TPTP Problem File: SLH0016^1.p
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- 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 : Finite_Fields/0003_Finite_Fields_Preliminary_Results/prob_00451_016259__18051542_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1647 ( 160 unt; 367 typ; 0 def)
% Number of atoms : 4580 (1198 equ; 0 cnn)
% Maximal formula atoms : 14 ( 3 avg)
% Number of connectives : 25473 ( 258 ~; 38 |; 43 &;22217 @)
% ( 0 <=>;2917 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 40 ( 39 usr)
% Number of type conns : 953 ( 953 >; 0 *; 0 +; 0 <<)
% Number of symbols : 329 ( 328 usr; 14 con; 0-4 aty)
% Number of variables : 3084 ( 40 ^;3015 !; 29 ?;3084 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:20:31.937
%------------------------------------------------------------------------------
% Could-be-implicit typings (39)
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thf(sy_c_Ring_Oring_Oadd_001tf__b_001tf__d,type,
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thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__c,type,
zero_a_c: partia8877618634411419171xt_a_c > a ).
thf(sy_c_Ring_Oring_Ozero_001tf__b_001tf__d,type,
zero_b_d: partia1897943568983147621xt_b_d > b ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__c,type,
ring_h2895973938487309445it_a_c: partia2670972154091845814t_unit > partia8877618634411419171xt_a_c > set_list_a_a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit_001tf__b_001tf__d,type,
ring_h108518356514633413it_b_d: partia2670972154091845814t_unit > partia1897943568983147621xt_b_d > set_list_a_b ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit_001tf__a_001tf__c,type,
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thf(sy_c_Ring_Oring__hom_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit_001tf__b_001tf__d,type,
ring_h8746557796359701252it_b_d: partia4026993951477142903t_unit > partia1897943568983147621xt_b_d > set_list_b_b ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__c_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_h5384962040972022021t_unit: partia8877618634411419171xt_a_c > partia2670972154091845814t_unit > set_a_list_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__c_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ring_Oring__hom_001tf__a_001tf__c_001tf__a_001tf__c,type,
ring_hom_a_c_a_c: partia8877618634411419171xt_a_c > partia8877618634411419171xt_a_c > set_a_a ).
thf(sy_c_Ring_Oring__hom_001tf__a_001tf__c_001tf__b_001tf__d,type,
ring_hom_a_c_b_d: partia8877618634411419171xt_a_c > partia1897943568983147621xt_b_d > set_a_b ).
thf(sy_c_Ring_Oring__hom_001tf__b_001tf__d_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ring_Oring__hom_001tf__b_001tf__d_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ring_Oring__hom_001tf__b_001tf__d_001tf__a_001tf__c,type,
ring_hom_b_d_a_c: partia1897943568983147621xt_b_d > partia8877618634411419171xt_a_c > set_b_a ).
thf(sy_c_Ring_Oring__hom_001tf__b_001tf__d_001tf__b_001tf__d,type,
ring_hom_b_d_b_d: partia1897943568983147621xt_b_d > partia1897943568983147621xt_b_d > set_b_b ).
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thf(sy_c_Ring_Osemiring_001tf__a_001tf__c,type,
semiring_a_c: partia8877618634411419171xt_a_c > $o ).
thf(sy_c_Ring_Osemiring_001tf__b_001tf__d,type,
semiring_b_d: partia1897943568983147621xt_b_d > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
ring_p8404492108403472412t_unit: partia5333488208502193986t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
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thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__b_J_J_001t__Product____Type__Ounit,type,
ring_p4916954238016387169t_unit: partia2587409828943155709t_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_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit,type,
ring_p5013149089651084391t_unit: partia4026993951477142903t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__c,type,
ring_p8803135361686045601in_a_c: partia8877618634411419171xt_a_c > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__b_001tf__d,type,
ring_p6015679779713369569in_b_d: partia1897943568983147621xt_b_d > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
ring_r5224476855413033410t_unit: partia5333488208502193986t_unit > list_list_list_a > $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_It__List__Olist_Itf__b_J_J_001t__Product____Type__Ounit,type,
ring_r4561388045816386823t_unit: partia2587409828943155709t_unit > list_list_b > $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__List__Olist_Itf__b_J_001t__Product____Type__Ounit,type,
ring_r7070601269410051085t_unit: partia4026993951477142903t_unit > list_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__c,type,
ring_r999134135267193927le_a_c: partia8877618634411419171xt_a_c > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__b_001tf__d,type,
ring_r7435050590149293703le_b_d: partia1897943568983147621xt_b_d > b > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
ring_r346321679897941977t_unit: partia5333488208502193986t_unit > list_list_list_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 ).
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ring_r415245522172713630t_unit: partia2587409828943155709t_unit > list_list_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit,type,
ring_r3344526403024810276t_unit: partia4026993951477142903t_unit > list_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__c,type,
ring_ring_prime_a_c: partia8877618634411419171xt_a_c > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__b_001tf__d,type,
ring_ring_prime_b_d: partia1897943568983147621xt_b_d > b > $o ).
thf(sy_c_Set_OCollect_001_062_Itf__a_Mtf__b_J,type,
collect_a_b: ( ( a > b ) > $o ) > set_a_b ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__b_J,type,
collect_list_b: ( list_b > $o ) > set_list_b ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
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__List__Olist_Itf__a_J_001tf__b,type,
image_list_a_b: ( list_a > b ) > set_list_a > set_b ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__b_J_001tf__a,type,
image_list_b_a: ( list_b > a ) > set_list_b > set_a ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__b_J_001tf__b,type,
image_list_b_b: ( list_b > b ) > set_list_b > set_b ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
image_a_b: ( a > b ) > set_a > set_b ).
thf(sy_c_Set_Oimage_001tf__b_001t__List__Olist_Itf__a_J,type,
image_b_list_a: ( b > list_a ) > set_b > set_list_a ).
thf(sy_c_Set_Oimage_001tf__b_001t__List__Olist_Itf__b_J,type,
image_b_list_b: ( b > list_b ) > set_b > set_list_b ).
thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
image_b_a: ( b > a ) > set_b > set_a ).
thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit,type,
subcri4677462317791934762t_unit: set_list_b > partia4026993951477142903t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit,type,
subdom4735476224308063485t_unit: set_list_b > partia4026993951477142903t_unit > $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_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit,type,
subfie7916738691610828529t_unit: set_list_b > partia4026993951477142903t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__c,type,
subfield_a_c: set_a > partia8877618634411419171xt_a_c > $o ).
thf(sy_c_Subrings_Osubfield_001tf__b_001tf__d,type,
subfield_b_d: set_b > partia1897943568983147621xt_b_d > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit,type,
subrin3833087656135479401t_unit: set_list_b > partia4026993951477142903t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__c,type,
subring_a_c: set_a > partia8877618634411419171xt_a_c > $o ).
thf(sy_c_Subrings_Osubring_001tf__b_001tf__d,type,
subring_b_d: set_b > partia1897943568983147621xt_b_d > $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__List__Olist_Itf__a_J_Mtf__b_J,type,
member_list_a_b: ( list_a > b ) > set_list_a_b > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__b_J_Mtf__a_J,type,
member_list_b_a: ( list_b > a ) > set_list_b_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__b_J_Mtf__b_J,type,
member_list_b_b: ( list_b > b ) > set_list_b_b > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__b_J_J,type,
member_a_list_b: ( a > list_b ) > set_a_list_b > $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_001_062_Itf__a_Mtf__b_J,type,
member_a_b: ( a > b ) > set_a_b > $o ).
thf(sy_c_member_001_062_Itf__b_Mt__List__Olist_Itf__a_J_J,type,
member_b_list_a: ( b > list_a ) > set_b_list_a > $o ).
thf(sy_c_member_001_062_Itf__b_Mt__List__Olist_Itf__b_J_J,type,
member_b_list_b: ( b > list_b ) > set_b_list_b > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__a_J,type,
member_b_a: ( b > a ) > set_b_a > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__b_J,type,
member_b_b: ( b > b ) > set_b_b > $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__List__Olist_Itf__b_J_J,type,
member_list_list_b: list_list_b > set_list_list_b > $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__List__Olist_Itf__b_J,type,
member_list_b: list_b > set_list_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_R,type,
r: partia8877618634411419171xt_a_c ).
thf(sy_v_S,type,
s: partia1897943568983147621xt_b_d ).
thf(sy_v_h,type,
h: a > b ).
thf(sy_v_x____,type,
x: list_a ).
thf(sy_v_xh____,type,
xh: a ).
thf(sy_v_xt____,type,
xt: list_a ).
% Relevant facts (1279)
thf(fact_0__092_060open_062xh_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
member_a @ xh @ ( partia778085601923319190xt_a_c @ r ) ).
% \<open>xh \<in> carrier R\<close>
thf(fact_1__092_060open_062xh_A_092_060noteq_062_A_092_060zero_062_092_060_094bsub_062R_092_060_094esub_062_092_060close_062,axiom,
( xh
!= ( zero_a_c @ r ) ) ).
% \<open>xh \<noteq> \<zero>\<^bsub>R\<^esub>\<close>
thf(fact_2_ds_Odomain__axioms,axiom,
domain_b_d @ s ).
% ds.domain_axioms
thf(fact_3_assms_I1_J,axiom,
member_a_b @ h @ ( ring_iso_a_c_b_d @ r @ s ) ).
% assms(1)
thf(fact_4_h__non__zero__iff,axiom,
! [X: a] :
( ( X
!= ( zero_a_c @ r ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( h @ X )
!= ( zero_b_d @ s ) ) ) ) ).
% h_non_zero_iff
thf(fact_5_ds_Ozero__is__prime_I1_J,axiom,
prime_b_ring_ext_b_d @ s @ ( zero_b_d @ s ) ).
% ds.zero_is_prime(1)
thf(fact_6_h_Ohom__zero,axiom,
( ( h @ ( zero_a_c @ r ) )
= ( zero_b_d @ s ) ) ).
% h.hom_zero
thf(fact_7_ds_Osemiring__axioms,axiom,
semiring_b_d @ s ).
% ds.semiring_axioms
thf(fact_8_local_OCons,axiom,
( x
= ( cons_a @ xh @ xt ) ) ).
% local.Cons
thf(fact_9_ds_Ozero__not__one,axiom,
( ( zero_b_d @ s )
!= ( one_b_ring_ext_b_d @ s ) ) ).
% ds.zero_not_one
thf(fact_10_ds_Osubring__props_I2_J,axiom,
! [K: set_b] :
( ( subfield_b_d @ K @ s )
=> ( member_b @ ( zero_b_d @ s ) @ K ) ) ).
% ds.subring_props(2)
thf(fact_11_h_Ois__abelian__group__hom,axiom,
abelia4987496640480023959_c_b_d @ r @ s @ h ).
% h.is_abelian_group_hom
thf(fact_12_ds_Ozero__divides,axiom,
! [A: b] :
( ( factor2325171414093416164xt_b_d @ s @ ( zero_b_d @ s ) @ A )
= ( A
= ( zero_b_d @ s ) ) ) ).
% ds.zero_divides
thf(fact_13_ds_Oabelian__monoid__axioms,axiom,
abelian_monoid_b_d @ s ).
% ds.abelian_monoid_axioms
thf(fact_14_ds_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_list_b,F: list_b > b] :
( ! [X2: list_b] :
( ( member_list_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_b_d @ s ) ) )
=> ( ( finsum_b_d_list_b @ s @ F @ A2 )
= ( zero_b_d @ s ) ) ) ).
% ds.add.finprod_one_eqI
thf(fact_15_ds_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_list_a,F: list_a > b] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_b_d @ s ) ) )
=> ( ( finsum_b_d_list_a @ s @ F @ A2 )
= ( zero_b_d @ s ) ) ) ).
% ds.add.finprod_one_eqI
thf(fact_16_ds_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_a_b,F: ( a > b ) > b] :
( ! [X2: a > b] :
( ( member_a_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_b_d @ s ) ) )
=> ( ( finsum_b_d_a_b @ s @ F @ A2 )
= ( zero_b_d @ s ) ) ) ).
% ds.add.finprod_one_eqI
thf(fact_17_ds_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_b,F: b > b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_b_d @ s ) ) )
=> ( ( finsum_b_d_b @ s @ F @ A2 )
= ( zero_b_d @ s ) ) ) ).
% ds.add.finprod_one_eqI
thf(fact_18_ds_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_a,F: a > b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_b_d @ s ) ) )
=> ( ( finsum_b_d_a @ s @ F @ A2 )
= ( zero_b_d @ s ) ) ) ).
% ds.add.finprod_one_eqI
thf(fact_19_h_Ohomh,axiom,
member_a_b @ h @ ( ring_hom_a_c_b_d @ r @ s ) ).
% h.homh
thf(fact_20_ds_Osubring__props_I3_J,axiom,
! [K: set_b] :
( ( subfield_b_d @ K @ s )
=> ( member_b @ ( one_b_ring_ext_b_d @ s ) @ K ) ) ).
% ds.subring_props(3)
thf(fact_21_dr_Oring__hom__restrict,axiom,
! [F: a > b,S: partia1897943568983147621xt_b_d,G: a > b] :
( ( member_a_b @ F @ ( ring_hom_a_c_b_d @ r @ S ) )
=> ( ! [R: a] :
( ( member_a @ R @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_a_b @ G @ ( ring_hom_a_c_b_d @ r @ S ) ) ) ) ).
% dr.ring_hom_restrict
thf(fact_22_dr_Oring__iso__restrict,axiom,
! [F: a > b,S: partia1897943568983147621xt_b_d,G: a > b] :
( ( member_a_b @ F @ ( ring_iso_a_c_b_d @ r @ S ) )
=> ( ! [R: a] :
( ( member_a @ R @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_a_b @ G @ ( ring_iso_a_c_b_d @ r @ S ) ) ) ) ).
% dr.ring_iso_restrict
thf(fact_23_dr_Odomain__axioms,axiom,
domain_a_c @ r ).
% dr.domain_axioms
thf(fact_24_dr_Ozero__closed,axiom,
member_a @ ( zero_a_c @ r ) @ ( partia778085601923319190xt_a_c @ r ) ).
% dr.zero_closed
thf(fact_25_ds_Oring__primeI,axiom,
! [P: b] :
( ( P
!= ( zero_b_d @ s ) )
=> ( ( prime_b_ring_ext_b_d @ s @ P )
=> ( ring_ring_prime_b_d @ s @ P ) ) ) ).
% ds.ring_primeI
thf(fact_26_dr_Oring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_ring_prime_a_c @ r @ P )
=> ( P
!= ( zero_a_c @ r ) ) ) ) ).
% dr.ring_primeE(1)
thf(fact_27_dr_Oonepideal,axiom,
principalideal_a_c @ ( partia778085601923319190xt_a_c @ r ) @ r ).
% dr.onepideal
thf(fact_28_dr_Oring__irreducibleE_I1_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_r999134135267193927le_a_c @ r @ R2 )
=> ( R2
!= ( zero_a_c @ r ) ) ) ) ).
% dr.ring_irreducibleE(1)
thf(fact_29_dr_Ozero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_c @ r @ ( zero_a_c @ r ) ).
% dr.zero_is_prime(1)
thf(fact_30_h_Ozero__closed,axiom,
member_b @ ( h @ ( zero_a_c @ r ) ) @ ( partia8782771468121683032xt_b_d @ s ) ).
% h.zero_closed
thf(fact_31_h_Ohom__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_b @ ( h @ X ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% h.hom_closed
thf(fact_32_ds_Oone__divides,axiom,
! [A: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( factor2325171414093416164xt_b_d @ s @ ( one_b_ring_ext_b_d @ s ) @ A ) ) ).
% ds.one_divides
thf(fact_33_ds_Odivides__zero,axiom,
! [A: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( factor2325171414093416164xt_b_d @ s @ A @ ( zero_b_d @ s ) ) ) ).
% ds.divides_zero
thf(fact_34_domain_Ozero__is__prime_I1_J,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( prime_1232919612140715622t_unit @ R3 @ ( zero_l347298301471573063t_unit @ R3 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_35_domain_Ozero__is__prime_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d] :
( ( domain_b_d @ R3 )
=> ( prime_b_ring_ext_b_d @ R3 @ ( zero_b_d @ R3 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_36_domain_Ozero__is__prime_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c] :
( ( domain_a_c @ R3 )
=> ( prime_a_ring_ext_a_c @ R3 @ ( zero_a_c @ R3 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_37_domain_Ozero__is__prime_I1_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( prime_2011924034616061926t_unit @ R3 @ ( zero_l4142658623432671053t_unit @ R3 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_38_domain_Ozero__is__prime_I1_J,axiom,
! [R3: partia4026993951477142903t_unit] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( prime_5961678782586786214t_unit @ R3 @ ( zero_l1056902381442676492t_unit @ R3 ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_39_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_40_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia4026993951477142903t_unit] :
( ( semiri9009524540797033698t_unit @ R3 )
=> ( member_list_b @ ( one_li3054569011217056637t_unit @ R3 ) @ ( partia1381092143316337258t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_41_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_42_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia1897943568983147621xt_b_d] :
( ( semiring_b_d @ R3 )
=> ( member_b @ ( one_b_ring_ext_b_d @ R3 ) @ ( partia8782771468121683032xt_b_d @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_43_semiring_Osemiring__simprules_I4_J,axiom,
! [R3: partia8877618634411419171xt_a_c] :
( ( semiring_a_c @ R3 )
=> ( member_a @ ( one_a_ring_ext_a_c @ R3 ) @ ( partia778085601923319190xt_a_c @ R3 ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_44_abelian__group__hom_Ozero__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia1897943568983147621xt_b_d,H2: b > b] :
( ( abelia2848305256108177367_d_b_d @ G2 @ H @ H2 )
=> ( member_b @ ( H2 @ ( zero_b_d @ G2 ) ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ).
% abelian_group_hom.zero_closed
thf(fact_45_abelian__group__hom_Ozero__closed,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia1897943568983147621xt_b_d,H2: a > b] :
( ( abelia4987496640480023959_c_b_d @ G2 @ H @ H2 )
=> ( member_b @ ( H2 @ ( zero_a_c @ G2 ) ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ).
% abelian_group_hom.zero_closed
thf(fact_46_abelian__group__hom_Ozero__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia8877618634411419171xt_a_c,H2: b > a] :
( ( abelia5635760838080853399_d_a_c @ G2 @ H @ H2 )
=> ( member_a @ ( H2 @ ( zero_b_d @ G2 ) ) @ ( partia778085601923319190xt_a_c @ H ) ) ) ).
% abelian_group_hom.zero_closed
thf(fact_47_abelian__group__hom_Ozero__closed,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia8877618634411419171xt_a_c,H2: a > a] :
( ( abelia7774952222452699991_c_a_c @ G2 @ H @ H2 )
=> ( member_a @ ( H2 @ ( zero_a_c @ G2 ) ) @ ( partia778085601923319190xt_a_c @ H ) ) ) ).
% abelian_group_hom.zero_closed
thf(fact_48_abelian__group__hom_Ozero__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia4026993951477142903t_unit,H2: b > list_b] :
( ( abelia4372663102398250333t_unit @ G2 @ H @ H2 )
=> ( member_list_b @ ( H2 @ ( zero_b_d @ G2 ) ) @ ( partia1381092143316337258t_unit @ H ) ) ) ).
% abelian_group_hom.zero_closed
thf(fact_49_abelian__group__hom_Ozero__closed,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia4026993951477142903t_unit,H2: a > list_b] :
( ( abelia7620252404543421213t_unit @ G2 @ H @ H2 )
=> ( member_list_b @ ( H2 @ ( zero_a_c @ G2 ) ) @ ( partia1381092143316337258t_unit @ H ) ) ) ).
% abelian_group_hom.zero_closed
thf(fact_50_abelian__group__hom_Ozero__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia2670972154091845814t_unit,H2: b > list_a] :
( ( abelia7458419344388244894t_unit @ G2 @ H @ H2 )
=> ( member_list_a @ ( H2 @ ( zero_b_d @ G2 ) ) @ ( partia5361259788508890537t_unit @ H ) ) ) ).
% abelian_group_hom.zero_closed
thf(fact_51_abelian__group__hom_Ozero__closed,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia2670972154091845814t_unit,H2: a > list_a] :
( ( abelia1482636609678639966t_unit @ G2 @ H @ H2 )
=> ( member_list_a @ ( H2 @ ( zero_a_c @ G2 ) ) @ ( partia5361259788508890537t_unit @ H ) ) ) ).
% abelian_group_hom.zero_closed
thf(fact_52_abelian__group__hom_Ozero__closed,axiom,
! [G2: partia2670972154091845814t_unit,H: partia1897943568983147621xt_b_d,H2: list_a > b] :
( ( abelia5429564962076027166it_b_d @ G2 @ H @ H2 )
=> ( member_b @ ( H2 @ ( zero_l4142658623432671053t_unit @ G2 ) ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ).
% abelian_group_hom.zero_closed
thf(fact_53_abelian__group__hom_Ozero__closed,axiom,
! [G2: partia4026993951477142903t_unit,H: partia1897943568983147621xt_b_d,H2: list_b > b] :
( ( abelia4844232365066319197it_b_d @ G2 @ H @ H2 )
=> ( member_b @ ( H2 @ ( zero_l1056902381442676492t_unit @ G2 ) ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ).
% abelian_group_hom.zero_closed
thf(fact_54_dr_Osemiring__axioms,axiom,
semiring_a_c @ r ).
% dr.semiring_axioms
thf(fact_55_dr_Ozero__not__one,axiom,
( ( zero_a_c @ r )
!= ( one_a_ring_ext_a_c @ r ) ) ).
% dr.zero_not_one
thf(fact_56_ds_Odivides__trans,axiom,
! [A: b,B: b,C: b] :
( ( factor2325171414093416164xt_b_d @ s @ A @ B )
=> ( ( factor2325171414093416164xt_b_d @ s @ B @ C )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( factor2325171414093416164xt_b_d @ s @ A @ C ) ) ) ) ).
% ds.divides_trans
thf(fact_57_ds_Oring__primeE_I1_J,axiom,
! [P: b] :
( ( member_b @ P @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_ring_prime_b_d @ s @ P )
=> ( P
!= ( zero_b_d @ s ) ) ) ) ).
% ds.ring_primeE(1)
thf(fact_58_dr_Oring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_ring_prime_a_c @ r @ P )
=> ( prime_a_ring_ext_a_c @ r @ P ) ) ) ).
% dr.ring_primeE(3)
thf(fact_59_ds_Oring__primeE_I3_J,axiom,
! [P: b] :
( ( member_b @ P @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_ring_prime_b_d @ s @ P )
=> ( prime_b_ring_ext_b_d @ s @ P ) ) ) ).
% ds.ring_primeE(3)
thf(fact_60_dr_Oring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_c @ r ) )
=> ( ( prime_a_ring_ext_a_c @ r @ P )
=> ( ring_ring_prime_a_c @ r @ P ) ) ) ).
% dr.ring_primeI
thf(fact_61_dr_Oone__closed,axiom,
member_a @ ( one_a_ring_ext_a_c @ r ) @ ( partia778085601923319190xt_a_c @ r ) ).
% dr.one_closed
thf(fact_62_ds_Ozero__closed,axiom,
member_b @ ( zero_b_d @ s ) @ ( partia8782771468121683032xt_b_d @ s ) ).
% ds.zero_closed
thf(fact_63_ds_Oone__closed,axiom,
member_b @ ( one_b_ring_ext_b_d @ s ) @ ( partia8782771468121683032xt_b_d @ s ) ).
% ds.one_closed
thf(fact_64_ds_Odivides__refl,axiom,
! [A: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( factor2325171414093416164xt_b_d @ s @ A @ A ) ) ).
% ds.divides_refl
thf(fact_65_that,axiom,
member_list_a @ x @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% that
thf(fact_66_mem__Collect__eq,axiom,
! [A: list_b,P2: list_b > $o] :
( ( member_list_b @ A @ ( collect_list_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A: list_a,P2: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A: a > b,P2: ( a > b ) > $o] :
( ( member_a_b @ A @ ( collect_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A: b,P2: b > $o] :
( ( member_b @ A @ ( collect_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A2: set_list_b] :
( ( collect_list_b
@ ^ [X3: list_b] : ( member_list_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_72_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_73_Collect__mem__eq,axiom,
! [A2: set_a_b] :
( ( collect_a_b
@ ^ [X3: a > b] : ( member_a_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X3: b] : ( member_b @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_76_h_Ohom__one,axiom,
( ( h @ ( one_a_ring_ext_a_c @ r ) )
= ( one_b_ring_ext_b_d @ s ) ) ).
% h.hom_one
thf(fact_77_ring__prime__def,axiom,
( ring_r5437400583859147359t_unit
= ( ^ [R4: partia2956882679547061052t_unit,A3: list_list_a] :
( ( A3
!= ( zero_l347298301471573063t_unit @ R4 ) )
& ( prime_1232919612140715622t_unit @ R4 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_78_ring__prime__def,axiom,
( ring_r3344526403024810276t_unit
= ( ^ [R4: partia4026993951477142903t_unit,A3: list_b] :
( ( A3
!= ( zero_l1056902381442676492t_unit @ R4 ) )
& ( prime_5961678782586786214t_unit @ R4 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_79_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R4: partia2670972154091845814t_unit,A3: list_a] :
( ( A3
!= ( zero_l4142658623432671053t_unit @ R4 ) )
& ( prime_2011924034616061926t_unit @ R4 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_80_ring__prime__def,axiom,
( ring_ring_prime_b_d
= ( ^ [R4: partia1897943568983147621xt_b_d,A3: b] :
( ( A3
!= ( zero_b_d @ R4 ) )
& ( prime_b_ring_ext_b_d @ R4 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_81_ring__prime__def,axiom,
( ring_ring_prime_a_c
= ( ^ [R4: partia8877618634411419171xt_a_c,A3: a] :
( ( A3
!= ( zero_a_c @ R4 ) )
& ( prime_a_ring_ext_a_c @ R4 @ A3 ) ) ) ) ).
% ring_prime_def
thf(fact_82_domain_Oring__irreducibleE_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,R2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r360171070648044744t_unit @ R3 @ R2 )
=> ( R2
!= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_83_domain_Oring__irreducibleE_I1_J,axiom,
! [R3: partia4026993951477142903t_unit,R2: list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( member_list_b @ R2 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( ring_r7070601269410051085t_unit @ R3 @ R2 )
=> ( R2
!= ( zero_l1056902381442676492t_unit @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_84_domain_Oring__irreducibleE_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r932985474545269838t_unit @ R3 @ R2 )
=> ( R2
!= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_85_domain_Oring__irreducibleE_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d,R2: b] :
( ( domain_b_d @ R3 )
=> ( ( member_b @ R2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( ring_r7435050590149293703le_b_d @ R3 @ R2 )
=> ( R2
!= ( zero_b_d @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_86_domain_Oring__irreducibleE_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c,R2: a] :
( ( domain_a_c @ R3 )
=> ( ( member_a @ R2 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( ring_r999134135267193927le_a_c @ R3 @ R2 )
=> ( R2
!= ( zero_a_c @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_87_domain_Oring__primeE_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r5437400583859147359t_unit @ R3 @ P )
=> ( P
!= ( zero_l347298301471573063t_unit @ R3 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_88_domain_Oring__primeE_I1_J,axiom,
! [R3: partia4026993951477142903t_unit,P: list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( ring_r3344526403024810276t_unit @ R3 @ P )
=> ( P
!= ( zero_l1056902381442676492t_unit @ R3 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_89_domain_Oring__primeE_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r6430282645014804837t_unit @ R3 @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_90_domain_Oring__primeE_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d,P: b] :
( ( domain_b_d @ R3 )
=> ( ( member_b @ P @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( ring_ring_prime_b_d @ R3 @ P )
=> ( P
!= ( zero_b_d @ R3 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_91_domain_Oring__primeE_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c,P: a] :
( ( domain_a_c @ R3 )
=> ( ( member_a @ P @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( ring_ring_prime_a_c @ R3 @ P )
=> ( P
!= ( zero_a_c @ R3 ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_92_domain_Oring__primeE_I3_J,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r5437400583859147359t_unit @ R3 @ P )
=> ( prime_1232919612140715622t_unit @ R3 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_93_domain_Oring__primeE_I3_J,axiom,
! [R3: partia4026993951477142903t_unit,P: list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( ring_r3344526403024810276t_unit @ R3 @ P )
=> ( prime_5961678782586786214t_unit @ R3 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_94_domain_Oring__primeE_I3_J,axiom,
! [R3: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r6430282645014804837t_unit @ R3 @ P )
=> ( prime_2011924034616061926t_unit @ R3 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_95_domain_Oring__primeE_I3_J,axiom,
! [R3: partia1897943568983147621xt_b_d,P: b] :
( ( domain_b_d @ R3 )
=> ( ( member_b @ P @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( ring_ring_prime_b_d @ R3 @ P )
=> ( prime_b_ring_ext_b_d @ R3 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_96_domain_Oring__primeE_I3_J,axiom,
! [R3: partia8877618634411419171xt_a_c,P: a] :
( ( domain_a_c @ R3 )
=> ( ( member_a @ P @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( ring_ring_prime_a_c @ R3 @ P )
=> ( prime_a_ring_ext_a_c @ R3 @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_97_abelian__group__hom_Ois__abelian__group__hom,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia1897943568983147621xt_b_d,H2: a > b] :
( ( abelia4987496640480023959_c_b_d @ G2 @ H @ H2 )
=> ( abelia4987496640480023959_c_b_d @ G2 @ H @ H2 ) ) ).
% abelian_group_hom.is_abelian_group_hom
thf(fact_98_ring__hom__closed,axiom,
! [H2: b > b,R3: partia1897943568983147621xt_b_d,S: partia1897943568983147621xt_b_d,X: b] :
( ( member_b_b @ H2 @ ( ring_hom_b_d_b_d @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_99_ring__hom__closed,axiom,
! [H2: b > a,R3: partia1897943568983147621xt_b_d,S: partia8877618634411419171xt_a_c,X: b] :
( ( member_b_a @ H2 @ ( ring_hom_b_d_a_c @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_100_ring__hom__closed,axiom,
! [H2: a > b,R3: partia8877618634411419171xt_a_c,S: partia1897943568983147621xt_b_d,X: a] :
( ( member_a_b @ H2 @ ( ring_hom_a_c_b_d @ R3 @ S ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_101_ring__hom__closed,axiom,
! [H2: a > a,R3: partia8877618634411419171xt_a_c,S: partia8877618634411419171xt_a_c,X: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_c_a_c @ R3 @ S ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_102_ring__hom__closed,axiom,
! [H2: list_b > b,R3: partia4026993951477142903t_unit,S: partia1897943568983147621xt_b_d,X: list_b] :
( ( member_list_b_b @ H2 @ ( ring_h8746557796359701252it_b_d @ R3 @ S ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_103_ring__hom__closed,axiom,
! [H2: list_b > a,R3: partia4026993951477142903t_unit,S: partia8877618634411419171xt_a_c,X: list_b] :
( ( member_list_b_a @ H2 @ ( ring_h2310641341477601476it_a_c @ R3 @ S ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_104_ring__hom__closed,axiom,
! [H2: list_a > b,R3: partia2670972154091845814t_unit,S: partia1897943568983147621xt_b_d,X: list_a] :
( ( member_list_a_b @ H2 @ ( ring_h108518356514633413it_b_d @ R3 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_105_ring__hom__closed,axiom,
! [H2: list_a > a,R3: partia2670972154091845814t_unit,S: partia8877618634411419171xt_a_c,X: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309445it_a_c @ R3 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_106_ring__hom__closed,axiom,
! [H2: b > list_b,R3: partia1897943568983147621xt_b_d,S: partia4026993951477142903t_unit,X: b] :
( ( member_b_list_b @ H2 @ ( ring_h8274988533691632388t_unit @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_list_b @ ( H2 @ X ) @ ( partia1381092143316337258t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_107_ring__hom__closed,axiom,
! [H2: b > list_a,R3: partia1897943568983147621xt_b_d,S: partia2670972154091845814t_unit,X: b] :
( ( member_b_list_a @ H2 @ ( ring_h2137372738826851141t_unit @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_hom_closed
thf(fact_108_ring__hom__one,axiom,
! [H2: b > b,R3: partia1897943568983147621xt_b_d,S: partia1897943568983147621xt_b_d] :
( ( member_b_b @ H2 @ ( ring_hom_b_d_b_d @ R3 @ S ) )
=> ( ( H2 @ ( one_b_ring_ext_b_d @ R3 ) )
= ( one_b_ring_ext_b_d @ S ) ) ) ).
% ring_hom_one
thf(fact_109_ring__hom__one,axiom,
! [H2: b > a,R3: partia1897943568983147621xt_b_d,S: partia8877618634411419171xt_a_c] :
( ( member_b_a @ H2 @ ( ring_hom_b_d_a_c @ R3 @ S ) )
=> ( ( H2 @ ( one_b_ring_ext_b_d @ R3 ) )
= ( one_a_ring_ext_a_c @ S ) ) ) ).
% ring_hom_one
thf(fact_110_ring__hom__one,axiom,
! [H2: a > b,R3: partia8877618634411419171xt_a_c,S: partia1897943568983147621xt_b_d] :
( ( member_a_b @ H2 @ ( ring_hom_a_c_b_d @ R3 @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_c @ R3 ) )
= ( one_b_ring_ext_b_d @ S ) ) ) ).
% ring_hom_one
thf(fact_111_ring__hom__one,axiom,
! [H2: a > a,R3: partia8877618634411419171xt_a_c,S: partia8877618634411419171xt_a_c] :
( ( member_a_a @ H2 @ ( ring_hom_a_c_a_c @ R3 @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_c @ R3 ) )
= ( one_a_ring_ext_a_c @ S ) ) ) ).
% ring_hom_one
thf(fact_112_ring__hom__one,axiom,
! [H2: b > list_a,R3: partia1897943568983147621xt_b_d,S: partia2670972154091845814t_unit] :
( ( member_b_list_a @ H2 @ ( ring_h2137372738826851141t_unit @ R3 @ S ) )
=> ( ( H2 @ ( one_b_ring_ext_b_d @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_113_ring__hom__one,axiom,
! [H2: b > list_b,R3: partia1897943568983147621xt_b_d,S: partia4026993951477142903t_unit] :
( ( member_b_list_b @ H2 @ ( ring_h8274988533691632388t_unit @ R3 @ S ) )
=> ( ( H2 @ ( one_b_ring_ext_b_d @ R3 ) )
= ( one_li3054569011217056637t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_114_ring__hom__one,axiom,
! [H2: a > list_a,R3: partia8877618634411419171xt_a_c,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_h5384962040972022021t_unit @ R3 @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_c @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_115_ring__hom__one,axiom,
! [H2: a > list_b,R3: partia8877618634411419171xt_a_c,S: partia4026993951477142903t_unit] :
( ( member_a_list_b @ H2 @ ( ring_h2299205798982027460t_unit @ R3 @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_c @ R3 ) )
= ( one_li3054569011217056637t_unit @ S ) ) ) ).
% ring_hom_one
thf(fact_116_ring__hom__one,axiom,
! [H2: list_a > b,R3: partia2670972154091845814t_unit,S: partia1897943568983147621xt_b_d] :
( ( member_list_a_b @ H2 @ ( ring_h108518356514633413it_b_d @ R3 @ S ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_b_ring_ext_b_d @ S ) ) ) ).
% ring_hom_one
thf(fact_117_ring__hom__one,axiom,
! [H2: list_a > a,R3: partia2670972154091845814t_unit,S: partia8877618634411419171xt_a_c] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309445it_a_c @ R3 @ S ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_a_ring_ext_a_c @ S ) ) ) ).
% ring_hom_one
thf(fact_118_abelian__group__hom_Ohom__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia1897943568983147621xt_b_d,H2: b > b,X: b] :
( ( abelia2848305256108177367_d_b_d @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ) ).
% abelian_group_hom.hom_closed
thf(fact_119_abelian__group__hom_Ohom__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia8877618634411419171xt_a_c,H2: b > a,X: b] :
( ( abelia5635760838080853399_d_a_c @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ H ) ) ) ) ).
% abelian_group_hom.hom_closed
thf(fact_120_abelian__group__hom_Ohom__closed,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia1897943568983147621xt_b_d,H2: a > b,X: a] :
( ( abelia4987496640480023959_c_b_d @ G2 @ H @ H2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ) ).
% abelian_group_hom.hom_closed
thf(fact_121_abelian__group__hom_Ohom__closed,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia8877618634411419171xt_a_c,H2: a > a,X: a] :
( ( abelia7774952222452699991_c_a_c @ G2 @ H @ H2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ H ) ) ) ) ).
% abelian_group_hom.hom_closed
thf(fact_122_abelian__group__hom_Ohom__closed,axiom,
! [G2: partia4026993951477142903t_unit,H: partia1897943568983147621xt_b_d,H2: list_b > b,X: list_b] :
( ( abelia4844232365066319197it_b_d @ G2 @ H @ H2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ) ).
% abelian_group_hom.hom_closed
thf(fact_123_abelian__group__hom_Ohom__closed,axiom,
! [G2: partia4026993951477142903t_unit,H: partia8877618634411419171xt_a_c,H2: list_b > a,X: list_b] :
( ( abelia7631687947038995229it_a_c @ G2 @ H @ H2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ H ) ) ) ) ).
% abelian_group_hom.hom_closed
thf(fact_124_abelian__group__hom_Ohom__closed,axiom,
! [G2: partia2670972154091845814t_unit,H: partia1897943568983147621xt_b_d,H2: list_a > b,X: list_a] :
( ( abelia5429564962076027166it_b_d @ G2 @ H @ H2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ) ).
% abelian_group_hom.hom_closed
thf(fact_125_abelian__group__hom_Ohom__closed,axiom,
! [G2: partia2670972154091845814t_unit,H: partia8877618634411419171xt_a_c,H2: list_a > a,X: list_a] :
( ( abelia8217020544048703198it_a_c @ G2 @ H @ H2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ H ) ) ) ) ).
% abelian_group_hom.hom_closed
thf(fact_126_abelian__group__hom_Ohom__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia4026993951477142903t_unit,H2: b > list_b,X: b] :
( ( abelia4372663102398250333t_unit @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( member_list_b @ ( H2 @ X ) @ ( partia1381092143316337258t_unit @ H ) ) ) ) ).
% abelian_group_hom.hom_closed
thf(fact_127_abelian__group__hom_Ohom__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia2670972154091845814t_unit,H2: b > list_a,X: b] :
( ( abelia7458419344388244894t_unit @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ H ) ) ) ) ).
% abelian_group_hom.hom_closed
thf(fact_128_abelian__group__hom_Ohom__zero,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia1897943568983147621xt_b_d,H2: b > b] :
( ( abelia2848305256108177367_d_b_d @ G2 @ H @ H2 )
=> ( ( H2 @ ( zero_b_d @ G2 ) )
= ( zero_b_d @ H ) ) ) ).
% abelian_group_hom.hom_zero
thf(fact_129_abelian__group__hom_Ohom__zero,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia8877618634411419171xt_a_c,H2: b > a] :
( ( abelia5635760838080853399_d_a_c @ G2 @ H @ H2 )
=> ( ( H2 @ ( zero_b_d @ G2 ) )
= ( zero_a_c @ H ) ) ) ).
% abelian_group_hom.hom_zero
thf(fact_130_abelian__group__hom_Ohom__zero,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia8877618634411419171xt_a_c,H2: a > a] :
( ( abelia7774952222452699991_c_a_c @ G2 @ H @ H2 )
=> ( ( H2 @ ( zero_a_c @ G2 ) )
= ( zero_a_c @ H ) ) ) ).
% abelian_group_hom.hom_zero
thf(fact_131_abelian__group__hom_Ohom__zero,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia1897943568983147621xt_b_d,H2: a > b] :
( ( abelia4987496640480023959_c_b_d @ G2 @ H @ H2 )
=> ( ( H2 @ ( zero_a_c @ G2 ) )
= ( zero_b_d @ H ) ) ) ).
