TPTP Problem File: SLH0206^1.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/0006_Formal_Polynomial_Derivatives/prob_00071_002055__18243460_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1595 ( 185 unt; 316 typ; 0 def)
% Number of atoms : 4921 (1479 equ; 0 cnn)
% Maximal formula atoms : 17 ( 3 avg)
% Number of connectives : 20838 ( 318 ~; 65 |; 85 &;17026 @)
% ( 0 <=>;3344 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 9 avg)
% Number of types : 40 ( 39 usr)
% Number of type conns : 733 ( 733 >; 0 *; 0 +; 0 <<)
% Number of symbols : 280 ( 277 usr; 19 con; 0-4 aty)
% Number of variables : 3718 ( 53 ^;3601 !; 64 ?;3718 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:21:54.996
%------------------------------------------------------------------------------
% Could-be-implicit typings (39)
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thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subcri8676831449680469861t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subcri7783154434480317835t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subdom561091866123308472t_unit: set_list_list_a > partia2956882679547061052t_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__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subdom3220114454046903646t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_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__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subfie4339374749748326226t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subrin3541368690557094692t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subrin5643252653130547402t_unit: set_set_list_a > partia7496981018696276118t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
bound_list_list_a: list_list_a > nat > ( nat > list_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bound_set_list_a: set_list_a > nat > ( nat > set_list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_lis8963924889346801084t_unit: partia2956882679547061052t_unit > set_nat_list_list_a ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_list_a ).
thf(sy_c_UnivPoly_Oup_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
up_set529185716248919906t_unit: partia7496981018696276118t_unit > set_nat_set_list_a ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__b,type,
up_a_b: partia2175431115845679010xt_a_b > set_nat_a ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member7168557129179038582list_a: ( list_list_a > list_a ) > set_li3422455791611400469list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_list_list_a_a: ( list_list_a > a ) > set_list_list_a_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8650753269014980122list_a: ( nat > list_list_a ) > set_nat_list_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
member_nat_list_a: ( nat > list_a ) > set_nat_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member491565700723299188list_a: ( nat > set_list_a ) > set_nat_set_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5524387281408368019list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_K,type,
k: set_a ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_d____,type,
d: list_a ).
thf(sy_v_f,type,
f: list_a ).
thf(sy_v_j____,type,
j: nat ).
thf(sy_v_k,type,
k2: nat ).
% Relevant facts (1271)
thf(fact_0_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1__092_060open_062local_Ocoeff_A_Ilocal_Onormalize_Ad_J_Ak_A_061_Alocal_Ocoeff_Ad_Ak_092_060close_062,axiom,
( ( coeff_a_b @ r @ ( normalize_a_b @ r @ d ) @ k2 )
= ( coeff_a_b @ r @ d @ k2 ) ) ).
% \<open>local.coeff (local.normalize d) k = local.coeff d k\<close>
thf(fact_2_calculation,axiom,
( ( coeff_a_b @ r @ ( formal4452980811800949548iv_a_b @ r @ f ) @ k2 )
= ( coeff_a_b @ r @ d @ k2 ) ) ).
% calculation
thf(fact_3_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_4_coeff__iff__length__cond,axiom,
! [P1: list_a,P2: list_a] :
( ( ( size_size_list_a @ P1 )
= ( size_size_list_a @ P2 ) )
=> ( ( P1 = P2 )
= ( ( coeff_a_b @ r @ P1 )
= ( coeff_a_b @ r @ P2 ) ) ) ) ).
% coeff_iff_length_cond
thf(fact_5_assms_I1_J,axiom,
subring_a_b @ k @ r ).
% assms(1)
thf(fact_6_coeff__iff__polynomial__cond,axiom,
! [K: set_a,P1: list_a,P2: list_a] :
( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ( P1 = P2 )
= ( ( coeff_a_b @ r @ P1 )
= ( coeff_a_b @ r @ P2 ) ) ) ) ) ).
% coeff_iff_polynomial_cond
thf(fact_7_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_8_normalize__coeff,axiom,
! [P: list_a] :
( ( coeff_a_b @ r @ P )
= ( coeff_a_b @ r @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_coeff
thf(fact_9_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_10_is__abelian__group,axiom,
abelian_group_a_b @ r ).
