ) @ H2 ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).
% add.one_in_subset
thf(fact_842_irreducible__imp__maximalideal,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r ) ) ) ).
% irreducible_imp_maximalideal
thf(fact_843_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_844_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_845_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_846_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_847_subset__Compl__singleton,axiom,
! [A2: set_list_list_a,B: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( uminus4049073354455507169list_a @ ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a ) ) )
= ( ~ ( member_list_list_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_848_subset__Compl__singleton,axiom,
! [A2: set_nat_list_a,B: nat > list_a] :
( ( ord_le2145805922479659755list_a @ A2 @ ( uminus8475698304107272002list_a @ ( insert_nat_list_a @ B @ bot_bo3806784159821827511list_a ) ) )
= ( ~ ( member_nat_list_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_849_subset__Compl__singleton,axiom,
! [A2: set_nat_a,B: nat > a] :
( ( ord_le871467723717165285_nat_a @ A2 @ ( uminus5121639726568051900_nat_a @ ( insert_nat_a @ B @ bot_bot_set_nat_a ) ) )
= ( ~ ( member_nat_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_850_subset__Compl__singleton,axiom,
! [A2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_851_subset__Compl__singleton,axiom,
! [A2: set_list_a,B: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( uminus7925729386456332763list_a @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( ~ ( member_list_a @ B @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_852_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_853_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_854_DiffE,axiom,
! [C: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A2 @ B2 ) )
=> ~ ( ( member_list_list_a @ C @ A2 )
=> ( member_list_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_855_DiffE,axiom,
! [C: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( minus_4169782841487898290list_a @ A2 @ B2 ) )
=> ~ ( ( member_nat_list_a @ C @ A2 )
=> ( member_nat_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_856_DiffE,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ~ ( ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_857_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_858_DiffE,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ~ ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_859_DiffD1,axiom,
! [C: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A2 @ B2 ) )
=> ( member_list_list_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_860_DiffD1,axiom,
! [C: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( minus_4169782841487898290list_a @ A2 @ B2 ) )
=> ( member_nat_list_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_861_DiffD1,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ( member_nat_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_862_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_863_DiffD1,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ( member_list_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_864_DiffD2,axiom,
! [C: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A2 @ B2 ) )
=> ~ ( member_list_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_865_DiffD2,axiom,
! [C: nat > list_a,A2: set_nat_list_a,B2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( minus_4169782841487898290list_a @ A2 @ B2 ) )
=> ~ ( member_nat_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_866_DiffD2,axiom,
! [C: nat > a,A2: set_nat_a,B2: set_nat_a] :
( ( member_nat_a @ C @ ( minus_490503922182417452_nat_a @ A2 @ B2 ) )
=> ~ ( member_nat_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_867_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_868_DiffD2,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ~ ( member_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_869_ring_Olagrange__basis__polynomial_Ocong,axiom,
lagran2649660974587678107al_a_b = lagran2649660974587678107al_a_b ).
% ring.lagrange_basis_polynomial.cong
thf(fact_870_maximalideal_Ois__maximalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( maximalideal_a_b @ I2 @ R ) ) ).
% maximalideal.is_maximalideal
thf(fact_871_maximalideal_Ois__maximalideal,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( maxima6585700282301356660t_unit @ I2 @ R )
=> ( maxima6585700282301356660t_unit @ I2 @ R ) ) ).
% maximalideal.is_maximalideal
thf(fact_872_maximalideal_OI__notcarr,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( maxima6585700282301356660t_unit @ I2 @ R )
=> ( ( partia5361259788508890537t_unit @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_873_maximalideal_OI__notcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_874_maximalideal_OI__notcarr,axiom,
! [I2: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( maxima7552488817642790894t_unit @ I2 @ R )
=> ( ( partia2464479390973590831t_unit @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_875_subset__Compl__self__eq,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_876_subset__Compl__self__eq,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( uminus7925729386456332763list_a @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_Compl_self_eq
thf(fact_877_Compl__insert,axiom,
! [X: a,A2: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X @ A2 ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_878_Compl__insert,axiom,
! [X: list_a,A2: set_list_a] :
( ( uminus7925729386456332763list_a @ ( insert_list_a @ X @ A2 ) )
= ( minus_646659088055828811list_a @ ( uminus7925729386456332763list_a @ A2 ) @ ( insert_list_a @ X @ bot_bot_set_list_a ) ) ) ).
% Compl_insert
thf(fact_879_field_Ozeromaximalideal,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( maxima2253313296322093082t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_880_field_Ozeromaximalideal,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( maxima3875439991530298004t_unit @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_881_field_Ozeromaximalideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_882_field_Ozeromaximalideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_883_field_Olagrange__poly,axiom,
! [R: partia6043505979758434576t_unit,S2: set_set_a,X: set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( finite_finite_set_a @ S2 )
=> ( ( ord_le3724670747650509150_set_a @ S2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ X @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ S2 ) )
=> ( member_list_set_a @ ( lagran5736318333021047625t_unit @ R @ S2 @ X ) @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ) ) ).