% abelian_group_hom.hom_zero
thf(fact_132_abelian__group__hom_Ohom__zero,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia2670972154091845814t_unit,H2: b > list_a] :
( ( abelia7458419344388244894t_unit @ G2 @ H @ H2 )
=> ( ( H2 @ ( zero_b_d @ G2 ) )
= ( zero_l4142658623432671053t_unit @ H ) ) ) ).
% abelian_group_hom.hom_zero
thf(fact_133_abelian__group__hom_Ohom__zero,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia4026993951477142903t_unit,H2: b > list_b] :
( ( abelia4372663102398250333t_unit @ G2 @ H @ H2 )
=> ( ( H2 @ ( zero_b_d @ G2 ) )
= ( zero_l1056902381442676492t_unit @ H ) ) ) ).
% abelian_group_hom.hom_zero
thf(fact_134_abelian__group__hom_Ohom__zero,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia2670972154091845814t_unit,H2: a > list_a] :
( ( abelia1482636609678639966t_unit @ G2 @ H @ H2 )
=> ( ( H2 @ ( zero_a_c @ G2 ) )
= ( zero_l4142658623432671053t_unit @ H ) ) ) ).
% abelian_group_hom.hom_zero
thf(fact_135_abelian__group__hom_Ohom__zero,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia4026993951477142903t_unit,H2: a > list_b] :
( ( abelia7620252404543421213t_unit @ G2 @ H @ H2 )
=> ( ( H2 @ ( zero_a_c @ G2 ) )
= ( zero_l1056902381442676492t_unit @ H ) ) ) ).
% abelian_group_hom.hom_zero
thf(fact_136_abelian__group__hom_Ohom__zero,axiom,
! [G2: partia2670972154091845814t_unit,H: partia1897943568983147621xt_b_d,H2: list_a > b] :
( ( abelia5429564962076027166it_b_d @ G2 @ H @ H2 )
=> ( ( H2 @ ( zero_l4142658623432671053t_unit @ G2 ) )
= ( zero_b_d @ H ) ) ) ).
% abelian_group_hom.hom_zero
thf(fact_137_abelian__group__hom_Ohom__zero,axiom,
! [G2: partia2670972154091845814t_unit,H: partia8877618634411419171xt_a_c,H2: list_a > a] :
( ( abelia8217020544048703198it_a_c @ G2 @ H @ H2 )
=> ( ( H2 @ ( zero_l4142658623432671053t_unit @ G2 ) )
= ( zero_a_c @ H ) ) ) ).
% abelian_group_hom.hom_zero
thf(fact_138_semiring_Oaxioms_I1_J,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( abelia3641329199688042803t_unit @ R3 ) ) ).
% semiring.axioms(1)
thf(fact_139_semiring_Oaxioms_I1_J,axiom,
! [R3: partia4026993951477142903t_unit] :
( ( semiri9009524540797033698t_unit @ R3 )
=> ( abelia6363847436574302712t_unit @ R3 ) ) ).
% semiring.axioms(1)
thf(fact_140_semiring_Oaxioms_I1_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( abelia226231641709521465t_unit @ R3 ) ) ).
% semiring.axioms(1)
thf(fact_141_semiring_Oaxioms_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d] :
( ( semiring_b_d @ R3 )
=> ( abelian_monoid_b_d @ R3 ) ) ).
% semiring.axioms(1)
thf(fact_142_semiring_Oaxioms_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c] :
( ( semiring_a_c @ R3 )
=> ( abelian_monoid_a_c @ R3 ) ) ).
% semiring.axioms(1)
thf(fact_143_domain_Ozero__not__one,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( zero_l347298301471573063t_unit @ R3 )
!= ( one_li8234411390022467901t_unit @ R3 ) ) ) ).
% domain.zero_not_one
thf(fact_144_domain_Ozero__not__one,axiom,
! [R3: partia1897943568983147621xt_b_d] :
( ( domain_b_d @ R3 )
=> ( ( zero_b_d @ R3 )
!= ( one_b_ring_ext_b_d @ R3 ) ) ) ).
% domain.zero_not_one
thf(fact_145_domain_Ozero__not__one,axiom,
! [R3: partia8877618634411419171xt_a_c] :
( ( domain_a_c @ R3 )
=> ( ( zero_a_c @ R3 )
!= ( one_a_ring_ext_a_c @ R3 ) ) ) ).
% domain.zero_not_one
thf(fact_146_domain_Ozero__not__one,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( zero_l4142658623432671053t_unit @ R3 )
!= ( one_li8328186300101108157t_unit @ R3 ) ) ) ).
% domain.zero_not_one
thf(fact_147_domain_Ozero__not__one,axiom,
! [R3: partia4026993951477142903t_unit] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( zero_l1056902381442676492t_unit @ R3 )
!= ( one_li3054569011217056637t_unit @ R3 ) ) ) ).
% domain.zero_not_one
thf(fact_148_domain_Oone__not__zero,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( one_li8234411390022467901t_unit @ R3 )
!= ( zero_l347298301471573063t_unit @ R3 ) ) ) ).
% domain.one_not_zero
thf(fact_149_domain_Oone__not__zero,axiom,
! [R3: partia1897943568983147621xt_b_d] :
( ( domain_b_d @ R3 )
=> ( ( one_b_ring_ext_b_d @ R3 )
!= ( zero_b_d @ R3 ) ) ) ).
% domain.one_not_zero
thf(fact_150_domain_Oone__not__zero,axiom,
! [R3: partia8877618634411419171xt_a_c] :
( ( domain_a_c @ R3 )
=> ( ( one_a_ring_ext_a_c @ R3 )
!= ( zero_a_c @ R3 ) ) ) ).
% domain.one_not_zero
thf(fact_151_domain_Oone__not__zero,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( one_li8328186300101108157t_unit @ R3 )
!= ( zero_l4142658623432671053t_unit @ R3 ) ) ) ).
% domain.one_not_zero
thf(fact_152_domain_Oone__not__zero,axiom,
! [R3: partia4026993951477142903t_unit] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( one_li3054569011217056637t_unit @ R3 )
!= ( zero_l1056902381442676492t_unit @ R3 ) ) ) ).
% domain.one_not_zero
thf(fact_153_abelian__monoidE_I2_J,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( abelia3641329199688042803t_unit @ R3 )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ).
% abelian_monoidE(2)
thf(fact_154_abelian__monoidE_I2_J,axiom,
! [R3: partia4026993951477142903t_unit] :
( ( abelia6363847436574302712t_unit @ R3 )
=> ( member_list_b @ ( zero_l1056902381442676492t_unit @ R3 ) @ ( partia1381092143316337258t_unit @ R3 ) ) ) ).
% abelian_monoidE(2)
thf(fact_155_abelian__monoidE_I2_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ R3 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ).
% abelian_monoidE(2)
thf(fact_156_abelian__monoidE_I2_J,axiom,
! [R3: partia1897943568983147621xt_b_d] :
( ( abelian_monoid_b_d @ R3 )
=> ( member_b @ ( zero_b_d @ R3 ) @ ( partia8782771468121683032xt_b_d @ R3 ) ) ) ).
% abelian_monoidE(2)
thf(fact_157_abelian__monoidE_I2_J,axiom,
! [R3: partia8877618634411419171xt_a_c] :
( ( abelian_monoid_a_c @ R3 )
=> ( member_a @ ( zero_a_c @ R3 ) @ ( partia778085601923319190xt_a_c @ R3 ) ) ) ).
% abelian_monoidE(2)
thf(fact_158_abelian__monoid_Ozero__closed,axiom,
! [G2: partia2956882679547061052t_unit] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ G2 ) @ ( partia2464479390973590831t_unit @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_159_abelian__monoid_Ozero__closed,axiom,
! [G2: partia4026993951477142903t_unit] :
( ( abelia6363847436574302712t_unit @ G2 )
=> ( member_list_b @ ( zero_l1056902381442676492t_unit @ G2 ) @ ( partia1381092143316337258t_unit @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_160_abelian__monoid_Ozero__closed,axiom,
! [G2: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ G2 ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_161_abelian__monoid_Ozero__closed,axiom,
! [G2: partia1897943568983147621xt_b_d] :
( ( abelian_monoid_b_d @ G2 )
=> ( member_b @ ( zero_b_d @ G2 ) @ ( partia8782771468121683032xt_b_d @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_162_abelian__monoid_Ozero__closed,axiom,
! [G2: partia8877618634411419171xt_a_c] :
( ( abelian_monoid_a_c @ G2 )
=> ( member_a @ ( zero_a_c @ G2 ) @ ( partia778085601923319190xt_a_c @ G2 ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_163_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_164_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia4026993951477142903t_unit] :
( ( semiri9009524540797033698t_unit @ R3 )
=> ( member_list_b @ ( zero_l1056902381442676492t_unit @ R3 ) @ ( partia1381092143316337258t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_165_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_166_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia1897943568983147621xt_b_d] :
( ( semiring_b_d @ R3 )
=> ( member_b @ ( zero_b_d @ R3 ) @ ( partia8782771468121683032xt_b_d @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_167_semiring_Osemiring__simprules_I2_J,axiom,
! [R3: partia8877618634411419171xt_a_c] :
( ( semiring_a_c @ R3 )
=> ( member_a @ ( zero_a_c @ R3 ) @ ( partia778085601923319190xt_a_c @ R3 ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_168_ds_Oisgcd__divides__l,axiom,
! [A: b,B: b] :
( ( factor2325171414093416164xt_b_d @ s @ A @ B )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( isgcd_b_ring_ext_b_d @ s @ A @ A @ B ) ) ) ) ).
% ds.isgcd_divides_l
thf(fact_169_ds_Oisgcd__divides__r,axiom,
! [B: b,A: b] :
( ( factor2325171414093416164xt_b_d @ s @ B @ A )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( isgcd_b_ring_ext_b_d @ s @ B @ A @ B ) ) ) ) ).
% ds.isgcd_divides_r
thf(fact_170_ds_Oonepideal,axiom,
principalideal_b_d @ ( partia8782771468121683032xt_b_d @ s ) @ s ).
% ds.onepideal
thf(fact_171_ds_Oring__irreducibleE_I1_J,axiom,
! [R2: b] :
( ( member_b @ R2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_r7435050590149293703le_b_d @ s @ R2 )
=> ( R2
!= ( zero_b_d @ s ) ) ) ) ).
% ds.ring_irreducibleE(1)
thf(fact_172_h_Oinj__on__domain,axiom,
( ( inj_on_a_b @ h @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( domain_b_d @ s )
=> ( domain_a_c @ r ) ) ) ).
% h.inj_on_domain
thf(fact_173_h__img,axiom,
( ( image_a_b @ h @ ( partia778085601923319190xt_a_c @ r ) )
= ( partia8782771468121683032xt_b_d @ s ) ) ).
% h_img
thf(fact_174_dr_Ocgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia778085601923319190xt_a_c @ r ) )
=> ( principalideal_a_c @ ( cgenid547466214215511830xt_a_c @ r @ I ) @ r ) ) ).
% dr.cgenideal_is_principalideal
thf(fact_175_h_Oa__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_b @ ( h @ ( a_inv_a_c @ r @ X ) ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% h.a_inv_closed
thf(fact_176_dr_Oring__irreducibleE_I2_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_r999134135267193927le_a_c @ r @ R2 )
=> ( irredu6211895651204806704xt_a_c @ r @ R2 ) ) ) ).
% dr.ring_irreducibleE(2)
thf(fact_177_ds_Osubring__props_I1_J,axiom,
! [K: set_b] :
( ( subfield_b_d @ K @ s )
=> ( ord_less_eq_set_b @ K @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% ds.subring_props(1)
thf(fact_178_dr_Oabelian__monoid__axioms,axiom,
abelian_monoid_a_c @ r ).
% dr.abelian_monoid_axioms
thf(fact_179_pdr_Oabelian__monoid__axioms,axiom,
abelia226231641709521465t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.abelian_monoid_axioms
thf(fact_180_pdr_Oring__primeI,axiom,
! [P: list_a] :
( ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P ) ) ) ).
% pdr.ring_primeI
thf(fact_181_pdr_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.semiring_axioms
thf(fact_182_pdr_Odomain__axioms,axiom,
domain6553523120543210313t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.domain_axioms
thf(fact_183_pdr_Ozero__is__prime_I1_J,axiom,
prime_2011924034616061926t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% pdr.zero_is_prime(1)
thf(fact_184_pdr_Ozero__not__one,axiom,
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.zero_not_one
thf(fact_185_dr_Ocgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ I @ ( cgenid547466214215511830xt_a_c @ r @ I ) ) ) ).
% dr.cgenideal_self
thf(fact_186_pdr_Oring__irreducibleE_I1_J,axiom,
! [R2: list_a] :
( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ R2 )
=> ( R2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.ring_irreducibleE(1)
thf(fact_187_pdr_Oring__primeE_I3_J,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P )
=> ( prime_2011924034616061926t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P ) ) ) ).
% pdr.ring_primeE(3)
thf(fact_188_pdr_Oring__primeE_I1_J,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.ring_primeE(1)
thf(fact_189_h__inj,axiom,
inj_on_a_b @ h @ ( partia778085601923319190xt_a_c @ r ) ).
% h_inj
thf(fact_190_dr_Oadd_Oinv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ ( a_inv_a_c @ r @ X ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% dr.add.inv_closed
thf(fact_191_dr_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( a_inv_a_c @ r @ ( a_inv_a_c @ r @ X ) )
= X ) ) ).
% dr.minus_minus
thf(fact_192_dr_Ominus__zero,axiom,
( ( a_inv_a_c @ r @ ( zero_a_c @ r ) )
= ( zero_a_c @ r ) ) ).
% dr.minus_zero
thf(fact_193_pdr_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% pdr.zero_closed
thf(fact_194_pdr_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% pdr.one_closed
thf(fact_195_dr_Oadd_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( a_inv_a_c @ r @ X )
= ( zero_a_c @ r ) )
= ( X
= ( zero_a_c @ r ) ) ) ) ).
% dr.add.inv_eq_1_iff
thf(fact_196_abelian__group__hom_Ohom__a__inv,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia1897943568983147621xt_b_d,H2: b > b,X: b] :
( ( abelia2848305256108177367_d_b_d @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( H2 @ ( a_inv_b_d @ G2 @ X ) )
= ( a_inv_b_d @ H @ ( H2 @ X ) ) ) ) ) ).
% abelian_group_hom.hom_a_inv
thf(fact_197_abelian__group__hom_Ohom__a__inv,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia8877618634411419171xt_a_c,H2: b > a,X: b] :
( ( abelia5635760838080853399_d_a_c @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( H2 @ ( a_inv_b_d @ G2 @ X ) )
= ( a_inv_a_c @ H @ ( H2 @ X ) ) ) ) ) ).
% abelian_group_hom.hom_a_inv
thf(fact_198_abelian__group__hom_Ohom__a__inv,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia8877618634411419171xt_a_c,H2: a > a,X: a] :
( ( abelia7774952222452699991_c_a_c @ G2 @ H @ H2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( H2 @ ( a_inv_a_c @ G2 @ X ) )
= ( a_inv_a_c @ H @ ( H2 @ X ) ) ) ) ) ).
% abelian_group_hom.hom_a_inv
thf(fact_199_abelian__group__hom_Ohom__a__inv,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia1897943568983147621xt_b_d,H2: a > b,X: a] :
( ( abelia4987496640480023959_c_b_d @ G2 @ H @ H2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( H2 @ ( a_inv_a_c @ G2 @ X ) )
= ( a_inv_b_d @ H @ ( H2 @ X ) ) ) ) ) ).
% abelian_group_hom.hom_a_inv
thf(fact_200_abelian__group__hom_Ohom__a__inv,axiom,
! [G2: partia4026993951477142903t_unit,H: partia1897943568983147621xt_b_d,H2: list_b > b,X: list_b] :
( ( abelia4844232365066319197it_b_d @ G2 @ H @ H2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( H2 @ ( a_inv_5858964851304622612t_unit @ G2 @ X ) )
= ( a_inv_b_d @ H @ ( H2 @ X ) ) ) ) ) ).
% abelian_group_hom.hom_a_inv
thf(fact_201_abelian__group__hom_Ohom__a__inv,axiom,
! [G2: partia4026993951477142903t_unit,H: partia8877618634411419171xt_a_c,H2: list_b > a,X: list_b] :
( ( abelia7631687947038995229it_a_c @ G2 @ H @ H2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( H2 @ ( a_inv_5858964851304622612t_unit @ G2 @ X ) )
= ( a_inv_a_c @ H @ ( H2 @ X ) ) ) ) ) ).
% abelian_group_hom.hom_a_inv
thf(fact_202_abelian__group__hom_Ohom__a__inv,axiom,
! [G2: partia2670972154091845814t_unit,H: partia1897943568983147621xt_b_d,H2: list_a > b,X: list_a] :
( ( abelia5429564962076027166it_b_d @ G2 @ H @ H2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( H2 @ ( a_inv_8944721093294617173t_unit @ G2 @ X ) )
= ( a_inv_b_d @ H @ ( H2 @ X ) ) ) ) ) ).
% abelian_group_hom.hom_a_inv
thf(fact_203_abelian__group__hom_Ohom__a__inv,axiom,
! [G2: partia2670972154091845814t_unit,H: partia8877618634411419171xt_a_c,H2: list_a > a,X: list_a] :
( ( abelia8217020544048703198it_a_c @ G2 @ H @ H2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( H2 @ ( a_inv_8944721093294617173t_unit @ G2 @ X ) )
= ( a_inv_a_c @ H @ ( H2 @ X ) ) ) ) ) ).
% abelian_group_hom.hom_a_inv
thf(fact_204_abelian__group__hom_Ohom__a__inv,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia4026993951477142903t_unit,H2: b > list_b,X: b] :
( ( abelia4372663102398250333t_unit @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( H2 @ ( a_inv_b_d @ G2 @ X ) )
= ( a_inv_5858964851304622612t_unit @ H @ ( H2 @ X ) ) ) ) ) ).
% abelian_group_hom.hom_a_inv
thf(fact_205_abelian__group__hom_Ohom__a__inv,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia2670972154091845814t_unit,H2: b > list_a,X: b] :
( ( abelia7458419344388244894t_unit @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( H2 @ ( a_inv_b_d @ G2 @ X ) )
= ( a_inv_8944721093294617173t_unit @ H @ ( H2 @ X ) ) ) ) ) ).
% abelian_group_hom.hom_a_inv
thf(fact_206_abelian__group__hom_Oa__inv__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia1897943568983147621xt_b_d,H2: b > b,X: b] :
( ( abelia2848305256108177367_d_b_d @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( member_b @ ( H2 @ ( a_inv_b_d @ G2 @ X ) ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ) ).
% abelian_group_hom.a_inv_closed
thf(fact_207_abelian__group__hom_Oa__inv__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia8877618634411419171xt_a_c,H2: b > a,X: b] :
( ( abelia5635760838080853399_d_a_c @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( member_a @ ( H2 @ ( a_inv_b_d @ G2 @ X ) ) @ ( partia778085601923319190xt_a_c @ H ) ) ) ) ).
% abelian_group_hom.a_inv_closed
thf(fact_208_abelian__group__hom_Oa__inv__closed,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia1897943568983147621xt_b_d,H2: a > b,X: a] :
( ( abelia4987496640480023959_c_b_d @ G2 @ H @ H2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( member_b @ ( H2 @ ( a_inv_a_c @ G2 @ X ) ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ) ).
% abelian_group_hom.a_inv_closed
thf(fact_209_abelian__group__hom_Oa__inv__closed,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia8877618634411419171xt_a_c,H2: a > a,X: a] :
( ( abelia7774952222452699991_c_a_c @ G2 @ H @ H2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( member_a @ ( H2 @ ( a_inv_a_c @ G2 @ X ) ) @ ( partia778085601923319190xt_a_c @ H ) ) ) ) ).
% abelian_group_hom.a_inv_closed
thf(fact_210_abelian__group__hom_Oa__inv__closed,axiom,
! [G2: partia4026993951477142903t_unit,H: partia1897943568983147621xt_b_d,H2: list_b > b,X: list_b] :
( ( abelia4844232365066319197it_b_d @ G2 @ H @ H2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( member_b @ ( H2 @ ( a_inv_5858964851304622612t_unit @ G2 @ X ) ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ) ).
% abelian_group_hom.a_inv_closed
thf(fact_211_abelian__group__hom_Oa__inv__closed,axiom,
! [G2: partia4026993951477142903t_unit,H: partia8877618634411419171xt_a_c,H2: list_b > a,X: list_b] :
( ( abelia7631687947038995229it_a_c @ G2 @ H @ H2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( member_a @ ( H2 @ ( a_inv_5858964851304622612t_unit @ G2 @ X ) ) @ ( partia778085601923319190xt_a_c @ H ) ) ) ) ).
% abelian_group_hom.a_inv_closed
thf(fact_212_abelian__group__hom_Oa__inv__closed,axiom,
! [G2: partia2670972154091845814t_unit,H: partia1897943568983147621xt_b_d,H2: list_a > b,X: list_a] :
( ( abelia5429564962076027166it_b_d @ G2 @ H @ H2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( member_b @ ( H2 @ ( a_inv_8944721093294617173t_unit @ G2 @ X ) ) @ ( partia8782771468121683032xt_b_d @ H ) ) ) ) ).
% abelian_group_hom.a_inv_closed
thf(fact_213_abelian__group__hom_Oa__inv__closed,axiom,
! [G2: partia2670972154091845814t_unit,H: partia8877618634411419171xt_a_c,H2: list_a > a,X: list_a] :
( ( abelia8217020544048703198it_a_c @ G2 @ H @ H2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( member_a @ ( H2 @ ( a_inv_8944721093294617173t_unit @ G2 @ X ) ) @ ( partia778085601923319190xt_a_c @ H ) ) ) ) ).
% abelian_group_hom.a_inv_closed
thf(fact_214_abelian__group__hom_Oa__inv__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia4026993951477142903t_unit,H2: b > list_b,X: b] :
( ( abelia4372663102398250333t_unit @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( member_list_b @ ( H2 @ ( a_inv_b_d @ G2 @ X ) ) @ ( partia1381092143316337258t_unit @ H ) ) ) ) ).
% abelian_group_hom.a_inv_closed
thf(fact_215_abelian__group__hom_Oa__inv__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia2670972154091845814t_unit,H2: b > list_a,X: b] :
( ( abelia7458419344388244894t_unit @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( member_list_a @ ( H2 @ ( a_inv_b_d @ G2 @ X ) ) @ ( partia5361259788508890537t_unit @ H ) ) ) ) ).
% abelian_group_hom.a_inv_closed
thf(fact_216_ring__irreducible__def,axiom,
( ring_r360171070648044744t_unit
= ( ^ [R4: partia2956882679547061052t_unit,A3: list_list_a] :
( ( A3
!= ( zero_l347298301471573063t_unit @ R4 ) )
& ( irredu4439051761327310013t_unit @ R4 @ A3 ) ) ) ) ).
% ring_irreducible_def
thf(fact_217_ring__irreducible__def,axiom,
( ring_r7070601269410051085t_unit
= ( ^ [R4: partia4026993951477142903t_unit,A3: list_b] :
( ( A3
!= ( zero_l1056902381442676492t_unit @ R4 ) )
& ( irredu8180679162501400317t_unit @ R4 @ A3 ) ) ) ) ).
% ring_irreducible_def
thf(fact_218_ring__irreducible__def,axiom,
( ring_r932985474545269838t_unit
= ( ^ [R4: partia2670972154091845814t_unit,A3: list_a] :
( ( A3
!= ( zero_l4142658623432671053t_unit @ R4 ) )
& ( irredu4230924414530676029t_unit @ R4 @ A3 ) ) ) ) ).
% ring_irreducible_def
thf(fact_219_ring__irreducible__def,axiom,
( ring_r7435050590149293703le_b_d
= ( ^ [R4: partia1897943568983147621xt_b_d,A3: b] :
( ( A3
!= ( zero_b_d @ R4 ) )
& ( irredu320915990819274225xt_b_d @ R4 @ A3 ) ) ) ) ).
% ring_irreducible_def
thf(fact_220_ring__irreducible__def,axiom,
( ring_r999134135267193927le_a_c
= ( ^ [R4: partia8877618634411419171xt_a_c,A3: a] :
( ( A3
!= ( zero_a_c @ R4 ) )
& ( irredu6211895651204806704xt_a_c @ R4 @ A3 ) ) ) ) ).
% ring_irreducible_def
thf(fact_221_domain_Oring__irreducibleE_I2_J,axiom,
! [R3: partia2956882679547061052t_unit,R2: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r360171070648044744t_unit @ R3 @ R2 )
=> ( irredu4439051761327310013t_unit @ R3 @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_222_domain_Oring__irreducibleE_I2_J,axiom,
! [R3: partia4026993951477142903t_unit,R2: list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( member_list_b @ R2 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( ring_r7070601269410051085t_unit @ R3 @ R2 )
=> ( irredu8180679162501400317t_unit @ R3 @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_223_domain_Oring__irreducibleE_I2_J,axiom,
! [R3: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r932985474545269838t_unit @ R3 @ R2 )
=> ( irredu4230924414530676029t_unit @ R3 @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_224_domain_Oring__irreducibleE_I2_J,axiom,
! [R3: partia1897943568983147621xt_b_d,R2: b] :
( ( domain_b_d @ R3 )
=> ( ( member_b @ R2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( ring_r7435050590149293703le_b_d @ R3 @ R2 )
=> ( irredu320915990819274225xt_b_d @ R3 @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_225_domain_Oring__irreducibleE_I2_J,axiom,
! [R3: partia8877618634411419171xt_a_c,R2: a] :
( ( domain_a_c @ R3 )
=> ( ( member_a @ R2 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( ring_r999134135267193927le_a_c @ R3 @ R2 )
=> ( irredu6211895651204806704xt_a_c @ R3 @ R2 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_226_pdr_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.onepideal
thf(fact_227_h_Oimg__is__subfield_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_c @ K @ r )
=> ( ( ( one_b_ring_ext_b_d @ s )
!= ( zero_b_d @ s ) )
=> ( subfield_b_d @ ( image_a_b @ h @ K ) @ s ) ) ) ).
% h.img_is_subfield(2)
thf(fact_228_h_Oimg__is__subfield_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_c @ K @ r )
=> ( ( ( one_b_ring_ext_b_d @ s )
!= ( zero_b_d @ s ) )
=> ( inj_on_a_b @ h @ K ) ) ) ).
% h.img_is_subfield(1)
thf(fact_229_ds_Oa__lcos__mult__one,axiom,
! [M: set_b] :
( ( ord_less_eq_set_b @ M @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( a_l_coset_b_d @ s @ ( zero_b_d @ s ) @ M )
= M ) ) ).
% ds.a_lcos_mult_one
thf(fact_230_pdr_Ocarrier__is__subcring,axiom,
subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.carrier_is_subcring
thf(fact_231_ds_Oto__contain__is__to__divide,axiom,
! [A: b,B: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ord_less_eq_set_b @ ( cgenid3879858590684755159xt_b_d @ s @ B ) @ ( cgenid3879858590684755159xt_b_d @ s @ A ) )
= ( factor2325171414093416164xt_b_d @ s @ A @ B ) ) ) ) ).
% ds.to_contain_is_to_divide
thf(fact_232_ds_Ocgenideal__is__principalideal,axiom,
! [I: b] :
( ( member_b @ I @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( principalideal_b_d @ ( cgenid3879858590684755159xt_b_d @ s @ I ) @ s ) ) ).
% ds.cgenideal_is_principalideal
thf(fact_233_ds_Osubalgebra__in__carrier,axiom,
! [K: set_b,V: set_b] :
( ( embedd6240069993967058123ra_b_d @ K @ V @ s )
=> ( ord_less_eq_set_b @ V @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% ds.subalgebra_in_carrier
thf(fact_234_ds_Ocarrier__is__subalgebra,axiom,
! [K: set_b] :
( ( ord_less_eq_set_b @ K @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( embedd6240069993967058123ra_b_d @ K @ ( partia8782771468121683032xt_b_d @ s ) @ s ) ) ).
% ds.carrier_is_subalgebra
thf(fact_235_ds_Osubring__props_I5_J,axiom,
! [K: set_b,H2: b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_b @ H2 @ K )
=> ( member_b @ ( a_inv_b_d @ s @ H2 ) @ K ) ) ) ).
% ds.subring_props(5)
thf(fact_236_dr_Osubring__props_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_c @ K @ r )
=> ( member_a @ ( zero_a_c @ r ) @ K ) ) ).
% dr.subring_props(2)
thf(fact_237_dr_Osubring__props_I3_J,axiom,
! [K: set_a] :
( ( subfield_a_c @ K @ r )
=> ( member_a @ ( one_a_ring_ext_a_c @ r ) @ K ) ) ).
% dr.subring_props(3)
thf(fact_238_dr_Osubring__props_I5_J,axiom,
! [K: set_a,H2: a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( a_inv_a_c @ r @ H2 ) @ K ) ) ) ).
% dr.subring_props(5)
thf(fact_239_ds_Ocgenideal__self,axiom,
! [I: b] :
( ( member_b @ I @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ I @ ( cgenid3879858590684755159xt_b_d @ s @ I ) ) ) ).
% ds.cgenideal_self
thf(fact_240_ds_Oring__irreducibleE_I2_J,axiom,
! [R2: b] :
( ( member_b @ R2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_r7435050590149293703le_b_d @ s @ R2 )
=> ( irredu320915990819274225xt_b_d @ s @ R2 ) ) ) ).
% ds.ring_irreducibleE(2)
thf(fact_241_ds_Oa__l__coset__subset__G,axiom,
! [H: set_b,X: b] :
( ( ord_less_eq_set_b @ H @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ord_less_eq_set_b @ ( a_l_coset_b_d @ s @ X @ H ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% ds.a_l_coset_subset_G
thf(fact_242_pdr_Oring__irreducibleE_I2_J,axiom,
! [R2: list_a] :
( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ R2 )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ R2 ) ) ) ).
% pdr.ring_irreducibleE(2)
thf(fact_243_dr_Opprime__iff__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ K ) @ P ) ) ) ) ).
% dr.pprime_iff_pirreducible
thf(fact_244_ds_Oadd_Oinv__closed,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ ( a_inv_b_d @ s @ X ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% ds.add.inv_closed
thf(fact_245_ds_Ominus__minus,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( a_inv_b_d @ s @ ( a_inv_b_d @ s @ X ) )
= X ) ) ).
% ds.minus_minus
thf(fact_246_ds_Ominus__zero,axiom,
( ( a_inv_b_d @ s @ ( zero_b_d @ s ) )
= ( zero_b_d @ s ) ) ).
% ds.minus_zero
thf(fact_247_ds_Oadd_Oinv__eq__1__iff,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( a_inv_b_d @ s @ X )
= ( zero_b_d @ s ) )
= ( X
= ( zero_b_d @ s ) ) ) ) ).
% ds.add.inv_eq_1_iff
thf(fact_248_pdr_Ominus__minus,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) )
= X ) ) ).
% pdr.minus_minus
thf(fact_249_pdr_Oadd_Oinv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.add.inv_closed
thf(fact_250_pdr_Ominus__zero,axiom,
( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.minus_zero
thf(fact_251_pdr_Oadd_Oinv__eq__1__iff,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( X
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.add.inv_eq_1_iff
thf(fact_252_h_Ohom__a__inv,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( h @ ( a_inv_a_c @ r @ X ) )
= ( a_inv_b_d @ s @ ( h @ X ) ) ) ) ).
% h.hom_a_inv
thf(fact_253_dr_Ouniv__poly__is__principal,axiom,
! [K: set_a] :
( ( subfield_a_c @ K @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ).
% dr.univ_poly_is_principal
thf(fact_254_dr_Olong__division__a__inv_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_c @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ K ) @ ( polynomial_pdiv_a_c @ r @ P @ Q ) ) ) ) ) ) ).
% dr.long_division_a_inv(1)
thf(fact_255_dr_Olong__division__closed_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_c @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ) ) ) ).
% dr.long_division_closed(1)
thf(fact_256_domain_Opprime__iff__pirreducible,axiom,
! [R3: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( subfie4546268998243038636t_unit @ K @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( ring_r346321679897941977t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) @ P )
= ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_257_domain_Opprime__iff__pirreducible,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b,P: list_list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( subfie7916738691610828529t_unit @ K @ R3 )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( ring_r415245522172713630t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) @ P )
= ( ring_r4561388045816386823t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_258_domain_Opprime__iff__pirreducible,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_259_domain_Opprime__iff__pirreducible,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b,P: list_b] :
( ( domain_b_d @ R3 )
=> ( ( subfield_b_d @ K @ R3 )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ R3 @ K ) @ P )
= ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ R3 @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_260_domain_Opprime__iff__pirreducible,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a,P: list_a] :
( ( domain_a_c @ R3 )
=> ( ( subfield_a_c @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ R3 @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ R3 @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_261_a__coset__hom_I1_J,axiom,
! [H2: b > b,R3: partia1897943568983147621xt_b_d,S: partia1897943568983147621xt_b_d,I2: set_b,A: b] :
( ( member_b_b @ H2 @ ( ring_hom_b_d_b_d @ R3 @ S ) )
=> ( ( ord_less_eq_set_b @ I2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( image_b_b @ H2 @ ( a_l_coset_b_d @ R3 @ A @ I2 ) )
= ( a_l_coset_b_d @ S @ ( H2 @ A ) @ ( image_b_b @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_262_a__coset__hom_I1_J,axiom,
! [H2: b > a,R3: partia1897943568983147621xt_b_d,S: partia8877618634411419171xt_a_c,I2: set_b,A: b] :
( ( member_b_a @ H2 @ ( ring_hom_b_d_a_c @ R3 @ S ) )
=> ( ( ord_less_eq_set_b @ I2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( image_b_a @ H2 @ ( a_l_coset_b_d @ R3 @ A @ I2 ) )
= ( a_l_coset_a_c @ S @ ( H2 @ A ) @ ( image_b_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_263_a__coset__hom_I1_J,axiom,
! [H2: a > b,R3: partia8877618634411419171xt_a_c,S: partia1897943568983147621xt_b_d,I2: set_a,A: a] :
( ( member_a_b @ H2 @ ( ring_hom_a_c_b_d @ R3 @ S ) )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( image_a_b @ H2 @ ( a_l_coset_a_c @ R3 @ A @ I2 ) )
= ( a_l_coset_b_d @ S @ ( H2 @ A ) @ ( image_a_b @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_264_a__coset__hom_I1_J,axiom,
! [H2: a > a,R3: partia8877618634411419171xt_a_c,S: partia8877618634411419171xt_a_c,I2: set_a,A: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_c_a_c @ R3 @ S ) )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( image_a_a @ H2 @ ( a_l_coset_a_c @ R3 @ A @ I2 ) )
= ( a_l_coset_a_c @ S @ ( H2 @ A ) @ ( image_a_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_265_a__coset__hom_I1_J,axiom,
! [H2: list_b > b,R3: partia4026993951477142903t_unit,S: partia1897943568983147621xt_b_d,I2: set_list_b,A: list_b] :
( ( member_list_b_b @ H2 @ ( ring_h8746557796359701252it_b_d @ R3 @ S ) )
=> ( ( ord_le8932221534207217157list_b @ I2 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( image_list_b_b @ H2 @ ( a_l_co3923087131696239825t_unit @ R3 @ A @ I2 ) )
= ( a_l_coset_b_d @ S @ ( H2 @ A ) @ ( image_list_b_b @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_266_a__coset__hom_I1_J,axiom,
! [H2: list_b > a,R3: partia4026993951477142903t_unit,S: partia8877618634411419171xt_a_c,I2: set_list_b,A: list_b] :
( ( member_list_b_a @ H2 @ ( ring_h2310641341477601476it_a_c @ R3 @ S ) )
=> ( ( ord_le8932221534207217157list_b @ I2 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( image_list_b_a @ H2 @ ( a_l_co3923087131696239825t_unit @ R3 @ A @ I2 ) )
= ( a_l_coset_a_c @ S @ ( H2 @ A ) @ ( image_list_b_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_267_a__coset__hom_I1_J,axiom,
! [H2: list_a > b,R3: partia2670972154091845814t_unit,S: partia1897943568983147621xt_b_d,I2: set_list_a,A: list_a] :
( ( member_list_a_b @ H2 @ ( ring_h108518356514633413it_b_d @ R3 @ S ) )
=> ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( image_list_a_b @ H2 @ ( a_l_co7008843373686234386t_unit @ R3 @ A @ I2 ) )
= ( a_l_coset_b_d @ S @ ( H2 @ A ) @ ( image_list_a_b @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_268_a__coset__hom_I1_J,axiom,
! [H2: list_a > a,R3: partia2670972154091845814t_unit,S: partia8877618634411419171xt_a_c,I2: set_list_a,A: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309445it_a_c @ R3 @ S ) )
=> ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( image_list_a_a @ H2 @ ( a_l_co7008843373686234386t_unit @ R3 @ A @ I2 ) )
= ( a_l_coset_a_c @ S @ ( H2 @ A ) @ ( image_list_a_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_269_a__coset__hom_I1_J,axiom,
! [H2: b > list_b,R3: partia1897943568983147621xt_b_d,S: partia4026993951477142903t_unit,I2: set_b,A: b] :
( ( member_b_list_b @ H2 @ ( ring_h8274988533691632388t_unit @ R3 @ S ) )
=> ( ( ord_less_eq_set_b @ I2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( image_b_list_b @ H2 @ ( a_l_coset_b_d @ R3 @ A @ I2 ) )
= ( a_l_co3923087131696239825t_unit @ S @ ( H2 @ A ) @ ( image_b_list_b @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_270_a__coset__hom_I1_J,axiom,
! [H2: b > list_a,R3: partia1897943568983147621xt_b_d,S: partia2670972154091845814t_unit,I2: set_b,A: b] :
( ( member_b_list_a @ H2 @ ( ring_h2137372738826851141t_unit @ R3 @ S ) )
=> ( ( ord_less_eq_set_b @ I2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( image_b_list_a @ H2 @ ( a_l_coset_b_d @ R3 @ A @ I2 ) )
= ( a_l_co7008843373686234386t_unit @ S @ ( H2 @ A ) @ ( image_b_list_a @ H2 @ I2 ) ) ) ) ) ) ).
% a_coset_hom(1)
thf(fact_271_dr_OpprimeE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% dr.pprimeE(1)
thf(fact_272_ds_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_b,E: set_b,V: set_b] :
( ( subfield_b_d @ K @ s )
=> ( ( embedd5921307093240156728on_b_d @ s @ K @ E )
=> ( ( embedd6240069993967058123ra_b_d @ K @ V @ s )
=> ( ( ord_less_eq_set_b @ V @ E )
=> ( embedd5921307093240156728on_b_d @ s @ K @ V ) ) ) ) ) ).
% ds.subalbegra_incl_imp_finite_dimension
thf(fact_273_pdr_Ocgenideal__is__principalideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ I ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.cgenideal_is_principalideal
thf(fact_274_pdr_Osum__zero__eq__neg,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( X
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y ) ) ) ) ) ).
% pdr.sum_zero_eq_neg
thf(fact_275_dr_Onormalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V2: a,Va: list_a] :
( X
!= ( cons_a @ V2 @ Va ) ) ) ).
% dr.normalize.cases
thf(fact_276_dr_Oa__l__coset__subset__G,axiom,
! [H: set_a,X: a] :
( ( ord_less_eq_set_a @ H @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_c @ r @ X @ H ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% dr.a_l_coset_subset_G
thf(fact_277_pds_Opprime__iff__pirreducible,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ring_r415245522172713630t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P )
= ( ring_r4561388045816386823t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P ) ) ) ) ).