% is_abelian_group
thf(fact_11__092_060open_062local_Ocoeff_A_Ipderiv_Af_J_Ak_A_061_Alocal_Ocoeff_A_Ilocal_Onormalize_Ad_J_Ak_092_060close_062,axiom,
( ( coeff_a_b @ r @ ( formal4452980811800949548iv_a_b @ r @ f ) @ k2 )
= ( coeff_a_b @ r @ ( normalize_a_b @ r @ d ) @ k2 ) ) ).
% \<open>local.coeff (pderiv f) k = local.coeff (local.normalize d) k\<close>
thf(fact_12_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_13_normalize__polynomial,axiom,
! [K: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K @ P )
=> ( ( normalize_a_b @ r @ P )
= P ) ) ).
% normalize_polynomial
thf(fact_14_c,axiom,
ord_less_nat @ k2 @ ( size_size_list_a @ d ) ).
% c
thf(fact_15_subdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subdomain_a_b @ H @ r ) ) ).
% subdomainI'
thf(fact_16_poly__coeff__in__carrier,axiom,
! [K: set_a,P: list_a,I: nat] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( member_a @ ( coeff_a_b @ r @ P @ I ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_coeff_in_carrier
thf(fact_17_var__closed_I2_J,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( polynomial_a_b @ r @ K @ ( var_a_b @ r ) ) ) ).
% var_closed(2)
thf(fact_18_ring_Onormalize__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ P )
= ( coeff_a_b @ R @ ( normalize_a_b @ R @ P ) ) ) ) ).
% ring.normalize_coeff
thf(fact_19_ring_Onormalize__coeff,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ P )
= ( coeff_6360649920519955023t_unit @ R @ ( normal637505603836502915t_unit @ R @ P ) ) ) ) ).
% ring.normalize_coeff
thf(fact_20_ring_Ocoeff__iff__polynomial__cond,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P1: list_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K @ P1 )
=> ( ( polyno1315193887021588240t_unit @ R @ K @ P2 )
=> ( ( P1 = P2 )
= ( ( coeff_6360649920519955023t_unit @ R @ P1 )
= ( coeff_6360649920519955023t_unit @ R @ P2 ) ) ) ) ) ) ).
% ring.coeff_iff_polynomial_cond
thf(fact_21_ring_Ocoeff__iff__polynomial__cond,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P1: list_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K @ P1 )
=> ( ( polynomial_a_b @ R @ K @ P2 )
=> ( ( P1 = P2 )
= ( ( coeff_a_b @ R @ P1 )
= ( coeff_a_b @ R @ P2 ) ) ) ) ) ) ).
% ring.coeff_iff_polynomial_cond
thf(fact_22_ring_Onormalize__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( polynomial_a_b @ R @ K @ P )
=> ( ( normalize_a_b @ R @ P )
= P ) ) ) ).
% ring.normalize_polynomial
thf(fact_23_ring_Onormalize__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K @ P )
=> ( ( normal637505603836502915t_unit @ R @ P )
= P ) ) ) ).
% ring.normalize_polynomial
thf(fact_24_normalize__length__le,axiom,
! [P: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) ) ).
% normalize_length_le
thf(fact_25_ring_Ocoeff__iff__length__cond,axiom,
! [R: partia2670972154091845814t_unit,P1: list_list_a,P2: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( size_s349497388124573686list_a @ P1 )
= ( size_s349497388124573686list_a @ P2 ) )
=> ( ( P1 = P2 )
= ( ( coeff_6360649920519955023t_unit @ R @ P1 )
= ( coeff_6360649920519955023t_unit @ R @ P2 ) ) ) ) ) ).
% ring.coeff_iff_length_cond
thf(fact_26_ring_Ocoeff__iff__length__cond,axiom,
! [R: partia2175431115845679010xt_a_b,P1: list_a,P2: list_a] :
( ( ring_a_b @ R )
=> ( ( ( size_size_list_a @ P1 )
= ( size_size_list_a @ P2 ) )
=> ( ( P1 = P2 )
= ( ( coeff_a_b @ R @ P1 )
= ( coeff_a_b @ R @ P2 ) ) ) ) ) ).