% field.lagrange_poly
thf(fact_884_field_Olagrange__poly,axiom,
! [R: partia7496981018696276118t_unit,S2: set_set_list_a,X: set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( finite5282473924520328461list_a @ S2 )
=> ( ( ord_le8877086941679407844list_a @ S2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ X @ ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ S2 ) )
=> ( member5524387281408368019list_a @ ( lagran8541024212194239043t_unit @ R @ S2 @ X ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ) ) ) ).
% field.lagrange_poly
thf(fact_885_field_Olagrange__poly,axiom,
! [R: partia2670972154091845814t_unit,S2: set_list_a,X: list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( finite_finite_list_a @ S2 )
=> ( ( ord_le8861187494160871172list_a @ S2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ X @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ S2 ) )
=> ( member_list_list_a @ ( lagran6985349428869127715t_unit @ R @ S2 @ X ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ) ).
% field.lagrange_poly
thf(fact_886_field_Olagrange__poly,axiom,
! [R: partia2956882679547061052t_unit,S2: set_list_list_a,X: list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( finite1660835950917165235list_a @ S2 )
=> ( ( ord_le8488217952732425610list_a @ S2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ X @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ S2 ) )
=> ( member5342144027231129785list_a @ ( lagran8662613185911980061t_unit @ R @ S2 @ X ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ) ).
% field.lagrange_poly
thf(fact_887_field_Olagrange__poly,axiom,
! [R: partia2175431115845679010xt_a_b,S2: set_a,X: a] :
( ( field_a_b @ R )
=> ( ( finite_finite_a @ S2 )
=> ( ( ord_less_eq_set_a @ S2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ S2 ) )
=> ( member_list_a @ ( lagran2649660974587678107al_a_b @ R @ S2 @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ) ).
% field.lagrange_poly
thf(fact_888_r_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 ) ) ) ) ).
% r.zeromaximalideal_eq_field
thf(fact_889_r_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 ) ) ) ) ).
% r.zeromaximalideal_fieldI
thf(fact_890_primeideal__iff__prime,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeideal_iff_prime
thf(fact_891_field__iff__prime,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
= ( ring_ring_prime_a_b @ r @ A ) ) ) ).
% field_iff_prime
thf(fact_892_ring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI
thf(fact_893_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_894_maximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ r )
=> ( primeideal_a_b @ I2 @ r ) ) ).
% maximalideal_prime
thf(fact_895_Compl__iff,axiom,
! [C: list_a,A2: set_list_a] :
( ( member_list_a @ C @ ( uminus7925729386456332763list_a @ A2 ) )
= ( ~ ( member_list_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_896_Compl__iff,axiom,
! [C: a,A2: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
= ( ~ ( member_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_897_Compl__iff,axiom,
! [C: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ C @ ( uminus4049073354455507169list_a @ A2 ) )
= ( ~ ( member_list_list_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_898_Compl__iff,axiom,
! [C: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( uminus8475698304107272002list_a @ A2 ) )
= ( ~ ( member_nat_list_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_899_Compl__iff,axiom,
! [C: nat > a,A2: set_nat_a] :
( ( member_nat_a @ C @ ( uminus5121639726568051900_nat_a @ A2 ) )
= ( ~ ( member_nat_a @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_900_ComplI,axiom,
! [C: list_a,A2: set_list_a] :
( ~ ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ ( uminus7925729386456332763list_a @ A2 ) ) ) ).
% ComplI
thf(fact_901_ComplI,axiom,
! [C: a,A2: set_a] :
( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% ComplI
thf(fact_902_ComplI,axiom,
! [C: list_list_a,A2: set_list_list_a] :
( ~ ( member_list_list_a @ C @ A2 )
=> ( member_list_list_a @ C @ ( uminus4049073354455507169list_a @ A2 ) ) ) ).
% ComplI
thf(fact_903_ComplI,axiom,
! [C: nat > list_a,A2: set_nat_list_a] :
( ~ ( member_nat_list_a @ C @ A2 )
=> ( member_nat_list_a @ C @ ( uminus8475698304107272002list_a @ A2 ) ) ) ).
% ComplI
thf(fact_904_ComplI,axiom,
! [C: nat > a,A2: set_nat_a] :
( ~ ( member_nat_a @ C @ A2 )
=> ( member_nat_a @ C @ ( uminus5121639726568051900_nat_a @ A2 ) ) ) ).
% ComplI
thf(fact_905_ring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).
% ring_primeE(3)
thf(fact_906_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_907_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_908_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_909_ComplD,axiom,
! [C: list_a,A2: set_list_a] :
( ( member_list_a @ C @ ( uminus7925729386456332763list_a @ A2 ) )
=> ~ ( member_list_a @ C @ A2 ) ) ).
% ComplD
thf(fact_910_ComplD,axiom,
! [C: a,A2: set_a] :
( ( member_a @ C @ ( uminus_uminus_set_a @ A2 ) )
=> ~ ( member_a @ C @ A2 ) ) ).
% ComplD
thf(fact_911_ComplD,axiom,
! [C: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ C @ ( uminus4049073354455507169list_a @ A2 ) )
=> ~ ( member_list_list_a @ C @ A2 ) ) ).
% ComplD
thf(fact_912_ComplD,axiom,
! [C: nat > list_a,A2: set_nat_list_a] :
( ( member_nat_list_a @ C @ ( uminus8475698304107272002list_a @ A2 ) )
=> ~ ( member_nat_list_a @ C @ A2 ) ) ).