% pds.pprime_iff_pirreducible
thf(fact_278_pds_Osubring__props_I1_J,axiom,
! [K: set_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ord_le8932221534207217157list_b @ K @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.subring_props(1)
thf(fact_279_pds_Oa__lcos__mult__one,axiom,
! [M: set_list_b] :
( ( ord_le8932221534207217157list_b @ M @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( a_l_co3923087131696239825t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ M )
= M ) ) ).
% pds.a_lcos_mult_one
thf(fact_280_pds_Oa__l__coset__subset__G,axiom,
! [H: set_list_b,X: list_b] :
( ( ord_le8932221534207217157list_b @ H @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ord_le8932221534207217157list_b @ ( a_l_co3923087131696239825t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ H ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.a_l_coset_subset_G
thf(fact_281_pds_Osubring__props_I2_J,axiom,
! [K: set_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( member_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ K ) ) ).
% pds.subring_props(2)
thf(fact_282_pds_Osubring__props_I3_J,axiom,
! [K: set_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( member_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ K ) ) ).
% pds.subring_props(3)
thf(fact_283_pds_Osubring__props_I5_J,axiom,
! [K: set_list_b,H2: list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_b @ H2 @ K )
=> ( member_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H2 ) @ K ) ) ) ).
% pds.subring_props(5)
thf(fact_284_pds_Oring__irreducibleE_I1_J,axiom,
! [R2: list_b] :
( ( member_list_b @ R2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ R2 )
=> ( R2
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.ring_irreducibleE(1)
thf(fact_285_pds_Oabelian__monoid__axioms,axiom,
abelia6363847436574302712t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.abelian_monoid_axioms
thf(fact_286_pds_Oring__primeI,axiom,
! [P: list_b] :
( ( P
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( prime_5961678782586786214t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P )
=> ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P ) ) ) ).
% pds.ring_primeI
thf(fact_287_pds_Oring__primeE_I1_J,axiom,
! [P: list_b] :
( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P )
=> ( P
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.ring_primeE(1)
thf(fact_288_pds_Oring__primeE_I3_J,axiom,
! [P: list_b] :
( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P )
=> ( prime_5961678782586786214t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P ) ) ) ).
% pds.ring_primeE(3)
thf(fact_289_pds_Osemiring__axioms,axiom,
semiri9009524540797033698t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.semiring_axioms
thf(fact_290_pds_Odomain__axioms,axiom,
domain3467766878553215752t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.domain_axioms
thf(fact_291_pds_Ozero__is__prime_I1_J,axiom,
prime_5961678782586786214t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% pds.zero_is_prime(1)
thf(fact_292_pds_Ozero__not__one,axiom,
( ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.zero_not_one
thf(fact_293_pds_Oring__irreducibleE_I2_J,axiom,
! [R2: list_b] :
( ( member_list_b @ R2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ R2 )
=> ( irredu8180679162501400317t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ R2 ) ) ) ).
% pds.ring_irreducibleE(2)
thf(fact_294_ds_Opprime__iff__pirreducible,axiom,
! [K: set_b,P: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ K ) @ P )
= ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ K ) @ P ) ) ) ) ).
% ds.pprime_iff_pirreducible
thf(fact_295_pdr_Opprime__iff__pirreducible,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P ) ) ) ) ).
% pdr.pprime_iff_pirreducible
thf(fact_296_dr_Oa__lcos__mult__one,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( a_l_coset_a_c @ r @ ( zero_a_c @ r ) @ M )
= M ) ) ).
% dr.a_lcos_mult_one
thf(fact_297_dr_Osubring__props_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_c @ K @ r )
=> ( ord_less_eq_set_a @ K @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% dr.subring_props(1)
thf(fact_298_ds_Otelescopic__base__dim_I1_J,axiom,
! [K: set_b,F2: set_b,E: set_b] :
( ( subfield_b_d @ K @ s )
=> ( ( subfield_b_d @ F2 @ s )
=> ( ( embedd5921307093240156728on_b_d @ s @ K @ F2 )
=> ( ( embedd5921307093240156728on_b_d @ s @ F2 @ E )
=> ( embedd5921307093240156728on_b_d @ s @ K @ E ) ) ) ) ) ).
% ds.telescopic_base_dim(1)
thf(fact_299_pdr_Osubring__props_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.subring_props(1)
thf(fact_300_pdr_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.a_l_coset_subset_G
thf(fact_301_pdr_Osubring__props_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ K ) ) ).
% pdr.subring_props(2)
thf(fact_302_pdr_Osubring__props_I7_J,axiom,
! [K: set_list_a,H1: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H22 @ K )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H1 @ H22 ) @ K ) ) ) ) ).
% pdr.subring_props(7)
thf(fact_303_pdr_Osubring__props_I3_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ K ) ) ).
% pdr.subring_props(3)
thf(fact_304_pdr_Osubring__props_I5_J,axiom,
! [K: set_list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_a @ H2 @ K )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H2 ) @ K ) ) ) ).
% pdr.subring_props(5)
thf(fact_305_ds_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_b,E: set_b] :
( ( subfield_b_d @ K @ s )
=> ( ( embedd5921307093240156728on_b_d @ s @ K @ E )
=> ( embedd6240069993967058123ra_b_d @ K @ E @ s ) ) ) ).
% ds.finite_dimension_imp_subalgebra
thf(fact_306_pdr_Oa__lcos__mult__one,axiom,
! [M: set_list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ M )
= M ) ) ).
% pdr.a_lcos_mult_one
thf(fact_307_pdr_Oa__lcos__m__assoc,axiom,
! [M: set_list_a,G: list_a,H2: list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H2 @ M ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ G @ H2 ) @ M ) ) ) ) ) ).
% pdr.a_lcos_m_assoc
thf(fact_308_pdr_Oadd_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% pdr.add.l_cancel
thf(fact_309_pdr_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% pdr.add.m_assoc
thf(fact_310_pdr_Oadd_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ X ) ) ) ) ).
% pdr.add.m_comm
thf(fact_311_pdr_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% pdr.add.m_lcomm
thf(fact_312_pdr_Oadd_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% pdr.add.r_cancel
thf(fact_313_pdr_Ocgenideal__self,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ I ) ) ) ).
% pdr.cgenideal_self
thf(fact_314_pdr_Oadd_Oinv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.add.inv_comm
thf(fact_315_pdr_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.add.l_inv_ex
thf(fact_316_pdr_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.add.one_unique
thf(fact_317_pdr_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.add.r_inv_ex
thf(fact_318_pdr_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% pdr.minus_unique
thf(fact_319_pdr_Oa__transpose__inv,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y )
= Z )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% pdr.a_transpose_inv
thf(fact_320_pdr_Oadd_Oinv__mult__group,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) ) ) ) ) ).
% pdr.add.inv_mult_group
thf(fact_321_pdr_Oadd_Oinv__solve__left,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( A
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B ) @ C ) )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ A ) ) ) ) ) ) ).
% pdr.add.inv_solve_left
thf(fact_322_pdr_Oadd_Oinv__solve__left_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B ) @ C )
= A )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ A ) ) ) ) ) ) ).
% pdr.add.inv_solve_left'
thf(fact_323_pdr_Oadd_Oinv__solve__right,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( A
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C ) ) )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ C ) ) ) ) ) ) ).
% pdr.add.inv_solve_right
thf(fact_324_pdr_Oadd_Oinv__solve__right_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C ) )
= A )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ C ) ) ) ) ) ) ).
% pdr.add.inv_solve_right'
thf(fact_325_pdr_Ominus__add,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y ) ) ) ) ) ).
% pdr.minus_add
thf(fact_326_pdr_Or__neg1,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) )
= Y ) ) ) ).
% pdr.r_neg1
thf(fact_327_pdr_Or__neg2,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ Y ) )
= Y ) ) ) ).
% pdr.r_neg2
thf(fact_328_dr_Olong__division__add_I1_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_c @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ K ) @ ( polynomial_pdiv_a_c @ r @ A @ Q ) @ ( polynomial_pdiv_a_c @ r @ B @ Q ) ) ) ) ) ) ) ).
% dr.long_division_add(1)
thf(fact_329_dr_Olong__division__zero_I1_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_c @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% dr.long_division_zero(1)
thf(fact_330_pdr_Ol__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.l_neg
thf(fact_331_pdr_Ominus__equality,axiom,
! [Y: list_a,X: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X )
= Y ) ) ) ) ).
% pdr.minus_equality
thf(fact_332_pdr_Or__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.r_neg
thf(fact_333_pds_Ozero__closed,axiom,
member_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% pds.zero_closed
thf(fact_334_pds_Oone__closed,axiom,
member_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% pds.one_closed
thf(fact_335_pds_Ominus__zero,axiom,
( ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.minus_zero
thf(fact_336_pds_Ominus__minus,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) )
= X ) ) ).
% pds.minus_minus
thf(fact_337_pds_Oadd_Oinv__eq__1__iff,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( X
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.add.inv_eq_1_iff
thf(fact_338_pds_Oadd_Oinv__closed,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.add.inv_closed
thf(fact_339_pdr_Oadd_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.add.m_closed
thf(fact_340_pdr_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% pdr.add.right_cancel
thf(fact_341_pdr_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ A )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.add.l_cancel_one
thf(fact_342_pdr_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ A ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.add.l_cancel_one'
thf(fact_343_pdr_Oadd_Or__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ X )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.add.r_cancel_one
thf(fact_344_pdr_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ X ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.add.r_cancel_one'
thf(fact_345_pdr_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ X )
= X ) ) ).
% pdr.l_zero
thf(fact_346_pdr_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= X ) ) ).
% pdr.r_zero
thf(fact_347_ring_Opdiv_Ocong,axiom,
polynomial_pdiv_a_c = polynomial_pdiv_a_c ).
% ring.pdiv.cong
thf(fact_348_ring_Opdiv_Ocong,axiom,
polynomial_pdiv_b_d = polynomial_pdiv_b_d ).
% ring.pdiv.cong
thf(fact_349_ring_Opdiv_Ocong,axiom,
polyno5893782122288709345t_unit = polyno5893782122288709345t_unit ).
% ring.pdiv.cong
thf(fact_350_ring_Opdiv_Ocong,axiom,
polyno2808025880298714784t_unit = polyno2808025880298714784t_unit ).
% ring.pdiv.cong
thf(fact_351_ring__iso__memE_I3_J,axiom,
! [H2: b > b,R3: partia1897943568983147621xt_b_d,S: partia1897943568983147621xt_b_d,X: b,Y: b] :
( ( member_b_b @ H2 @ ( ring_iso_b_d_b_d @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( H2 @ ( add_b_d @ R3 @ X @ Y ) )
= ( add_b_d @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_352_ring__iso__memE_I3_J,axiom,
! [H2: b > a,R3: partia1897943568983147621xt_b_d,S: partia8877618634411419171xt_a_c,X: b,Y: b] :
( ( member_b_a @ H2 @ ( ring_iso_b_d_a_c @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( H2 @ ( add_b_d @ R3 @ X @ Y ) )
= ( add_a_c @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_353_ring__iso__memE_I3_J,axiom,
! [H2: a > a,R3: partia8877618634411419171xt_a_c,S: partia8877618634411419171xt_a_c,X: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_c_a_c @ R3 @ S ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( H2 @ ( add_a_c @ R3 @ X @ Y ) )
= ( add_a_c @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_354_ring__iso__memE_I3_J,axiom,
! [H2: a > b,R3: partia8877618634411419171xt_a_c,S: partia1897943568983147621xt_b_d,X: a,Y: a] :
( ( member_a_b @ H2 @ ( ring_iso_a_c_b_d @ R3 @ S ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( H2 @ ( add_a_c @ R3 @ X @ Y ) )
= ( add_b_d @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_355_ring__iso__memE_I3_J,axiom,
! [H2: list_b > b,R3: partia4026993951477142903t_unit,S: partia1897943568983147621xt_b_d,X: list_b,Y: list_b] :
( ( member_list_b_b @ H2 @ ( ring_i3676047618198725658it_b_d @ R3 @ S ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( H2 @ ( add_li4567129529168622413t_unit @ R3 @ X @ Y ) )
= ( add_b_d @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_356_ring__iso__memE_I3_J,axiom,
! [H2: list_b > a,R3: partia4026993951477142903t_unit,S: partia8877618634411419171xt_a_c,X: list_b,Y: list_b] :
( ( member_list_b_a @ H2 @ ( ring_i6463503200171401690it_a_c @ R3 @ S ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( H2 @ ( add_li4567129529168622413t_unit @ R3 @ X @ Y ) )
= ( add_a_c @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_357_ring__iso__memE_I3_J,axiom,
! [H2: list_a > b,R3: partia2670972154091845814t_unit,S: partia1897943568983147621xt_b_d,X: list_a,Y: list_a] :
( ( member_list_a_b @ H2 @ ( ring_i4261380215208433627it_b_d @ R3 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R3 @ X @ Y ) )
= ( add_b_d @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_358_ring__iso__memE_I3_J,axiom,
! [H2: list_a > a,R3: partia2670972154091845814t_unit,S: partia8877618634411419171xt_a_c,X: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109659it_a_c @ R3 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R3 @ X @ Y ) )
= ( add_a_c @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_359_ring__iso__memE_I3_J,axiom,
! [H2: b > list_b,R3: partia1897943568983147621xt_b_d,S: partia4026993951477142903t_unit,X: b,Y: b] :
( ( member_b_list_b @ H2 @ ( ring_i3204478355530656794t_unit @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( H2 @ ( add_b_d @ R3 @ X @ Y ) )
= ( add_li4567129529168622413t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_360_ring__iso__memE_I3_J,axiom,
! [H2: b > list_a,R3: partia1897943568983147621xt_b_d,S: partia2670972154091845814t_unit,X: b,Y: b] :
( ( member_b_list_a @ H2 @ ( ring_i6290234597520651355t_unit @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( H2 @ ( add_b_d @ R3 @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_361_domain_Olong__division__add_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,K: set_list_list_a,A: list_list_list_a,B: list_list_list_a,Q: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( subfie4546268998243038636t_unit @ K @ R3 )
=> ( ( member5342144027231129785list_a @ A @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( member5342144027231129785list_a @ B @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( polyno4115915122720352731t_unit @ R3 @ ( add_li5162926044081146114t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) @ A @ B ) @ Q )
= ( add_li5162926044081146114t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) @ ( polyno4115915122720352731t_unit @ R3 @ A @ Q ) @ ( polyno4115915122720352731t_unit @ R3 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_362_domain_Olong__division__add_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_c @ R3 )
=> ( ( subfield_a_c @ K @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( polynomial_pdiv_a_c @ R3 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ R3 @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ R3 @ K ) @ ( polynomial_pdiv_a_c @ R3 @ A @ Q ) @ ( polynomial_pdiv_a_c @ R3 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_363_domain_Olong__division__add_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b,A: list_b,B: list_b,Q: list_b] :
( ( domain_b_d @ R3 )
=> ( ( subfield_b_d @ K @ R3 )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( polynomial_pdiv_b_d @ R3 @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ R3 @ K ) @ A @ B ) @ Q )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ R3 @ K ) @ ( polynomial_pdiv_b_d @ R3 @ A @ Q ) @ ( polynomial_pdiv_b_d @ R3 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_364_domain_Olong__division__add_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R3 @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( polyno5893782122288709345t_unit @ R3 @ A @ Q ) @ ( polyno5893782122288709345t_unit @ R3 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_365_domain_Olong__division__add_I1_J,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b,A: list_list_b,B: list_list_b,Q: list_list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( subfie7916738691610828529t_unit @ K @ R3 )
=> ( ( member_list_list_b @ A @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_b @ B @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( polyno2808025880298714784t_unit @ R3 @ ( add_li4375960627168867399t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) @ A @ B ) @ Q )
= ( add_li4375960627168867399t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) @ ( polyno2808025880298714784t_unit @ R3 @ A @ Q ) @ ( polyno2808025880298714784t_unit @ R3 @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_366_domain_Olong__division__zero_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( subfie4546268998243038636t_unit @ K @ R3 )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( polyno4115915122720352731t_unit @ R3 @ nil_list_list_a @ Q )
= nil_list_list_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_367_domain_Olong__division__zero_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a,Q: list_a] :
( ( domain_a_c @ R3 )
=> ( ( subfield_a_c @ K @ R3 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( polynomial_pdiv_a_c @ R3 @ nil_a @ Q )
= nil_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_368_domain_Olong__division__zero_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b,Q: list_b] :
( ( domain_b_d @ R3 )
=> ( ( subfield_b_d @ K @ R3 )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( polynomial_pdiv_b_d @ R3 @ nil_b @ Q )
= nil_b ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_369_domain_Olong__division__zero_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R3 @ nil_list_a @ Q )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_370_domain_Olong__division__zero_I1_J,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b,Q: list_list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( subfie7916738691610828529t_unit @ K @ R3 )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( polyno2808025880298714784t_unit @ R3 @ nil_list_b @ Q )
= nil_list_b ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_371_ring__hom__add,axiom,
! [H2: b > b,R3: partia1897943568983147621xt_b_d,S: partia1897943568983147621xt_b_d,X: b,Y: b] :
( ( member_b_b @ H2 @ ( ring_hom_b_d_b_d @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( H2 @ ( add_b_d @ R3 @ X @ Y ) )
= ( add_b_d @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_372_ring__hom__add,axiom,
! [H2: b > a,R3: partia1897943568983147621xt_b_d,S: partia8877618634411419171xt_a_c,X: b,Y: b] :
( ( member_b_a @ H2 @ ( ring_hom_b_d_a_c @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( H2 @ ( add_b_d @ R3 @ X @ Y ) )
= ( add_a_c @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_373_ring__hom__add,axiom,
! [H2: a > a,R3: partia8877618634411419171xt_a_c,S: partia8877618634411419171xt_a_c,X: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_hom_a_c_a_c @ R3 @ S ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( H2 @ ( add_a_c @ R3 @ X @ Y ) )
= ( add_a_c @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_374_ring__hom__add,axiom,
! [H2: a > b,R3: partia8877618634411419171xt_a_c,S: partia1897943568983147621xt_b_d,X: a,Y: a] :
( ( member_a_b @ H2 @ ( ring_hom_a_c_b_d @ R3 @ S ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( H2 @ ( add_a_c @ R3 @ X @ Y ) )
= ( add_b_d @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_375_ring__hom__add,axiom,
! [H2: list_b > b,R3: partia4026993951477142903t_unit,S: partia1897943568983147621xt_b_d,X: list_b,Y: list_b] :
( ( member_list_b_b @ H2 @ ( ring_h8746557796359701252it_b_d @ R3 @ S ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( H2 @ ( add_li4567129529168622413t_unit @ R3 @ X @ Y ) )
= ( add_b_d @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_376_ring__hom__add,axiom,
! [H2: list_b > a,R3: partia4026993951477142903t_unit,S: partia8877618634411419171xt_a_c,X: list_b,Y: list_b] :
( ( member_list_b_a @ H2 @ ( ring_h2310641341477601476it_a_c @ R3 @ S ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( H2 @ ( add_li4567129529168622413t_unit @ R3 @ X @ Y ) )
= ( add_a_c @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_377_ring__hom__add,axiom,
! [H2: list_a > b,R3: partia2670972154091845814t_unit,S: partia1897943568983147621xt_b_d,X: list_a,Y: list_a] :
( ( member_list_a_b @ H2 @ ( ring_h108518356514633413it_b_d @ R3 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R3 @ X @ Y ) )
= ( add_b_d @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_378_ring__hom__add,axiom,
! [H2: list_a > a,R3: partia2670972154091845814t_unit,S: partia8877618634411419171xt_a_c,X: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_h2895973938487309445it_a_c @ R3 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R3 @ X @ Y ) )
= ( add_a_c @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_379_ring__hom__add,axiom,
! [H2: b > list_b,R3: partia1897943568983147621xt_b_d,S: partia4026993951477142903t_unit,X: b,Y: b] :
( ( member_b_list_b @ H2 @ ( ring_h8274988533691632388t_unit @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( H2 @ ( add_b_d @ R3 @ X @ Y ) )
= ( add_li4567129529168622413t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_380_ring__hom__add,axiom,
! [H2: b > list_a,R3: partia1897943568983147621xt_b_d,S: partia2670972154091845814t_unit,X: b,Y: b] :
( ( member_b_list_a @ H2 @ ( ring_h2137372738826851141t_unit @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( H2 @ ( add_b_d @ R3 @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_hom_add
thf(fact_381_abelian__monoidE_I5_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ X @ Y )
= ( add_li174743652000525320t_unit @ R3 @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_382_abelian__monoidE_I5_J,axiom,
! [R3: partia4026993951477142903t_unit,X: list_b,Y: list_b] :
( ( abelia6363847436574302712t_unit @ R3 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ X @ Y )
= ( add_li4567129529168622413t_unit @ R3 @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_383_abelian__monoidE_I5_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ X @ Y )
= ( add_li7652885771158616974t_unit @ R3 @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_384_abelian__monoidE_I5_J,axiom,
! [R3: partia1897943568983147621xt_b_d,X: b,Y: b] :
( ( abelian_monoid_b_d @ R3 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ X @ Y )
= ( add_b_d @ R3 @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_385_abelian__monoidE_I5_J,axiom,
! [R3: partia8877618634411419171xt_a_c,X: a,Y: a] :
( ( abelian_monoid_a_c @ R3 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ X @ Y )
= ( add_a_c @ R3 @ Y @ X ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_386_abelian__monoidE_I3_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ ( add_li174743652000525320t_unit @ R3 @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R3 @ X @ ( add_li174743652000525320t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_387_abelian__monoidE_I3_J,axiom,
! [R3: partia4026993951477142903t_unit,X: list_b,Y: list_b,Z: list_b] :
( ( abelia6363847436574302712t_unit @ R3 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ ( add_li4567129529168622413t_unit @ R3 @ X @ Y ) @ Z )
= ( add_li4567129529168622413t_unit @ R3 @ X @ ( add_li4567129529168622413t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_388_abelian__monoidE_I3_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ ( add_li7652885771158616974t_unit @ R3 @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R3 @ X @ ( add_li7652885771158616974t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_389_abelian__monoidE_I3_J,axiom,
! [R3: partia1897943568983147621xt_b_d,X: b,Y: b,Z: b] :
( ( abelian_monoid_b_d @ R3 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ ( add_b_d @ R3 @ X @ Y ) @ Z )
= ( add_b_d @ R3 @ X @ ( add_b_d @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_390_abelian__monoidE_I3_J,axiom,
! [R3: partia8877618634411419171xt_a_c,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_c @ R3 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ ( add_a_c @ R3 @ X @ Y ) @ Z )
= ( add_a_c @ R3 @ X @ ( add_a_c @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_391_abelian__monoidE_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R3 @ X @ Y ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_392_abelian__monoidE_I1_J,axiom,
! [R3: partia4026993951477142903t_unit,X: list_b,Y: list_b] :
( ( abelia6363847436574302712t_unit @ R3 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( member_list_b @ ( add_li4567129529168622413t_unit @ R3 @ X @ Y ) @ ( partia1381092143316337258t_unit @ R3 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_393_abelian__monoidE_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R3 @ X @ Y ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_394_abelian__monoidE_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d,X: b,Y: b] :
( ( abelian_monoid_b_d @ R3 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_b @ ( add_b_d @ R3 @ X @ Y ) @ ( partia8782771468121683032xt_b_d @ R3 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_395_abelian__monoidE_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c,X: a,Y: a] :
( ( abelian_monoid_a_c @ R3 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( member_a @ ( add_a_c @ R3 @ X @ Y ) @ ( partia778085601923319190xt_a_c @ R3 ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_396_abelian__monoid_Oa__comm,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( add_li174743652000525320t_unit @ G2 @ X @ Y )
= ( add_li174743652000525320t_unit @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_397_abelian__monoid_Oa__comm,axiom,
! [G2: partia4026993951477142903t_unit,X: list_b,Y: list_b] :
( ( abelia6363847436574302712t_unit @ G2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( add_li4567129529168622413t_unit @ G2 @ X @ Y )
= ( add_li4567129529168622413t_unit @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_398_abelian__monoid_Oa__comm,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( add_li7652885771158616974t_unit @ G2 @ X @ Y )
= ( add_li7652885771158616974t_unit @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_399_abelian__monoid_Oa__comm,axiom,
! [G2: partia1897943568983147621xt_b_d,X: b,Y: b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( add_b_d @ G2 @ X @ Y )
= ( add_b_d @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_400_abelian__monoid_Oa__comm,axiom,
! [G2: partia8877618634411419171xt_a_c,X: a,Y: a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( add_a_c @ G2 @ X @ Y )
= ( add_a_c @ G2 @ Y @ X ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_401_abelian__monoid_Oa__assoc,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( add_li174743652000525320t_unit @ G2 @ ( add_li174743652000525320t_unit @ G2 @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ G2 @ X @ ( add_li174743652000525320t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_402_abelian__monoid_Oa__assoc,axiom,
! [G2: partia4026993951477142903t_unit,X: list_b,Y: list_b,Z: list_b] :
( ( abelia6363847436574302712t_unit @ G2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( add_li4567129529168622413t_unit @ G2 @ ( add_li4567129529168622413t_unit @ G2 @ X @ Y ) @ Z )
= ( add_li4567129529168622413t_unit @ G2 @ X @ ( add_li4567129529168622413t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_403_abelian__monoid_Oa__assoc,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( add_li7652885771158616974t_unit @ G2 @ ( add_li7652885771158616974t_unit @ G2 @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ G2 @ X @ ( add_li7652885771158616974t_unit @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_404_abelian__monoid_Oa__assoc,axiom,
! [G2: partia1897943568983147621xt_b_d,X: b,Y: b,Z: b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( add_b_d @ G2 @ ( add_b_d @ G2 @ X @ Y ) @ Z )
= ( add_b_d @ G2 @ X @ ( add_b_d @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_405_abelian__monoid_Oa__assoc,axiom,
! [G2: partia8877618634411419171xt_a_c,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( add_a_c @ G2 @ ( add_a_c @ G2 @ X @ Y ) @ Z )
= ( add_a_c @ G2 @ X @ ( add_a_c @ G2 @ Y @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_406_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( add_li174743652000525320t_unit @ G2 @ X @ ( add_li174743652000525320t_unit @ G2 @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ G2 @ Y @ ( add_li174743652000525320t_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_407_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia4026993951477142903t_unit,X: list_b,Y: list_b,Z: list_b] :
( ( abelia6363847436574302712t_unit @ G2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( add_li4567129529168622413t_unit @ G2 @ X @ ( add_li4567129529168622413t_unit @ G2 @ Y @ Z ) )
= ( add_li4567129529168622413t_unit @ G2 @ Y @ ( add_li4567129529168622413t_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_408_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( add_li7652885771158616974t_unit @ G2 @ X @ ( add_li7652885771158616974t_unit @ G2 @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ G2 @ Y @ ( add_li7652885771158616974t_unit @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_409_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia1897943568983147621xt_b_d,X: b,Y: b,Z: b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( add_b_d @ G2 @ X @ ( add_b_d @ G2 @ Y @ Z ) )
= ( add_b_d @ G2 @ Y @ ( add_b_d @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_410_abelian__monoid_Oa__lcomm,axiom,
! [G2: partia8877618634411419171xt_a_c,X: a,Y: a,Z: a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( add_a_c @ G2 @ X @ ( add_a_c @ G2 @ Y @ Z ) )
= ( add_a_c @ G2 @ Y @ ( add_a_c @ G2 @ X @ Z ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_411_abelian__monoid_Oa__closed,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ G2 @ X @ Y ) @ ( partia2464479390973590831t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_412_abelian__monoid_Oa__closed,axiom,
! [G2: partia4026993951477142903t_unit,X: list_b,Y: list_b] :
( ( abelia6363847436574302712t_unit @ G2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( member_list_b @ ( add_li4567129529168622413t_unit @ G2 @ X @ Y ) @ ( partia1381092143316337258t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_413_abelian__monoid_Oa__closed,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ G2 @ X @ Y ) @ ( partia5361259788508890537t_unit @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_414_abelian__monoid_Oa__closed,axiom,
! [G2: partia1897943568983147621xt_b_d,X: b,Y: b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( member_b @ ( add_b_d @ G2 @ X @ Y ) @ ( partia8782771468121683032xt_b_d @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_415_abelian__monoid_Oa__closed,axiom,
! [G2: partia8877618634411419171xt_a_c,X: a,Y: a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( member_a @ ( add_a_c @ G2 @ X @ Y ) @ ( partia778085601923319190xt_a_c @ G2 ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_416_principal__domain_Oaxioms_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d] :
( ( ring_p6015679779713369569in_b_d @ R3 )
=> ( domain_b_d @ R3 ) ) ).
% principal_domain.axioms(1)
thf(fact_417_principal__domain_Oaxioms_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c] :
( ( ring_p8803135361686045601in_a_c @ R3 )
=> ( domain_a_c @ R3 ) ) ).
% principal_domain.axioms(1)
thf(fact_418_principal__domain_Oaxioms_I1_J,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( ring_p8098905331641078952t_unit @ R3 )
=> ( domain6553523120543210313t_unit @ R3 ) ) ).
% principal_domain.axioms(1)
thf(fact_419_principal__domain_Oaxioms_I1_J,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( ring_p715737262848045090t_unit @ R3 )
=> ( domain7810152921033798211t_unit @ R3 ) ) ).
% principal_domain.axioms(1)
thf(fact_420_principal__domain_Oaxioms_I1_J,axiom,
! [R3: partia4026993951477142903t_unit] :
( ( ring_p5013149089651084391t_unit @ R3 )
=> ( domain3467766878553215752t_unit @ R3 ) ) ).
% principal_domain.axioms(1)
thf(fact_421_principal__domain_Oaxioms_I1_J,axiom,
! [R3: partia2587409828943155709t_unit] :
( ( ring_p4916954238016387169t_unit @ R3 )
=> ( domain2787997859347364482t_unit @ R3 ) ) ).
% principal_domain.axioms(1)
thf(fact_422_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ X @ ( add_li174743652000525320t_unit @ R3 @ Y @ Z ) )
= ( add_li174743652000525320t_unit @ R3 @ Y @ ( add_li174743652000525320t_unit @ R3 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_423_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia4026993951477142903t_unit,X: list_b,Y: list_b,Z: list_b] :
( ( semiri9009524540797033698t_unit @ R3 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ X @ ( add_li4567129529168622413t_unit @ R3 @ Y @ Z ) )
= ( add_li4567129529168622413t_unit @ R3 @ Y @ ( add_li4567129529168622413t_unit @ R3 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_424_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ X @ ( add_li7652885771158616974t_unit @ R3 @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ R3 @ Y @ ( add_li7652885771158616974t_unit @ R3 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_425_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia1897943568983147621xt_b_d,X: b,Y: b,Z: b] :
( ( semiring_b_d @ R3 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ X @ ( add_b_d @ R3 @ Y @ Z ) )
= ( add_b_d @ R3 @ Y @ ( add_b_d @ R3 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_426_semiring_Osemiring__simprules_I12_J,axiom,
! [R3: partia8877618634411419171xt_a_c,X: a,Y: a,Z: a] :
( ( semiring_a_c @ R3 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ X @ ( add_a_c @ R3 @ Y @ Z ) )
= ( add_a_c @ R3 @ Y @ ( add_a_c @ R3 @ X @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_427_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ X @ Y )
= ( add_li174743652000525320t_unit @ R3 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_428_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia4026993951477142903t_unit,X: list_b,Y: list_b] :
( ( semiri9009524540797033698t_unit @ R3 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ X @ Y )
= ( add_li4567129529168622413t_unit @ R3 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_429_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ X @ Y )
= ( add_li7652885771158616974t_unit @ R3 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_430_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia1897943568983147621xt_b_d,X: b,Y: b] :
( ( semiring_b_d @ R3 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ X @ Y )
= ( add_b_d @ R3 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_431_semiring_Osemiring__simprules_I7_J,axiom,
! [R3: partia8877618634411419171xt_a_c,X: a,Y: a] :
( ( semiring_a_c @ R3 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ X @ Y )
= ( add_a_c @ R3 @ Y @ X ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_432_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a,Z: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Z @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ ( add_li174743652000525320t_unit @ R3 @ X @ Y ) @ Z )
= ( add_li174743652000525320t_unit @ R3 @ X @ ( add_li174743652000525320t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_433_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia4026993951477142903t_unit,X: list_b,Y: list_b,Z: list_b] :
( ( semiri9009524540797033698t_unit @ R3 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ ( add_li4567129529168622413t_unit @ R3 @ X @ Y ) @ Z )
= ( add_li4567129529168622413t_unit @ R3 @ X @ ( add_li4567129529168622413t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_434_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a,Y: list_a,Z: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ ( add_li7652885771158616974t_unit @ R3 @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ R3 @ X @ ( add_li7652885771158616974t_unit @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_435_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia1897943568983147621xt_b_d,X: b,Y: b,Z: b] :
( ( semiring_b_d @ R3 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ ( add_b_d @ R3 @ X @ Y ) @ Z )
= ( add_b_d @ R3 @ X @ ( add_b_d @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_436_semiring_Osemiring__simprules_I5_J,axiom,
! [R3: partia8877618634411419171xt_a_c,X: a,Y: a,Z: a] :
( ( semiring_a_c @ R3 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ ( add_a_c @ R3 @ X @ Y ) @ Z )
= ( add_a_c @ R3 @ X @ ( add_a_c @ R3 @ Y @ Z ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_437_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a,Y: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R3 @ X @ Y ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_438_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia4026993951477142903t_unit,X: list_b,Y: list_b] :
( ( semiri9009524540797033698t_unit @ R3 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( member_list_b @ ( add_li4567129529168622413t_unit @ R3 @ X @ Y ) @ ( partia1381092143316337258t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_439_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a,Y: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R3 @ X @ Y ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_440_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d,X: b,Y: b] :
( ( semiring_b_d @ R3 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_b @ ( add_b_d @ R3 @ X @ Y ) @ ( partia8782771468121683032xt_b_d @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_441_semiring_Osemiring__simprules_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c,X: a,Y: a] :
( ( semiring_a_c @ R3 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( member_a @ ( add_a_c @ R3 @ X @ Y ) @ ( partia778085601923319190xt_a_c @ R3 ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_442_abelian__group__hom_Ohom__add,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia1897943568983147621xt_b_d,H2: b > b,X: b,Y: b] :
( ( abelia2848305256108177367_d_b_d @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( H2 @ ( add_b_d @ G2 @ X @ Y ) )
= ( add_b_d @ H @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% abelian_group_hom.hom_add
thf(fact_443_abelian__group__hom_Ohom__add,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia8877618634411419171xt_a_c,H2: b > a,X: b,Y: b] :
( ( abelia5635760838080853399_d_a_c @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( H2 @ ( add_b_d @ G2 @ X @ Y ) )
= ( add_a_c @ H @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% abelian_group_hom.hom_add
thf(fact_444_abelian__group__hom_Ohom__add,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia8877618634411419171xt_a_c,H2: a > a,X: a,Y: a] :
( ( abelia7774952222452699991_c_a_c @ G2 @ H @ H2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( H2 @ ( add_a_c @ G2 @ X @ Y ) )
= ( add_a_c @ H @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% abelian_group_hom.hom_add
thf(fact_445_abelian__group__hom_Ohom__add,axiom,
! [G2: partia8877618634411419171xt_a_c,H: partia1897943568983147621xt_b_d,H2: a > b,X: a,Y: a] :
( ( abelia4987496640480023959_c_b_d @ G2 @ H @ H2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( H2 @ ( add_a_c @ G2 @ X @ Y ) )
= ( add_b_d @ H @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% abelian_group_hom.hom_add
thf(fact_446_abelian__group__hom_Ohom__add,axiom,
! [G2: partia4026993951477142903t_unit,H: partia1897943568983147621xt_b_d,H2: list_b > b,X: list_b,Y: list_b] :
( ( abelia4844232365066319197it_b_d @ G2 @ H @ H2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( H2 @ ( add_li4567129529168622413t_unit @ G2 @ X @ Y ) )
= ( add_b_d @ H @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% abelian_group_hom.hom_add
thf(fact_447_abelian__group__hom_Ohom__add,axiom,
! [G2: partia4026993951477142903t_unit,H: partia8877618634411419171xt_a_c,H2: list_b > a,X: list_b,Y: list_b] :
( ( abelia7631687947038995229it_a_c @ G2 @ H @ H2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( H2 @ ( add_li4567129529168622413t_unit @ G2 @ X @ Y ) )
= ( add_a_c @ H @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% abelian_group_hom.hom_add
thf(fact_448_abelian__group__hom_Ohom__add,axiom,
! [G2: partia2670972154091845814t_unit,H: partia1897943568983147621xt_b_d,H2: list_a > b,X: list_a,Y: list_a] :
( ( abelia5429564962076027166it_b_d @ G2 @ H @ H2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ G2 @ X @ Y ) )
= ( add_b_d @ H @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% abelian_group_hom.hom_add
thf(fact_449_abelian__group__hom_Ohom__add,axiom,
! [G2: partia2670972154091845814t_unit,H: partia8877618634411419171xt_a_c,H2: list_a > a,X: list_a,Y: list_a] :
( ( abelia8217020544048703198it_a_c @ G2 @ H @ H2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ G2 @ X @ Y ) )
= ( add_a_c @ H @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% abelian_group_hom.hom_add
thf(fact_450_abelian__group__hom_Ohom__add,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia4026993951477142903t_unit,H2: b > list_b,X: b,Y: b] :
( ( abelia4372663102398250333t_unit @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( H2 @ ( add_b_d @ G2 @ X @ Y ) )
= ( add_li4567129529168622413t_unit @ H @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% abelian_group_hom.hom_add
thf(fact_451_abelian__group__hom_Ohom__add,axiom,
! [G2: partia1897943568983147621xt_b_d,H: partia2670972154091845814t_unit,H2: b > list_a,X: b,Y: b] :
( ( abelia7458419344388244894t_unit @ G2 @ H @ H2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( H2 @ ( add_b_d @ G2 @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ H @ ( H2 @ X ) @ ( H2 @ Y ) ) ) ) ) ) ).
% abelian_group_hom.hom_add
thf(fact_452_abelian__monoidE_I4_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a] :
( ( abelia3641329199688042803t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ ( zero_l347298301471573063t_unit @ R3 ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_453_abelian__monoidE_I4_J,axiom,
! [R3: partia4026993951477142903t_unit,X: list_b] :
( ( abelia6363847436574302712t_unit @ R3 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ ( zero_l1056902381442676492t_unit @ R3 ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_454_abelian__monoidE_I4_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ ( zero_l4142658623432671053t_unit @ R3 ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_455_abelian__monoidE_I4_J,axiom,
! [R3: partia1897943568983147621xt_b_d,X: b] :
( ( abelian_monoid_b_d @ R3 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ ( zero_b_d @ R3 ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_456_abelian__monoidE_I4_J,axiom,
! [R3: partia8877618634411419171xt_a_c,X: a] :
( ( abelian_monoid_a_c @ R3 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ ( zero_a_c @ R3 ) @ X )
= X ) ) ) ).
% abelian_monoidE(4)
thf(fact_457_abelian__monoid_Ol__zero,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( add_li174743652000525320t_unit @ G2 @ ( zero_l347298301471573063t_unit @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_458_abelian__monoid_Ol__zero,axiom,
! [G2: partia4026993951477142903t_unit,X: list_b] :
( ( abelia6363847436574302712t_unit @ G2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( add_li4567129529168622413t_unit @ G2 @ ( zero_l1056902381442676492t_unit @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_459_abelian__monoid_Ol__zero,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( add_li7652885771158616974t_unit @ G2 @ ( zero_l4142658623432671053t_unit @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_460_abelian__monoid_Ol__zero,axiom,
! [G2: partia1897943568983147621xt_b_d,X: b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( add_b_d @ G2 @ ( zero_b_d @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_461_abelian__monoid_Ol__zero,axiom,
! [G2: partia8877618634411419171xt_a_c,X: a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( add_a_c @ G2 @ ( zero_a_c @ G2 ) @ X )
= X ) ) ) ).