% ComplD
thf(fact_913_ComplD,axiom,
! [C: nat > a,A2: set_nat_a] :
( ( member_nat_a @ C @ ( uminus5121639726568051900_nat_a @ A2 ) )
=> ~ ( member_nat_a @ C @ A2 ) ) ).
% ComplD
thf(fact_914_primeideal_Oprimeideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( primeideal_a_b @ I2 @ R ) ) ).
% primeideal.primeideal
thf(fact_915_primeideal_Oprimeideal,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( primei6309817859076077608t_unit @ I2 @ R ) ) ).
% primeideal.primeideal
thf(fact_916_primeideal_OI__notcarr,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( ( partia5361259788508890537t_unit @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_917_primeideal_OI__notcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_918_primeideal_OI__notcarr,axiom,
! [I2: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( primei2288432046033540002t_unit @ I2 @ R )
=> ( ( partia2464479390973590831t_unit @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_919_principal__domain_Ofield__iff__prime,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ A @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( field_1540243473349940225t_unit @ ( factRi7259693425559269476t_unit @ R @ ( cgenid24865672677839267t_unit @ R @ A ) ) )
= ( ring_r5437400583859147359t_unit @ R @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_920_principal__domain_Ofield__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) ) )
= ( ring_ring_prime_a_b @ R @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_921_principal__domain_Ofield__iff__prime,axiom,
! [R: partia2670972154091845814t_unit,A: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( field_26233345952514695t_unit @ ( factRi3329376332477095402t_unit @ R @ ( cgenid9131348535277946915t_unit @ R @ A ) ) )
= ( ring_r6430282645014804837t_unit @ R @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_922_r_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 ) ) ) ) ).
% r.maximalideal_prime
thf(fact_923_domain__iff__prime,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
= ( ring_ring_prime_a_b @ r @ A ) ) ) ).
% domain_iff_prime
thf(fact_924_r_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 ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( 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 @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.field_intro2
thf(fact_925_r_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 ) ) ) )
=> ( ! [A5: list_a] :
( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A5
!= ( 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 ) ) @ A5 @ 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 ) ) ) ) ) ).
% r.cring_fieldI2
thf(fact_926_r_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 ) ) ) ) ) ) ).
% r.m_assoc
thf(fact_927_r_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 ) ) ) ) ).
% r.m_comm
thf(fact_928_r_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 ) ) ) ) ) ) ).
% r.m_lcomm
thf(fact_929_r_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 ) ) ) ) ) ).
% r.Units_closed
thf(fact_930_r_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% r.l_distr
thf(fact_931_r_Or__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ Y ) ) ) ) ) ) ).
% r.r_distr
thf(fact_932_r_Ol__minus,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) ) ) ) ).
% r.l_minus
thf(fact_933_r_Or__minus,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) ) ) ) ).
% r.r_minus
thf(fact_934_r_Oprod__unit__l,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% r.prod_unit_l
thf(fact_935_r_Oprod__unit__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% r.prod_unit_r
thf(fact_936_r_Ounit__factor,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% r.unit_factor
thf(fact_937_r_Oinv__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% r.inv_unique
thf(fact_938_r_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.one_unique
thf(fact_939_r_Oadd__pow__ldistr__int,axiom,
! [A: list_a,B: list_a,K2: int] :
( ( 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 ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A ) @ B )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% r.add_pow_ldistr_int
thf(fact_940_r_Oadd__pow__rdistr__int,axiom,
! [A: list_a,B: list_a,K2: int] :
( ( 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 ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ B ) )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% r.add_pow_rdistr_int
thf(fact_941_r_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 ) ) ) ) ) ) ) ).
% r.Units_inv_comm
thf(fact_942_r_Oideal__eq__carrier__iff,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( 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 ) ) @ A ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.ideal_eq_carrier_iff
thf(fact_943_r_Oline__extension__mem__iff,axiom,
! [U: list_a,K: set_list_a,A: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) )
= ( ? [X3: list_a] :
( ( member_list_a @ X3 @ K )
& ? [Y5: list_a] :
( ( member_list_a @ Y5 @ E )
& ( U
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ A ) @ Y5 ) ) ) ) ) ) ).
% r.line_extension_mem_iff
thf(fact_944_r_OUnits__l__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.Units_l_inv_ex
thf(fact_945_r_OUnits__r__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.Units_r_inv_ex
thf(fact_946_r_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 ) ) ) ) ).
% r.cring_fieldI
thf(fact_947_r_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 ) ) ) ) ) ) ).
% r.m_closed
thf(fact_948_r_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 ) ) ) ) ) ).
% r.finite_ring_finite_units
thf(fact_949_r_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 ) ) ) ) ) ) ).
% r.Units_m_closed
thf(fact_950_r_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 ) ) ) ).
% r.Units_one_closed
thf(fact_951_r_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 ) ) ) ) ) ).
% r.l_null
thf(fact_952_r_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 ) ) ) ) ) ).
% r.r_null
thf(fact_953_r_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 ) ) ) ) ) ).
% r.Units_l_cancel
thf(fact_954_r_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 ) ) ).
% r.l_one
thf(fact_955_r_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 ) ) ).
% r.r_one
thf(fact_956_r_OUnits__minus__one__closed,axiom,
member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% r.Units_minus_one_closed
thf(fact_957_r_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 ) ) ).
% r.FactRing_zeroideal(2)
thf(fact_958_r_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 ) ) ).
% r.FactRing_zeroideal(1)
thf(fact_959_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia6043505979758434576t_unit,R3: set_a,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R3 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( R3
= ( mult_s7930653359683758801t_unit @ R @ A @ B ) )
=> ( ( member_set_a @ A @ ( units_2471184348132832486t_unit @ R ) )
| ( member_set_a @ B @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_960_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( R3
= ( mult_l4853965630390486993t_unit @ R @ A @ B ) )
=> ( ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ R ) )
| ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_961_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ R @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_962_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ R @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_963_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia6043505979758434576t_unit,R3: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R3 )
=> ~ ( member_set_a @ R3 @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_964_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R3 )
=> ~ ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_965_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R3 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_966_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R3 )
=> ~ ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_967_domain_Ointegral,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_968_domain_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_969_domain_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_970_domain_Ointegral,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( A
= ( zero_l347298301471573063t_unit @ R ) )
| ( B
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_971_domain_Om__lcancel,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( A
!= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( mult_s7930653359683758801t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_972_domain_Om__lcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( mult_l7073676228092353617t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_973_domain_Om__lcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( mult_a_ring_ext_a_b @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_974_domain_Om__lcancel,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( A
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( mult_l4853965630390486993t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_975_domain_Om__rcancel,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( A
!= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ B @ A )
= ( mult_s7930653359683758801t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_976_domain_Om__rcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ B @ A )
= ( mult_l7073676228092353617t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_977_domain_Om__rcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ B @ A )
= ( mult_a_ring_ext_a_b @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_978_domain_Om__rcancel,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( A
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ B @ A )
= ( mult_l4853965630390486993t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_979_domain_Ointegral__iff,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
= ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_980_domain_Ointegral__iff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_981_domain_Ointegral__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
= ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_982_domain_Ointegral__iff,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( zero_l347298301471573063t_unit @ R ) )
= ( ( A
= ( zero_l347298301471573063t_unit @ R ) )
| ( B
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_983_r_OsubdomainI,axiom,
! [H2: set_list_a] :
( ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H2 )
=> ( ( member_list_a @ H22 @ H2 )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( H1
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( H22
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.subdomainI
thf(fact_984_r_Oring__primeI,axiom,
! [P2: list_a] :
( ( P2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ).
% r.ring_primeI
thf(fact_985_r_Omonoid__cancelI,axiom,
( ! [A5: list_a,B6: list_a,C5: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C5 @ A5 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C5 @ B6 ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A5 = B6 ) ) ) ) )
=> ( ! [A5: list_a,B6: list_a,C5: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ C5 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B6 @ C5 ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A5 = B6 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.monoid_cancelI
thf(fact_986_is__root__poly__mult__imp__is__root,axiom,
! [P2: list_a,Q2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q2 ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q2 @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_987_r_Osubfield__m__inv__simprule,axiom,
! [K: set_list_a,K2: list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A ) @ K )
=> ( member_list_a @ A @ K ) ) ) ) ) ).
% r.subfield_m_inv_simprule
thf(fact_988_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_989_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_990_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_991_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_992_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_993_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_994_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_995_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_996_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_997_m__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_998_m__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_999_integral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_1000_local_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_1001_r__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_1002_l__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_1003_Units__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_1004_Units__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_1005_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_1006_inv__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_1007_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_1008_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_1009_ideal__eq__carrier__iff,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_1010_ring__irreducibleE_I5_J,axiom,
! [R3: a,A: a,B: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_1011_ring__irreducibleE_I4_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_1012_square__eq__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ r ) )
| ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% square_eq_one
thf(fact_1013_r_Osubring__props_I7_J,axiom,
! [K: set_list_a,H12: list_a,H23: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H12 @ K )
=> ( ( member_list_a @ H23 @ K )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H23 ) @ K ) ) ) ) ).
% r.subring_props(7)
thf(fact_1014_r_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 ) ) ).
% r.subring_props(2)
thf(fact_1015_r_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 ) ) ) ) ).
% r.subring_props(6)
thf(fact_1016_r_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 ) ) ).
% r.subring_props(4)
thf(fact_1017_r_Osubring__props_I5_J,axiom,
! [K: set_list_a,H: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H @ K )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ K ) ) ) ).
% r.subring_props(5)
thf(fact_1018_r_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 ) ) ).
% r.subring_props(3)
thf(fact_1019_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_1020_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A5
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A5 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_1021_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_1022_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_1023_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_1024_r_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 ) ) ) ) ) ).
% r.subring_props(1)
thf(fact_1025_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_1026_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_1027_ring__irreducibleI,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A5: a,B6: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B6 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ r @ A5 @ B6 ) )
=> ( ( member_a @ A5 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B6 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R3 ) ) ) ) ).
% ring_irreducibleI
thf(fact_1028_r_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 ) ) ) ) ).
% r.zeroprimeideal_domainI
thf(fact_1029_r_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 ) ) ) ) ).