% abelian_monoid.l_zero
thf(fact_462_abelian__monoid_Or__zero,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( add_li174743652000525320t_unit @ G2 @ X @ ( zero_l347298301471573063t_unit @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_463_abelian__monoid_Or__zero,axiom,
! [G2: partia4026993951477142903t_unit,X: list_b] :
( ( abelia6363847436574302712t_unit @ G2 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( add_li4567129529168622413t_unit @ G2 @ X @ ( zero_l1056902381442676492t_unit @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_464_abelian__monoid_Or__zero,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( add_li7652885771158616974t_unit @ G2 @ X @ ( zero_l4142658623432671053t_unit @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_465_abelian__monoid_Or__zero,axiom,
! [G2: partia1897943568983147621xt_b_d,X: b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( add_b_d @ G2 @ X @ ( zero_b_d @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_466_abelian__monoid_Or__zero,axiom,
! [G2: partia8877618634411419171xt_a_c,X: a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( add_a_c @ G2 @ X @ ( zero_a_c @ G2 ) )
= X ) ) ) ).
% abelian_monoid.r_zero
thf(fact_467_abelian__monoid_Ominus__unique,axiom,
! [G2: partia2956882679547061052t_unit,Y: list_list_a,X: list_list_a,Y2: list_list_a] :
( ( abelia3641329199688042803t_unit @ G2 )
=> ( ( ( add_li174743652000525320t_unit @ G2 @ Y @ X )
= ( zero_l347298301471573063t_unit @ G2 ) )
=> ( ( ( add_li174743652000525320t_unit @ G2 @ X @ Y2 )
= ( zero_l347298301471573063t_unit @ G2 ) )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_468_abelian__monoid_Ominus__unique,axiom,
! [G2: partia4026993951477142903t_unit,Y: list_b,X: list_b,Y2: list_b] :
( ( abelia6363847436574302712t_unit @ G2 )
=> ( ( ( add_li4567129529168622413t_unit @ G2 @ Y @ X )
= ( zero_l1056902381442676492t_unit @ G2 ) )
=> ( ( ( add_li4567129529168622413t_unit @ G2 @ X @ Y2 )
= ( zero_l1056902381442676492t_unit @ G2 ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_469_abelian__monoid_Ominus__unique,axiom,
! [G2: partia2670972154091845814t_unit,Y: list_a,X: list_a,Y2: list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( ( add_li7652885771158616974t_unit @ G2 @ Y @ X )
= ( zero_l4142658623432671053t_unit @ G2 ) )
=> ( ( ( add_li7652885771158616974t_unit @ G2 @ X @ Y2 )
= ( zero_l4142658623432671053t_unit @ G2 ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_470_abelian__monoid_Ominus__unique,axiom,
! [G2: partia1897943568983147621xt_b_d,Y: b,X: b,Y2: b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( ( add_b_d @ G2 @ Y @ X )
= ( zero_b_d @ G2 ) )
=> ( ( ( add_b_d @ G2 @ X @ Y2 )
= ( zero_b_d @ G2 ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_471_abelian__monoid_Ominus__unique,axiom,
! [G2: partia8877618634411419171xt_a_c,Y: a,X: a,Y2: a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( ( add_a_c @ G2 @ Y @ X )
= ( zero_a_c @ G2 ) )
=> ( ( ( add_a_c @ G2 @ X @ Y2 )
= ( zero_a_c @ G2 ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ G2 ) )
=> ( Y = Y2 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_472_abelian__monoidI,axiom,
! [R3: partia2956882679547061052t_unit] :
( ! [X2: list_list_a,Y3: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Y3 ) @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ! [X2: list_list_a,Y3: list_list_a,Z2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ ( add_li174743652000525320t_unit @ R3 @ X2 @ Y3 ) @ Z2 )
= ( add_li174743652000525320t_unit @ R3 @ X2 @ ( add_li174743652000525320t_unit @ R3 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ ( zero_l347298301471573063t_unit @ R3 ) @ X2 )
= X2 ) )
=> ( ! [X2: list_list_a,Y3: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ X2 @ Y3 )
= ( add_li174743652000525320t_unit @ R3 @ Y3 @ X2 ) ) ) )
=> ( abelia3641329199688042803t_unit @ R3 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_473_abelian__monoidI,axiom,
! [R3: partia4026993951477142903t_unit] :
( ! [X2: list_b,Y3: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y3 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( member_list_b @ ( add_li4567129529168622413t_unit @ R3 @ X2 @ Y3 ) @ ( partia1381092143316337258t_unit @ R3 ) ) ) )
=> ( ( member_list_b @ ( zero_l1056902381442676492t_unit @ R3 ) @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ! [X2: list_b,Y3: list_b,Z2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y3 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Z2 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ ( add_li4567129529168622413t_unit @ R3 @ X2 @ Y3 ) @ Z2 )
= ( add_li4567129529168622413t_unit @ R3 @ X2 @ ( add_li4567129529168622413t_unit @ R3 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ ( zero_l1056902381442676492t_unit @ R3 ) @ X2 )
= X2 ) )
=> ( ! [X2: list_b,Y3: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( member_list_b @ Y3 @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ X2 @ Y3 )
= ( add_li4567129529168622413t_unit @ R3 @ Y3 @ X2 ) ) ) )
=> ( abelia6363847436574302712t_unit @ R3 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_474_abelian__monoidI,axiom,
! [R3: partia2670972154091845814t_unit] :
( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y3 ) @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ! [X2: list_a,Y3: list_a,Z2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y3 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R3 @ X2 @ ( add_li7652885771158616974t_unit @ R3 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ ( zero_l4142658623432671053t_unit @ R3 ) @ X2 )
= X2 ) )
=> ( ! [X2: list_a,Y3: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ X2 @ Y3 )
= ( add_li7652885771158616974t_unit @ R3 @ Y3 @ X2 ) ) ) )
=> ( abelia226231641709521465t_unit @ R3 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_475_abelian__monoidI,axiom,
! [R3: partia1897943568983147621xt_b_d] :
( ! [X2: b,Y3: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y3 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_b @ ( add_b_d @ R3 @ X2 @ Y3 ) @ ( partia8782771468121683032xt_b_d @ R3 ) ) ) )
=> ( ( member_b @ ( zero_b_d @ R3 ) @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ! [X2: b,Y3: b,Z2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y3 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Z2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ ( add_b_d @ R3 @ X2 @ Y3 ) @ Z2 )
= ( add_b_d @ R3 @ X2 @ ( add_b_d @ R3 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ ( zero_b_d @ R3 ) @ X2 )
= X2 ) )
=> ( ! [X2: b,Y3: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( member_b @ Y3 @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ X2 @ Y3 )
= ( add_b_d @ R3 @ Y3 @ X2 ) ) ) )
=> ( abelian_monoid_b_d @ R3 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_476_abelian__monoidI,axiom,
! [R3: partia8877618634411419171xt_a_c] :
( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y3 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( member_a @ ( add_a_c @ R3 @ X2 @ Y3 ) @ ( partia778085601923319190xt_a_c @ R3 ) ) ) )
=> ( ( member_a @ ( zero_a_c @ R3 ) @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ! [X2: a,Y3: a,Z2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y3 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Z2 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ ( add_a_c @ R3 @ X2 @ Y3 ) @ Z2 )
= ( add_a_c @ R3 @ X2 @ ( add_a_c @ R3 @ Y3 @ Z2 ) ) ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ ( zero_a_c @ R3 ) @ X2 )
= X2 ) )
=> ( ! [X2: a,Y3: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( member_a @ Y3 @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ X2 @ Y3 )
= ( add_a_c @ R3 @ Y3 @ X2 ) ) ) )
=> ( abelian_monoid_a_c @ R3 ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_477_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ X @ ( zero_l347298301471573063t_unit @ R3 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_478_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia4026993951477142903t_unit,X: list_b] :
( ( semiri9009524540797033698t_unit @ R3 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ X @ ( zero_l1056902381442676492t_unit @ R3 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_479_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ X @ ( zero_l4142658623432671053t_unit @ R3 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_480_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia1897943568983147621xt_b_d,X: b] :
( ( semiring_b_d @ R3 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ X @ ( zero_b_d @ R3 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_481_semiring_Osemiring__simprules_I11_J,axiom,
! [R3: partia8877618634411419171xt_a_c,X: a] :
( ( semiring_a_c @ R3 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ X @ ( zero_a_c @ R3 ) )
= X ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_482_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia2956882679547061052t_unit,X: list_list_a] :
( ( semiri2265994252334843677t_unit @ R3 )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( add_li174743652000525320t_unit @ R3 @ ( zero_l347298301471573063t_unit @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_483_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia4026993951477142903t_unit,X: list_b] :
( ( semiri9009524540797033698t_unit @ R3 )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( add_li4567129529168622413t_unit @ R3 @ ( zero_l1056902381442676492t_unit @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_484_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia2670972154091845814t_unit,X: list_a] :
( ( semiri2871908745932252451t_unit @ R3 )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( add_li7652885771158616974t_unit @ R3 @ ( zero_l4142658623432671053t_unit @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_485_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia1897943568983147621xt_b_d,X: b] :
( ( semiring_b_d @ R3 )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( add_b_d @ R3 @ ( zero_b_d @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_486_semiring_Osemiring__simprules_I6_J,axiom,
! [R3: partia8877618634411419171xt_a_c,X: a] :
( ( semiring_a_c @ R3 )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( add_a_c @ R3 @ ( zero_a_c @ R3 ) @ X )
= X ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_487_domain_OpprimeE_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( subfie4546268998243038636t_unit @ K @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( ring_r346321679897941977t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) @ P )
=> ( P != nil_list_list_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_488_domain_OpprimeE_I1_J,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b,P: list_list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( subfie7916738691610828529t_unit @ K @ R3 )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( ring_r415245522172713630t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) @ P )
=> ( P != nil_list_b ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_489_domain_OpprimeE_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_490_domain_OpprimeE_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b,P: list_b] :
( ( domain_b_d @ R3 )
=> ( ( subfield_b_d @ K @ R3 )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ R3 @ K ) @ P )
=> ( P != nil_b ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_491_domain_OpprimeE_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a,P: list_a] :
( ( domain_a_c @ R3 )
=> ( ( subfield_a_c @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ R3 @ K ) @ P )
=> ( P != nil_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_492_domain_Olong__division__closed_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a,Q: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( subfie4546268998243038636t_unit @ K @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( member5342144027231129785list_a @ ( polyno4115915122720352731t_unit @ R3 @ P @ Q ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_493_domain_Olong__division__closed_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_c @ R3 )
=> ( ( subfield_a_c @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_c @ R3 @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_494_domain_Olong__division__closed_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b,P: list_b,Q: list_b] :
( ( domain_b_d @ R3 )
=> ( ( subfield_b_d @ K @ R3 )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( member_list_b @ ( polynomial_pdiv_b_d @ R3 @ P @ Q ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_495_domain_Olong__division__closed_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ R3 @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_496_domain_Olong__division__closed_I1_J,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( subfie7916738691610828529t_unit @ K @ R3 )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( member_list_list_b @ ( polyno2808025880298714784t_unit @ R3 @ P @ Q ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_497_domain_Olong__division__a__inv_I1_J,axiom,
! [R3: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a,Q: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( subfie4546268998243038636t_unit @ K @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( polyno4115915122720352731t_unit @ R3 @ ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) @ P ) @ Q )
= ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) @ ( polyno4115915122720352731t_unit @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_498_domain_Olong__division__a__inv_I1_J,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_c @ R3 )
=> ( ( subfield_a_c @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( polynomial_pdiv_a_c @ R3 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ R3 @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ R3 @ K ) @ ( polynomial_pdiv_a_c @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_499_domain_Olong__division__a__inv_I1_J,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b,P: list_b,Q: list_b] :
( ( domain_b_d @ R3 )
=> ( ( subfield_b_d @ K @ R3 )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( polynomial_pdiv_b_d @ R3 @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ R3 @ K ) @ P ) @ Q )
= ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ R3 @ K ) @ ( polynomial_pdiv_b_d @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_500_domain_Olong__division__a__inv_I1_J,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R3 @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ P ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) @ ( polyno5893782122288709345t_unit @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_501_domain_Olong__division__a__inv_I1_J,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( subfie7916738691610828529t_unit @ K @ R3 )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( polyno2808025880298714784t_unit @ R3 @ ( a_inv_2010862973944421262t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) @ P ) @ Q )
= ( a_inv_2010862973944421262t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) @ ( polyno2808025880298714784t_unit @ R3 @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_502_principal__domain_Oprimeness__condition,axiom,
! [R3: partia2587409828943155709t_unit,P: list_list_b] :
( ( ring_p4916954238016387169t_unit @ R3 )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ R3 ) )
=> ( ( ring_r4561388045816386823t_unit @ R3 @ P )
= ( ring_r415245522172713630t_unit @ R3 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_503_principal__domain_Oprimeness__condition,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_a] :
( ( ring_p715737262848045090t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( ring_r360171070648044744t_unit @ R3 @ P )
= ( ring_r5437400583859147359t_unit @ R3 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_504_principal__domain_Oprimeness__condition,axiom,
! [R3: partia4026993951477142903t_unit,P: list_b] :
( ( ring_p5013149089651084391t_unit @ R3 )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( ring_r7070601269410051085t_unit @ R3 @ P )
= ( ring_r3344526403024810276t_unit @ R3 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_505_principal__domain_Oprimeness__condition,axiom,
! [R3: partia2670972154091845814t_unit,P: list_a] :
( ( ring_p8098905331641078952t_unit @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( ring_r932985474545269838t_unit @ R3 @ P )
= ( ring_r6430282645014804837t_unit @ R3 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_506_principal__domain_Oprimeness__condition,axiom,
! [R3: partia1897943568983147621xt_b_d,P: b] :
( ( ring_p6015679779713369569in_b_d @ R3 )
=> ( ( member_b @ P @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( ring_r7435050590149293703le_b_d @ R3 @ P )
= ( ring_ring_prime_b_d @ R3 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_507_principal__domain_Oprimeness__condition,axiom,
! [R3: partia8877618634411419171xt_a_c,P: a] :
( ( ring_p8803135361686045601in_a_c @ R3 )
=> ( ( member_a @ P @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( ring_r999134135267193927le_a_c @ R3 @ P )
= ( ring_ring_prime_a_c @ R3 @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_508_ring__iso__memE_I1_J,axiom,
! [H2: b > b,R3: partia1897943568983147621xt_b_d,S: partia1897943568983147621xt_b_d,X: b] :
( ( member_b_b @ H2 @ ( ring_iso_b_d_b_d @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_509_ring__iso__memE_I1_J,axiom,
! [H2: b > a,R3: partia1897943568983147621xt_b_d,S: partia8877618634411419171xt_a_c,X: b] :
( ( member_b_a @ H2 @ ( ring_iso_b_d_a_c @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_510_ring__iso__memE_I1_J,axiom,
! [H2: a > b,R3: partia8877618634411419171xt_a_c,S: partia1897943568983147621xt_b_d,X: a] :
( ( member_a_b @ H2 @ ( ring_iso_a_c_b_d @ R3 @ S ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_511_ring__iso__memE_I1_J,axiom,
! [H2: a > a,R3: partia8877618634411419171xt_a_c,S: partia8877618634411419171xt_a_c,X: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_c_a_c @ R3 @ S ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_512_ring__iso__memE_I1_J,axiom,
! [H2: list_b > b,R3: partia4026993951477142903t_unit,S: partia1897943568983147621xt_b_d,X: list_b] :
( ( member_list_b_b @ H2 @ ( ring_i3676047618198725658it_b_d @ R3 @ S ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_513_ring__iso__memE_I1_J,axiom,
! [H2: list_b > a,R3: partia4026993951477142903t_unit,S: partia8877618634411419171xt_a_c,X: list_b] :
( ( member_list_b_a @ H2 @ ( ring_i6463503200171401690it_a_c @ R3 @ S ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_514_ring__iso__memE_I1_J,axiom,
! [H2: list_a > b,R3: partia2670972154091845814t_unit,S: partia1897943568983147621xt_b_d,X: list_a] :
( ( member_list_a_b @ H2 @ ( ring_i4261380215208433627it_b_d @ R3 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_b @ ( H2 @ X ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_515_ring__iso__memE_I1_J,axiom,
! [H2: list_a > a,R3: partia2670972154091845814t_unit,S: partia8877618634411419171xt_a_c,X: list_a] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109659it_a_c @ R3 @ S ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( member_a @ ( H2 @ X ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_516_ring__iso__memE_I1_J,axiom,
! [H2: b > list_b,R3: partia1897943568983147621xt_b_d,S: partia4026993951477142903t_unit,X: b] :
( ( member_b_list_b @ H2 @ ( ring_i3204478355530656794t_unit @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_list_b @ ( H2 @ X ) @ ( partia1381092143316337258t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_517_ring__iso__memE_I1_J,axiom,
! [H2: b > list_a,R3: partia1897943568983147621xt_b_d,S: partia2670972154091845814t_unit,X: b] :
( ( member_b_list_a @ H2 @ ( ring_i6290234597520651355t_unit @ R3 @ S ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( member_list_a @ ( H2 @ X ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_518_ring__iso__memE_I4_J,axiom,
! [H2: b > b,R3: partia1897943568983147621xt_b_d,S: partia1897943568983147621xt_b_d] :
( ( member_b_b @ H2 @ ( ring_iso_b_d_b_d @ R3 @ S ) )
=> ( ( H2 @ ( one_b_ring_ext_b_d @ R3 ) )
= ( one_b_ring_ext_b_d @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_519_ring__iso__memE_I4_J,axiom,
! [H2: b > a,R3: partia1897943568983147621xt_b_d,S: partia8877618634411419171xt_a_c] :
( ( member_b_a @ H2 @ ( ring_iso_b_d_a_c @ R3 @ S ) )
=> ( ( H2 @ ( one_b_ring_ext_b_d @ R3 ) )
= ( one_a_ring_ext_a_c @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_520_ring__iso__memE_I4_J,axiom,
! [H2: a > a,R3: partia8877618634411419171xt_a_c,S: partia8877618634411419171xt_a_c] :
( ( member_a_a @ H2 @ ( ring_iso_a_c_a_c @ R3 @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_c @ R3 ) )
= ( one_a_ring_ext_a_c @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_521_ring__iso__memE_I4_J,axiom,
! [H2: a > b,R3: partia8877618634411419171xt_a_c,S: partia1897943568983147621xt_b_d] :
( ( member_a_b @ H2 @ ( ring_iso_a_c_b_d @ R3 @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_c @ R3 ) )
= ( one_b_ring_ext_b_d @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_522_ring__iso__memE_I4_J,axiom,
! [H2: b > list_a,R3: partia1897943568983147621xt_b_d,S: partia2670972154091845814t_unit] :
( ( member_b_list_a @ H2 @ ( ring_i6290234597520651355t_unit @ R3 @ S ) )
=> ( ( H2 @ ( one_b_ring_ext_b_d @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_523_ring__iso__memE_I4_J,axiom,
! [H2: b > list_b,R3: partia1897943568983147621xt_b_d,S: partia4026993951477142903t_unit] :
( ( member_b_list_b @ H2 @ ( ring_i3204478355530656794t_unit @ R3 @ S ) )
=> ( ( H2 @ ( one_b_ring_ext_b_d @ R3 ) )
= ( one_li3054569011217056637t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_524_ring__iso__memE_I4_J,axiom,
! [H2: a > list_a,R3: partia8877618634411419171xt_a_c,S: partia2670972154091845814t_unit] :
( ( member_a_list_a @ H2 @ ( ring_i314451862811046427t_unit @ R3 @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_c @ R3 ) )
= ( one_li8328186300101108157t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_525_ring__iso__memE_I4_J,axiom,
! [H2: a > list_b,R3: partia8877618634411419171xt_a_c,S: partia4026993951477142903t_unit] :
( ( member_a_list_b @ H2 @ ( ring_i6452067657675827674t_unit @ R3 @ S ) )
=> ( ( H2 @ ( one_a_ring_ext_a_c @ R3 ) )
= ( one_li3054569011217056637t_unit @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_526_ring__iso__memE_I4_J,axiom,
! [H2: list_a > b,R3: partia2670972154091845814t_unit,S: partia1897943568983147621xt_b_d] :
( ( member_list_a_b @ H2 @ ( ring_i4261380215208433627it_b_d @ R3 @ S ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_b_ring_ext_b_d @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_527_ring__iso__memE_I4_J,axiom,
! [H2: list_a > a,R3: partia2670972154091845814t_unit,S: partia8877618634411419171xt_a_c] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109659it_a_c @ R3 @ S ) )
=> ( ( H2 @ ( one_li8328186300101108157t_unit @ R3 ) )
= ( one_a_ring_ext_a_c @ S ) ) ) ).
% ring_iso_memE(4)
thf(fact_528_pdr_Ouniv__poly__is__principal,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ).
% pdr.univ_poly_is_principal
thf(fact_529_ds_Ouniv__poly__is__principal,axiom,
! [K: set_b] :
( ( subfield_b_d @ K @ s )
=> ( ring_p5013149089651084391t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ).
% ds.univ_poly_is_principal
thf(fact_530_pds_Ouniv__poly__is__principal,axiom,
! [K: set_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ring_p4916954238016387169t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ).
% pds.univ_poly_is_principal
thf(fact_531_dr_Oexists__unique__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X2: produc9164743771328383783list_a] :
( ( polyno2806191415236617129es_a_c @ r @ P @ Q @ X2 )
& ! [Y4: produc9164743771328383783list_a] :
( ( polyno2806191415236617129es_a_c @ r @ P @ Q @ Y4 )
=> ( Y4 = X2 ) ) ) ) ) ) ) ).
% dr.exists_unique_long_division
thf(fact_532_pds_Oonepideal,axiom,
princi5701163198563039320t_unit @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.onepideal
thf(fact_533_dr_Omonic__degree__one__root__condition,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( polyno4133073214067823461ot_a_c @ r @ ( cons_a @ ( one_a_ring_ext_a_c @ r ) @ ( cons_a @ ( a_inv_a_c @ r @ A ) @ nil_a ) ) @ B )
= ( A = B ) ) ) ).
% dr.monic_degree_one_root_condition
thf(fact_534_h_Oimg__is__subalgebra,axiom,
! [K: set_a,V: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( embedd9027525575939734155ra_a_c @ K @ V @ r )
=> ( embedd6240069993967058123ra_b_d @ ( image_a_b @ h @ K ) @ ( image_a_b @ h @ V ) @ s ) ) ) ).
% h.img_is_subalgebra
thf(fact_535_pdr_Ominus__eq,axiom,
! [X: list_a,Y: list_a] :
( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y ) ) ) ).
% pdr.minus_eq
thf(fact_536_univ__poly__zero__closed,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b] : ( member_list_list_b @ nil_list_b @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_537_univ__poly__zero__closed,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_538_univ__poly__zero__closed,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b] : ( member_list_b @ nil_b @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_539_univ__poly__zero__closed,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_540_dr_Oadd_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_c @ r @ A @ C )
= ( add_a_c @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( A = B ) ) ) ) ) ).
% dr.add.r_cancel
thf(fact_541_dr_Oadd_Om__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ X @ ( add_a_c @ r @ Y @ Z ) )
= ( add_a_c @ r @ Y @ ( add_a_c @ r @ X @ Z ) ) ) ) ) ) ).
% dr.add.m_lcomm
thf(fact_542_dr_Oadd_Om__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ X @ Y )
= ( add_a_c @ r @ Y @ X ) ) ) ) ).
% dr.add.m_comm
thf(fact_543_dr_Oadd_Om__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ ( add_a_c @ r @ X @ Y ) @ Z )
= ( add_a_c @ r @ X @ ( add_a_c @ r @ Y @ Z ) ) ) ) ) ) ).
% dr.add.m_assoc
thf(fact_544_dr_Oadd_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_c @ r @ C @ A )
= ( add_a_c @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( A = B ) ) ) ) ) ).
% dr.add.l_cancel
thf(fact_545_ds_Oadd_Ol__cancel,axiom,
! [C: b,A: b,B: b] :
( ( ( add_b_d @ s @ C @ A )
= ( add_b_d @ s @ C @ B ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( A = B ) ) ) ) ) ).
% ds.add.l_cancel
thf(fact_546_ds_Oadd_Om__assoc,axiom,
! [X: b,Y: b,Z: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ ( add_b_d @ s @ X @ Y ) @ Z )
= ( add_b_d @ s @ X @ ( add_b_d @ s @ Y @ Z ) ) ) ) ) ) ).
% ds.add.m_assoc
thf(fact_547_ds_Oadd_Om__comm,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ X @ Y )
= ( add_b_d @ s @ Y @ X ) ) ) ) ).
% ds.add.m_comm
thf(fact_548_ds_Oadd_Om__lcomm,axiom,
! [X: b,Y: b,Z: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ X @ ( add_b_d @ s @ Y @ Z ) )
= ( add_b_d @ s @ Y @ ( add_b_d @ s @ X @ Z ) ) ) ) ) ) ).
% ds.add.m_lcomm
thf(fact_549_ds_Oadd_Or__cancel,axiom,
! [A: b,C: b,B: b] :
( ( ( add_b_d @ s @ A @ C )
= ( add_b_d @ s @ B @ C ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( A = B ) ) ) ) ) ).
% ds.add.r_cancel
thf(fact_550_dr_Osubring__props_I7_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( add_a_c @ r @ H1 @ H22 ) @ K ) ) ) ) ).
% dr.subring_props(7)
thf(fact_551_ds_Osubring__props_I7_J,axiom,
! [K: set_b,H1: b,H22: b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_b @ H1 @ K )
=> ( ( member_b @ H22 @ K )
=> ( member_b @ ( add_b_d @ s @ H1 @ H22 ) @ K ) ) ) ) ).
% ds.subring_props(7)
thf(fact_552_dr_Ominus__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( add_a_c @ r @ Y @ X )
= ( zero_a_c @ r ) )
=> ( ( ( add_a_c @ r @ X @ Y2 )
= ( zero_a_c @ r ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% dr.minus_unique
thf(fact_553_dr_Oadd_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
& ( ( add_a_c @ r @ X @ X2 )
= ( zero_a_c @ r ) ) ) ) ).
% dr.add.r_inv_ex
thf(fact_554_dr_Oadd_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_a_c @ r ) ) ) ) ).
% dr.add.one_unique
thf(fact_555_dr_Oadd_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
& ( ( add_a_c @ r @ X2 @ X )
= ( zero_a_c @ r ) ) ) ) ).
% dr.add.l_inv_ex
thf(fact_556_dr_Oadd_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_c @ r @ X @ Y )
= ( zero_a_c @ r ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ Y @ X )
= ( zero_a_c @ r ) ) ) ) ) ).
% dr.add.inv_comm
thf(fact_557_ds_Oadd_Oinv__comm,axiom,
! [X: b,Y: b] :
( ( ( add_b_d @ s @ X @ Y )
= ( zero_b_d @ s ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ Y @ X )
= ( zero_b_d @ s ) ) ) ) ) ).
% ds.add.inv_comm
thf(fact_558_ds_Oadd_Ol__inv__ex,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ? [X2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
& ( ( add_b_d @ s @ X2 @ X )
= ( zero_b_d @ s ) ) ) ) ).
% ds.add.l_inv_ex
thf(fact_559_ds_Oadd_Oone__unique,axiom,
! [U: b] :
( ( member_b @ U @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ! [X2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_b_d @ s ) ) ) ) ).
% ds.add.one_unique
thf(fact_560_ds_Oadd_Or__inv__ex,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ? [X2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
& ( ( add_b_d @ s @ X @ X2 )
= ( zero_b_d @ s ) ) ) ) ).
% ds.add.r_inv_ex
thf(fact_561_ds_Ominus__unique,axiom,
! [Y: b,X: b,Y2: b] :
( ( ( add_b_d @ s @ Y @ X )
= ( zero_b_d @ s ) )
=> ( ( ( add_b_d @ s @ X @ Y2 )
= ( zero_b_d @ s ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% ds.minus_unique
thf(fact_562_dr_Or__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ X @ ( add_a_c @ r @ ( a_inv_a_c @ r @ X ) @ Y ) )
= Y ) ) ) ).
% dr.r_neg2
thf(fact_563_dr_Or__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ ( a_inv_a_c @ r @ X ) @ ( add_a_c @ r @ X @ Y ) )
= Y ) ) ) ).
% dr.r_neg1
thf(fact_564_dr_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( a_inv_a_c @ r @ ( add_a_c @ r @ X @ Y ) )
= ( add_a_c @ r @ ( a_inv_a_c @ r @ X ) @ ( a_inv_a_c @ r @ Y ) ) ) ) ) ).
% dr.minus_add
thf(fact_565_dr_Oadd_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( add_a_c @ r @ B @ ( a_inv_a_c @ r @ C ) )
= A )
= ( B
= ( add_a_c @ r @ A @ C ) ) ) ) ) ) ).
% dr.add.inv_solve_right'
thf(fact_566_dr_Oadd_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( A
= ( add_a_c @ r @ B @ ( a_inv_a_c @ r @ C ) ) )
= ( B
= ( add_a_c @ r @ A @ C ) ) ) ) ) ) ).
% dr.add.inv_solve_right
thf(fact_567_dr_Oadd_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( add_a_c @ r @ ( a_inv_a_c @ r @ B ) @ C )
= A )
= ( C
= ( add_a_c @ r @ B @ A ) ) ) ) ) ) ).
% dr.add.inv_solve_left'
thf(fact_568_dr_Oadd_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( A
= ( add_a_c @ r @ ( a_inv_a_c @ r @ B ) @ C ) )
= ( C
= ( add_a_c @ r @ B @ A ) ) ) ) ) ) ).
% dr.add.inv_solve_left
thf(fact_569_dr_Oadd_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( a_inv_a_c @ r @ ( add_a_c @ r @ X @ Y ) )
= ( add_a_c @ r @ ( a_inv_a_c @ r @ Y ) @ ( a_inv_a_c @ r @ X ) ) ) ) ) ).
% dr.add.inv_mult_group
thf(fact_570_dr_Oa__transpose__inv,axiom,
! [X: a,Y: a,Z: a] :
( ( ( add_a_c @ r @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ ( a_inv_a_c @ r @ X ) @ Z )
= Y ) ) ) ) ) ).
% dr.a_transpose_inv
thf(fact_571_ds_Oa__transpose__inv,axiom,
! [X: b,Y: b,Z: b] :
( ( ( add_b_d @ s @ X @ Y )
= Z )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ ( a_inv_b_d @ s @ X ) @ Z )
= Y ) ) ) ) ) ).
% ds.a_transpose_inv
thf(fact_572_ds_Oadd_Oinv__mult__group,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( a_inv_b_d @ s @ ( add_b_d @ s @ X @ Y ) )
= ( add_b_d @ s @ ( a_inv_b_d @ s @ Y ) @ ( a_inv_b_d @ s @ X ) ) ) ) ) ).
% ds.add.inv_mult_group
thf(fact_573_ds_Oadd_Oinv__solve__left,axiom,
! [A: b,B: b,C: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( A
= ( add_b_d @ s @ ( a_inv_b_d @ s @ B ) @ C ) )
= ( C
= ( add_b_d @ s @ B @ A ) ) ) ) ) ) ).
% ds.add.inv_solve_left
thf(fact_574_ds_Oadd_Oinv__solve__left_H,axiom,
! [A: b,B: b,C: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( add_b_d @ s @ ( a_inv_b_d @ s @ B ) @ C )
= A )
= ( C
= ( add_b_d @ s @ B @ A ) ) ) ) ) ) ).
% ds.add.inv_solve_left'
thf(fact_575_ds_Oadd_Oinv__solve__right,axiom,
! [A: b,B: b,C: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( A
= ( add_b_d @ s @ B @ ( a_inv_b_d @ s @ C ) ) )
= ( B
= ( add_b_d @ s @ A @ C ) ) ) ) ) ) ).
% ds.add.inv_solve_right
thf(fact_576_ds_Oadd_Oinv__solve__right_H,axiom,
! [A: b,B: b,C: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( add_b_d @ s @ B @ ( a_inv_b_d @ s @ C ) )
= A )
= ( B
= ( add_b_d @ s @ A @ C ) ) ) ) ) ) ).
% ds.add.inv_solve_right'
thf(fact_577_ds_Ominus__add,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( a_inv_b_d @ s @ ( add_b_d @ s @ X @ Y ) )
= ( add_b_d @ s @ ( a_inv_b_d @ s @ X ) @ ( a_inv_b_d @ s @ Y ) ) ) ) ) ).
% ds.minus_add
thf(fact_578_ds_Or__neg1,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ ( a_inv_b_d @ s @ X ) @ ( add_b_d @ s @ X @ Y ) )
= Y ) ) ) ).
% ds.r_neg1
thf(fact_579_ds_Or__neg2,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ X @ ( add_b_d @ s @ ( a_inv_b_d @ s @ X ) @ Y ) )
= Y ) ) ) ).
% ds.r_neg2
thf(fact_580_pds_Oadd_Ol__cancel,axiom,
! [C: list_b,A: list_b,B: list_b] :
( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C @ A )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C @ B ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( A = B ) ) ) ) ) ).
% pds.add.l_cancel
thf(fact_581_pds_Oadd_Om__assoc,axiom,
! [X: list_b,Y: list_b,Z: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) @ Z )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ Z ) ) ) ) ) ) ).
% pds.add.m_assoc
thf(fact_582_pds_Oadd_Om__comm,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ X ) ) ) ) ).
% pds.add.m_comm
thf(fact_583_pds_Oadd_Om__lcomm,axiom,
! [X: list_b,Y: list_b,Z: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ Z ) )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Z ) ) ) ) ) ) ).
% pds.add.m_lcomm
thf(fact_584_pds_Oadd_Or__cancel,axiom,
! [A: list_b,C: list_b,B: list_b] :
( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ C )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ C ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( A = B ) ) ) ) ) ).
% pds.add.r_cancel
thf(fact_585_pds_Osubring__props_I7_J,axiom,
! [K: set_list_b,H1: list_b,H22: list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_b @ H1 @ K )
=> ( ( member_list_b @ H22 @ K )
=> ( member_list_b @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H1 @ H22 ) @ K ) ) ) ) ).
% pds.subring_props(7)
thf(fact_586_ds_Olong__division__add_I1_J,axiom,
! [K: set_b,A: list_b,B: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( polynomial_pdiv_b_d @ s @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ K ) @ A @ B ) @ Q )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ K ) @ ( polynomial_pdiv_b_d @ s @ A @ Q ) @ ( polynomial_pdiv_b_d @ s @ B @ Q ) ) ) ) ) ) ) ).
% ds.long_division_add(1)
thf(fact_587_ds_Olong__division__zero_I1_J,axiom,
! [K: set_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( polynomial_pdiv_b_d @ s @ nil_b @ Q )
= nil_b ) ) ) ).
% ds.long_division_zero(1)
thf(fact_588_ds_Olong__division__closed_I1_J,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( member_list_b @ ( polynomial_pdiv_b_d @ s @ P @ Q ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ) ) ) ).
% ds.long_division_closed(1)
thf(fact_589_dr_Osum__zero__eq__neg,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( add_a_c @ r @ X @ Y )
= ( zero_a_c @ r ) )
=> ( X
= ( a_inv_a_c @ r @ Y ) ) ) ) ) ).
% dr.sum_zero_eq_neg
thf(fact_590_dr_Or__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ X @ ( a_inv_a_c @ r @ X ) )
= ( zero_a_c @ r ) ) ) ).
% dr.r_neg
thf(fact_591_dr_Ominus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_c @ r @ Y @ X )
= ( zero_a_c @ r ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( a_inv_a_c @ r @ X )
= Y ) ) ) ) ).
% dr.minus_equality
thf(fact_592_dr_Ol__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ ( a_inv_a_c @ r @ X ) @ X )
= ( zero_a_c @ r ) ) ) ).
% dr.l_neg
thf(fact_593_ds_Ol__neg,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ ( a_inv_b_d @ s @ X ) @ X )
= ( zero_b_d @ s ) ) ) ).
% ds.l_neg
thf(fact_594_ds_Ominus__equality,axiom,
! [Y: b,X: b] :
( ( ( add_b_d @ s @ Y @ X )
= ( zero_b_d @ s ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( a_inv_b_d @ s @ X )
= Y ) ) ) ) ).
% ds.minus_equality
thf(fact_595_ds_Or__neg,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ X @ ( a_inv_b_d @ s @ X ) )
= ( zero_b_d @ s ) ) ) ).
% ds.r_neg
thf(fact_596_ds_Osum__zero__eq__neg,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( add_b_d @ s @ X @ Y )
= ( zero_b_d @ s ) )
=> ( X
= ( a_inv_b_d @ s @ Y ) ) ) ) ) ).
% ds.sum_zero_eq_neg
thf(fact_597_dr_Oa__lcos__m__assoc,axiom,
! [M: set_a,G: a,H2: a] :
( ( ord_less_eq_set_a @ M @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ G @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ H2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( a_l_coset_a_c @ r @ G @ ( a_l_coset_a_c @ r @ H2 @ M ) )
= ( a_l_coset_a_c @ r @ ( add_a_c @ r @ G @ H2 ) @ M ) ) ) ) ) ).
% dr.a_lcos_m_assoc
thf(fact_598_dr_Osubalgebra__in__carrier,axiom,
! [K: set_a,V: set_a] :
( ( embedd9027525575939734155ra_a_c @ K @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% dr.subalgebra_in_carrier
thf(fact_599_dr_Ocarrier__is__subalgebra,axiom,
! [K: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia778085601923319190xt_a_c @ r ) )
=> ( embedd9027525575939734155ra_a_c @ K @ ( partia778085601923319190xt_a_c @ r ) @ r ) ) ).
% dr.carrier_is_subalgebra
thf(fact_600_ds_Oa__lcos__m__assoc,axiom,
! [M: set_b,G: b,H2: b] :
( ( ord_less_eq_set_b @ M @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ G @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ H2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( a_l_coset_b_d @ s @ G @ ( a_l_coset_b_d @ s @ H2 @ M ) )
= ( a_l_coset_b_d @ s @ ( add_b_d @ s @ G @ H2 ) @ M ) ) ) ) ) ).
% ds.a_lcos_m_assoc
thf(fact_601_pds_Oadd_Oinv__comm,axiom,
! [X: list_b,Y: list_b] :
( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ X )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.add.inv_comm
thf(fact_602_pds_Oadd_Ol__inv__ex,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ? [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ X )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.add.l_inv_ex
thf(fact_603_pds_Oadd_Oone__unique,axiom,
! [U: list_b] :
( ( member_list_b @ U @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ! [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.add.one_unique
thf(fact_604_pds_Oadd_Or__inv__ex,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ? [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ X2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.add.r_inv_ex
thf(fact_605_pds_Ominus__unique,axiom,
! [Y: list_b,X: list_b,Y2: list_b] :
( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ X )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% pds.minus_unique
thf(fact_606_pds_Oa__transpose__inv,axiom,
! [X: list_b,Y: list_b,Z: list_b] :
( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y )
= Z )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% pds.a_transpose_inv
thf(fact_607_pds_Oadd_Oinv__mult__group,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) ) ) ) ) ).
% pds.add.inv_mult_group
thf(fact_608_pds_Oadd_Oinv__solve__left,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( A
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B ) @ C ) )
= ( C
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ A ) ) ) ) ) ) ).
% pds.add.inv_solve_left
thf(fact_609_pds_Oadd_Oinv__solve__left_H,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B ) @ C )
= A )
= ( C
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ A ) ) ) ) ) ) ).
% pds.add.inv_solve_left'
thf(fact_610_pds_Oadd_Oinv__solve__right,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( A
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C ) ) )
= ( B
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ C ) ) ) ) ) ) ).