% r.domain_eq_zeroprimeideal
thf(fact_1030_r_Oline__extension__smult__closed,axiom,
! [K: set_list_a,E: set_list_a,A: list_a,K2: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K3: list_a,V2: list_a] :
( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ V2 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) ) ) ) ) ) ) ) ).
% r.line_extension_smult_closed
thf(fact_1031_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_1032_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_1033_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_1034_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_1035_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_1036_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_1037_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_1038_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_1039_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_1040_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_1041_r_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 ) ) ) ) ) ) ).
% r.subfield_m_inv(3)
thf(fact_1042_r_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 ) ) ) ) ) ) ).
% r.subfield_m_inv(2)
thf(fact_1043_r_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_list_a,E: set_list_a,V: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ V @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ V ) ) ) ) ) ).
% r.subalbegra_incl_imp_finite_dimension
thf(fact_1044_carrier__is__subfield,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subfield
thf(fact_1045_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_1046_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_1047_subring__props_I7_J,axiom,
! [K: set_a,H12: a,H23: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H12 @ K )
=> ( ( member_a @ H23 @ K )
=> ( member_a @ ( add_a_b @ r @ H12 @ H23 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_1048_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_1049_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1050_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_1051_subring__props_I5_J,axiom,
! [K: set_a,H: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H @ K )
=> ( member_a @ ( a_inv_a_b @ r @ H ) @ K ) ) ) ).
% subring_props(5)
thf(fact_1052_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_1053_pprime__iff__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_1054_r_Otelescopic__base__dim_I1_J,axiom,
! [K: set_list_a,F2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ F2 )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F2 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E ) ) ) ) ) ).
% r.telescopic_base_dim(1)
thf(fact_1055_r_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 ) ) ) ) ).
% r.inv_eq_imp_eq
thf(fact_1056_pprimeE_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_1057_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_1058_r_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 ) ) ) ) ) ) ).
% r.inv_eq_one_eq
thf(fact_1059_r_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.finite_dimension_imp_subalgebra
thf(fact_1060_univ__poly__infinite__dimension,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ~ ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ K ) @ ( image_a_list_a @ ( poly_of_const_a_b @ r ) @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% univ_poly_infinite_dimension
thf(fact_1061_subfield__m__inv__simprule,axiom,
! [K: set_a,K2: a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ A ) @ K )
=> ( member_a @ A @ K ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_1062_r_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 ) ) ) ) ).
% r.comm_inv_char
thf(fact_1063_r_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 ) ) ) ) ) ).
% r.inv_char
thf(fact_1064_r_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 ) ) ) ) ) ) ).
% r.inv_unique'
thf(fact_1065_r_Oinv__eq__neg__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 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( X
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% r.inv_eq_neg_one_eq
thf(fact_1066_r_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 ) ) ) ) ) ).
% r.subfield_m_inv(1)
thf(fact_1067_r_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 ) ) ) ) ) ).
% r.Units_inv_Units
thf(fact_1068_r_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 ) ) ).
% r.Units_inv_inv
thf(fact_1069_r_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 ) ) ) ) ).
% r.inv_one
thf(fact_1070_r_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 ) ) ) ) ) ).
% r.Units_inv_closed
thf(fact_1071_r_Oinv__neg__one,axiom,
( ( m_inv_2802811658206063947t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.inv_neg_one
thf(fact_1072_r_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 ) ) ) ) ) ).
% r.Units_l_inv
thf(fact_1073_r_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 ) ) ) ) ) ).
% r.Units_r_inv
thf(fact_1074_long__division__a__inv_I1_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_1075_long__division__add_I1_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ A @ Q2 ) @ ( polynomial_pdiv_a_b @ r @ B @ Q2 ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_1076_rupture__is__field__iff__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_1077_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_1078_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_1079_r_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 ) ) ) ) ) ) ) ).
% r.poly_of_const_in_carrier
thf(fact_1080_r_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 ) ) ) ).
% r.univ_poly_subfield_of_consts
thf(fact_1081_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_1082_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_1083_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_1084_inv__eq__self,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( X
= ( m_inv_a_ring_ext_a_b @ r @ X ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ r ) )
| ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% inv_eq_self
thf(fact_1085_inv__eq__neg__one__eq,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( ( m_inv_a_ring_ext_a_b @ r @ X )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% inv_eq_neg_one_eq
thf(fact_1086_long__division__closed_I1_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_1087_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_1088_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_1089_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_1090_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_1091_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_1092_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_1093_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_1094_inv__neg__one,axiom,
( ( m_inv_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) )
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ).
% inv_neg_one
thf(fact_1095_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_1096_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_1097_pdiv__pmod,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_1098_r_Ospace__subgroup__props_I6_J,axiom,
! [K: set_list_a,N2: nat,E: set_list_a,K2: list_a,A: 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 ) ) @ N2 @ 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 @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A ) @ E )
=> ( member_list_a @ A @ E ) ) ) ) ) ) ).
% r.space_subgroup_props(6)
thf(fact_1099_r_OsubringI,axiom,
! [H2: set_list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ H2 )
=> ( ! [H3: list_a] :
( ( member_list_a @ H3 @ H2 )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H3 ) @ H2 ) )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H2 )
=> ( ( member_list_a @ H22 @ H2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 ) @ H2 ) ) )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H2 )
=> ( ( member_list_a @ H22 @ H2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 ) @ H2 ) ) )
=> ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% r.subringI
thf(fact_1100_r_Ocarrier__is__subring,axiom,
subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% r.carrier_is_subring
thf(fact_1101_r_Odimension__is__inj,axiom,
! [K: set_list_a,N2: 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 ) ) @ N2 @ K @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M2 @ K @ E )
=> ( N2 = M2 ) ) ) ) ).
% r.dimension_is_inj
thf(fact_1102_long__division__closed_I2_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_1103_r_Ofinite__dimensionE_H,axiom,
! [K: set_list_a,E: set_list_a] :
( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ~ ! [N3: nat] :
~ ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ K @ E ) ) ).
% r.finite_dimensionE'
thf(fact_1104_r_Ofinite__dimensionI,axiom,
! [N2: nat,K: set_list_a,E: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N2 @ K @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E ) ) ).
% r.finite_dimensionI
thf(fact_1105_r_Ofinite__dimension__def,axiom,
! [K: set_list_a,E: set_list_a] :
( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
= ( ? [N4: nat] : ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N4 @ K @ E ) ) ) ).
% r.finite_dimension_def
thf(fact_1106_r_OsubcringI_H,axiom,
! [H2: set_list_a] :
( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% r.subcringI'
thf(fact_1107_r_Ospace__subgroup__props_I3_J,axiom,
! [K: set_list_a,N2: nat,E: set_list_a,V1: list_a,V22: 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 ) ) @ N2 @ K @ E )
=> ( ( member_list_a @ V1 @ E )
=> ( ( member_list_a @ V22 @ E )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ V1 @ V22 ) @ E ) ) ) ) ) ).
% r.space_subgroup_props(3)
thf(fact_1108_r_Ospace__subgroup__props_I2_J,axiom,
! [K: set_list_a,N2: 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 ) ) @ N2 @ K @ E )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ E ) ) ) ).
% r.space_subgroup_props(2)
thf(fact_1109_r_Ospace__subgroup__props_I5_J,axiom,
! [K: set_list_a,N2: nat,E: set_list_a,K2: list_a,V3: 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 ) ) @ N2 @ K @ E )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ V3 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ V3 ) @ E ) ) ) ) ) ).
% r.space_subgroup_props(5)
thf(fact_1110_long__division__add_I2_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ A @ Q2 ) @ ( polynomial_pmod_a_b @ r @ B @ Q2 ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_1111_long__division__add__iff,axiom,
! [K: set_a,A: list_a,B: list_a,C: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
= ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ C ) @ Q2 )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ C ) @ Q2 ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_1112_r_Ospace__subgroup__props_I4_J,axiom,
! [K: set_list_a,N2: nat,E: set_list_a,V3: 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 ) ) @ N2 @ K @ E )
=> ( ( member_list_a @ V3 @ E )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ V3 ) @ E ) ) ) ) ).
% r.space_subgroup_props(4)
thf(fact_1113_long__division__a__inv_I2_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_1114_r_Ounique__dimension,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ? [X2: nat] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ K @ E )
& ! [Y6: nat] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y6 @ K @ E )
=> ( Y6 = X2 ) ) ) ) ) ).
% r.unique_dimension
thf(fact_1115_r_OsubcringI,axiom,
! [H2: set_list_a] :
( ( subrin6918843898125473962t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H2 )
=> ( ( member_list_a @ H22 @ H2 )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H22 @ H1 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.subcringI
thf(fact_1116_r_Ospace__subgroup__props_I1_J,axiom,
! [K: set_list_a,N2: 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 ) ) @ N2 @ K @ E )
=> ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.space_subgroup_props(1)
thf(fact_1117_r_Ocarrier__polynomial__shell,axiom,
! [K: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_list_a @ P2 @ ( 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 ) ) ) ) ) ) ) ) ).
% r.carrier_polynomial_shell
thf(fact_1118_r_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 ) ) ) ) ).
% r.dimension_zero
thf(fact_1119_r_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 ) ) ).
% r.zero_dim
thf(fact_1120_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_1121_univ__poly__not__field,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_not_field
thf(fact_1122_univ__poly__is__domain,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_domain
thf(fact_1123_univ__poly__a__inv__consistent,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_1124_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q2: list_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q2 )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q2 ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_1125_pirreducibleE_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_1126_pirreducibleE_I3_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a,R3: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R3 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ) ) ) ).
% pirreducibleE(3)
thf(fact_1127_subringI,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H2 )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ H2 ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ H2 ) ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ H2 ) ) )
=> ( subring_a_b @ H2 @ r ) ) ) ) ) ) ).
% subringI
thf(fact_1128_subfieldI_H,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ! [K3: a] :
( ( member_a @ K3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( member_a @ ( m_inv_a_ring_ext_a_b @ r @ K3 ) @ K ) )
=> ( subfield_a_b @ K @ r ) ) ) ).
% subfieldI'
thf(fact_1129_carrier__polynomial__shell,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_1130_r_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 ) ) )
| ? [V4: 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 @ V4 @ E2 ) )
& ( member_list_a @ V4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ~ ( member_list_a @ V4 @ E2 )
& ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N4 @ K4 @ E2 ) ) ) ) ).