% pds.add.inv_solve_right
thf(fact_611_pds_Oadd_Oinv__solve__right_H,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C ) )
= A )
= ( B
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ C ) ) ) ) ) ) ).
% pds.add.inv_solve_right'
thf(fact_612_pds_Ominus__add,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y ) ) ) ) ) ).
% pds.minus_add
thf(fact_613_pds_Or__neg1,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) )
= Y ) ) ) ).
% pds.r_neg1
thf(fact_614_pds_Or__neg2,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ Y ) )
= Y ) ) ) ).
% pds.r_neg2
thf(fact_615_ds_Olong__division__a__inv_I1_J,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( polynomial_pdiv_b_d @ s @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ K ) @ P ) @ Q )
= ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ K ) @ ( polynomial_pdiv_b_d @ s @ P @ Q ) ) ) ) ) ) ).
% ds.long_division_a_inv(1)
thf(fact_616_ds_OpprimeE_I1_J,axiom,
! [K: set_b,P: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ K ) @ P )
=> ( P != nil_b ) ) ) ) ).
% ds.pprimeE(1)
thf(fact_617_pds_Ol__neg,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ X )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.l_neg
thf(fact_618_pds_Ominus__equality,axiom,
! [Y: list_b,X: list_b] :
( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ X )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X )
= Y ) ) ) ) ).
% pds.minus_equality
thf(fact_619_pds_Or__neg,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.r_neg
thf(fact_620_pds_Osum__zero__eq__neg,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( X
= ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y ) ) ) ) ) ).
% pds.sum_zero_eq_neg
thf(fact_621_pdr_Olong__division__add_I1_J,axiom,
! [K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ Q ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ Q ) ) ) ) ) ) ) ).
% pdr.long_division_add(1)
thf(fact_622_pdr_Olong__division__zero_I1_J,axiom,
! [K: set_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ nil_list_a @ Q )
= nil_list_a ) ) ) ).
% pdr.long_division_zero(1)
thf(fact_623_pdr_Olong__division__a__inv_I1_J,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) ) ) ) ) ) ).
% pdr.long_division_a_inv(1)
thf(fact_624_pdr_Olong__division__closed_I1_J,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ) ) ) ).
% pdr.long_division_closed(1)
thf(fact_625_pds_Olong__division__add_I1_J,axiom,
! [K: set_list_b,A: list_list_b,B: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ A @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ B @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( polyno2808025880298714784t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( add_li4375960627168867399t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ A @ B ) @ Q )
= ( add_li4375960627168867399t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ ( polyno2808025880298714784t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ Q ) @ ( polyno2808025880298714784t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ Q ) ) ) ) ) ) ) ).
% pds.long_division_add(1)
thf(fact_626_pds_Olong__division__zero_I1_J,axiom,
! [K: set_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( polyno2808025880298714784t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ nil_list_b @ Q )
= nil_list_b ) ) ) ).
% pds.long_division_zero(1)
thf(fact_627_pds_Olong__division__a__inv_I1_J,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( polyno2808025880298714784t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_2010862973944421262t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P ) @ Q )
= ( a_inv_2010862973944421262t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ ( polyno2808025880298714784t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) ) ) ) ) ) ).
% pds.long_division_a_inv(1)
thf(fact_628_pds_Olong__division__closed_I1_J,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( member_list_list_b @ ( polyno2808025880298714784t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ) ) ) ).
% pds.long_division_closed(1)
thf(fact_629_pds_Oa__lcos__m__assoc,axiom,
! [M: set_list_b,G: list_b,H2: list_b] :
( ( ord_le8932221534207217157list_b @ M @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ G @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ H2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( a_l_co3923087131696239825t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ G @ ( a_l_co3923087131696239825t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H2 @ M ) )
= ( a_l_co3923087131696239825t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ G @ H2 ) @ M ) ) ) ) ) ).
% pds.a_lcos_m_assoc
thf(fact_630_pdr_OpprimeE_I1_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ).
% pdr.pprimeE(1)
thf(fact_631_pds_OpprimeE_I1_J,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ring_r415245522172713630t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P )
=> ( P != nil_list_b ) ) ) ) ).
% pds.pprimeE(1)
thf(fact_632_dr_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( add_a_c @ r @ Y @ X )
= ( add_a_c @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% dr.add.right_cancel
thf(fact_633_dr_Oadd_Om__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ ( add_a_c @ r @ X @ Y ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% dr.add.m_closed
thf(fact_634_ds_Oadd_Om__closed,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ ( add_b_d @ s @ X @ Y ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% ds.add.m_closed
thf(fact_635_ds_Oadd_Oright__cancel,axiom,
! [X: b,Y: b,Z: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( add_b_d @ s @ Y @ X )
= ( add_b_d @ s @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% ds.add.right_cancel
thf(fact_636_dr_Or__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ X @ ( zero_a_c @ r ) )
= X ) ) ).
% dr.r_zero
thf(fact_637_dr_Ol__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( add_a_c @ r @ ( zero_a_c @ r ) @ X )
= X ) ) ).
% dr.l_zero
thf(fact_638_dr_Oadd_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( X
= ( add_a_c @ r @ A @ X ) )
= ( A
= ( zero_a_c @ r ) ) ) ) ) ).
% dr.add.r_cancel_one'
thf(fact_639_dr_Oadd_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( add_a_c @ r @ A @ X )
= X )
= ( A
= ( zero_a_c @ r ) ) ) ) ) ).
% dr.add.r_cancel_one
thf(fact_640_dr_Oadd_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( X
= ( add_a_c @ r @ X @ A ) )
= ( A
= ( zero_a_c @ r ) ) ) ) ) ).
% dr.add.l_cancel_one'
thf(fact_641_dr_Oadd_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( add_a_c @ r @ X @ A )
= X )
= ( A
= ( zero_a_c @ r ) ) ) ) ) ).
% dr.add.l_cancel_one
thf(fact_642_ds_Oadd_Ol__cancel__one,axiom,
! [X: b,A: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( add_b_d @ s @ X @ A )
= X )
= ( A
= ( zero_b_d @ s ) ) ) ) ) ).
% ds.add.l_cancel_one
thf(fact_643_ds_Oadd_Ol__cancel__one_H,axiom,
! [X: b,A: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( X
= ( add_b_d @ s @ X @ A ) )
= ( A
= ( zero_b_d @ s ) ) ) ) ) ).
% ds.add.l_cancel_one'
thf(fact_644_ds_Oadd_Or__cancel__one,axiom,
! [X: b,A: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( add_b_d @ s @ A @ X )
= X )
= ( A
= ( zero_b_d @ s ) ) ) ) ) ).
% ds.add.r_cancel_one
thf(fact_645_ds_Oadd_Or__cancel__one_H,axiom,
! [X: b,A: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( X
= ( add_b_d @ s @ A @ X ) )
= ( A
= ( zero_b_d @ s ) ) ) ) ) ).
% ds.add.r_cancel_one'
thf(fact_646_ds_Ol__zero,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ ( zero_b_d @ s ) @ X )
= X ) ) ).
% ds.l_zero
thf(fact_647_ds_Or__zero,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( add_b_d @ s @ X @ ( zero_b_d @ s ) )
= X ) ) ).
% ds.r_zero
thf(fact_648_pds_Oadd_Om__closed,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.add.m_closed
thf(fact_649_pds_Oadd_Oright__cancel,axiom,
! [X: list_b,Y: list_b,Z: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ X )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% pds.add.right_cancel
thf(fact_650_pds_Oadd_Ol__cancel__one,axiom,
! [X: list_b,A: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ A )
= X )
= ( A
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.add.l_cancel_one
thf(fact_651_pds_Oadd_Ol__cancel__one_H,axiom,
! [X: list_b,A: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( X
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ A ) )
= ( A
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.add.l_cancel_one'
thf(fact_652_pds_Oadd_Or__cancel__one,axiom,
! [X: list_b,A: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ X )
= X )
= ( A
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.add.r_cancel_one
thf(fact_653_pds_Oadd_Or__cancel__one_H,axiom,
! [X: list_b,A: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( X
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ X ) )
= ( A
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.add.r_cancel_one'
thf(fact_654_pds_Ol__zero,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ X )
= X ) ) ).
% pds.l_zero
thf(fact_655_pds_Or__zero,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= X ) ) ).
% pds.r_zero
thf(fact_656_h_Ohom__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( h @ ( add_a_c @ r @ X @ Y ) )
= ( add_b_d @ s @ ( h @ X ) @ ( h @ Y ) ) ) ) ) ).
% h.hom_add
thf(fact_657_pdr_Ominus__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.minus_closed
thf(fact_658_pdr_Or__right__minus__eq,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( A = B ) ) ) ) ).
% pdr.r_right_minus_eq
thf(fact_659_ring_Ois__root_Ocong,axiom,
polyno4133073214067823461ot_a_c = polyno4133073214067823461ot_a_c ).
% ring.is_root.cong
thf(fact_660_ring_Ois__root_Ocong,axiom,
polyno1345617632095147429ot_b_d = polyno1345617632095147429ot_b_d ).
% ring.is_root.cong
thf(fact_661_ring_Ois__root_Ocong,axiom,
polyno6951661231331188332t_unit = polyno6951661231331188332t_unit ).
% ring.is_root.cong
thf(fact_662_ring_Ois__root_Ocong,axiom,
polyno3865904989341193771t_unit = polyno3865904989341193771t_unit ).
% ring.is_root.cong
thf(fact_663_ring_Olong__divides_Ocong,axiom,
polyno2806191415236617129es_a_c = polyno2806191415236617129es_a_c ).
% ring.long_divides.cong
thf(fact_664_ring_Olong__divides_Ocong,axiom,
polyno3861286681177809007t_unit = polyno3861286681177809007t_unit ).
% ring.long_divides.cong
thf(fact_665_ring_Olong__divides_Ocong,axiom,
polyno6947042923167803568t_unit = polyno6947042923167803568t_unit ).
% ring.long_divides.cong
thf(fact_666_ring_Olong__divides_Ocong,axiom,
polyno18735833263941097es_b_d = polyno18735833263941097es_b_d ).
% ring.long_divides.cong
thf(fact_667_a__minus__def,axiom,
( a_minu2241224857956505934t_unit
= ( ^ [R4: partia2956882679547061052t_unit,X3: list_list_a,Y5: list_list_a] : ( add_li174743652000525320t_unit @ R4 @ X3 @ ( a_inv_7033018035630854991t_unit @ R4 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_668_a__minus__def,axiom,
( a_minu898264511480707987t_unit
= ( ^ [R4: partia4026993951477142903t_unit,X3: list_b,Y5: list_b] : ( add_li4567129529168622413t_unit @ R4 @ X3 @ ( a_inv_5858964851304622612t_unit @ R4 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_669_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R4: partia2670972154091845814t_unit,X3: list_a,Y5: list_a] : ( add_li7652885771158616974t_unit @ R4 @ X3 @ ( a_inv_8944721093294617173t_unit @ R4 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_670_a__minus__def,axiom,
( a_minus_b_d
= ( ^ [R4: partia1897943568983147621xt_b_d,X3: b,Y5: b] : ( add_b_d @ R4 @ X3 @ ( a_inv_b_d @ R4 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_671_a__minus__def,axiom,
( a_minus_a_c
= ( ^ [R4: partia8877618634411419171xt_a_c,X3: a,Y5: a] : ( add_a_c @ R4 @ X3 @ ( a_inv_a_c @ R4 @ Y5 ) ) ) ) ).
% a_minus_def
thf(fact_672_domain_Oexists__unique__long__division,axiom,
! [R3: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a,Q: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( subfie4546268998243038636t_unit @ K @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) )
=> ( ( Q != nil_list_list_a )
=> ? [X2: produc3789376428941379879list_a] :
( ( polyno8316157972930027562t_unit @ R3 @ P @ Q @ X2 )
& ! [Y4: produc3789376428941379879list_a] :
( ( polyno8316157972930027562t_unit @ R3 @ P @ Q @ Y4 )
=> ( Y4 = X2 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_673_domain_Oexists__unique__long__division,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_c @ R3 )
=> ( ( subfield_a_c @ K @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X2: produc9164743771328383783list_a] :
( ( polyno2806191415236617129es_a_c @ R3 @ P @ Q @ X2 )
& ! [Y4: produc9164743771328383783list_a] :
( ( polyno2806191415236617129es_a_c @ R3 @ P @ Q @ Y4 )
=> ( Y4 = X2 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_674_domain_Oexists__unique__long__division,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( subfie7916738691610828529t_unit @ K @ R3 )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) )
=> ( ( Q != nil_list_b )
=> ? [X2: produc4603089062067288551list_b] :
( ( polyno3861286681177809007t_unit @ R3 @ P @ Q @ X2 )
& ! [Y4: produc4603089062067288551list_b] :
( ( polyno3861286681177809007t_unit @ R3 @ P @ Q @ Y4 )
=> ( Y4 = X2 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_675_domain_Oexists__unique__long__division,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) )
=> ( ( Q != nil_list_a )
=> ? [X2: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R3 @ P @ Q @ X2 )
& ! [Y4: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R3 @ P @ Q @ Y4 )
=> ( Y4 = X2 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_676_domain_Oexists__unique__long__division,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b,P: list_b,Q: list_b] :
( ( domain_b_d @ R3 )
=> ( ( subfield_b_d @ K @ R3 )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ K ) ) )
=> ( ( Q != nil_b )
=> ? [X2: produc3963297410138542439list_b] :
( ( polyno18735833263941097es_b_d @ R3 @ P @ Q @ X2 )
& ! [Y4: produc3963297410138542439list_b] :
( ( polyno18735833263941097es_b_d @ R3 @ P @ Q @ Y4 )
=> ( Y4 = X2 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_677_univ__poly__zero,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ R3 @ K ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_678_univ__poly__zero,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b] :
( ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ R3 @ K ) )
= nil_b ) ).
% univ_poly_zero
thf(fact_679_univ__poly__zero,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b] :
( ( zero_l4548515276639915142t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) )
= nil_list_b ) ).
% univ_poly_zero
thf(fact_680_univ__poly__zero,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_681_domain_Ouniv__poly__is__principal,axiom,
! [R3: partia2956882679547061052t_unit,K: set_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( subfie4546268998243038636t_unit @ K @ R3 )
=> ( ring_p8404492108403472412t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_682_domain_Ouniv__poly__is__principal,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a] :
( ( domain_a_c @ R3 )
=> ( ( subfield_a_c @ K @ R3 )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_c @ R3 @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_683_domain_Ouniv__poly__is__principal,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( subfie1779122896746047282t_unit @ K @ R3 )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_684_domain_Ouniv__poly__is__principal,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b] :
( ( domain_b_d @ R3 )
=> ( ( subfield_b_d @ K @ R3 )
=> ( ring_p5013149089651084391t_unit @ ( univ_poly_b_d @ R3 @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_685_domain_Ouniv__poly__is__principal,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( subfie7916738691610828529t_unit @ K @ R3 )
=> ( ring_p4916954238016387169t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_686_univ__poly__one,axiom,
! [R3: partia8877618634411419171xt_a_c,K: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ R3 @ K ) )
= ( cons_a @ ( one_a_ring_ext_a_c @ R3 ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_687_univ__poly__one,axiom,
! [R3: partia1897943568983147621xt_b_d,K: set_b] :
( ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ R3 @ K ) )
= ( cons_b @ ( one_b_ring_ext_b_d @ R3 ) @ nil_b ) ) ).
% univ_poly_one
thf(fact_688_univ__poly__one,axiom,
! [R3: partia4026993951477142903t_unit,K: set_list_b] :
( ( one_li965687310408830589t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ K ) )
= ( cons_list_b @ ( one_li3054569011217056637t_unit @ R3 ) @ nil_list_b ) ) ).
% univ_poly_one
thf(fact_689_univ__poly__one,axiom,
! [R3: partia2670972154091845814t_unit,K: set_list_a] :
( ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K ) )
= ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ nil_list_a ) ) ).
% univ_poly_one
thf(fact_690_pds_Oexists__unique__long__division,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( Q != nil_list_b )
=> ? [X2: produc4603089062067288551list_b] :
( ( polyno3861286681177809007t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q @ X2 )
& ! [Y4: produc4603089062067288551list_b] :
( ( polyno3861286681177809007t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q @ Y4 )
=> ( Y4 = X2 ) ) ) ) ) ) ) ).
% pds.exists_unique_long_division
thf(fact_691_pdr_Oexists__unique__long__division,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( Q != nil_list_a )
=> ? [X2: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q @ X2 )
& ! [Y4: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q @ Y4 )
=> ( Y4 = X2 ) ) ) ) ) ) ) ).
% pdr.exists_unique_long_division
thf(fact_692_ds_Oexists__unique__long__division,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( Q != nil_b )
=> ? [X2: produc3963297410138542439list_b] :
( ( polyno18735833263941097es_b_d @ s @ P @ Q @ X2 )
& ! [Y4: produc3963297410138542439list_b] :
( ( polyno18735833263941097es_b_d @ s @ P @ Q @ Y4 )
=> ( Y4 = X2 ) ) ) ) ) ) ) ).
% ds.exists_unique_long_division
thf(fact_693_h_Oinfinite__dimension__hom,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_c @ K @ r )
=> ( ( ( one_b_ring_ext_b_d @ s )
!= ( zero_b_d @ s ) )
=> ( ( inj_on_a_b @ h @ E )
=> ( ( embedd9027525575939734155ra_a_c @ K @ E @ r )
=> ( ~ ( embedd8708762675212832760on_a_c @ r @ K @ E )
=> ~ ( embedd5921307093240156728on_b_d @ s @ ( image_a_b @ h @ K ) @ ( image_a_b @ h @ E ) ) ) ) ) ) ) ).
% h.infinite_dimension_hom
thf(fact_694_pdr_Oadd_Oone__in__subset,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( H != bot_bot_set_list_a )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 ) @ H ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ H )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Xa ) @ H ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ H ) ) ) ) ) ).
% pdr.add.one_in_subset
thf(fact_695_dr_Ois__root__imp__pdivides,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( polyno4133073214067823461ot_a_c @ r @ P @ X )
=> ( polyno5814909790663948099es_a_c @ r @ ( cons_a @ ( one_a_ring_ext_a_c @ r ) @ ( cons_a @ ( a_inv_a_c @ r @ X ) @ nil_a ) ) @ P ) ) ) ).
% dr.is_root_imp_pdivides
thf(fact_696_pds_Oadd_Oone__in__subset,axiom,
! [H: set_list_b] :
( ( ord_le8932221534207217157list_b @ H @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( H != bot_bot_set_list_b )
=> ( ! [X2: list_b] :
( ( member_list_b @ X2 @ H )
=> ( member_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 ) @ H ) )
=> ( ! [X2: list_b] :
( ( member_list_b @ X2 @ H )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ H )
=> ( member_list_b @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Xa ) @ H ) ) )
=> ( member_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ H ) ) ) ) ) ).
% pds.add.one_in_subset
thf(fact_697_pds_Onormalize_Ocases,axiom,
! [X: list_list_b] :
( ( X != nil_list_b )
=> ~ ! [V2: list_b,Va: list_list_b] :
( X
!= ( cons_list_b @ V2 @ Va ) ) ) ).
% pds.normalize.cases
thf(fact_698_pdr_Onormalize_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ~ ! [V2: list_a,Va: list_list_a] :
( X
!= ( cons_list_a @ V2 @ Va ) ) ) ).
% pdr.normalize.cases
thf(fact_699_ds_Onormalize_Ocases,axiom,
! [X: list_b] :
( ( X != nil_b )
=> ~ ! [V2: b,Va: list_b] :
( X
!= ( cons_b @ V2 @ Va ) ) ) ).
% ds.normalize.cases
thf(fact_700_dr_Ominus__eq,axiom,
! [X: a,Y: a] :
( ( a_minus_a_c @ r @ X @ Y )
= ( add_a_c @ r @ X @ ( a_inv_a_c @ r @ Y ) ) ) ).
% dr.minus_eq
thf(fact_701_ds_Ominus__eq,axiom,
! [X: b,Y: b] :
( ( a_minus_b_d @ s @ X @ Y )
= ( add_b_d @ s @ X @ ( a_inv_b_d @ s @ Y ) ) ) ).
% ds.minus_eq
thf(fact_702_dr_Ozero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948099es_a_c @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% dr.zero_pdivides
thf(fact_703_dr_Ozero__pdivides__zero,axiom,
polyno5814909790663948099es_a_c @ r @ nil_a @ nil_a ).
% dr.zero_pdivides_zero
thf(fact_704_dr_Otelescopic__base__dim_I1_J,axiom,
! [K: set_a,F2: set_a,E: set_a] :
( ( subfield_a_c @ K @ r )
=> ( ( subfield_a_c @ F2 @ r )
=> ( ( embedd8708762675212832760on_a_c @ r @ K @ F2 )
=> ( ( embedd8708762675212832760on_a_c @ r @ F2 @ E )
=> ( embedd8708762675212832760on_a_c @ r @ K @ E ) ) ) ) ) ).
% dr.telescopic_base_dim(1)
thf(fact_705_dr_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_a,E: set_a] :
( ( subfield_a_c @ K @ r )
=> ( ( embedd8708762675212832760on_a_c @ r @ K @ E )
=> ( embedd9027525575939734155ra_a_c @ K @ E @ r ) ) ) ).
% dr.finite_dimension_imp_subalgebra
thf(fact_706_pdr_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= bot_bot_set_list_a ) ).
% pdr.carrier_not_empty
thf(fact_707_pds_Ocarrier__not__empty,axiom,
( ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= bot_bot_set_list_b ) ).
% pds.carrier_not_empty
thf(fact_708_pds_Ominus__eq,axiom,
! [X: list_b,Y: list_b] :
( ( a_minu898264511480707987t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y ) ) ) ).
% pds.minus_eq
thf(fact_709_pdr_Osubring__props_I4_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( K != bot_bot_set_list_a ) ) ).
% pdr.subring_props(4)
thf(fact_710_pds_Osubring__props_I4_J,axiom,
! [K: set_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( K != bot_bot_set_list_b ) ) ).
% pds.subring_props(4)
thf(fact_711_ds_Omonic__degree__one__root__condition,axiom,
! [A: b,B: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( polyno1345617632095147429ot_b_d @ s @ ( cons_b @ ( one_b_ring_ext_b_d @ s ) @ ( cons_b @ ( a_inv_b_d @ s @ A ) @ nil_b ) ) @ B )
= ( A = B ) ) ) ).
% ds.monic_degree_one_root_condition
thf(fact_712_dr_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_a,E: set_a,V: set_a] :
( ( subfield_a_c @ K @ r )
=> ( ( embedd8708762675212832760on_a_c @ r @ K @ E )
=> ( ( embedd9027525575939734155ra_a_c @ K @ V @ r )
=> ( ( ord_less_eq_set_a @ V @ E )
=> ( embedd8708762675212832760on_a_c @ r @ K @ V ) ) ) ) ) ).
% dr.subalbegra_incl_imp_finite_dimension
thf(fact_713_pdr_Omonic__degree__one__root__condition,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A ) @ nil_list_a ) ) @ B )
= ( A = B ) ) ) ).
% pdr.monic_degree_one_root_condition
thf(fact_714_pds_Omonic__degree__one__root__condition,axiom,
! [A: list_b,B: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( cons_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( cons_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A ) @ nil_list_b ) ) @ B )
= ( A = B ) ) ) ).
% pds.monic_degree_one_root_condition
thf(fact_715_dr_Opdivides__imp__is__root,axiom,
! [P: list_a,X: a] :
( ( P != nil_a )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( polyno5814909790663948099es_a_c @ r @ ( cons_a @ ( one_a_ring_ext_a_c @ r ) @ ( cons_a @ ( a_inv_a_c @ r @ X ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823461ot_a_c @ r @ P @ X ) ) ) ) ).
% dr.pdivides_imp_is_root
thf(fact_716_dr_Ominus__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ ( a_minus_a_c @ r @ X @ Y ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% dr.minus_closed
thf(fact_717_ds_Ominus__closed,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ ( a_minus_b_d @ s @ X @ Y ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% ds.minus_closed
thf(fact_718_dr_Or__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( a_minus_a_c @ r @ A @ B )
= ( zero_a_c @ r ) )
= ( A = B ) ) ) ) ).
% dr.r_right_minus_eq
thf(fact_719_ds_Or__right__minus__eq,axiom,
! [A: b,B: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( a_minus_b_d @ s @ A @ B )
= ( zero_b_d @ s ) )
= ( A = B ) ) ) ) ).
% ds.r_right_minus_eq
thf(fact_720_pds_Ominus__closed,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( a_minu898264511480707987t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.minus_closed
thf(fact_721_ds_Ofinsum__empty,axiom,
! [F: list_a > b] :
( ( finsum_b_d_list_a @ s @ F @ bot_bot_set_list_a )
= ( zero_b_d @ s ) ) ).
% ds.finsum_empty
thf(fact_722_ds_Ofinsum__empty,axiom,
! [F: list_b > b] :
( ( finsum_b_d_list_b @ s @ F @ bot_bot_set_list_b )
= ( zero_b_d @ s ) ) ).
% ds.finsum_empty
thf(fact_723_ds_Ofinsum__empty,axiom,
! [F: a > b] :
( ( finsum_b_d_a @ s @ F @ bot_bot_set_a )
= ( zero_b_d @ s ) ) ).
% ds.finsum_empty
thf(fact_724_ds_Ofinsum__empty,axiom,
! [F: b > b] :
( ( finsum_b_d_b @ s @ F @ bot_bot_set_b )
= ( zero_b_d @ s ) ) ).
% ds.finsum_empty
thf(fact_725_pds_Or__right__minus__eq,axiom,
! [A: list_b,B: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( a_minu898264511480707987t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( A = B ) ) ) ) ).
% pds.r_right_minus_eq
thf(fact_726_domain_Ozero__pdivides__zero,axiom,
! [R3: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( polyno4453881341673752516t_unit @ R3 @ nil_list_list_a @ nil_list_list_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_727_domain_Ozero__pdivides__zero,axiom,
! [R3: partia8877618634411419171xt_a_c] :
( ( domain_a_c @ R3 )
=> ( polyno5814909790663948099es_a_c @ R3 @ nil_a @ nil_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_728_domain_Ozero__pdivides__zero,axiom,
! [R3: partia4026993951477142903t_unit] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( polyno4931040496010026249t_unit @ R3 @ nil_list_b @ nil_list_b ) ) ).
% domain.zero_pdivides_zero
thf(fact_729_domain_Ozero__pdivides__zero,axiom,
! [R3: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( polyno8016796738000020810t_unit @ R3 @ nil_list_a @ nil_list_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_730_domain_Ozero__pdivides__zero,axiom,
! [R3: partia1897943568983147621xt_b_d] :
( ( domain_b_d @ R3 )
=> ( polyno3027454208691272067es_b_d @ R3 @ nil_b @ nil_b ) ) ).
% domain.zero_pdivides_zero
thf(fact_731_domain_Ozero__pdivides,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( polyno4453881341673752516t_unit @ R3 @ nil_list_list_a @ P )
= ( P = nil_list_list_a ) ) ) ).
% domain.zero_pdivides
thf(fact_732_domain_Ozero__pdivides,axiom,
! [R3: partia8877618634411419171xt_a_c,P: list_a] :
( ( domain_a_c @ R3 )
=> ( ( polyno5814909790663948099es_a_c @ R3 @ nil_a @ P )
= ( P = nil_a ) ) ) ).
% domain.zero_pdivides
thf(fact_733_domain_Ozero__pdivides,axiom,
! [R3: partia4026993951477142903t_unit,P: list_list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( polyno4931040496010026249t_unit @ R3 @ nil_list_b @ P )
= ( P = nil_list_b ) ) ) ).
% domain.zero_pdivides
thf(fact_734_domain_Ozero__pdivides,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( polyno8016796738000020810t_unit @ R3 @ nil_list_a @ P )
= ( P = nil_list_a ) ) ) ).
% domain.zero_pdivides
thf(fact_735_domain_Ozero__pdivides,axiom,
! [R3: partia1897943568983147621xt_b_d,P: list_b] :
( ( domain_b_d @ R3 )
=> ( ( polyno3027454208691272067es_b_d @ R3 @ nil_b @ P )
= ( P = nil_b ) ) ) ).
% domain.zero_pdivides
thf(fact_736_abelian__monoid_Ofinsum__empty,axiom,
! [G2: partia1897943568983147621xt_b_d,F: a > b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( finsum_b_d_a @ G2 @ F @ bot_bot_set_a )
= ( zero_b_d @ G2 ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_737_abelian__monoid_Ofinsum__empty,axiom,
! [G2: partia8877618634411419171xt_a_c,F: a > a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( finsum_a_c_a @ G2 @ F @ bot_bot_set_a )
= ( zero_a_c @ G2 ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_738_abelian__monoid_Ofinsum__empty,axiom,
! [G2: partia1897943568983147621xt_b_d,F: b > b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( finsum_b_d_b @ G2 @ F @ bot_bot_set_b )
= ( zero_b_d @ G2 ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_739_abelian__monoid_Ofinsum__empty,axiom,
! [G2: partia8877618634411419171xt_a_c,F: b > a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( finsum_a_c_b @ G2 @ F @ bot_bot_set_b )
= ( zero_a_c @ G2 ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_740_abelian__monoid_Ofinsum__empty,axiom,
! [G2: partia1897943568983147621xt_b_d,F: list_a > b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( finsum_b_d_list_a @ G2 @ F @ bot_bot_set_list_a )
= ( zero_b_d @ G2 ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_741_abelian__monoid_Ofinsum__empty,axiom,
! [G2: partia8877618634411419171xt_a_c,F: list_a > a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( finsum_a_c_list_a @ G2 @ F @ bot_bot_set_list_a )
= ( zero_a_c @ G2 ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_742_abelian__monoid_Ofinsum__empty,axiom,
! [G2: partia1897943568983147621xt_b_d,F: list_b > b] :
( ( abelian_monoid_b_d @ G2 )
=> ( ( finsum_b_d_list_b @ G2 @ F @ bot_bot_set_list_b )
= ( zero_b_d @ G2 ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_743_abelian__monoid_Ofinsum__empty,axiom,
! [G2: partia8877618634411419171xt_a_c,F: list_b > a] :
( ( abelian_monoid_a_c @ G2 )
=> ( ( finsum_a_c_list_b @ G2 @ F @ bot_bot_set_list_b )
= ( zero_a_c @ G2 ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_744_abelian__monoid_Ofinsum__empty,axiom,
! [G2: partia4026993951477142903t_unit,F: a > list_b] :
( ( abelia6363847436574302712t_unit @ G2 )
=> ( ( finsum6239823004737640281unit_a @ G2 @ F @ bot_bot_set_a )
= ( zero_l1056902381442676492t_unit @ G2 ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_745_abelian__monoid_Ofinsum__empty,axiom,
! [G2: partia2670972154091845814t_unit,F: a > list_a] :
( ( abelia226231641709521465t_unit @ G2 )
=> ( ( finsum7322697649718157656unit_a @ G2 @ F @ bot_bot_set_a )
= ( zero_l4142658623432671053t_unit @ G2 ) ) ) ).
% abelian_monoid.finsum_empty
thf(fact_746_domain_Opdivides__imp__is__root,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( P != nil_list_list_a )
=> ( ( member_list_list_a @ X @ ( partia2464479390973590831t_unit @ R3 ) )
=> ( ( polyno4453881341673752516t_unit @ R3 @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ X ) @ nil_list_list_a ) ) @ P )
=> ( polyno5142720416380192742t_unit @ R3 @ P @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_747_domain_Opdivides__imp__is__root,axiom,
! [R3: partia8877618634411419171xt_a_c,P: list_a,X: a] :
( ( domain_a_c @ R3 )
=> ( ( P != nil_a )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ R3 ) )
=> ( ( polyno5814909790663948099es_a_c @ R3 @ ( cons_a @ ( one_a_ring_ext_a_c @ R3 ) @ ( cons_a @ ( a_inv_a_c @ R3 @ X ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823461ot_a_c @ R3 @ P @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_748_domain_Opdivides__imp__is__root,axiom,
! [R3: partia1897943568983147621xt_b_d,P: list_b,X: b] :
( ( domain_b_d @ R3 )
=> ( ( P != nil_b )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ R3 ) )
=> ( ( polyno3027454208691272067es_b_d @ R3 @ ( cons_b @ ( one_b_ring_ext_b_d @ R3 ) @ ( cons_b @ ( a_inv_b_d @ R3 @ X ) @ nil_b ) ) @ P )
=> ( polyno1345617632095147429ot_b_d @ R3 @ P @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_749_domain_Opdivides__imp__is__root,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( P != nil_list_a )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ R3 ) )
=> ( ( polyno8016796738000020810t_unit @ R3 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ X ) @ nil_list_a ) ) @ P )
=> ( polyno6951661231331188332t_unit @ R3 @ P @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_750_domain_Opdivides__imp__is__root,axiom,
! [R3: partia4026993951477142903t_unit,P: list_list_b,X: list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( P != nil_list_b )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ R3 ) )
=> ( ( polyno4931040496010026249t_unit @ R3 @ ( cons_list_b @ ( one_li3054569011217056637t_unit @ R3 ) @ ( cons_list_b @ ( a_inv_5858964851304622612t_unit @ R3 @ X ) @ nil_list_b ) ) @ P )
=> ( polyno3865904989341193771t_unit @ R3 @ P @ X ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_751_domain_Ois__root__imp__pdivides,axiom,
! [R3: partia2956882679547061052t_unit,P: list_list_list_a,X: list_list_a] :
( ( domain7810152921033798211t_unit @ R3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) ) )
=> ( ( polyno5142720416380192742t_unit @ R3 @ P @ X )
=> ( polyno4453881341673752516t_unit @ R3 @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R3 @ X ) @ nil_list_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_752_domain_Ois__root__imp__pdivides,axiom,
! [R3: partia8877618634411419171xt_a_c,P: list_a,X: a] :
( ( domain_a_c @ R3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ R3 @ ( partia778085601923319190xt_a_c @ R3 ) ) ) )
=> ( ( polyno4133073214067823461ot_a_c @ R3 @ P @ X )
=> ( polyno5814909790663948099es_a_c @ R3 @ ( cons_a @ ( one_a_ring_ext_a_c @ R3 ) @ ( cons_a @ ( a_inv_a_c @ R3 @ X ) @ nil_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_753_domain_Ois__root__imp__pdivides,axiom,
! [R3: partia1897943568983147621xt_b_d,P: list_b,X: b] :
( ( domain_b_d @ R3 )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ R3 @ ( partia8782771468121683032xt_b_d @ R3 ) ) ) )
=> ( ( polyno1345617632095147429ot_b_d @ R3 @ P @ X )
=> ( polyno3027454208691272067es_b_d @ R3 @ ( cons_b @ ( one_b_ring_ext_b_d @ R3 ) @ ( cons_b @ ( a_inv_b_d @ R3 @ X ) @ nil_b ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_754_domain_Ois__root__imp__pdivides,axiom,
! [R3: partia2670972154091845814t_unit,P: list_list_a,X: list_a] :
( ( domain6553523120543210313t_unit @ R3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R3 @ P @ X )
=> ( polyno8016796738000020810t_unit @ R3 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R3 @ X ) @ nil_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_755_domain_Ois__root__imp__pdivides,axiom,
! [R3: partia4026993951477142903t_unit,P: list_list_b,X: list_b] :
( ( domain3467766878553215752t_unit @ R3 )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ R3 @ ( partia1381092143316337258t_unit @ R3 ) ) ) )
=> ( ( polyno3865904989341193771t_unit @ R3 @ P @ X )
=> ( polyno4931040496010026249t_unit @ R3 @ ( cons_list_b @ ( one_li3054569011217056637t_unit @ R3 ) @ ( cons_list_b @ ( a_inv_5858964851304622612t_unit @ R3 @ X ) @ nil_list_b ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_756_pds_Ois__root__imp__pdivides,axiom,
! [P: list_list_b,X: list_b] :
( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ X )
=> ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( cons_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( cons_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ nil_list_b ) ) @ P ) ) ) ).
% pds.is_root_imp_pdivides
thf(fact_757_pdr_Ois__root__imp__pdivides,axiom,
! [P: list_list_a,X: list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ X )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ nil_list_a ) ) @ P ) ) ) ).
% pdr.is_root_imp_pdivides
thf(fact_758_pds_Oexists__long__division,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( Q != nil_list_b )
=> ~ ! [B2: list_list_b] :
( ( member_list_list_b @ B2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ! [R: list_list_b] :
( ( member_list_list_b @ R @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ~ ( polyno3861286681177809007t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q @ ( produc8181704880241203287list_b @ B2 @ R ) ) ) ) ) ) ) ) ).
% pds.exists_long_division
thf(fact_759_pdr_Oexists__long__division,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( Q != nil_list_a )
=> ~ ! [B2: list_list_a] :
( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ! [R: list_list_a] :
( ( member_list_list_a @ R @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ~ ( polyno6947042923167803568t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q @ ( produc8696003437204565271list_a @ B2 @ R ) ) ) ) ) ) ) ) ).
% pdr.exists_long_division
thf(fact_760_pds_Olong__division__a__inv_I2_J,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_2010862973944421262t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P ) @ Q )
= ( a_inv_2010862973944421262t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) ) ) ) ) ) ).
% pds.long_division_a_inv(2)
thf(fact_761_pds_Opoly__add_Ocases,axiom,
! [X: produc4603089062067288551list_b] :
~ ! [P1: list_list_b,P22: list_list_b] :
( X
!= ( produc8181704880241203287list_b @ P1 @ P22 ) ) ).
% pds.poly_add.cases
thf(fact_762_pdr_Opoly__add_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
~ ! [P1: list_list_a,P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ P1 @ P22 ) ) ).
% pdr.poly_add.cases
thf(fact_763_dr_Ocarrier__not__empty,axiom,
( ( partia778085601923319190xt_a_c @ r )
!= bot_bot_set_a ) ).
% dr.carrier_not_empty
thf(fact_764_ds_Ocarrier__not__empty,axiom,
( ( partia8782771468121683032xt_b_d @ s )
!= bot_bot_set_b ) ).
% ds.carrier_not_empty
thf(fact_765_dr_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_list_b,F: list_b > a] :
( ! [X2: list_b] :
( ( member_list_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_a_c @ r ) ) )
=> ( ( finsum_a_c_list_b @ r @ F @ A2 )
= ( zero_a_c @ r ) ) ) ).
% dr.add.finprod_one_eqI
thf(fact_766_dr_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_list_a,F: list_a > a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_a_c @ r ) ) )
=> ( ( finsum_a_c_list_a @ r @ F @ A2 )
= ( zero_a_c @ r ) ) ) ).
% dr.add.finprod_one_eqI
thf(fact_767_dr_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_a_b,F: ( a > b ) > a] :
( ! [X2: a > b] :
( ( member_a_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_a_c @ r ) ) )
=> ( ( finsum_a_c_a_b @ r @ F @ A2 )
= ( zero_a_c @ r ) ) ) ).
% dr.add.finprod_one_eqI
thf(fact_768_dr_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_b,F: b > a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_a_c @ r ) ) )
=> ( ( finsum_a_c_b @ r @ F @ A2 )
= ( zero_a_c @ r ) ) ) ).
% dr.add.finprod_one_eqI
thf(fact_769_dr_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_a,F: a > a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_a_c @ r ) ) )
=> ( ( finsum_a_c_a @ r @ F @ A2 )
= ( zero_a_c @ r ) ) ) ).
% dr.add.finprod_one_eqI
thf(fact_770_ds_Ozero__pdivides,axiom,
! [P: list_b] :
( ( polyno3027454208691272067es_b_d @ s @ nil_b @ P )
= ( P = nil_b ) ) ).
% ds.zero_pdivides
thf(fact_771_ds_Ozero__pdivides__zero,axiom,
polyno3027454208691272067es_b_d @ s @ nil_b @ nil_b ).