% r.dimension.simps
thf(fact_1131_r_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 ) ) )
=> ~ ! [V2: list_a,E3: set_list_a,N3: nat] :
( ( A1
= ( suc @ N3 ) )
=> ( ( A32
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A22 @ V2 @ E3 ) )
=> ( ( member_list_a @ V2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( member_list_a @ V2 @ E3 )
=> ~ ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N3 @ A22 @ E3 ) ) ) ) ) ) ) ).
% r.dimension.cases
thf(fact_1132_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_1133_subcringI_H,axiom,
! [H2: set_a] :
( ( subring_a_b @ H2 @ r )
=> ( subcring_a_b @ H2 @ r ) ) ).
% subcringI'
thf(fact_1134_subdomainI_H,axiom,
! [H2: set_a] :
( ( subring_a_b @ H2 @ r )
=> ( subdomain_a_b @ H2 @ r ) ) ).
% subdomainI'
thf(fact_1135_subcringI,axiom,
! [H2: set_a] :
( ( subring_a_b @ H2 @ r )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( mult_a_ring_ext_a_b @ r @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ r @ H22 @ H1 ) ) ) )
=> ( subcring_a_b @ H2 @ r ) ) ) ).
% subcringI
thf(fact_1136_subdomainI,axiom,
! [H2: set_a] :
( ( subcring_a_b @ H2 @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H2 )
=> ( ( member_a @ H22 @ H2 )
=> ( ( ( 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 @ H2 @ r ) ) ) ) ).
% subdomainI
thf(fact_1137_r_OSuc__dim,axiom,
! [V3: list_a,E: set_list_a,N2: nat,K: set_list_a] :
( ( member_list_a @ V3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( member_list_a @ V3 @ E )
=> ( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N2 @ K @ E )
=> ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( suc @ N2 ) @ K @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ V3 @ E ) ) ) ) ) ).
% r.Suc_dim
thf(fact_1138_r_Odimension__backwards,axiom,
! [K: set_list_a,N2: 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 @ N2 ) @ K @ E )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ? [E4: set_list_a] :
( ( embedd3793949463769647726t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N2 @ K @ E4 )
& ~ ( member_list_a @ X2 @ E4 )
& ( E
= ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ X2 @ E4 ) ) ) ) ) ) ).
% r.dimension_backwards
thf(fact_1139_euclidean__function,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q3: a,R4: a] :
( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B @ Q3 ) @ R4 ) )
& ( ( R4
= ( zero_a_b @ r ) )
| ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ) ) ).
% euclidean_function
thf(fact_1140_euclidean__domainI,axiom,
! [Phi: a > nat] :
( ! [A5: a,B6: a] :
( ( member_a @ A5 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B6 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q4: a,R5: a] :
( ( member_a @ Q4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A5
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B6 @ Q4 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ r ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B6 ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).
% euclidean_domainI
thf(fact_1141_r_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_3240872107759947550t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.order_gt_0_iff_finite
thf(fact_1142_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% order_gt_0_iff_finite
thf(fact_1143_r_OboundD__carrier,axiom,
! [N2: nat,F: nat > list_a,M2: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N2 @ F )
=> ( ( ord_less_nat @ N2 @ M2 )
=> ( member_list_a @ ( F @ M2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% r.boundD_carrier
thf(fact_1144_alg__mult__gt__zero__iff__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1145_boundD__carrier,axiom,
! [N2: nat,F: nat > a,M2: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N2 @ F )
=> ( ( ord_less_nat @ N2 @ M2 )
=> ( member_a @ ( F @ M2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_1146_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1147_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1148_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1149_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1150_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1151_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1152_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1153_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1154_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1155_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1156_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1157_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1158_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1159_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1160_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1161_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1162_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1163_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1164_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1165_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1166_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1167_le__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1168_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1169_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1170_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_1171_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1172_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1173_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1174_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1175_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1176_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1177_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1178_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1179_diff__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1180_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1181_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1182_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_1183_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1184_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1185_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1186_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1187_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1188_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1189_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_1190_nat__arith_Osuc1,axiom,
! [A2: nat,K2: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1191_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1192_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1193_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K2: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1194_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K2 @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1195_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1196_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1197_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1198_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1199_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1200_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1201_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_1202_less__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_1203_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1204_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1205_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1206_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1207_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1208_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1209_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1210_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1211_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1212_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1213_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1214_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_1215_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_1216_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1217_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q3: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1218_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1219_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1220_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M4: nat,N4: nat] :
? [K5: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M4 @ K5 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1221_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1222_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_1223_pirreducibleI,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R4 ) )
=> ( ( member_list_a @ Q3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_1224_r_Obound__upD,axiom,
! [F: nat > list_a] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [N3: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N3 @ F ) ) ).
% r.bound_upD
thf(fact_1225_long__division__zero_I2_J,axiom,
! [K: set_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q2 )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_1226_pirreducibleE_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_1227_long__division__zero_I1_J,axiom,
! [K: set_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q2 )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_1228_pprimeE_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_1229_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1230_add__leE,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K2 @ N2 ) ) ) ).
% add_leE
thf(fact_1231_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1232_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1233_add__leD1,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1234_add__leD2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N2 )
=> ( ord_less_eq_nat @ K2 @ N2 ) ) ).
% add_leD2
thf(fact_1235_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1236_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1237_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1238_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1239_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1240_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N4: nat] :
? [K5: nat] :
( N4
= ( plus_plus_nat @ M4 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1241_exists__unique__long__division,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ? [X2: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ X2 )
& ! [Y6: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ Y6 )
=> ( Y6 = X2 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_1242_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ r ) @ N3 @ F ) ) ).
% bound_upD
thf(fact_1243_long__divisionE,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_1244_long__divisionI,axiom,
! [K: set_a,P2: list_a,Q2: list_a,B: list_a,R3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B @ R3 ) )
=> ( ( produc6837034575241423639list_a @ B @ R3 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q2 ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_1245_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_1246_exists__long__division,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ~ ! [B6: list_a] :
( ( member_list_a @ B6 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P2 @ Q2 @ ( produc6837034575241423639list_a @ B6 @ R4 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_1247_pprimeI,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q3 )
| ( polyno5814909790663948098es_a_b @ r @ P2 @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ) ) ).
% pprimeI
thf(fact_1248_r_Oirreducible__prod__rI,axiom,
! [A: list_a,B: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( 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 ) ) @ A @ B ) ) ) ) ) ) ).
% r.irreducible_prod_rI
thf(fact_1249_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_1250_zero__pdivides,axiom,
! [P2: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P2 )
= ( P2 = nil_a ) ) ).
% zero_pdivides
thf(fact_1251_pdivides__zero,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P2 @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_1252_r_Oirreducible__prod__lI,axiom,
! [B: list_a,A: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( 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 ) ) @ A @ B ) ) ) ) ) ) ).
% r.irreducible_prod_lI
thf(fact_1253_pmod__zero__iff__pdivides,axiom,
! [K: set_a,P2: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P2 @ Q2 )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q2 @ P2 ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_1254_pprimeE_I3_J,axiom,
! [K: set_a,P2: list_a,Q2: list_a,R3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
| ( polyno5814909790663948098es_a_b @ r @ P2 @ R3 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_1255_same__pmod__iff__pdivides,axiom,
! [K: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
= ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
= ( polyno5814909790663948098es_a_b @ r @ Q2 @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_1256_pdivides__imp__splitted,axiom,
! [P2: list_a,Q2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q2 )
=> ( polyno8329700637149614481ed_a_b @ r @ P2 ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_1257_is__root__imp__pdivides,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 ) ) ) ).
% is_root_imp_pdivides
thf(fact_1258_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V2: a,Va: list_a] :
( X
!= ( cons_a @ V2 @ Va ) ) ) ).
% normalize.cases
thf(fact_1259_combine_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [K3: a,Ks: list_a,U2: a,Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ K3 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_1260_poly__mult_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V2: a,Va: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V2 @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_1261_ring__irreducibleE_I2_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( irredu6211895646901577903xt_a_b @ r @ R3 ) ) ) ).
% ring_irreducibleE(2)
thf(fact_1262_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_1263_irreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irredu6211895646901577903xt_a_b @ r @ B )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_1264_irreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_1265_monic__degree__one__root__condition,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) @ B )
= ( A = B ) ) ) ).
% monic_degree_one_root_condition
thf(fact_1266_pdivides__imp__is__root,axiom,
! [P2: list_a,X: a] :
( ( P2 != nil_a )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 )
=> ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_1267_alg__multE_I2_J,axiom,
! [X: a,P2: list_a,N2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N2 ) @ P2 )
=> ( ord_less_eq_nat @ N2 @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_1268_le__alg__mult__imp__pdivides,axiom,
! [X: a,P2: list_a,N2: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N2 @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N2 ) @ P2 ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_1269_r_OUnits__pow__closed,axiom,
! [X: list_a,D2: nat] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ D2 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.Units_pow_closed
thf(fact_1270_r_Onat__pow__zero,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% r.nat_pow_zero
thf(fact_1271_polynomial__pow__not__zero,axiom,
! [P2: list_a,N2: nat] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ N2 )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_1272_r_Ogroup__commutes__pow,axiom,
! [X: list_a,Y: list_a,N2: nat] :
( ( ( 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 ) )
=> ( ( 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 ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N2 ) @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N2 ) ) ) ) ) ) ).
% r.group_commutes_pow
thf(fact_1273_r_Onat__pow__comm,axiom,
! [X: list_a,N2: nat,M2: nat] :
( ( 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 ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M2 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N2 ) ) ) ) ).
% r.nat_pow_comm
thf(fact_1274_r_Onat__pow__distrib,axiom,
! [X: list_a,Y: list_a,N2: 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 ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ N2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N2 ) ) ) ) ) ).
% r.nat_pow_distrib
thf(fact_1275_r_Opow__mult__distrib,axiom,
! [X: list_a,Y: list_a,N2: nat] :
( ( ( 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 ) )
=> ( ( 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 ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ N2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N2 ) ) ) ) ) ) ).
% r.pow_mult_distrib
thf(fact_1276_r_Omonic__degree__one__root__condition,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) @ nil_list_a ) ) @ B )
= ( A = B ) ) ) ).
% r.monic_degree_one_root_condition
% Conjectures (1)
thf(conj_0,conjecture,
member_list_a @ ( lagran1063865941317790773te_a_b @ r @ s @ f ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
%------------------------------------------------------------------------------