% ds.zero_pdivides_zero
thf(fact_772_dr_Osubring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_c @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% dr.subring_props(4)
thf(fact_773_ds_Osubring__props_I4_J,axiom,
! [K: set_b] :
( ( subfield_b_d @ K @ s )
=> ( K != bot_bot_set_b ) ) ).
% ds.subring_props(4)
thf(fact_774_pdr_Ocombine_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [K2: list_a,Ks: list_list_a,U2: list_a,Us: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ K2 @ Ks ) @ ( cons_list_a @ U2 @ Us ) ) )
=> ( ! [Us: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Us ) )
=> ~ ! [Ks: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).
% pdr.combine.cases
thf(fact_775_pdr_Opoly__mult_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
=> ~ ! [V2: list_a,Va: list_list_a,P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ V2 @ Va ) @ P22 ) ) ) ).
% pdr.poly_mult.cases
thf(fact_776_pds_Ocombine_Ocases,axiom,
! [X: produc4603089062067288551list_b] :
( ! [K2: list_b,Ks: list_list_b,U2: list_b,Us: list_list_b] :
( X
!= ( produc8181704880241203287list_b @ ( cons_list_b @ K2 @ Ks ) @ ( cons_list_b @ U2 @ Us ) ) )
=> ( ! [Us: list_list_b] :
( X
!= ( produc8181704880241203287list_b @ nil_list_b @ Us ) )
=> ~ ! [Ks: list_list_b] :
( X
!= ( produc8181704880241203287list_b @ Ks @ nil_list_b ) ) ) ) ).
% pds.combine.cases
thf(fact_777_pds_Opoly__mult_Ocases,axiom,
! [X: produc4603089062067288551list_b] :
( ! [P22: list_list_b] :
( X
!= ( produc8181704880241203287list_b @ nil_list_b @ P22 ) )
=> ~ ! [V2: list_b,Va: list_list_b,P22: list_list_b] :
( X
!= ( produc8181704880241203287list_b @ ( cons_list_b @ V2 @ Va ) @ P22 ) ) ) ).
% pds.poly_mult.cases
thf(fact_778_pdr_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_list_b,F: list_b > list_a] :
( ! [X2: list_b] :
( ( member_list_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) )
=> ( ( finsum8721804984859891807list_b @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F @ A2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.add.finprod_one_eqI
thf(fact_779_pdr_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_list_a,F: list_a > list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) )
=> ( ( finsum8721804980556663006list_a @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F @ A2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.add.finprod_one_eqI
thf(fact_780_pdr_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_a_b,F: ( a > b ) > list_a] :
( ! [X2: a > b] :
( ( member_a_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) )
=> ( ( finsum4909627266889545996it_a_b @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F @ A2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.add.finprod_one_eqI
thf(fact_781_pdr_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_b,F: b > list_a] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) )
=> ( ( finsum7322697649718157657unit_b @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F @ A2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.add.finprod_one_eqI
thf(fact_782_pdr_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_a,F: a > list_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) )
=> ( ( finsum7322697649718157656unit_a @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F @ A2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.add.finprod_one_eqI
thf(fact_783_pds_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_list_b,F: list_b > list_b] :
( ! [X2: list_b] :
( ( member_list_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) )
=> ( ( finsum6087063263485057504list_b @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F @ A2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.add.finprod_one_eqI
thf(fact_784_pds_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_list_a,F: list_a > list_b] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) )
=> ( ( finsum6087063259181828703list_a @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F @ A2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.add.finprod_one_eqI
thf(fact_785_pds_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_a_b,F: ( a > b ) > list_b] :
( ! [X2: a > b] :
( ( member_a_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) )
=> ( ( finsum4265744524993377995it_a_b @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F @ A2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.add.finprod_one_eqI
thf(fact_786_pds_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_b,F: b > list_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) )
=> ( ( finsum6239823004737640282unit_b @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F @ A2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.add.finprod_one_eqI
thf(fact_787_pds_Oadd_Ofinprod__one__eqI,axiom,
! [A2: set_a,F: a > list_b] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( F @ X2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) )
=> ( ( finsum6239823004737640281unit_a @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F @ A2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.add.finprod_one_eqI
thf(fact_788_pdr_Ozero__pdivides__zero,axiom,
polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ nil_list_a @ nil_list_a ).
% pdr.zero_pdivides_zero
thf(fact_789_pdr_Ozero__pdivides,axiom,
! [P: list_list_a] :
( ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ nil_list_a @ P )
= ( P = nil_list_a ) ) ).
% pdr.zero_pdivides
thf(fact_790_pds_Ozero__pdivides,axiom,
! [P: list_list_b] :
( ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ nil_list_b @ P )
= ( P = nil_list_b ) ) ).
% pds.zero_pdivides
thf(fact_791_pds_Ozero__pdivides__zero,axiom,
polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ nil_list_b @ nil_list_b ).
% pds.zero_pdivides_zero
thf(fact_792_dr_Oadd_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H )
=> ( member_a @ ( a_inv_a_c @ r @ X2 ) @ H ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ H )
=> ! [Xa: a] :
( ( member_a @ Xa @ H )
=> ( member_a @ ( add_a_c @ r @ X2 @ Xa ) @ H ) ) )
=> ( member_a @ ( zero_a_c @ r ) @ H ) ) ) ) ) ).
% dr.add.one_in_subset
thf(fact_793_ds_Oadd_Oone__in__subset,axiom,
! [H: set_b] :
( ( ord_less_eq_set_b @ H @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( H != bot_bot_set_b )
=> ( ! [X2: b] :
( ( member_b @ X2 @ H )
=> ( member_b @ ( a_inv_b_d @ s @ X2 ) @ H ) )
=> ( ! [X2: b] :
( ( member_b @ X2 @ H )
=> ! [Xa: b] :
( ( member_b @ Xa @ H )
=> ( member_b @ ( add_b_d @ s @ X2 @ Xa ) @ H ) ) )
=> ( member_b @ ( zero_b_d @ s ) @ H ) ) ) ) ) ).
% ds.add.one_in_subset
thf(fact_794_pds_Olong__division__closed_I2_J,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( member_list_list_b @ ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ) ) ) ).
% pds.long_division_closed(2)
thf(fact_795_ds_Opdivides__imp__is__root,axiom,
! [P: list_b,X: b] :
( ( P != nil_b )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( polyno3027454208691272067es_b_d @ s @ ( cons_b @ ( one_b_ring_ext_b_d @ s ) @ ( cons_b @ ( a_inv_b_d @ s @ X ) @ nil_b ) ) @ P )
=> ( polyno1345617632095147429ot_b_d @ s @ P @ X ) ) ) ) ).
% ds.pdivides_imp_is_root
thf(fact_796_pds_Olong__division__zero_I2_J,axiom,
! [K: set_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ nil_list_b @ Q )
= nil_list_b ) ) ) ).
% pds.long_division_zero(2)
thf(fact_797_pds_Osame__pmod__iff__pdivides,axiom,
! [K: set_list_b,A: list_list_b,B: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ A @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ B @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ Q )
= ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ Q ) )
= ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Q @ ( a_minu6442441833124848013t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ A @ B ) ) ) ) ) ) ) ).
% pds.same_pmod_iff_pdivides
thf(fact_798_pds_Olong__division__add__iff,axiom,
! [K: set_list_b,A: list_list_b,B: list_list_b,C: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ A @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ B @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ C @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ Q )
= ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ Q ) )
= ( ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( add_li4375960627168867399t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ A @ C ) @ Q )
= ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( add_li4375960627168867399t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% pds.long_division_add_iff
thf(fact_799_pds_Olong__division__add_I2_J,axiom,
! [K: set_list_b,A: list_list_b,B: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ A @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ B @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( add_li4375960627168867399t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ A @ B ) @ Q )
= ( add_li4375960627168867399t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ Q ) @ ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ Q ) ) ) ) ) ) ) ).
% pds.long_division_add(2)
thf(fact_800_ds_Ois__root__imp__pdivides,axiom,
! [P: list_b,X: b] :
( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( polyno1345617632095147429ot_b_d @ s @ P @ X )
=> ( polyno3027454208691272067es_b_d @ s @ ( cons_b @ ( one_b_ring_ext_b_d @ s ) @ ( cons_b @ ( a_inv_b_d @ s @ X ) @ nil_b ) ) @ P ) ) ) ).
% ds.is_root_imp_pdivides
thf(fact_801_pds_Opmod__zero__iff__pdivides,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q )
= nil_list_b )
= ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Q @ P ) ) ) ) ) ).
% pds.pmod_zero_iff_pdivides
thf(fact_802_pdr_Opdivides__imp__is__root,axiom,
! [P: list_list_a,X: list_a] :
( ( P != nil_list_a )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ nil_list_a ) ) @ P )
=> ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ X ) ) ) ) ).
% pdr.pdivides_imp_is_root
thf(fact_803_pds_Opdivides__imp__is__root,axiom,
! [P: list_list_b,X: list_b] :
( ( P != nil_list_b )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( cons_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( cons_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ nil_list_b ) ) @ P )
=> ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ X ) ) ) ) ).
% pds.pdivides_imp_is_root
thf(fact_804_pds_Olong__divisionE,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( Q != nil_list_b )
=> ( polyno3861286681177809007t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q @ ( produc8181704880241203287list_b @ ( polyno2808025880298714784t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) @ ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) ) ) ) ) ) ) ).
% pds.long_divisionE
thf(fact_805_pds_Olong__divisionI,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b,B: list_list_b,R2: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( Q != nil_list_b )
=> ( ( polyno3861286681177809007t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q @ ( produc8181704880241203287list_b @ B @ R2 ) )
=> ( ( produc8181704880241203287list_b @ B @ R2 )
= ( produc8181704880241203287list_b @ ( polyno2808025880298714784t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) @ ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) ) ) ) ) ) ) ) ).
% pds.long_divisionI
thf(fact_806_dr_Ofinsum__empty,axiom,
! [F: list_a > a] :
( ( finsum_a_c_list_a @ r @ F @ bot_bot_set_list_a )
= ( zero_a_c @ r ) ) ).
% dr.finsum_empty
thf(fact_807_dr_Ofinsum__empty,axiom,
! [F: list_b > a] :
( ( finsum_a_c_list_b @ r @ F @ bot_bot_set_list_b )
= ( zero_a_c @ r ) ) ).
% dr.finsum_empty
thf(fact_808_dr_Ofinsum__empty,axiom,
! [F: a > a] :
( ( finsum_a_c_a @ r @ F @ bot_bot_set_a )
= ( zero_a_c @ r ) ) ).
% dr.finsum_empty
thf(fact_809_dr_Ofinsum__empty,axiom,
! [F: b > a] :
( ( finsum_a_c_b @ r @ F @ bot_bot_set_b )
= ( zero_a_c @ r ) ) ).
% dr.finsum_empty
thf(fact_810_pdr_Ofinsum__empty,axiom,
! [F: list_a > list_a] :
( ( finsum8721804980556663006list_a @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F @ bot_bot_set_list_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.finsum_empty
thf(fact_811_pdr_Ofinsum__empty,axiom,
! [F: list_b > list_a] :
( ( finsum8721804984859891807list_b @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F @ bot_bot_set_list_b )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.finsum_empty
thf(fact_812_pdr_Ofinsum__empty,axiom,
! [F: a > list_a] :
( ( finsum7322697649718157656unit_a @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F @ bot_bot_set_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.finsum_empty
thf(fact_813_pdr_Ofinsum__empty,axiom,
! [F: b > list_a] :
( ( finsum7322697649718157657unit_b @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F @ bot_bot_set_b )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.finsum_empty
thf(fact_814_pds_Ofinsum__empty,axiom,
! [F: list_a > list_b] :
( ( finsum6087063259181828703list_a @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F @ bot_bot_set_list_a )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.finsum_empty
thf(fact_815_pds_Ofinsum__empty,axiom,
! [F: list_b > list_b] :
( ( finsum6087063263485057504list_b @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F @ bot_bot_set_list_b )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.finsum_empty
thf(fact_816_pds_Ofinsum__empty,axiom,
! [F: a > list_b] :
( ( finsum6239823004737640281unit_a @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F @ bot_bot_set_a )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.finsum_empty
thf(fact_817_pds_Ofinsum__empty,axiom,
! [F: b > list_b] :
( ( finsum6239823004737640282unit_b @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F @ bot_bot_set_b )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.finsum_empty
thf(fact_818_ring_Opmod_Ocong,axiom,
polynomial_pmod_a_c = polynomial_pmod_a_c ).
% ring.pmod.cong
thf(fact_819_ring_Opmod_Ocong,axiom,
polynomial_pmod_b_d = polynomial_pmod_b_d ).
% ring.pmod.cong
thf(fact_820_pdr_Olong__divisionI,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a,B: list_list_a,R2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q @ ( produc8696003437204565271list_a @ B @ R2 ) )
=> ( ( produc8696003437204565271list_a @ B @ R2 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) ) ) ) ) ) ) ) ).
% pdr.long_divisionI
thf(fact_821_pdr_Olong__divisionE,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( Q != nil_list_a )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) ) ) ) ) ) ) ).
% pdr.long_divisionE
thf(fact_822_pds_Opdiv__pmod,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( P
= ( add_li4375960627168867399t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ Q @ ( polyno2808025880298714784t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) ) @ ( polyno7865366480153646481t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) ) ) ) ) ) ).
% pds.pdiv_pmod
thf(fact_823_pdr_Opmod__zero__iff__pdivides,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Q @ P ) ) ) ) ) ).
% pdr.pmod_zero_iff_pdivides
thf(fact_824_pds_OpprimeE_I3_J,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b,R2: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ring_r415245522172713630t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ R2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ Q @ R2 ) )
=> ( ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q )
| ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ R2 ) ) ) ) ) ) ) ) ).
% pds.pprimeE(3)
thf(fact_825_dr_Ocombine_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [K2: a,Ks: list_a,U2: a,Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ K2 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% dr.combine.cases
thf(fact_826_dr_Opoly__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 ) ) ) ).
% dr.poly_mult.cases
thf(fact_827_ds_Opoly__mult_Ocases,axiom,
! [X: produc3963297410138542439list_b] :
( ! [P22: list_b] :
( X
!= ( produc1564554178308465111list_b @ nil_b @ P22 ) )
=> ~ ! [V2: b,Va: list_b,P22: list_b] :
( X
!= ( produc1564554178308465111list_b @ ( cons_b @ V2 @ Va ) @ P22 ) ) ) ).
% ds.poly_mult.cases
thf(fact_828_ds_Ocombine_Ocases,axiom,
! [X: produc3963297410138542439list_b] :
( ! [K2: b,Ks: list_b,U2: b,Us: list_b] :
( X
!= ( produc1564554178308465111list_b @ ( cons_b @ K2 @ Ks ) @ ( cons_b @ U2 @ Us ) ) )
=> ( ! [Us: list_b] :
( X
!= ( produc1564554178308465111list_b @ nil_b @ Us ) )
=> ~ ! [Ks: list_b] :
( X
!= ( produc1564554178308465111list_b @ Ks @ nil_b ) ) ) ) ).
% ds.combine.cases
thf(fact_829_dr_Olong__division__closed_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_c @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ) ) ) ).
% dr.long_division_closed(2)
thf(fact_830_ds_Olong__division__closed_I2_J,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( member_list_b @ ( polynomial_pmod_b_d @ s @ P @ Q ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ) ) ) ).
% ds.long_division_closed(2)
thf(fact_831_dr_Olong__division__add_I2_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( polynomial_pmod_a_c @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ K ) @ ( polynomial_pmod_a_c @ r @ A @ Q ) @ ( polynomial_pmod_a_c @ r @ B @ Q ) ) ) ) ) ) ) ).
% dr.long_division_add(2)
thf(fact_832_dr_Olong__division__add__iff,axiom,
! [K: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_c @ r @ A @ Q )
= ( polynomial_pmod_a_c @ r @ B @ Q ) )
= ( ( polynomial_pmod_a_c @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ K ) @ A @ C ) @ Q )
= ( polynomial_pmod_a_c @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% dr.long_division_add_iff
thf(fact_833_ds_Olong__division__add_I2_J,axiom,
! [K: set_b,A: list_b,B: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( polynomial_pmod_b_d @ s @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ K ) @ A @ B ) @ Q )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ K ) @ ( polynomial_pmod_b_d @ s @ A @ Q ) @ ( polynomial_pmod_b_d @ s @ B @ Q ) ) ) ) ) ) ) ).
% ds.long_division_add(2)
thf(fact_834_ds_Olong__division__add__iff,axiom,
! [K: set_b,A: list_b,B: list_b,C: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ( polynomial_pmod_b_d @ s @ A @ Q )
= ( polynomial_pmod_b_d @ s @ B @ Q ) )
= ( ( polynomial_pmod_b_d @ s @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ K ) @ A @ C ) @ Q )
= ( polynomial_pmod_b_d @ s @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% ds.long_division_add_iff
thf(fact_835_dr_Olong__division__zero_I2_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( polynomial_pmod_a_c @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% dr.long_division_zero(2)
thf(fact_836_ds_Olong__division__zero_I2_J,axiom,
! [K: set_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( polynomial_pmod_b_d @ s @ nil_b @ Q )
= nil_b ) ) ) ).
% ds.long_division_zero(2)
thf(fact_837_dr_Olong__division__a__inv_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( polynomial_pmod_a_c @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ K ) @ ( polynomial_pmod_a_c @ r @ P @ Q ) ) ) ) ) ) ).
% dr.long_division_a_inv(2)
thf(fact_838_ds_Olong__division__a__inv_I2_J,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( polynomial_pmod_b_d @ s @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ K ) @ P ) @ Q )
= ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ K ) @ ( polynomial_pmod_b_d @ s @ P @ Q ) ) ) ) ) ) ).
% ds.long_division_a_inv(2)
thf(fact_839_dr_Opmod__zero__iff__pdivides,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_c @ r @ P @ Q )
= nil_a )
= ( polyno5814909790663948099es_a_c @ r @ Q @ P ) ) ) ) ) ).
% dr.pmod_zero_iff_pdivides
thf(fact_840_ds_Opmod__zero__iff__pdivides,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ( polynomial_pmod_b_d @ s @ P @ Q )
= nil_b )
= ( polyno3027454208691272067es_b_d @ s @ Q @ P ) ) ) ) ) ).
% ds.pmod_zero_iff_pdivides
thf(fact_841_dr_Oexists__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B2: list_a] :
( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ! [R: list_a] :
( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ~ ( polyno2806191415236617129es_a_c @ r @ P @ Q @ ( produc6837034575241423639list_a @ B2 @ R ) ) ) ) ) ) ) ) ).
% dr.exists_long_division
thf(fact_842_ds_Oexists__long__division,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( Q != nil_b )
=> ~ ! [B2: list_b] :
( ( member_list_b @ B2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ! [R: list_b] :
( ( member_list_b @ R @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ~ ( polyno18735833263941097es_b_d @ s @ P @ Q @ ( produc1564554178308465111list_b @ B2 @ R ) ) ) ) ) ) ) ) ).
% ds.exists_long_division
thf(fact_843_dr_Osame__pmod__iff__pdivides,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_c @ r @ A @ Q )
= ( polynomial_pmod_a_c @ r @ B @ Q ) )
= ( polyno5814909790663948099es_a_c @ r @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_c @ r @ K ) @ A @ B ) ) ) ) ) ) ) ).
% dr.same_pmod_iff_pdivides
thf(fact_844_ds_Osame__pmod__iff__pdivides,axiom,
! [K: set_b,A: list_b,B: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ( polynomial_pmod_b_d @ s @ A @ Q )
= ( polynomial_pmod_b_d @ s @ B @ Q ) )
= ( polyno3027454208691272067es_b_d @ s @ Q @ ( a_minu898264511480707987t_unit @ ( univ_poly_b_d @ s @ K ) @ A @ B ) ) ) ) ) ) ) ).
% ds.same_pmod_iff_pdivides
thf(fact_845_pdr_Olong__division__closed_I2_J,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ) ) ) ).
% pdr.long_division_closed(2)
thf(fact_846_dr_Olong__divisionE,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617129es_a_c @ r @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_c @ r @ P @ Q ) @ ( polynomial_pmod_a_c @ r @ P @ Q ) ) ) ) ) ) ) ).
% dr.long_divisionE
thf(fact_847_dr_Olong__divisionI,axiom,
! [K: set_a,P: list_a,Q: list_a,B: list_a,R2: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617129es_a_c @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_c @ r @ P @ Q ) @ ( polynomial_pmod_a_c @ r @ P @ Q ) ) ) ) ) ) ) ) ).
% dr.long_divisionI
thf(fact_848_ds_Olong__divisionI,axiom,
! [K: set_b,P: list_b,Q: list_b,B: list_b,R2: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( Q != nil_b )
=> ( ( polyno18735833263941097es_b_d @ s @ P @ Q @ ( produc1564554178308465111list_b @ B @ R2 ) )
=> ( ( produc1564554178308465111list_b @ B @ R2 )
= ( produc1564554178308465111list_b @ ( polynomial_pdiv_b_d @ s @ P @ Q ) @ ( polynomial_pmod_b_d @ s @ P @ Q ) ) ) ) ) ) ) ) ).
% ds.long_divisionI
thf(fact_849_ds_Olong__divisionE,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( Q != nil_b )
=> ( polyno18735833263941097es_b_d @ s @ P @ Q @ ( produc1564554178308465111list_b @ ( polynomial_pdiv_b_d @ s @ P @ Q ) @ ( polynomial_pmod_b_d @ s @ P @ Q ) ) ) ) ) ) ) ).
% ds.long_divisionE
thf(fact_850_pdr_Olong__division__zero_I2_J,axiom,
! [K: set_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ nil_list_a @ Q )
= nil_list_a ) ) ) ).
% pdr.long_division_zero(2)
thf(fact_851_pdr_Osame__pmod__iff__pdivides,axiom,
! [K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ Q )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ Q ) )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Q @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ A @ B ) ) ) ) ) ) ) ).
% pdr.same_pmod_iff_pdivides
thf(fact_852_pds_Ois__root__poly__mult__imp__is__root,axiom,
! [P: list_list_b,Q: list_list_b,X: list_b] :
( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ P @ Q ) @ X )
=> ( ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ X )
| ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Q @ X ) ) ) ) ) ).
% pds.is_root_poly_mult_imp_is_root
thf(fact_853_pdr_Olong__division__add_I2_J,axiom,
! [K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ Q ) ) ) ) ) ) ) ).
% pdr.long_division_add(2)
thf(fact_854_pdr_Olong__division__add__iff,axiom,
! [K: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ Q )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ Q ) )
= ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ A @ C ) @ Q )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% pdr.long_division_add_iff
thf(fact_855_pdr_Olong__division__a__inv_I2_J,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) ) ) ) ) ) ).
% pdr.long_division_a_inv(2)
thf(fact_856_pdr_Opdiv__pmod,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ Q @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) ) ) ) ) ) ).
% pdr.pdiv_pmod
thf(fact_857_pds_OpprimeI,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( P != nil_list_b )
=> ( ~ ( member_list_list_b @ P @ ( units_6858163862972288294t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ! [Q2: list_list_b,R: list_list_b] :
( ( member_list_list_b @ Q2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ R @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ Q2 @ R ) )
=> ( ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q2 )
| ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ R ) ) ) ) )
=> ( ring_r415245522172713630t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P ) ) ) ) ) ) ).
% pds.pprimeI
thf(fact_858_pdr_OpprimeE_I3_J,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a,R2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ Q @ R2 ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q )
| ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ R2 ) ) ) ) ) ) ) ) ).
% pdr.pprimeE(3)
thf(fact_859_ds_Opoly__add_Ocases,axiom,
! [X: produc3963297410138542439list_b] :
~ ! [P1: list_b,P22: list_b] :
( X
!= ( produc1564554178308465111list_b @ P1 @ P22 ) ) ).
% ds.poly_add.cases
thf(fact_860_dr_Opoly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% dr.poly_add.cases
thf(fact_861_dr_Odivides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151074478948643xt_a_c @ r @ A @ B )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( factor8216151074478948643xt_a_c @ r @ A @ ( mult_a_ring_ext_a_c @ r @ B @ C ) ) ) ) ) ).
% dr.divides_prod_r
thf(fact_862_dr_Odivides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( factor8216151074478948643xt_a_c @ r @ A @ B )
=> ( factor8216151074478948643xt_a_c @ r @ A @ ( mult_a_ring_ext_a_c @ r @ C @ B ) ) ) ) ) ) ).
% dr.divides_prod_l
thf(fact_863_dr_Odivides__mult,axiom,
! [A: a,C: a,B: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( factor8216151074478948643xt_a_c @ r @ A @ B )
=> ( factor8216151074478948643xt_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ C @ A ) @ ( mult_a_ring_ext_a_c @ r @ C @ B ) ) ) ) ) ).
% dr.divides_mult
thf(fact_864_dr_Odivides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151074478948643xt_a_c @ r @ A @ B )
=> ( ( factor8216151074478948643xt_a_c @ r @ B @ C )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( factor8216151074478948643xt_a_c @ r @ A @ C ) ) ) ) ).
% dr.divides_trans
thf(fact_865_dr_Om__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ X @ ( mult_a_ring_ext_a_c @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_c @ r @ Y @ ( mult_a_ring_ext_a_c @ r @ X @ Z ) ) ) ) ) ) ).
% dr.m_lcomm
thf(fact_866_dr_Om__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ X @ Y )
= ( mult_a_ring_ext_a_c @ r @ Y @ X ) ) ) ) ).
% dr.m_comm
thf(fact_867_dr_Om__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_c @ r @ X @ ( mult_a_ring_ext_a_c @ r @ Y @ Z ) ) ) ) ) ) ).
% dr.m_assoc
thf(fact_868_ds_Om__assoc,axiom,
! [X: b,Y: b,Z: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ X @ Y ) @ Z )
= ( mult_b_ring_ext_b_d @ s @ X @ ( mult_b_ring_ext_b_d @ s @ Y @ Z ) ) ) ) ) ) ).
% ds.m_assoc
thf(fact_869_ds_Om__comm,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ X @ Y )
= ( mult_b_ring_ext_b_d @ s @ Y @ X ) ) ) ) ).
% ds.m_comm
thf(fact_870_ds_Om__lcomm,axiom,
! [X: b,Y: b,Z: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ X @ ( mult_b_ring_ext_b_d @ s @ Y @ Z ) )
= ( mult_b_ring_ext_b_d @ s @ Y @ ( mult_b_ring_ext_b_d @ s @ X @ Z ) ) ) ) ) ) ).
% ds.m_lcomm
thf(fact_871_dr_Ozero__divides,axiom,
! [A: a] :
( ( factor8216151074478948643xt_a_c @ r @ ( zero_a_c @ r ) @ A )
= ( A
= ( zero_a_c @ r ) ) ) ).
% dr.zero_divides
thf(fact_872_dr_Osubring__props_I6_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_c @ r @ H1 @ H22 ) @ K ) ) ) ) ).
% dr.subring_props(6)
thf(fact_873_ds_Osubring__props_I6_J,axiom,
! [K: set_b,H1: b,H22: b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_b @ H1 @ K )
=> ( ( member_b @ H22 @ K )
=> ( member_b @ ( mult_b_ring_ext_b_d @ s @ H1 @ H22 ) @ K ) ) ) ) ).
% ds.subring_props(6)
thf(fact_874_dr_Odivides__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( factor8216151074478948643xt_a_c @ r @ A @ ( zero_a_c @ r ) ) ) ).
% dr.divides_zero
thf(fact_875_dr_Om__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( mult_a_ring_ext_a_c @ r @ B @ A )
= ( mult_a_ring_ext_a_c @ r @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% dr.m_rcancel
thf(fact_876_dr_Om__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( mult_a_ring_ext_a_c @ r @ A @ B )
= ( mult_a_ring_ext_a_c @ r @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% dr.m_lcancel
thf(fact_877_dr_Ointegral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( mult_a_ring_ext_a_c @ r @ A @ B )
= ( zero_a_c @ r ) )
= ( ( A
= ( zero_a_c @ r ) )
| ( B
= ( zero_a_c @ r ) ) ) ) ) ) ).
% dr.integral_iff
thf(fact_878_dr_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_c @ r @ A @ B )
= ( zero_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( A
= ( zero_a_c @ r ) )
| ( B
= ( zero_a_c @ r ) ) ) ) ) ) ).
% dr.integral
thf(fact_879_ds_Ointegral,axiom,
! [A: b,B: b] :
( ( ( mult_b_ring_ext_b_d @ s @ A @ B )
= ( zero_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( A
= ( zero_b_d @ s ) )
| ( B
= ( zero_b_d @ s ) ) ) ) ) ) ).
% ds.integral
thf(fact_880_ds_Ointegral__iff,axiom,
! [A: b,B: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( mult_b_ring_ext_b_d @ s @ A @ B )
= ( zero_b_d @ s ) )
= ( ( A
= ( zero_b_d @ s ) )
| ( B
= ( zero_b_d @ s ) ) ) ) ) ) ).
% ds.integral_iff
thf(fact_881_ds_Om__lcancel,axiom,
! [A: b,B: b,C: b] :
( ( A
!= ( zero_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( mult_b_ring_ext_b_d @ s @ A @ B )
= ( mult_b_ring_ext_b_d @ s @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% ds.m_lcancel
thf(fact_882_ds_Om__rcancel,axiom,
! [A: b,B: b,C: b] :
( ( A
!= ( zero_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( mult_b_ring_ext_b_d @ s @ B @ A )
= ( mult_b_ring_ext_b_d @ s @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% ds.m_rcancel
thf(fact_883_dr_Or__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ Z @ ( add_a_c @ r @ X @ Y ) )
= ( add_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ Z @ X ) @ ( mult_a_ring_ext_a_c @ r @ Z @ Y ) ) ) ) ) ) ).
% dr.r_distr
thf(fact_884_dr_Ol__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ ( add_a_c @ r @ X @ Y ) @ Z )
= ( add_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ X @ Z ) @ ( mult_a_ring_ext_a_c @ r @ Y @ Z ) ) ) ) ) ) ).
% dr.l_distr
thf(fact_885_ds_Ol__distr,axiom,
! [X: b,Y: b,Z: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ ( add_b_d @ s @ X @ Y ) @ Z )
= ( add_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ X @ Z ) @ ( mult_b_ring_ext_b_d @ s @ Y @ Z ) ) ) ) ) ) ).
% ds.l_distr
thf(fact_886_ds_Or__distr,axiom,
! [X: b,Y: b,Z: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ Z @ ( add_b_d @ s @ X @ Y ) )
= ( add_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ Z @ X ) @ ( mult_b_ring_ext_b_d @ s @ Z @ Y ) ) ) ) ) ) ).
% ds.r_distr
thf(fact_887_ds_Oinv__unique,axiom,
! [Y: b,X: b,Y2: b] :
( ( ( mult_b_ring_ext_b_d @ s @ Y @ X )
= ( one_b_ring_ext_b_d @ s ) )
=> ( ( ( mult_b_ring_ext_b_d @ s @ X @ Y2 )
= ( one_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% ds.inv_unique
thf(fact_888_ds_Oone__unique,axiom,
! [U: b] :
( ( member_b @ U @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ! [X2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ U @ X2 )
= X2 ) )
=> ( U
= ( one_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.one_unique
thf(fact_889_dr_Oone__divides,axiom,
! [A: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( factor8216151074478948643xt_a_c @ r @ ( one_a_ring_ext_a_c @ r ) @ A ) ) ).
% dr.one_divides
thf(fact_890_dr_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ U @ X2 )
= X2 ) )
=> ( U
= ( one_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.one_unique
thf(fact_891_dr_Oinv__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_c @ r @ Y @ X )
= ( one_a_ring_ext_a_c @ r ) )
=> ( ( ( mult_a_ring_ext_a_c @ r @ X @ Y2 )
= ( one_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% dr.inv_unique
thf(fact_892_dr_Or__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ X @ ( a_inv_a_c @ r @ Y ) )
= ( a_inv_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ X @ Y ) ) ) ) ) ).
% dr.r_minus
thf(fact_893_dr_Ol__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ ( a_inv_a_c @ r @ X ) @ Y )
= ( a_inv_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ X @ Y ) ) ) ) ) ).
% dr.l_minus
thf(fact_894_ds_Ol__minus,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ ( a_inv_b_d @ s @ X ) @ Y )
= ( a_inv_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ X @ Y ) ) ) ) ) ).
% ds.l_minus
thf(fact_895_ds_Or__minus,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ X @ ( a_inv_b_d @ s @ Y ) )
= ( a_inv_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ X @ Y ) ) ) ) ) ).
% ds.r_minus
thf(fact_896_ds_Odivides__mult,axiom,
! [A: b,C: b,B: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( factor2325171414093416164xt_b_d @ s @ A @ B )
=> ( factor2325171414093416164xt_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ C @ A ) @ ( mult_b_ring_ext_b_d @ s @ C @ B ) ) ) ) ) ).
% ds.divides_mult
thf(fact_897_ds_Odivides__prod__l,axiom,
! [A: b,B: b,C: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( factor2325171414093416164xt_b_d @ s @ A @ B )
=> ( factor2325171414093416164xt_b_d @ s @ A @ ( mult_b_ring_ext_b_d @ s @ C @ B ) ) ) ) ) ) ).
% ds.divides_prod_l
thf(fact_898_ds_Odivides__prod__r,axiom,
! [A: b,B: b,C: b] :
( ( factor2325171414093416164xt_b_d @ s @ A @ B )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( factor2325171414093416164xt_b_d @ s @ A @ ( mult_b_ring_ext_b_d @ s @ B @ C ) ) ) ) ) ).
% ds.divides_prod_r
thf(fact_899_pdr_Odivides__prod__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ C ) ) ) ) ) ).
% pdr.divides_prod_r
thf(fact_900_pdr_Odivides__prod__l,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C @ B ) ) ) ) ) ) ).
% pdr.divides_prod_l
thf(fact_901_pdr_Odivides__mult,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C @ B ) ) ) ) ) ).
% pdr.divides_mult
thf(fact_902_pdr_Odivides__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ C )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ C ) ) ) ) ).
% pdr.divides_trans
thf(fact_903_pdr_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% pdr.m_lcomm
thf(fact_904_pdr_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ X ) ) ) ) ).
% pdr.m_comm
thf(fact_905_pdr_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% pdr.m_assoc
thf(fact_906_pds_Odivides__mult,axiom,
! [A: list_b,C: list_b,B: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C @ A ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C @ B ) ) ) ) ) ).
% pds.divides_mult
thf(fact_907_pds_Odivides__prod__l,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C @ B ) ) ) ) ) ) ).
% pds.divides_prod_l
thf(fact_908_pds_Odivides__prod__r,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ C ) ) ) ) ) ).
% pds.divides_prod_r
thf(fact_909_pds_Odivides__trans,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
=> ( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ C )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ C ) ) ) ) ).
% pds.divides_trans
thf(fact_910_pds_Om__assoc,axiom,
! [X: list_b,Y: list_b,Z: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) @ Z )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ Z ) ) ) ) ) ) ).
% pds.m_assoc
thf(fact_911_pds_Om__comm,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ X ) ) ) ) ).
% pds.m_comm
thf(fact_912_pds_Om__lcomm,axiom,
! [X: list_b,Y: list_b,Z: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ Z ) )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Z ) ) ) ) ) ) ).
% pds.m_lcomm
thf(fact_913_pdr_Ozero__divides,axiom,
! [A: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ A )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.zero_divides
thf(fact_914_pds_Ozero__divides,axiom,
! [A: list_b] :
( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ A )
= ( A
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.zero_divides
thf(fact_915_pdr_Osubring__props_I6_J,axiom,
! [K: set_list_a,H1: list_a,H22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H22 @ K )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H1 @ H22 ) @ K ) ) ) ) ).
% pdr.subring_props(6)
thf(fact_916_pds_Osubring__props_I6_J,axiom,
! [K: set_list_b,H1: list_b,H22: list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_b @ H1 @ K )
=> ( ( member_list_b @ H22 @ K )
=> ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H1 @ H22 ) @ K ) ) ) ) ).
% pds.subring_props(6)
thf(fact_917_ds_Osquare__eq__one,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( mult_b_ring_ext_b_d @ s @ X @ X )
= ( one_b_ring_ext_b_d @ s ) )
=> ( ( X
= ( one_b_ring_ext_b_d @ s ) )
| ( X
= ( a_inv_b_d @ s @ ( one_b_ring_ext_b_d @ s ) ) ) ) ) ) ).
% ds.square_eq_one
thf(fact_918_dr_Osquare__eq__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( mult_a_ring_ext_a_c @ r @ X @ X )
= ( one_a_ring_ext_a_c @ r ) )
=> ( ( X
= ( one_a_ring_ext_a_c @ r ) )
| ( X
= ( a_inv_a_c @ r @ ( one_a_ring_ext_a_c @ r ) ) ) ) ) ) ).
% dr.square_eq_one
thf(fact_919_dr_Oto__contain__is__to__divide,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ord_less_eq_set_a @ ( cgenid547466214215511830xt_a_c @ r @ B ) @ ( cgenid547466214215511830xt_a_c @ r @ A ) )
= ( factor8216151074478948643xt_a_c @ r @ A @ B ) ) ) ) ).
% dr.to_contain_is_to_divide
thf(fact_920_pdr_Odivides__zero,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.divides_zero
thf(fact_921_pdr_Om__rcancel,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( A
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% pdr.m_rcancel
thf(fact_922_pdr_Om__lcancel,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( A
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% pdr.m_lcancel
thf(fact_923_pdr_Ointegral__iff,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
| ( B
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.integral_iff
thf(fact_924_pdr_Ointegral,axiom,
! [A: list_a,B: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
| ( B
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.integral
thf(fact_925_pds_Odivides__zero,axiom,
! [A: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.divides_zero
thf(fact_926_pds_Ointegral,axiom,
! [A: list_b,B: list_b] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( A
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
| ( B
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.integral
thf(fact_927_pds_Ointegral__iff,axiom,
! [A: list_b,B: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( ( A
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
| ( B
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.integral_iff
thf(fact_928_pds_Om__lcancel,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( A
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% pds.m_lcancel
thf(fact_929_pds_Om__rcancel,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( A
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ A )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% pds.m_rcancel
thf(fact_930_pdr_Or__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Z @ Y ) ) ) ) ) ) ).
% pdr.r_distr
thf(fact_931_pdr_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% pdr.l_distr
thf(fact_932_pds_Ol__distr,axiom,
! [X: list_b,Y: list_b,Z: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) @ Z )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Z ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ Z ) ) ) ) ) ) ).
% pds.l_distr
thf(fact_933_pds_Or__distr,axiom,
! [X: list_b,Y: list_b,Z: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Z @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Z @ X ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Z @ Y ) ) ) ) ) ) ).
% pds.r_distr
thf(fact_934_pdr_Oone__divides,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ A ) ) ).
% pdr.one_divides
thf(fact_935_pdr_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.one_unique
thf(fact_936_pdr_Oinv__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% pdr.inv_unique
thf(fact_937_pds_Oone__divides,axiom,
! [A: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ A ) ) ).
% pds.one_divides
thf(fact_938_pds_Oinv__unique,axiom,
! [Y: list_b,X: list_b,Y2: list_b] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ X )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y2 )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% pds.inv_unique
thf(fact_939_pds_Oone__unique,axiom,
! [U: list_b] :
( ( member_list_b @ U @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ! [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ U @ X2 )
= X2 ) )
=> ( U
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.one_unique
thf(fact_940_pdr_Or__minus,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) ) ) ) ) ).
% pdr.r_minus
thf(fact_941_pdr_Ol__minus,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ Y )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) ) ) ) ) ).
% pdr.l_minus
thf(fact_942_pds_Ol__minus,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ Y )
= ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) ) ) ) ) ).
% pds.l_minus
thf(fact_943_pds_Or__minus,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y ) )
= ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) ) ) ) ) ).
% pds.r_minus
thf(fact_944_dr_Ois__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( polyno4133073214067823461ot_a_c @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) @ X )
=> ( ( polyno4133073214067823461ot_a_c @ r @ P @ X )
| ( polyno4133073214067823461ot_a_c @ r @ Q @ X ) ) ) ) ) ).
% dr.is_root_poly_mult_imp_is_root
thf(fact_945_ds_Ois__root__poly__mult__imp__is__root,axiom,
! [P: list_b,Q: list_b,X: b] :
( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( polyno1345617632095147429ot_b_d @ s @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) @ X )
=> ( ( polyno1345617632095147429ot_b_d @ s @ P @ X )
| ( polyno1345617632095147429ot_b_d @ s @ Q @ X ) ) ) ) ) ).
% ds.is_root_poly_mult_imp_is_root
thf(fact_946_dr_Opdivides__iff__shell,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( polyno5814909790663948099es_a_c @ r @ P @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ K ) @ P @ Q ) ) ) ) ) ).
% dr.pdivides_iff_shell
thf(fact_947_ds_Opdivides__iff__shell,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( polyno3027454208691272067es_b_d @ s @ P @ Q )
= ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ K ) @ P @ Q ) ) ) ) ) ).
% ds.pdivides_iff_shell
thf(fact_948_pdr_Osquare__eq__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( X
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
| ( X
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.square_eq_one
thf(fact_949_pds_Osquare__eq__one,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ X )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( X
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
| ( X
= ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.square_eq_one
thf(fact_950_pdr_Oto__contain__is__to__divide,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A ) )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B ) ) ) ) ).
% pdr.to_contain_is_to_divide
thf(fact_951_dr_OpprimeE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( polyno5814909790663948099es_a_c @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ K ) @ Q @ R2 ) )
=> ( ( polyno5814909790663948099es_a_c @ r @ P @ Q )
| ( polyno5814909790663948099es_a_c @ r @ P @ R2 ) ) ) ) ) ) ) ) ).
% dr.pprimeE(3)
thf(fact_952_ds_OpprimeE_I3_J,axiom,
! [K: set_b,P: list_b,Q: list_b,R2: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ K ) @ P )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ R2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( polyno3027454208691272067es_b_d @ s @ P @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ K ) @ Q @ R2 ) )
=> ( ( polyno3027454208691272067es_b_d @ s @ P @ Q )
| ( polyno3027454208691272067es_b_d @ s @ P @ R2 ) ) ) ) ) ) ) ) ).
% ds.pprimeE(3)
thf(fact_953_pdr_Opdivides__iff__shell,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q )
= ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ Q ) ) ) ) ) ).
% pdr.pdivides_iff_shell
thf(fact_954_pds_Opdivides__iff__shell,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q )
= ( factor8908767930780902896t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ Q ) ) ) ) ) ).
% pds.pdivides_iff_shell
thf(fact_955_dr_Opdiv__pmod,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ K ) @ Q @ ( polynomial_pdiv_a_c @ r @ P @ Q ) ) @ ( polynomial_pmod_a_c @ r @ P @ Q ) ) ) ) ) ) ).
% dr.pdiv_pmod
thf(fact_956_ds_Opdiv__pmod,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( P
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ K ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ K ) @ Q @ ( polynomial_pdiv_b_d @ s @ P @ Q ) ) @ ( polynomial_pmod_b_d @ s @ P @ Q ) ) ) ) ) ) ).
% ds.pdiv_pmod
thf(fact_957_pdr_Ois__root__poly__mult__imp__is__root,axiom,
! [P: list_list_a,Q: list_list_a,X: list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ P @ Q ) @ X )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ X )
| ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Q @ X ) ) ) ) ) ).
% pdr.is_root_poly_mult_imp_is_root
thf(fact_958_pds_OpprimeE_I2_J,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ring_r415245522172713630t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P )
=> ~ ( member_list_list_b @ P @ ( units_6858163862972288294t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ) ) ) ).
% pds.pprimeE(2)
thf(fact_959_dr_Odivides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151074478948643xt_a_c @ r @ A @ B )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( factor8216151074478948643xt_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ A @ C ) @ ( mult_a_ring_ext_a_c @ r @ B @ C ) ) ) ) ) ) ).
% dr.divides_mult_rI
thf(fact_960_dr_Odivides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151074478948643xt_a_c @ r @ A @ B )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( factor8216151074478948643xt_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ C @ A ) @ ( mult_a_ring_ext_a_c @ r @ C @ B ) ) ) ) ) ).
% dr.divides_mult_lI
thf(fact_961_dr_Odivides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( factor8216151074478948643xt_a_c @ r @ A @ A ) ) ).
% dr.divides_refl
thf(fact_962_dr_Om__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_c @ r @ X @ Y ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% dr.m_closed
thf(fact_963_ds_Om__closed,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ ( mult_b_ring_ext_b_d @ s @ X @ Y ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% ds.m_closed
thf(fact_964_dr_Or__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ X @ ( zero_a_c @ r ) )
= ( zero_a_c @ r ) ) ) ).
% dr.r_null
thf(fact_965_dr_Ol__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ ( zero_a_c @ r ) @ X )
= ( zero_a_c @ r ) ) ) ).
% dr.l_null
thf(fact_966_ds_Ol__null,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ ( zero_b_d @ s ) @ X )
= ( zero_b_d @ s ) ) ) ).
% ds.l_null
thf(fact_967_ds_Or__null,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ X @ ( zero_b_d @ s ) )
= ( zero_b_d @ s ) ) ) ).
% ds.r_null
thf(fact_968_ds_Ol__one,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ ( one_b_ring_ext_b_d @ s ) @ X )
= X ) ) ).
% ds.l_one
thf(fact_969_ds_Or__one,axiom,
! [X: b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ X @ ( one_b_ring_ext_b_d @ s ) )
= X ) ) ).
% ds.r_one
thf(fact_970_dr_Or__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ X @ ( one_a_ring_ext_a_c @ r ) )
= X ) ) ).
% dr.r_one
thf(fact_971_dr_Ol__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ ( one_a_ring_ext_a_c @ r ) @ X )
= X ) ) ).
% dr.l_one
thf(fact_972_ds_Odivides__mult__lI,axiom,
! [A: b,B: b,C: b] :
( ( factor2325171414093416164xt_b_d @ s @ A @ B )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( factor2325171414093416164xt_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ C @ A ) @ ( mult_b_ring_ext_b_d @ s @ C @ B ) ) ) ) ) ).
% ds.divides_mult_lI
thf(fact_973_ds_Odivides__mult__rI,axiom,
! [A: b,B: b,C: b] :
( ( factor2325171414093416164xt_b_d @ s @ A @ B )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( factor2325171414093416164xt_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ A @ C ) @ ( mult_b_ring_ext_b_d @ s @ B @ C ) ) ) ) ) ) ).
% ds.divides_mult_rI
thf(fact_974_pdr_Odivides__mult__rI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ C ) ) ) ) ) ) ).
% pdr.divides_mult_rI
thf(fact_975_pdr_Odivides__mult__lI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C @ B ) ) ) ) ) ).
% pdr.divides_mult_lI
thf(fact_976_pdr_Odivides__refl,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ A ) ) ).
% pdr.divides_refl
thf(fact_977_pdr_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.m_closed
thf(fact_978_pds_Odivides__mult__lI,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C @ A ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C @ B ) ) ) ) ) ).
% pds.divides_mult_lI
thf(fact_979_pds_Odivides__mult__rI,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ C ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ C ) ) ) ) ) ) ).
% pds.divides_mult_rI
thf(fact_980_pds_Odivides__refl,axiom,
! [A: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ A ) ) ).
% pds.divides_refl
thf(fact_981_pds_Om__closed,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.m_closed
thf(fact_982_pdr_Or__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.r_null
thf(fact_983_pdr_Ol__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.l_null
thf(fact_984_pds_Ol__null,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ X )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.l_null
thf(fact_985_pds_Or__null,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.r_null
thf(fact_986_pdr_Or__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= X ) ) ).
% pdr.r_one
thf(fact_987_pdr_Ol__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ X )
= X ) ) ).
% pdr.l_one
thf(fact_988_pds_Ol__one,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ X )
= X ) ) ).
% pds.l_one
thf(fact_989_pds_Or__one,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= X ) ) ).
% pds.r_one
thf(fact_990_h_Ohom__mult,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( h @ ( mult_a_ring_ext_a_c @ r @ X @ Y ) )
= ( mult_b_ring_ext_b_d @ s @ ( h @ X ) @ ( h @ Y ) ) ) ) ) ).
% h.hom_mult
thf(fact_991_pds_Oisgcd__divides__l,axiom,
! [A: list_b,B: list_b] :
( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( isgcd_5068363846668152615t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ A @ B ) ) ) ) ).
% pds.isgcd_divides_l
thf(fact_992_pds_Oisgcd__divides__r,axiom,
! [B: list_b,A: list_b] :
( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ A )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( isgcd_5068363846668152615t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ A @ B ) ) ) ) ).
% pds.isgcd_divides_r
thf(fact_993_pdr_Oisgcd__divides__r,axiom,
! [B: list_a,A: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ A )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ A @ B ) ) ) ) ).
% pdr.isgcd_divides_r
thf(fact_994_pdr_Oisgcd__divides__l,axiom,
! [A: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ A @ B ) ) ) ) ).
% pdr.isgcd_divides_l
thf(fact_995_dr_OUnits__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_c @ r ) )
=> ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% dr.Units_closed
thf(fact_996_ds_OUnits__closed,axiom,
! [X: b] :
( ( member_b @ X @ ( units_b_ring_ext_b_d @ s ) )
=> ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% ds.Units_closed
thf(fact_997_dr_Oprod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_c @ r @ A @ B ) @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) ) ).
% dr.prod_unit_l
thf(fact_998_dr_Oprod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_c @ r @ A @ B ) @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) ) ).
% dr.prod_unit_r
thf(fact_999_dr_Ounit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_c @ r @ A @ B ) @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) ).
% dr.unit_factor
thf(fact_1000_ds_Ounit__factor,axiom,
! [A: b,B: b] :
( ( member_b @ ( mult_b_ring_ext_b_d @ s @ A @ B ) @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ A @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) ).
% ds.unit_factor
thf(fact_1001_ds_Oprod__unit__r,axiom,
! [A: b,B: b] :
( ( member_b @ ( mult_b_ring_ext_b_d @ s @ A @ B ) @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ B @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ A @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) ) ).
% ds.prod_unit_r
thf(fact_1002_ds_Oprod__unit__l,axiom,
! [A: b,B: b] :
( ( member_b @ ( mult_b_ring_ext_b_d @ s @ A @ B ) @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ B @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) ) ).
% ds.prod_unit_l
thf(fact_1003_dr_Odivides__unit,axiom,
! [A: a,U: a] :
( ( factor8216151074478948643xt_a_c @ r @ A @ U )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_c @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) ).
% dr.divides_unit
thf(fact_1004_dr_Ounit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( factor8216151074478948643xt_a_c @ r @ U @ A ) ) ) ).
% dr.unit_divides
thf(fact_1005_dr_Oisgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor8216151074478948643xt_a_c @ r @ A @ B )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( isgcd_a_ring_ext_a_c @ r @ A @ A @ B ) ) ) ) ).
% dr.isgcd_divides_l
thf(fact_1006_dr_Oisgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor8216151074478948643xt_a_c @ r @ B @ A )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( isgcd_a_ring_ext_a_c @ r @ B @ A @ B ) ) ) ) ).
% dr.isgcd_divides_r
thf(fact_1007_ds_Ounit__divides,axiom,
! [U: b,A: b] :
( ( member_b @ U @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( factor2325171414093416164xt_b_d @ s @ U @ A ) ) ) ).
% ds.unit_divides
thf(fact_1008_ds_Odivides__unit,axiom,
! [A: b,U: b] :
( ( factor2325171414093416164xt_b_d @ s @ A @ U )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ U @ ( units_b_ring_ext_b_d @ s ) )
=> ( member_b @ A @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) ).
% ds.divides_unit
thf(fact_1009_ds_OUnits__inv__comm,axiom,
! [X: b,Y: b] :
( ( ( mult_b_ring_ext_b_d @ s @ X @ Y )
= ( one_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ X @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ Y @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ Y @ X )
= ( one_b_ring_ext_b_d @ s ) ) ) ) ) ).
% ds.Units_inv_comm
thf(fact_1010_dr_OUnits__inv__comm,axiom,
! [X: a,Y: a] :
( ( ( mult_a_ring_ext_a_c @ r @ X @ Y )
= ( one_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ X @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ Y @ X )
= ( one_a_ring_ext_a_c @ r ) ) ) ) ) ).
% dr.Units_inv_comm
thf(fact_1011_dr_Oideal__eq__carrier__iff,axiom,
! [A: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( partia778085601923319190xt_a_c @ r )
= ( cgenid547466214215511830xt_a_c @ r @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.ideal_eq_carrier_iff
thf(fact_1012_ds_Oideal__eq__carrier__iff,axiom,
! [A: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( partia8782771468121683032xt_b_d @ s )
= ( cgenid3879858590684755159xt_b_d @ s @ A ) )
= ( member_b @ A @ ( units_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.ideal_eq_carrier_iff
thf(fact_1013_dr_Oring__irreducibleE_I4_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_r999134135267193927le_a_c @ r @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.ring_irreducibleE(4)
thf(fact_1014_ds_Oring__irreducibleE_I4_J,axiom,
! [R2: b] :
( ( member_b @ R2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_r7435050590149293703le_b_d @ s @ R2 )
=> ~ ( member_b @ R2 @ ( units_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.ring_irreducibleE(4)
thf(fact_1015_pdr_OUnits__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.Units_closed
thf(fact_1016_pds_OUnits__closed,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.Units_closed
thf(fact_1017_ds_OUnits__l__inv__ex,axiom,
! [X: b] :
( ( member_b @ X @ ( units_b_ring_ext_b_d @ s ) )
=> ? [X2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
& ( ( mult_b_ring_ext_b_d @ s @ X2 @ X )
= ( one_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.Units_l_inv_ex
thf(fact_1018_ds_OUnits__r__inv__ex,axiom,
! [X: b] :
( ( member_b @ X @ ( units_b_ring_ext_b_d @ s ) )
=> ? [X2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
& ( ( mult_b_ring_ext_b_d @ s @ X @ X2 )
= ( one_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.Units_r_inv_ex
thf(fact_1019_dr_OUnits__l__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_c @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
& ( ( mult_a_ring_ext_a_c @ r @ X2 @ X )
= ( one_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.Units_l_inv_ex
thf(fact_1020_dr_OUnits__r__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_c @ r ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
& ( ( mult_a_ring_ext_a_c @ r @ X @ X2 )
= ( one_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.Units_r_inv_ex
thf(fact_1021_ds_OUnit__eq__dividesone,axiom,
! [U: b] :
( ( member_b @ U @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ U @ ( units_b_ring_ext_b_d @ s ) )
= ( factor2325171414093416164xt_b_d @ s @ U @ ( one_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.Unit_eq_dividesone
thf(fact_1022_ds_Odivides__one,axiom,
! [A: b] :
( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( factor2325171414093416164xt_b_d @ s @ A @ ( one_b_ring_ext_b_d @ s ) )
= ( member_b @ A @ ( units_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.divides_one
thf(fact_1023_dr_OUnit__eq__dividesone,axiom,
! [U: a] :
( ( member_a @ U @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_c @ r ) )
= ( factor8216151074478948643xt_a_c @ r @ U @ ( one_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.Unit_eq_dividesone
thf(fact_1024_dr_Odivides__one,axiom,
! [A: a] :
( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( factor8216151074478948643xt_a_c @ r @ A @ ( one_a_ring_ext_a_c @ r ) )
= ( member_a @ A @ ( units_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.divides_one
thf(fact_1025_dr_Oirreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irredu6211895651204806704xt_a_c @ r @ B )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( irredu6211895651204806704xt_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ A @ B ) ) ) ) ) ) ).
% dr.irreducible_prod_lI
thf(fact_1026_dr_Oirreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu6211895651204806704xt_a_c @ r @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( irredu6211895651204806704xt_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ A @ B ) ) ) ) ) ) ).
% dr.irreducible_prod_rI
thf(fact_1027_ds_Oirreducible__prod__rI,axiom,
! [A: b,B: b] :
( ( irredu320915990819274225xt_b_d @ s @ A )
=> ( ( member_b @ B @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( irredu320915990819274225xt_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ A @ B ) ) ) ) ) ) ).
% ds.irreducible_prod_rI
thf(fact_1028_ds_Oirreducible__prod__lI,axiom,
! [B: b,A: b] :
( ( irredu320915990819274225xt_b_d @ s @ B )
=> ( ( member_b @ A @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( irredu320915990819274225xt_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ A @ B ) ) ) ) ) ) ).
% ds.irreducible_prod_lI
thf(fact_1029_dr_Oring__irreducibleE_I5_J,axiom,
! [R2: a,A: a,B: a] :
( ( member_a @ R2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_r999134135267193927le_a_c @ r @ R2 )
=> ( ( member_a @ A @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_c @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_c @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) ) ) ) ).
% dr.ring_irreducibleE(5)
thf(fact_1030_ds_Oring__irreducibleE_I5_J,axiom,
! [R2: b,A: b,B: b] :
( ( member_b @ R2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_r7435050590149293703le_b_d @ s @ R2 )
=> ( ( member_b @ A @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( R2
= ( mult_b_ring_ext_b_d @ s @ A @ B ) )
=> ( ( member_b @ A @ ( units_b_ring_ext_b_d @ s ) )
| ( member_b @ B @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) ) ) ) ).
% ds.ring_irreducibleE(5)
thf(fact_1031_pdr_Oprod__unit__l,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.prod_unit_l
thf(fact_1032_pdr_Oprod__unit__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.prod_unit_r
thf(fact_1033_pdr_Ounit__factor,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.unit_factor
thf(fact_1034_pds_Ounit__factor,axiom,
! [A: list_b,B: list_b] :
( ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ A @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.unit_factor
thf(fact_1035_pds_Oprod__unit__r,axiom,
! [A: list_b,B: list_b] :
( ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ A @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.prod_unit_r
thf(fact_1036_pds_Oprod__unit__l,axiom,
! [A: list_b,B: list_b] :
( ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ B @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.prod_unit_l
thf(fact_1037_pdr_OUnits__inv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.Units_inv_comm
thf(fact_1038_pds_OUnits__inv__comm,axiom,
! [X: list_b,Y: list_b] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ X @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ X )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.Units_inv_comm
thf(fact_1039_pdr_Odivides__unit,axiom,
! [A: list_a,U: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ U )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.divides_unit
thf(fact_1040_pdr_Ounit__divides,axiom,
! [U: list_a,A: list_a] :
( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ U @ A ) ) ) ).
% pdr.unit_divides
thf(fact_1041_pds_Ounit__divides,axiom,
! [U: list_b,A: list_b] :
( ( member_list_b @ U @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ U @ A ) ) ) ).
% pds.unit_divides
thf(fact_1042_pds_Odivides__unit,axiom,
! [A: list_b,U: list_b] :
( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ U )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ U @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ A @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.divides_unit
thf(fact_1043_pdr_Oideal__eq__carrier__iff,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.ideal_eq_carrier_iff
thf(fact_1044_pdr_Oring__irreducibleE_I4_J,axiom,
! [R2: list_a] :
( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ R2 )
=> ~ ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.ring_irreducibleE(4)
thf(fact_1045_pds_Oring__irreducibleE_I4_J,axiom,
! [R2: list_b] :
( ( member_list_b @ R2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ R2 )
=> ~ ( member_list_b @ R2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.ring_irreducibleE(4)
thf(fact_1046_dr_OpprimeE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ) ) ) ).
% dr.pprimeE(2)
thf(fact_1047_ds_OpprimeE_I2_J,axiom,
! [K: set_b,P: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ K ) @ P )
=> ~ ( member_list_b @ P @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ) ) ) ).
% ds.pprimeE(2)
thf(fact_1048_pdr_OUnits__l__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.Units_l_inv_ex
thf(fact_1049_pdr_OUnits__r__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.Units_r_inv_ex
thf(fact_1050_pds_OUnits__r__inv__ex,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ? [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ X2 )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.Units_r_inv_ex
thf(fact_1051_pds_OUnits__l__inv__ex,axiom,
! [X: list_b] :
( ( member_list_b @ X @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ? [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ X )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.Units_l_inv_ex
thf(fact_1052_pdr_OUnit__eq__dividesone,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ U @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.Unit_eq_dividesone
thf(fact_1053_pdr_Odivides__one,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.divides_one
thf(fact_1054_pds_Odivides__one,axiom,
! [A: list_b] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( member_list_b @ A @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.divides_one
thf(fact_1055_pds_OUnit__eq__dividesone,axiom,
! [U: list_b] :
( ( member_list_b @ U @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ U @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ U @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.Unit_eq_dividesone
thf(fact_1056_pdr_Oring__irreducibleE_I5_J,axiom,
! [R2: list_a,A: list_a,B: list_a] :
( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( R2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ) ) ).
% pdr.ring_irreducibleE(5)
thf(fact_1057_pds_Oring__irreducibleE_I5_J,axiom,
! [R2: list_b,A: list_b,B: list_b] :
( ( member_list_b @ R2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ R2 )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( R2
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B ) )
=> ( ( member_list_b @ A @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
| ( member_list_b @ B @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ) ) ).
% pds.ring_irreducibleE(5)
thf(fact_1058_pdr_Oirreducible__prod__lI,axiom,
! [B: list_a,A: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B ) ) ) ) ) ) ).
% pdr.irreducible_prod_lI
thf(fact_1059_pdr_Oirreducible__prod__rI,axiom,
! [A: list_a,B: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B ) ) ) ) ) ) ).
% pdr.irreducible_prod_rI
thf(fact_1060_pds_Oirreducible__prod__rI,axiom,
! [A: list_b,B: list_b] :
( ( irredu8180679162501400317t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A )
=> ( ( member_list_b @ B @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( irredu8180679162501400317t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B ) ) ) ) ) ) ).
% pds.irreducible_prod_rI
thf(fact_1061_pds_Oirreducible__prod__lI,axiom,
! [B: list_b,A: list_b] :
( ( irredu8180679162501400317t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B )
=> ( ( member_list_b @ A @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( irredu8180679162501400317t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B ) ) ) ) ) ) ).
% pds.irreducible_prod_lI
thf(fact_1062_pdr_OpprimeE_I2_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ) ) ) ).
% pdr.pprimeE(2)
thf(fact_1063_dr_OpprimeI,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ! [Q2: list_a,R: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( polyno5814909790663948099es_a_c @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ K ) @ Q2 @ R ) )
=> ( ( polyno5814909790663948099es_a_c @ r @ P @ Q2 )
| ( polyno5814909790663948099es_a_c @ r @ P @ R ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ K ) @ P ) ) ) ) ) ) ).
% dr.pprimeI
thf(fact_1064_ds_OpprimeI,axiom,
! [K: set_b,P: list_b] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( P != nil_b )
=> ( ~ ( member_list_b @ P @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ! [Q2: list_b,R: list_b] :
( ( member_list_b @ Q2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ R @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( polyno3027454208691272067es_b_d @ s @ P @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ K ) @ Q2 @ R ) )
=> ( ( polyno3027454208691272067es_b_d @ s @ P @ Q2 )
| ( polyno3027454208691272067es_b_d @ s @ P @ R ) ) ) ) )
=> ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ K ) @ P ) ) ) ) ) ) ).
% ds.pprimeI
thf(fact_1065_dr_OUnits__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_c @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_c @ r @ X @ Y ) @ ( units_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.Units_m_closed
thf(fact_1066_ds_OUnits__m__closed,axiom,
! [X: b,Y: b] :
( ( member_b @ X @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ Y @ ( units_b_ring_ext_b_d @ s ) )
=> ( member_b @ ( mult_b_ring_ext_b_d @ s @ X @ Y ) @ ( units_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.Units_m_closed
thf(fact_1067_ds_OUnits__one__closed,axiom,
member_b @ ( one_b_ring_ext_b_d @ s ) @ ( units_b_ring_ext_b_d @ s ) ).
% ds.Units_one_closed
thf(fact_1068_dr_OUnits__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_c @ r ) @ ( units_a_ring_ext_a_c @ r ) ).
% dr.Units_one_closed
thf(fact_1069_pdr_OpprimeI,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ! [Q2: list_list_a,R: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ Q2 @ R ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q2 )
| ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ R ) ) ) ) )
=> ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P ) ) ) ) ) ) ).
% pdr.pprimeI
thf(fact_1070_dr_OUnits__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( mult_a_ring_ext_a_c @ r @ X @ Y )
= ( mult_a_ring_ext_a_c @ r @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% dr.Units_l_cancel
thf(fact_1071_ds_OUnits__l__cancel,axiom,
! [X: b,Y: b,Z: b] :
( ( member_b @ X @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( mult_b_ring_ext_b_d @ s @ X @ Y )
= ( mult_b_ring_ext_b_d @ s @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% ds.Units_l_cancel
thf(fact_1072_ds_OUnits__minus__one__closed,axiom,
member_b @ ( a_inv_b_d @ s @ ( one_b_ring_ext_b_d @ s ) ) @ ( units_b_ring_ext_b_d @ s ) ).
% ds.Units_minus_one_closed
thf(fact_1073_dr_OUnits__minus__one__closed,axiom,
member_a @ ( a_inv_a_c @ r @ ( one_a_ring_ext_a_c @ r ) ) @ ( units_a_ring_ext_a_c @ r ) ).
% dr.Units_minus_one_closed
thf(fact_1074_pdr_OUnits__m__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.Units_m_closed
thf(fact_1075_pds_OUnits__m__closed,axiom,
! [X: list_b,Y: list_b] :
( ( member_list_b @ X @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.Units_m_closed
thf(fact_1076_pdr_OUnits__one__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% pdr.Units_one_closed
thf(fact_1077_pds_OUnits__one__closed,axiom,
member_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% pds.Units_one_closed
thf(fact_1078_pdr_OUnits__l__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% pdr.Units_l_cancel
thf(fact_1079_pds_OUnits__l__cancel,axiom,
! [X: list_b,Y: list_b,Z: list_b] :
( ( member_list_b @ X @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% pds.Units_l_cancel
thf(fact_1080_pdr_OUnits__minus__one__closed,axiom,
member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% pdr.Units_minus_one_closed
thf(fact_1081_pds_OUnits__minus__one__closed,axiom,
member_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% pds.Units_minus_one_closed
thf(fact_1082_pdr_OsubdomainI,axiom,
! [H: set_list_a] :
( ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ! [H12: list_a,H23: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H12 @ H23 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( H12
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
| ( H23
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.subdomainI
thf(fact_1083_pds_Omonoid__cancelI,axiom,
( ! [A4: list_b,B2: list_b,C2: list_b] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C2 @ A4 )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C2 @ B2 ) )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( A4 = B2 ) ) ) ) )
=> ( ! [A4: list_b,B2: list_b,C2: list_b] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ C2 )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B2 @ C2 ) )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( A4 = B2 ) ) ) ) )
=> ( monoid8253019609946410375t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.monoid_cancelI
thf(fact_1084_pdr_Omonoid__cancelI,axiom,
( ! [A4: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C2 @ A4 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C2 @ B2 ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( A4 = B2 ) ) ) ) )
=> ( ! [A4: list_a,B2: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B2 @ C2 ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( A4 = B2 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.monoid_cancelI
thf(fact_1085_pds_Oline__extension__smult__closed,axiom,
! [K: set_list_b,E: set_list_b,A: list_b,K3: list_b,U: list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ! [K2: list_b,V2: list_b] :
( ( member_list_b @ K2 @ K )
=> ( ( member_list_b @ V2 @ E )
=> ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K2 @ V2 ) @ E ) ) )
=> ( ( ord_le8932221534207217157list_b @ E @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ K3 @ K )
=> ( ( member_list_b @ U @ ( embedd2064902177841597106t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K @ A @ E ) )
=> ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K3 @ U ) @ ( embedd2064902177841597106t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K @ A @ E ) ) ) ) ) ) ) ) ).
% pds.line_extension_smult_closed
thf(fact_1086_pdr_Oline__extension__smult__closed,axiom,
! [K: set_list_a,E: set_list_a,A: list_a,K3: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ! [K2: list_a,V2: list_a] :
( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ V2 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K2 @ V2 ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K3 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K @ A @ E ) ) ) ) ) ) ) ) ).
% pdr.line_extension_smult_closed
thf(fact_1087_dr_Omonoid__cancelI,axiom,
( ! [A4: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_c @ r @ C2 @ A4 )
= ( mult_a_ring_ext_a_c @ r @ C2 @ B2 ) )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( A4 = B2 ) ) ) ) )
=> ( ! [A4: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_c @ r @ A4 @ C2 )
= ( mult_a_ring_ext_a_c @ r @ B2 @ C2 ) )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( A4 = B2 ) ) ) ) )
=> ( monoid5798828376123148986xt_a_c @ r ) ) ) ).
% dr.monoid_cancelI
thf(fact_1088_ds_Omonoid__cancelI,axiom,
( ! [A4: b,B2: b,C2: b] :
( ( ( mult_b_ring_ext_b_d @ s @ C2 @ A4 )
= ( mult_b_ring_ext_b_d @ s @ C2 @ B2 ) )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( A4 = B2 ) ) ) ) )
=> ( ! [A4: b,B2: b,C2: b] :
( ( ( mult_b_ring_ext_b_d @ s @ A4 @ C2 )
= ( mult_b_ring_ext_b_d @ s @ B2 @ C2 ) )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( A4 = B2 ) ) ) ) )
=> ( monoid9131220752592392315xt_b_d @ s ) ) ) ).
% ds.monoid_cancelI
thf(fact_1089_pdr_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_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.line_extension_in_carrier
thf(fact_1090_pds_Oline__extension__in__carrier,axiom,
! [K: set_list_b,A: list_b,E: set_list_b] :
( ( ord_le8932221534207217157list_b @ K @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ E @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ord_le8932221534207217157list_b @ ( embedd2064902177841597106t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K @ A @ E ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.line_extension_in_carrier
thf(fact_1091_pdr_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_c @ r @ ( partia778085601923319190xt_a_c @ 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_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X3 @ A ) @ Y5 ) ) ) ) ) ) ).
% pdr.line_extension_mem_iff
thf(fact_1092_pds_Oline__extension__mem__iff,axiom,
! [U: list_b,K: set_list_b,A: list_b,E: set_list_b] :
( ( member_list_b @ U @ ( embedd2064902177841597106t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K @ A @ E ) )
= ( ? [X3: list_b] :
( ( member_list_b @ X3 @ K )
& ? [Y5: list_b] :
( ( member_list_b @ Y5 @ E )
& ( U
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X3 @ A ) @ Y5 ) ) ) ) ) ) ).
% pds.line_extension_mem_iff
thf(fact_1093_pds_Ofactors__mult__single,axiom,
! [A: list_b,Fb: list_list_b,B: list_b] :
( ( irredu8180679162501400317t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A )
=> ( ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fb @ B )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( cons_list_b @ A @ Fb ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B ) ) ) ) ) ).
% pds.factors_mult_single
thf(fact_1094_pdr_Ofactors__mult__single,axiom,
! [A: list_a,Fb: list_list_a,B: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fb @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( cons_list_a @ A @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B ) ) ) ) ) ).
% pdr.factors_mult_single
thf(fact_1095_pds_OpirreducibleI,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( P != nil_list_b )
=> ( ~ ( member_list_list_b @ P @ ( units_6858163862972288294t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ! [Q2: list_list_b,R: list_list_b] :
( ( member_list_list_b @ Q2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ R @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( P
= ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ Q2 @ R ) )
=> ( ( member_list_list_b @ Q2 @ ( units_6858163862972288294t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
| ( member_list_list_b @ R @ ( units_6858163862972288294t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ) ) ) )
=> ( ring_r4561388045816386823t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P ) ) ) ) ) ) ).
% pds.pirreducibleI
thf(fact_1096_pdr_OpirreducibleI,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ! [Q2: list_list_a,R: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ Q2 @ R ) )
=> ( ( member_list_list_a @ Q2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
| ( member_list_list_a @ R @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P ) ) ) ) ) ) ).
% pdr.pirreducibleI
thf(fact_1097_pdr_Ocarrier__is__subring,axiom,
subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.carrier_is_subring
thf(fact_1098_pds_Ocarrier__is__subring,axiom,
subrin3833087656135479401t_unit @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.carrier_is_subring
thf(fact_1099_pdr_Ouniv__poly__is__abelian__monoid,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( abelia3641329199688042803t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ).
% pdr.univ_poly_is_abelian_monoid
thf(fact_1100_pdr_Ouniv__poly__is__domain,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ).
% pdr.univ_poly_is_domain
thf(fact_1101_pds_Ouniv__poly__is__abelian__monoid,axiom,
! [K: set_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( abelia7842546174856384882t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ).
% pds.univ_poly_is_abelian_monoid
thf(fact_1102_pds_Ouniv__poly__is__domain,axiom,
! [K: set_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( domain2787997859347364482t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ).
% pds.univ_poly_is_domain
thf(fact_1103_pdr_OsubcringI_H,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.subcringI'
thf(fact_1104_pdr_OsubdomainI_H,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( subdom7821232466298058046t_unit @ H @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.subdomainI'
thf(fact_1105_pdr_OsubcringI,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ! [H12: list_a,H23: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H12 @ H23 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H23 @ H12 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.subcringI
thf(fact_1106_pdr_Opdivides__zero,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ nil_list_a ) ) ) ).
% pdr.pdivides_zero
thf(fact_1107_pds_Opdivides__zero,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ nil_list_b ) ) ) ).
% pds.pdivides_zero
thf(fact_1108_pdr_Ouniv__poly__a__inv__consistent,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ P ) ) ) ) ).
% pdr.univ_poly_a_inv_consistent
thf(fact_1109_pds_Ouniv__poly__a__inv__consistent,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( a_inv_2010862973944421262t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P )
= ( a_inv_2010862973944421262t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ P ) ) ) ) ).
% pds.univ_poly_a_inv_consistent
thf(fact_1110_pdr_OpirreducibleE_I1_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ).
% pdr.pirreducibleE(1)
thf(fact_1111_pds_OpirreducibleE_I1_J,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ring_r4561388045816386823t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P )
=> ( P != nil_list_b ) ) ) ) ).
% pds.pirreducibleE(1)
thf(fact_1112_pdr_OpirreducibleE_I2_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ) ) ) ).
% pdr.pirreducibleE(2)
thf(fact_1113_pds_OpirreducibleE_I2_J,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ring_r4561388045816386823t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P )
=> ~ ( member_list_list_b @ P @ ( units_6858163862972288294t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ) ) ) ).
% pds.pirreducibleE(2)
thf(fact_1114_pdr_Ouniv__poly__a__minus__consistent,axiom,
! [K: set_list_a,Q: list_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ Q )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ P @ Q ) ) ) ) ).
% pdr.univ_poly_a_minus_consistent
thf(fact_1115_pds_Ouniv__poly__a__minus__consistent,axiom,
! [K: set_list_b,Q: list_list_b,P: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( a_minu6442441833124848013t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ Q )
= ( a_minu6442441833124848013t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ P @ Q ) ) ) ) ).
% pds.univ_poly_a_minus_consistent
thf(fact_1116_pdr_OpirreducibleE_I3_J,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a,R2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ Q @ R2 ) )
=> ( ( member_list_list_a @ Q @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
| ( member_list_list_a @ R2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ) ) ) ) ) ) ) ).
% pdr.pirreducibleE(3)
thf(fact_1117_pds_OpirreducibleE_I3_J,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b,R2: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ring_r4561388045816386823t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ R2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( P
= ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ Q @ R2 ) )
=> ( ( member_list_list_b @ Q @ ( units_6858163862972288294t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
| ( member_list_list_b @ R2 @ ( units_6858163862972288294t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ) ) ) ) ) ) ) ).
% pds.pirreducibleE(3)
thf(fact_1118_pdr_OsubringI,axiom,
! [H: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ H )
=> ( ! [H3: list_a] :
( ( member_list_a @ H3 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H3 ) @ H ) )
=> ( ! [H12: list_a,H23: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H12 @ H23 ) @ H ) ) )
=> ( ! [H12: list_a,H23: list_a] :
( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H12 @ H23 ) @ H ) ) )
=> ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.subringI
thf(fact_1119_pds_OsubringI,axiom,
! [H: set_list_b] :
( ( ord_le8932221534207217157list_b @ H @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ H )
=> ( ! [H3: list_b] :
( ( member_list_b @ H3 @ H )
=> ( member_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H3 ) @ H ) )
=> ( ! [H12: list_b,H23: list_b] :
( ( member_list_b @ H12 @ H )
=> ( ( member_list_b @ H23 @ H )
=> ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H12 @ H23 ) @ H ) ) )
=> ( ! [H12: list_b,H23: list_b] :
( ( member_list_b @ H12 @ H )
=> ( ( member_list_b @ H23 @ H )
=> ( member_list_b @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H12 @ H23 ) @ H ) ) )
=> ( subrin3833087656135479401t_unit @ H @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.subringI
thf(fact_1120_pdr_Ocarrier__polynomial__shell,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.carrier_polynomial_shell
thf(fact_1121_pds_Ocarrier__polynomial__shell,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.carrier_polynomial_shell
thf(fact_1122_pds_OsubdomainI_H,axiom,
! [H: set_list_b] :
( ( subrin3833087656135479401t_unit @ H @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( subdom4735476224308063485t_unit @ H @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.subdomainI'
thf(fact_1123_pds_Oconst__term__simprules__shell_I4_J,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( a_inv_2010862973944421262t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P ) )
= ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P ) ) ) ) ) ).
% pds.const_term_simprules_shell(4)
thf(fact_1124_dr_Ocarrier__is__subring,axiom,
subring_a_c @ ( partia778085601923319190xt_a_c @ r ) @ r ).
% dr.carrier_is_subring
thf(fact_1125_ds_Ocarrier__is__subring,axiom,
subring_b_d @ ( partia8782771468121683032xt_b_d @ s ) @ s ).
% ds.carrier_is_subring
thf(fact_1126_dr_Ouniv__poly__is__domain,axiom,
! [K: set_a] :
( ( subring_a_c @ K @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ).
% dr.univ_poly_is_domain
thf(fact_1127_ds_Ouniv__poly__is__domain,axiom,
! [K: set_b] :
( ( subring_b_d @ K @ s )
=> ( domain3467766878553215752t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ).
% ds.univ_poly_is_domain
thf(fact_1128_dr_Ouniv__poly__is__abelian__monoid,axiom,
! [K: set_a] :
( ( subring_a_c @ K @ r )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ).
% dr.univ_poly_is_abelian_monoid
thf(fact_1129_ds_Ouniv__poly__is__abelian__monoid,axiom,
! [K: set_b] :
( ( subring_b_d @ K @ s )
=> ( abelia6363847436574302712t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ).
% ds.univ_poly_is_abelian_monoid
thf(fact_1130_dr_Ouniv__poly__a__inv__consistent,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ K ) @ P )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P ) ) ) ) ).
% dr.univ_poly_a_inv_consistent
thf(fact_1131_ds_Ouniv__poly__a__inv__consistent,axiom,
! [K: set_b,P: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ K ) @ P )
= ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P ) ) ) ) ).
% ds.univ_poly_a_inv_consistent
thf(fact_1132_h_Oimg__is__subring,axiom,
! [K: set_a] :
( ( subring_a_c @ K @ r )
=> ( subring_b_d @ ( image_a_b @ h @ K ) @ s ) ) ).
% h.img_is_subring
thf(fact_1133_dr_Opdivides__zero,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( polyno5814909790663948099es_a_c @ r @ P @ nil_a ) ) ) ).
% dr.pdivides_zero
thf(fact_1134_ds_Opdivides__zero,axiom,
! [K: set_b,P: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( polyno3027454208691272067es_b_d @ s @ P @ nil_b ) ) ) ).
% ds.pdivides_zero
thf(fact_1135_dr_OpirreducibleE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% dr.pirreducibleE(1)
thf(fact_1136_ds_OpirreducibleE_I1_J,axiom,
! [K: set_b,P: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ K ) @ P )
=> ( P != nil_b ) ) ) ) ).
% ds.pirreducibleE(1)
thf(fact_1137_dr_Ouniv__poly__a__minus__consistent,axiom,
! [K: set_a,Q: list_a,P: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_c @ r @ K ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q ) ) ) ) ).
% dr.univ_poly_a_minus_consistent
thf(fact_1138_ds_Ouniv__poly__a__minus__consistent,axiom,
! [K: set_b,Q: list_b,P: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( a_minu898264511480707987t_unit @ ( univ_poly_b_d @ s @ K ) @ P @ Q )
= ( a_minu898264511480707987t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q ) ) ) ) ).
% ds.univ_poly_a_minus_consistent
thf(fact_1139_dr_OpirreducibleE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ) ) ) ).
% dr.pirreducibleE(2)
thf(fact_1140_ds_OpirreducibleE_I2_J,axiom,
! [K: set_b,P: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ K ) @ P )
=> ~ ( member_list_b @ P @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ) ) ) ).
% ds.pirreducibleE(2)
thf(fact_1141_pds_Oconst__term__not__zero,axiom,
! [P: list_list_b] :
( ( ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P )
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( P != nil_list_b ) ) ).
% pds.const_term_not_zero
thf(fact_1142_dr_OpirreducibleE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ K ) @ Q @ R2 ) )
=> ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ K ) ) )
| ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ) ) ) ) ) ) ) ).
% dr.pirreducibleE(3)
thf(fact_1143_ds_OpirreducibleE_I3_J,axiom,
! [K: set_b,P: list_b,Q: list_b,R2: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ K ) @ P )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ R2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( P
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ K ) @ Q @ R2 ) )
=> ( ( member_list_b @ Q @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ K ) ) )
| ( member_list_b @ R2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ) ) ) ) ) ) ) ).
% ds.pirreducibleE(3)
thf(fact_1144_ds_OsubringI,axiom,
! [H: set_b] :
( ( ord_less_eq_set_b @ H @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ ( one_b_ring_ext_b_d @ s ) @ H )
=> ( ! [H3: b] :
( ( member_b @ H3 @ H )
=> ( member_b @ ( a_inv_b_d @ s @ H3 ) @ H ) )
=> ( ! [H12: b,H23: b] :
( ( member_b @ H12 @ H )
=> ( ( member_b @ H23 @ H )
=> ( member_b @ ( mult_b_ring_ext_b_d @ s @ H12 @ H23 ) @ H ) ) )
=> ( ! [H12: b,H23: b] :
( ( member_b @ H12 @ H )
=> ( ( member_b @ H23 @ H )
=> ( member_b @ ( add_b_d @ s @ H12 @ H23 ) @ H ) ) )
=> ( subring_b_d @ H @ s ) ) ) ) ) ) ).
% ds.subringI
thf(fact_1145_dr_OsubringI,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_c @ r ) @ H )
=> ( ! [H3: a] :
( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_c @ r @ H3 ) @ H ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_c @ r @ H12 @ H23 ) @ H ) ) )
=> ( ! [H12: a,H23: a] :
( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( add_a_c @ r @ H12 @ H23 ) @ H ) ) )
=> ( subring_a_c @ H @ r ) ) ) ) ) ) ).
% dr.subringI
thf(fact_1146_pds_Oconst__term__simprules__shell_I1_J,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( member_list_b @ ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P ) @ K ) ) ) ).
% pds.const_term_simprules_shell(1)
thf(fact_1147_dr_OpirreducibleI,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ! [Q2: list_a,R: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ K ) @ Q2 @ R ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ K ) ) )
| ( member_list_a @ R @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ K ) @ P ) ) ) ) ) ) ).
% dr.pirreducibleI
thf(fact_1148_ds_OpirreducibleI,axiom,
! [K: set_b,P: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( P != nil_b )
=> ( ~ ( member_list_b @ P @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ! [Q2: list_b,R: list_b] :
( ( member_list_b @ Q2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ R @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( P
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ K ) @ Q2 @ R ) )
=> ( ( member_list_b @ Q2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ K ) ) )
| ( member_list_b @ R @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ) ) ) )
=> ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ K ) @ P ) ) ) ) ) ) ).
% ds.pirreducibleI
thf(fact_1149_pds_Oconst__term__simprules__shell_I3_J,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( add_li4375960627168867399t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ Q ) )
= ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P ) @ ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Q ) ) ) ) ) ) ).
% pds.const_term_simprules_shell(3)
thf(fact_1150_pds_Oconst__term__simprules__shell_I2_J,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ Q ) )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P ) @ ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Q ) ) ) ) ) ) ).
% pds.const_term_simprules_shell(2)
thf(fact_1151_dr_Ocarrier__polynomial__shell,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% dr.carrier_polynomial_shell
thf(fact_1152_ds_Ocarrier__polynomial__shell,axiom,
! [K: set_b,P: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% ds.carrier_polynomial_shell
thf(fact_1153_pds_OsubdomainI,axiom,
! [H: set_list_b] :
( ( subcri4677462317791934762t_unit @ H @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ! [H12: list_b,H23: list_b] :
( ( member_list_b @ H12 @ H )
=> ( ( member_list_b @ H23 @ H )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H12 @ H23 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( H12
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
| ( H23
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) )
=> ( subdom4735476224308063485t_unit @ H @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.subdomainI
thf(fact_1154_pdr_Oconst__term__simprules__shell_I4_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P ) ) ) ) ) ).
% pdr.const_term_simprules_shell(4)
thf(fact_1155_ds_Oconst__term__not__zero,axiom,
! [P: list_b] :
( ( ( const_term_b_d @ s @ P )
!= ( zero_b_d @ s ) )
=> ( P != nil_b ) ) ).
% ds.const_term_not_zero
thf(fact_1156_dr_Oconst__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_c @ r @ P )
!= ( zero_a_c @ r ) )
=> ( P != nil_a ) ) ).
% dr.const_term_not_zero
thf(fact_1157_dr_Oconst__term__simprules__shell_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( member_a @ ( const_term_a_c @ r @ P ) @ K ) ) ) ).
% dr.const_term_simprules_shell(1)
thf(fact_1158_ds_Oconst__term__simprules__shell_I1_J,axiom,
! [K: set_b,P: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( member_b @ ( const_term_b_d @ s @ P ) @ K ) ) ) ).
% ds.const_term_simprules_shell(1)
thf(fact_1159_pds_Ocarrier__is__subcring,axiom,
subcri4677462317791934762t_unit @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.carrier_is_subcring
thf(fact_1160_pds_OsubcringI_H,axiom,
! [H: set_list_b] :
( ( subrin3833087656135479401t_unit @ H @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( subcri4677462317791934762t_unit @ H @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.subcringI'
thf(fact_1161_dr_Oconst__term__simprules__shell_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( const_term_a_c @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ K ) @ P @ Q ) )
= ( add_a_c @ r @ ( const_term_a_c @ r @ P ) @ ( const_term_a_c @ r @ Q ) ) ) ) ) ) ).
% dr.const_term_simprules_shell(3)
thf(fact_1162_ds_Oconst__term__simprules__shell_I3_J,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( const_term_b_d @ s @ ( add_li4567129529168622413t_unit @ ( univ_poly_b_d @ s @ K ) @ P @ Q ) )
= ( add_b_d @ s @ ( const_term_b_d @ s @ P ) @ ( const_term_b_d @ s @ Q ) ) ) ) ) ) ).
% ds.const_term_simprules_shell(3)
thf(fact_1163_dr_Oconst__term__simprules__shell_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( const_term_a_c @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ K ) @ P @ Q ) )
= ( mult_a_ring_ext_a_c @ r @ ( const_term_a_c @ r @ P ) @ ( const_term_a_c @ r @ Q ) ) ) ) ) ) ).
% dr.const_term_simprules_shell(2)
thf(fact_1164_ds_Oconst__term__simprules__shell_I2_J,axiom,
! [K: set_b,P: list_b,Q: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( const_term_b_d @ s @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ K ) @ P @ Q ) )
= ( mult_b_ring_ext_b_d @ s @ ( const_term_b_d @ s @ P ) @ ( const_term_b_d @ s @ Q ) ) ) ) ) ) ).
% ds.const_term_simprules_shell(2)
thf(fact_1165_pdr_Oconst__term__not__zero,axiom,
! [P: list_list_a] :
( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( P != nil_list_a ) ) ).
% pdr.const_term_not_zero
thf(fact_1166_dr_Oconst__term__simprules__shell_I4_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( const_term_a_c @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ K ) @ P ) )
= ( a_inv_a_c @ r @ ( const_term_a_c @ r @ P ) ) ) ) ) ).
% dr.const_term_simprules_shell(4)
thf(fact_1167_ds_Oconst__term__simprules__shell_I4_J,axiom,
! [K: set_b,P: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( const_term_b_d @ s @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ K ) @ P ) )
= ( a_inv_b_d @ s @ ( const_term_b_d @ s @ P ) ) ) ) ) ).
% ds.const_term_simprules_shell(4)
thf(fact_1168_pds_OsubcringI,axiom,
! [H: set_list_b] :
( ( subrin3833087656135479401t_unit @ H @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ! [H12: list_b,H23: list_b] :
( ( member_list_b @ H12 @ H )
=> ( ( member_list_b @ H23 @ H )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H12 @ H23 )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H23 @ H12 ) ) ) )
=> ( subcri4677462317791934762t_unit @ H @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.subcringI
thf(fact_1169_pdr_Oconst__term__simprules__shell_I1_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P ) @ K ) ) ) ).
% pdr.const_term_simprules_shell(1)
thf(fact_1170_pdr_Oconst__term__simprules__shell_I3_J,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Q ) ) ) ) ) ) ).
% pdr.const_term_simprules_shell(3)
thf(fact_1171_pdr_Oconst__term__simprules__shell_I2_J,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Q ) ) ) ) ) ) ).
% pdr.const_term_simprules_shell(2)
thf(fact_1172_pds_Osubalgebra__in__carrier,axiom,
! [K: set_list_b,V: set_list_b] :
( ( embedd7906597418576622673t_unit @ K @ V @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ord_le8932221534207217157list_b @ V @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.subalgebra_in_carrier
thf(fact_1173_pds_Ocarrier__is__subalgebra,axiom,
! [K: set_list_b] :
( ( ord_le8932221534207217157list_b @ K @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( embedd7906597418576622673t_unit @ K @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.carrier_is_subalgebra
thf(fact_1174_pds_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_list_b,E: set_list_b,V: set_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( embedd7483416153302036030t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K @ E )
=> ( ( embedd7906597418576622673t_unit @ K @ V @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( ord_le8932221534207217157list_b @ V @ E )
=> ( embedd7483416153302036030t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K @ V ) ) ) ) ) ).
% pds.subalbegra_incl_imp_finite_dimension
thf(fact_1175_pdr_Osubalgebra__in__carrier,axiom,
! [K: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.subalgebra_in_carrier
thf(fact_1176_pds_Otelescopic__base__dim_I1_J,axiom,
! [K: set_list_b,F2: set_list_b,E: set_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( subfie7916738691610828529t_unit @ F2 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( embedd7483416153302036030t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K @ F2 )
=> ( ( embedd7483416153302036030t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F2 @ E )
=> ( embedd7483416153302036030t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K @ E ) ) ) ) ) ).
% pds.telescopic_base_dim(1)
thf(fact_1177_pdr_Ocarrier__is__subalgebra,axiom,
! [K: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.carrier_is_subalgebra
thf(fact_1178_pds_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_list_b,E: set_list_b] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( embedd7483416153302036030t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K @ E )
=> ( embedd7906597418576622673t_unit @ K @ E @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.finite_dimension_imp_subalgebra
thf(fact_1179_pdr_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_list_a,E: set_list_a,V: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K @ E )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ V @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K @ V ) ) ) ) ) ).
% pdr.subalbegra_incl_imp_finite_dimension
thf(fact_1180_pds_Opirreducible__pow__pdivides__iff,axiom,
! [K: set_list_b,P: list_list_b,Q: list_list_b,R2: list_list_b,N: nat] :
( ( subfie7916738691610828529t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ Q @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( member_list_list_b @ R2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ring_r4561388045816386823t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P )
=> ( ~ ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ Q )
=> ( ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ N ) @ ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ Q @ R2 ) )
= ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% pds.pirreducible_pow_pdivides_iff
thf(fact_1181_pdr_Otelescopic__base__dim_I1_J,axiom,
! [K: set_list_a,F2: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( subfie1779122896746047282t_unit @ F2 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K @ F2 )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F2 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K @ E ) ) ) ) ) ).
% pdr.telescopic_base_dim(1)
thf(fact_1182_pdr_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K @ E )
=> ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.finite_dimension_imp_subalgebra
thf(fact_1183_pds_Opolynomial__pow__not__zero,axiom,
! [P: list_list_b,N: nat] :
( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( P != nil_list_b )
=> ( ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ P @ N )
!= nil_list_b ) ) ) ).
% pds.polynomial_pow_not_zero
thf(fact_1184_pds_Osubring__polynomial__pow__not__zero,axiom,
! [K: set_list_b,P: list_list_b,N: nat] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( P != nil_list_b )
=> ( ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ N )
!= nil_list_b ) ) ) ) ).
% pds.subring_polynomial_pow_not_zero
thf(fact_1185_pds_Oalg__multE_I1_J,axiom,
! [X: list_b,P: list_list_b] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( P != nil_list_b )
=> ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ ( cons_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( cons_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ nil_list_b ) ) @ ( polyno1173882569968769117t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ X ) ) @ P ) ) ) ) ).
% pds.alg_multE(1)
thf(fact_1186_pds_Osubring__polynomial__pow__division,axiom,
! [K: set_list_b,P: list_list_b,N: nat,M2: nat] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( factor8908767930780902896t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ N ) @ ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ M2 ) ) ) ) ) ).
% pds.subring_polynomial_pow_division
thf(fact_1187_dr_OUnits__pow__closed,axiom,
! [X: a,D: nat] :
( ( member_a @ X @ ( units_a_ring_ext_a_c @ r ) )
=> ( member_a @ ( pow_a_6622632918335387297_c_nat @ r @ X @ D ) @ ( units_a_ring_ext_a_c @ r ) ) ) ).
% dr.Units_pow_closed
thf(fact_1188_ds_OUnits__pow__closed,axiom,
! [X: b,D: nat] :
( ( member_b @ X @ ( units_b_ring_ext_b_d @ s ) )
=> ( member_b @ ( pow_b_8026974530307897442_d_nat @ s @ X @ D ) @ ( units_b_ring_ext_b_d @ s ) ) ) ).
% ds.Units_pow_closed
thf(fact_1189_dr_Opow__non__zero,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( X
!= ( zero_a_c @ r ) )
=> ( ( pow_a_6622632918335387297_c_nat @ r @ X @ N )
!= ( zero_a_c @ r ) ) ) ) ).
% dr.pow_non_zero
thf(fact_1190_ds_Opow__non__zero,axiom,
! [X: b,N: nat] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( X
!= ( zero_b_d @ s ) )
=> ( ( pow_b_8026974530307897442_d_nat @ s @ X @ N )
!= ( zero_b_d @ s ) ) ) ) ).
% ds.pow_non_zero
thf(fact_1191_dr_Ogroup__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_c @ r @ X @ Y )
= ( mult_a_ring_ext_a_c @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ ( pow_a_6622632918335387297_c_nat @ r @ X @ N ) @ Y )
= ( mult_a_ring_ext_a_c @ r @ Y @ ( pow_a_6622632918335387297_c_nat @ r @ X @ N ) ) ) ) ) ) ).
% dr.group_commutes_pow
thf(fact_1192_dr_Onat__pow__comm,axiom,
! [X: a,N: nat,M2: nat] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ ( pow_a_6622632918335387297_c_nat @ r @ X @ N ) @ ( pow_a_6622632918335387297_c_nat @ r @ X @ M2 ) )
= ( mult_a_ring_ext_a_c @ r @ ( pow_a_6622632918335387297_c_nat @ r @ X @ M2 ) @ ( pow_a_6622632918335387297_c_nat @ r @ X @ N ) ) ) ) ).
% dr.nat_pow_comm
thf(fact_1193_dr_Onat__pow__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( pow_a_6622632918335387297_c_nat @ r @ ( mult_a_ring_ext_a_c @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_c @ r @ ( pow_a_6622632918335387297_c_nat @ r @ X @ N ) @ ( pow_a_6622632918335387297_c_nat @ r @ Y @ N ) ) ) ) ) ).
% dr.nat_pow_distrib
thf(fact_1194_dr_Opow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_c @ r @ X @ Y )
= ( mult_a_ring_ext_a_c @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( pow_a_6622632918335387297_c_nat @ r @ ( mult_a_ring_ext_a_c @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_c @ r @ ( pow_a_6622632918335387297_c_nat @ r @ X @ N ) @ ( pow_a_6622632918335387297_c_nat @ r @ Y @ N ) ) ) ) ) ) ).
% dr.pow_mult_distrib
thf(fact_1195_ds_Opow__mult__distrib,axiom,
! [X: b,Y: b,N: nat] :
( ( ( mult_b_ring_ext_b_d @ s @ X @ Y )
= ( mult_b_ring_ext_b_d @ s @ Y @ X ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( pow_b_8026974530307897442_d_nat @ s @ ( mult_b_ring_ext_b_d @ s @ X @ Y ) @ N )
= ( mult_b_ring_ext_b_d @ s @ ( pow_b_8026974530307897442_d_nat @ s @ X @ N ) @ ( pow_b_8026974530307897442_d_nat @ s @ Y @ N ) ) ) ) ) ) ).
% ds.pow_mult_distrib
thf(fact_1196_ds_Onat__pow__distrib,axiom,
! [X: b,Y: b,N: nat] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( pow_b_8026974530307897442_d_nat @ s @ ( mult_b_ring_ext_b_d @ s @ X @ Y ) @ N )
= ( mult_b_ring_ext_b_d @ s @ ( pow_b_8026974530307897442_d_nat @ s @ X @ N ) @ ( pow_b_8026974530307897442_d_nat @ s @ Y @ N ) ) ) ) ) ).
% ds.nat_pow_distrib
thf(fact_1197_ds_Onat__pow__comm,axiom,
! [X: b,N: nat,M2: nat] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ ( pow_b_8026974530307897442_d_nat @ s @ X @ N ) @ ( pow_b_8026974530307897442_d_nat @ s @ X @ M2 ) )
= ( mult_b_ring_ext_b_d @ s @ ( pow_b_8026974530307897442_d_nat @ s @ X @ M2 ) @ ( pow_b_8026974530307897442_d_nat @ s @ X @ N ) ) ) ) ).
% ds.nat_pow_comm
thf(fact_1198_ds_Ogroup__commutes__pow,axiom,
! [X: b,Y: b,N: nat] :
( ( ( mult_b_ring_ext_b_d @ s @ X @ Y )
= ( mult_b_ring_ext_b_d @ s @ Y @ X ) )
=> ( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ ( pow_b_8026974530307897442_d_nat @ s @ X @ N ) @ Y )
= ( mult_b_ring_ext_b_d @ s @ Y @ ( pow_b_8026974530307897442_d_nat @ s @ X @ N ) ) ) ) ) ) ).
% ds.group_commutes_pow
thf(fact_1199_pdr_OUnits__pow__closed,axiom,
! [X: list_a,D: nat] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ D ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.Units_pow_closed
thf(fact_1200_pds_OUnits__pow__closed,axiom,
! [X: list_b,D: nat] :
( ( member_list_b @ X @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ D ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.Units_pow_closed
thf(fact_1201_pdr_Opow__non__zero,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( X
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ N )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.pow_non_zero
thf(fact_1202_pds_Opow__non__zero,axiom,
! [X: list_b,N: nat] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( X
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ N )
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.pow_non_zero
thf(fact_1203_pdr_Ogroup__commutes__pow,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ N ) @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ N ) ) ) ) ) ) ).
% pdr.group_commutes_pow
thf(fact_1204_pdr_Onat__pow__comm,axiom,
! [X: list_a,N: nat,M2: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ M2 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ M2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ N ) ) ) ) ).
% pdr.nat_pow_comm
thf(fact_1205_pdr_Onat__pow__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ N ) ) ) ) ) ).
% pdr.nat_pow_distrib
thf(fact_1206_pdr_Opow__mult__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y @ N ) ) ) ) ) ) ).
% pdr.pow_mult_distrib
thf(fact_1207_pds_Opow__mult__distrib,axiom,
! [X: list_b,Y: list_b,N: nat] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ X ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) @ N )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ N ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ N ) ) ) ) ) ) ).
% pds.pow_mult_distrib
thf(fact_1208_pds_Onat__pow__distrib,axiom,
! [X: list_b,Y: list_b,N: nat] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y ) @ N )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ N ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ N ) ) ) ) ) ).
% pds.nat_pow_distrib
thf(fact_1209_pds_Onat__pow__comm,axiom,
! [X: list_b,N: nat,M2: nat] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ N ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ M2 ) )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ M2 ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ N ) ) ) ) ).
% pds.nat_pow_comm
thf(fact_1210_pds_Ogroup__commutes__pow,axiom,
! [X: list_b,Y: list_b,N: nat] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ Y )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ X ) )
=> ( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ N ) @ Y )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ N ) ) ) ) ) ) ).
% pds.group_commutes_pow
thf(fact_1211_dr_Opolynomial__pow__not__zero,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ N )
!= nil_a ) ) ) ).
% dr.polynomial_pow_not_zero
thf(fact_1212_ds_Opolynomial__pow__not__zero,axiom,
! [P: list_b,N: nat] :
( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( P != nil_b )
=> ( ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ N )
!= nil_b ) ) ) ).
% ds.polynomial_pow_not_zero
thf(fact_1213_h_Ohom__nat__pow,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( h @ ( pow_a_6622632918335387297_c_nat @ r @ X @ N ) )
= ( pow_b_8026974530307897442_d_nat @ s @ ( h @ X ) @ N ) ) ) ).
% h.hom_nat_pow
thf(fact_1214_dr_Osubring__polynomial__pow__not__zero,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ K ) @ P @ N )
!= nil_a ) ) ) ) ).
% dr.subring_polynomial_pow_not_zero
thf(fact_1215_ds_Osubring__polynomial__pow__not__zero,axiom,
! [K: set_b,P: list_b,N: nat] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( P != nil_b )
=> ( ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ K ) @ P @ N )
!= nil_b ) ) ) ) ).
% ds.subring_polynomial_pow_not_zero
thf(fact_1216_dr_Opolynomial__pow__division,axiom,
! [P: list_a,N: nat,M2: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( polyno5814909790663948099es_a_c @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ M2 ) ) ) ) ).
% dr.polynomial_pow_division
thf(fact_1217_ds_Opolynomial__pow__division,axiom,
! [P: list_b,N: nat,M2: nat] :
( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( polyno3027454208691272067es_b_d @ s @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ N ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ M2 ) ) ) ) ).
% ds.polynomial_pow_division
thf(fact_1218_dr_Osubring__polynomial__pow__division,axiom,
! [K: set_a,P: list_a,N: nat,M2: nat] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ K ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ K ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ K ) @ P @ M2 ) ) ) ) ) ).
% dr.subring_polynomial_pow_division
thf(fact_1219_ds_Osubring__polynomial__pow__division,axiom,
! [K: set_b,P: list_b,N: nat,M2: nat] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ K ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ K ) @ P @ N ) @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ K ) @ P @ M2 ) ) ) ) ) ).
% ds.subring_polynomial_pow_division
thf(fact_1220_pdr_Opolynomial__pow__not__zero,axiom,
! [P: list_list_a,N: nat] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ P @ N )
!= nil_list_a ) ) ) ).
% pdr.polynomial_pow_not_zero
thf(fact_1221_pdr_Osubring__polynomial__pow__not__zero,axiom,
! [K: set_list_a,P: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ N )
!= nil_list_a ) ) ) ) ).
% pdr.subring_polynomial_pow_not_zero
thf(fact_1222_dr_Opirreducible__pow__pdivides__iff,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a,N: nat] :
( ( subfield_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ K ) @ P )
=> ( ~ ( polyno5814909790663948099es_a_c @ r @ P @ Q )
=> ( ( polyno5814909790663948099es_a_c @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ K ) @ Q @ R2 ) )
= ( polyno5814909790663948099es_a_c @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% dr.pirreducible_pow_pdivides_iff
thf(fact_1223_ds_Opirreducible__pow__pdivides__iff,axiom,
! [K: set_b,P: list_b,Q: list_b,R2: list_b,N: nat] :
( ( subfield_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ Q @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( member_list_b @ R2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ K ) @ P )
=> ( ~ ( polyno3027454208691272067es_b_d @ s @ P @ Q )
=> ( ( polyno3027454208691272067es_b_d @ s @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ K ) @ P @ N ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ K ) @ Q @ R2 ) )
= ( polyno3027454208691272067es_b_d @ s @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% ds.pirreducible_pow_pdivides_iff
thf(fact_1224_pdr_Opolynomial__pow__division,axiom,
! [P: list_list_a,N: nat,M2: nat] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ P @ M2 ) ) ) ) ).
% pdr.polynomial_pow_division
thf(fact_1225_pdr_Osubring__polynomial__pow__division,axiom,
! [K: set_list_a,P: list_list_a,N: nat,M2: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ M2 ) ) ) ) ) ).
% pdr.subring_polynomial_pow_division
thf(fact_1226_dr_Onat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ ( pow_a_6622632918335387297_c_nat @ r @ X @ N ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% dr.nat_pow_closed
thf(fact_1227_ds_Onat__pow__closed,axiom,
! [X: b,N: nat] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ ( pow_b_8026974530307897442_d_nat @ s @ X @ N ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% ds.nat_pow_closed
thf(fact_1228_pds_Opolynomial__pow__division,axiom,
! [P: list_list_b,N: nat,M2: nat] :
( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ P @ N ) @ ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ P @ M2 ) ) ) ) ).
% pds.polynomial_pow_division
thf(fact_1229_ds_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_b_8026974530307897442_d_nat @ s @ ( one_b_ring_ext_b_d @ s ) @ N )
= ( one_b_ring_ext_b_d @ s ) ) ).
% ds.nat_pow_one
thf(fact_1230_dr_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_a_6622632918335387297_c_nat @ r @ ( one_a_ring_ext_a_c @ r ) @ N )
= ( one_a_ring_ext_a_c @ r ) ) ).
% dr.nat_pow_one
thf(fact_1231_pdr_Opirreducible__pow__pdivides__iff,axiom,
! [K: set_list_a,P: list_list_a,Q: list_list_a,R2: list_list_a,N: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P )
=> ( ~ ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ Q )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ Q @ R2 ) )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% pdr.pirreducible_pow_pdivides_iff
thf(fact_1232_pds_Ole__alg__mult__imp__pdivides,axiom,
! [X: list_b,P: list_list_b,N: nat] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno1173882569968769117t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ X ) )
=> ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ ( cons_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( cons_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ nil_list_b ) ) @ N ) @ P ) ) ) ) ).
% pds.le_alg_mult_imp_pdivides
thf(fact_1233_pds_Oalg__multE_I2_J,axiom,
! [X: list_b,P: list_list_b,N: nat] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( P != nil_list_b )
=> ( ( polyno4931040496010026249t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ ( cons_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( cons_list_b @ ( a_inv_5858964851304622612t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X ) @ nil_list_b ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno1173882569968769117t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P @ X ) ) ) ) ) ) ).
% pds.alg_multE(2)
thf(fact_1234_pdr_Onat__pow__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.nat_pow_closed
thf(fact_1235_pds_Onat__pow__closed,axiom,
! [X: list_b,N: nat] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ N ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.nat_pow_closed
thf(fact_1236_pdr_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ N )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.nat_pow_one
thf(fact_1237_pds_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ N )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.nat_pow_one
thf(fact_1238_pdr_Ole__alg__mult__imp__pdivides,axiom,
! [X: list_a,P: list_list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ X ) )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ nil_list_a ) ) @ N ) @ P ) ) ) ) ).
% pdr.le_alg_mult_imp_pdivides
thf(fact_1239_pdr_Oalg__multE_I2_J,axiom,
! [X: list_a,P: list_list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ nil_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ X ) ) ) ) ) ) ).
% pdr.alg_multE(2)
thf(fact_1240_ds_Oalg__multE_I1_J,axiom,
! [X: b,P: list_b] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( P != nil_b )
=> ( polyno3027454208691272067es_b_d @ s @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( cons_b @ ( one_b_ring_ext_b_d @ s ) @ ( cons_b @ ( a_inv_b_d @ s @ X ) @ nil_b ) ) @ ( polyno1634975279954809559lt_b_d @ s @ P @ X ) ) @ P ) ) ) ) ).
% ds.alg_multE(1)
thf(fact_1241_dr_Oalg__multE_I1_J,axiom,
! [X: a,P: list_a] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948099es_a_c @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_c @ r ) @ ( cons_a @ ( a_inv_a_c @ r @ X ) @ nil_a ) ) @ ( polyno4422430861927485591lt_a_c @ r @ P @ X ) ) @ P ) ) ) ) ).
% dr.alg_multE(1)
thf(fact_1242_ds_Ole__alg__mult__imp__pdivides,axiom,
! [X: b,P: list_b,N: nat] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno1634975279954809559lt_b_d @ s @ P @ X ) )
=> ( polyno3027454208691272067es_b_d @ s @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( cons_b @ ( one_b_ring_ext_b_d @ s ) @ ( cons_b @ ( a_inv_b_d @ s @ X ) @ nil_b ) ) @ N ) @ P ) ) ) ) ).
% ds.le_alg_mult_imp_pdivides
thf(fact_1243_ds_Oalg__multE_I2_J,axiom,
! [X: b,P: list_b,N: nat] :
( ( member_b @ X @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( P != nil_b )
=> ( ( polyno3027454208691272067es_b_d @ s @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( cons_b @ ( one_b_ring_ext_b_d @ s ) @ ( cons_b @ ( a_inv_b_d @ s @ X ) @ nil_b ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno1634975279954809559lt_b_d @ s @ P @ X ) ) ) ) ) ) ).
% ds.alg_multE(2)
thf(fact_1244_dr_Ole__alg__mult__imp__pdivides,axiom,
! [X: a,P: list_a,N: nat] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485591lt_a_c @ r @ P @ X ) )
=> ( polyno5814909790663948099es_a_c @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_c @ r ) @ ( cons_a @ ( a_inv_a_c @ r @ X ) @ nil_a ) ) @ N ) @ P ) ) ) ) ).
% dr.le_alg_mult_imp_pdivides
thf(fact_1245_dr_Oalg__multE_I2_J,axiom,
! [X: a,P: list_a,N: nat] :
( ( member_a @ X @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno5814909790663948099es_a_c @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_c @ r ) @ ( cons_a @ ( a_inv_a_c @ r @ X ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485591lt_a_c @ r @ P @ X ) ) ) ) ) ) ).
% dr.alg_multE(2)
thf(fact_1246_pdr_Oalg__multE_I1_J,axiom,
! [X: list_a,P: list_list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( P != nil_list_a )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P @ X ) ) @ P ) ) ) ) ).
% pdr.alg_multE(1)
thf(fact_1247_pds_Ovar__pow__closed,axiom,
! [K: set_list_b,N: nat] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( member_list_list_b @ ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ ( var_li5368196932703410780t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ N ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ) ).
% pds.var_pow_closed
thf(fact_1248_pdr_Ovar__pow__closed,axiom,
! [K: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ) ).
% pdr.var_pow_closed
thf(fact_1249_pdr_Ovar__closed_I1_J,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) ) ) ).
% pdr.var_closed(1)
thf(fact_1250_pds_Ovar__closed_I1_J,axiom,
! [K: set_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( member_list_list_b @ ( var_li5368196932703410780t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) ) ) ).
% pds.var_closed(1)
thf(fact_1251_pds_Ounitary__monom__eq__var__pow,axiom,
! [K: set_list_b,N: nat] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( monom_4360707845066158047t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ N )
= ( pow_li8767036347513248878it_nat @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ ( var_li5368196932703410780t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ N ) ) ) ).
% pds.unitary_monom_eq_var_pow
thf(fact_1252_pdr_Ounitary__monom__eq__var__pow,axiom,
! [K: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( monom_7446464087056152608t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ N )
= ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ N ) ) ) ).
% pdr.unitary_monom_eq_var_pow
thf(fact_1253_dr_Ovar__closed_I1_J,axiom,
! [K: set_a] :
( ( subring_a_c @ K @ r )
=> ( member_list_a @ ( var_a_c @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ) ).
% dr.var_closed(1)
thf(fact_1254_ds_Ovar__closed_I1_J,axiom,
! [K: set_b] :
( ( subring_b_d @ K @ s )
=> ( member_list_b @ ( var_b_d @ s ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ) ).
% ds.var_closed(1)
thf(fact_1255_dr_Ovar__pow__closed,axiom,
! [K: set_a,N: nat] :
( ( subring_a_c @ K @ r )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ K ) @ ( var_a_c @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) ) ) ).
% dr.var_pow_closed
thf(fact_1256_ds_Ovar__pow__closed,axiom,
! [K: set_b,N: nat] :
( ( subring_b_d @ K @ s )
=> ( member_list_b @ ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ K ) @ ( var_b_d @ s ) @ N ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) ) ) ).
% ds.var_pow_closed
thf(fact_1257_pds_Opoly__mult__var,axiom,
! [K: set_list_b,P: list_list_b] :
( ( subrin3833087656135479401t_unit @ K @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_list_b @ P @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) ) )
=> ( ( ( P = nil_list_b )
=> ( ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ ( var_li5368196932703410780t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= nil_list_b ) )
& ( ( P != nil_list_b )
=> ( ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ K ) @ P @ ( var_li5368196932703410780t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( append_list_b @ P @ ( cons_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ nil_list_b ) ) ) ) ) ) ) ).
% pds.poly_mult_var
thf(fact_1258_pdr_Opoly__mult__var,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) ) )
=> ( ( ( P = nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= nil_list_a ) )
& ( ( P != nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ K ) @ P @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ) ).
% pdr.poly_mult_var
thf(fact_1259_ds_Ounitary__monom__eq__var__pow,axiom,
! [K: set_b,N: nat] :
( ( subring_b_d @ K @ s )
=> ( ( monom_b_d @ s @ ( one_b_ring_ext_b_d @ s ) @ N )
= ( pow_li3934666113891012974it_nat @ ( univ_poly_b_d @ s @ K ) @ ( var_b_d @ s ) @ N ) ) ) ).
% ds.unitary_monom_eq_var_pow
thf(fact_1260_dr_Ounitary__monom__eq__var__pow,axiom,
! [K: set_a,N: nat] :
( ( subring_a_c @ K @ r )
=> ( ( monom_a_c @ r @ ( one_a_ring_ext_a_c @ r ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_c @ r @ K ) @ ( var_a_c @ r ) @ N ) ) ) ).
% dr.unitary_monom_eq_var_pow
thf(fact_1261_pds_Oconst__term__eq__last,axiom,
! [P: list_list_b,A: list_b] :
( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ P ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( append_list_b @ P @ ( cons_list_b @ A @ nil_list_b ) ) )
= A ) ) ) ).
% pds.const_term_eq_last
thf(fact_1262_pds_Oconst__term__explicit,axiom,
! [P: list_list_b,A: list_b] :
( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ P ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( P != nil_list_b )
=> ( ( ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P )
= A )
=> ~ ! [P3: list_list_b] :
( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ P3 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( P
!= ( append_list_b @ P3 @ ( cons_list_b @ A @ nil_list_b ) ) ) ) ) ) ) ).
% pds.const_term_explicit
thf(fact_1263_pds_Oconst__term__simprules_I1_J,axiom,
! [P: list_list_b] :
( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ P ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( const_3652410027514832260t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.const_term_simprules(1)
thf(fact_1264_pds_Ofactors__closed,axiom,
! [Fs: list_list_b,A: list_b] :
( ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fs @ A )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Fs ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.factors_closed
thf(fact_1265_pds_Ofactors__dividesI,axiom,
! [Fs: list_list_b,A: list_b,F: list_b] :
( ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fs @ A )
=> ( ( member_list_b @ F @ ( set_list_b2 @ Fs ) )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Fs ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor5707471399880574448t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ F @ A ) ) ) ) ).
% pds.factors_dividesI
thf(fact_1266_dr_Opoly__mult__var,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_c @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ K ) ) )
=> ( ( ( P = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ K ) @ P @ ( var_a_c @ r ) )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ K ) @ P @ ( var_a_c @ r ) )
= ( append_a @ P @ ( cons_a @ ( zero_a_c @ r ) @ nil_a ) ) ) ) ) ) ) ).
% dr.poly_mult_var
thf(fact_1267_ds_Opoly__mult__var,axiom,
! [K: set_b,P: list_b] :
( ( subring_b_d @ K @ s )
=> ( ( member_list_b @ P @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ K ) ) )
=> ( ( ( P = nil_b )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ K ) @ P @ ( var_b_d @ s ) )
= nil_b ) )
& ( ( P != nil_b )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ K ) @ P @ ( var_b_d @ s ) )
= ( append_b @ P @ ( cons_b @ ( zero_b_d @ s ) @ nil_b ) ) ) ) ) ) ) ).
% ds.poly_mult_var
thf(fact_1268_pds_Ofactors__mult,axiom,
! [Fa: list_list_b,A: list_b,Fb: list_list_b,B: list_b] :
( ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fa @ A )
=> ( ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fb @ B )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Fa ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Fb ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( append_list_b @ Fa @ Fb ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ B ) ) ) ) ) ) ).
% pds.factors_mult
thf(fact_1269_pds_Omonom__in__carrier,axiom,
! [A: list_b,N: nat] :
( ( member_list_b @ A @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ord_le8932221534207217157list_b @ ( set_list_b2 @ ( monom_4360707845066158047t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A @ N ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.monom_in_carrier
thf(fact_1270_pds_Oexp__base__closed,axiom,
! [X: list_b,N: nat] :
( ( member_list_b @ X @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ord_le8932221534207217157list_b @ ( set_list_b2 @ ( polyno437060639131926335t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ N ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.exp_base_closed
thf(fact_1271_pdr_Oconst__term__explicit,axiom,
! [P: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P )
= A )
=> ~ ! [P3: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( P
!= ( append_list_a @ P3 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ).
% pdr.const_term_explicit
thf(fact_1272_pdr_Oconst__term__simprules_I1_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.const_term_simprules(1)
thf(fact_1273_pdr_Ofactors__closed,axiom,
! [Fs: list_list_a,A: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fs @ A )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.factors_closed
thf(fact_1274_pdr_Ofactors__dividesI,axiom,
! [Fs: list_list_a,A: list_a,F: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fs @ A )
=> ( ( member_list_a @ F @ ( set_list_a2 @ Fs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ F @ A ) ) ) ) ).
% pdr.factors_dividesI
thf(fact_1275_pdr_Ofactors__mult,axiom,
! [Fa: list_list_a,A: list_a,Fb: list_list_a,B: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fa @ A )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fb @ B )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fa ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fb ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( append_list_a @ Fa @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ B ) ) ) ) ) ) ).
% pdr.factors_mult
thf(fact_1276_pdr_Oconst__term__eq__last,axiom,
! [P: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ A @ nil_list_a ) ) )
= A ) ) ) ).
% pdr.const_term_eq_last
thf(fact_1277_pdr_Omonom__in__carrier,axiom,
! [A: list_a,N: nat] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.monom_in_carrier
thf(fact_1278_pdr_Oexp__base__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.exp_base_closed
% Conjectures (1)
thf(conj_0,conjecture,
( ( h @ xh )
!= ( zero_b_d @ s ) ) ).
%------------------------------------------------------------------------------