% ring.coeff_iff_length_cond
thf(fact_27_ee__length,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( size_size_list_a @ As )
= ( size_size_list_a @ Bs ) ) ) ).
% ee_length
thf(fact_28_subcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subcring_a_b @ H @ r ) ) ).
% subcringI'
thf(fact_29_dense__repr__normalize,axiom,
! [P: list_a] :
( ( dense_repr_a_b @ r @ ( normalize_a_b @ r @ P ) )
= ( dense_repr_a_b @ r @ P ) ) ).
% dense_repr_normalize
thf(fact_30_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_31_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A2: a,B2: a] :
( ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A2 @ B2 ) )
=> ( ( member_a @ ( F @ A2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A2 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_32_associated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_33_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_34_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_35_carrier__polynomial,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P ) ) ) ).
% carrier_polynomial
thf(fact_36_monom__decomp,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( P
= ( poly_of_dense_a_b @ r @ ( dense_repr_a_b @ r @ P ) ) ) ) ) ).
% monom_decomp
thf(fact_37_ring_Odense__repr_Ocong,axiom,
dense_repr_a_b = dense_repr_a_b ).
% ring.dense_repr.cong
thf(fact_38_ring_Odense__repr__normalize,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( dense_5814815041220002634t_unit @ R @ ( normal637505603836502915t_unit @ R @ P ) )
= ( dense_5814815041220002634t_unit @ R @ P ) ) ) ).
% ring.dense_repr_normalize
thf(fact_39_ring_Odense__repr__normalize,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( dense_repr_a_b @ R @ ( normalize_a_b @ R @ P ) )
= ( dense_repr_a_b @ R @ P ) ) ) ).
% ring.dense_repr_normalize
thf(fact_40_ring_Onormalize__length__le,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ R @ P ) ) @ ( size_size_list_a @ P ) ) ) ).
% ring.normalize_length_le
thf(fact_41_ring_Onormalize__length__le,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ R @ P ) ) @ ( size_s349497388124573686list_a @ P ) ) ) ).
% ring.normalize_length_le
thf(fact_42_ring_Ocarrier__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( polynomial_a_b @ R @ K @ P )
=> ( polynomial_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) @ P ) ) ) ) ).
% ring.carrier_polynomial
thf(fact_43_ring_Ocarrier__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K @ P )
=> ( polyno1315193887021588240t_unit @ R @ ( partia5361259788508890537t_unit @ R ) @ P ) ) ) ) ).
% ring.carrier_polynomial
thf(fact_44_ring_Ocarrier__polynomial,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( polyno42001669949040266t_unit @ R @ K @ P )
=> ( polyno42001669949040266t_unit @ R @ ( partia2464479390973590831t_unit @ R ) @ P ) ) ) ) ).
% ring.carrier_polynomial
thf(fact_45_domain_Ovar__closed_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( polyno1315193887021588240t_unit @ R @ K @ ( var_li8453953174693405341t_unit @ R ) ) ) ) ).
% domain.var_closed(2)
thf(fact_46_domain_Ovar__closed_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( polynomial_a_b @ R @ K @ ( var_a_b @ R ) ) ) ) ).
% domain.var_closed(2)
thf(fact_47_ring_Opoly__coeff__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( polynomial_a_b @ R @ K @ P )
=> ( member_a @ ( coeff_a_b @ R @ P @ I ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.poly_coeff_in_carrier
thf(fact_48_ring_Opoly__coeff__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K @ P )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ R @ P @ I ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.poly_coeff_in_carrier
thf(fact_49_ring_Opoly__coeff__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a,I: nat] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( polyno42001669949040266t_unit @ R @ K @ P )
=> ( member_list_list_a @ ( coeff_1681977662540381769t_unit @ R @ P @ I ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% ring.poly_coeff_in_carrier
thf(fact_50_mem__Collect__eq,axiom,
! [A: a,P3: a > $o] :
( ( member_a @ A @ ( collect_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_51_mem__Collect__eq,axiom,
! [A: list_a,P3: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: nat > a,P3: ( nat > a ) > $o] :
( ( member_nat_a @ A @ ( collect_nat_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: list_list_a,P3: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A3: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A3: set_nat_a] :
( ( collect_nat_a
@ ^ [X: nat > a] : ( member_nat_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A3: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X: list_list_a] : ( member_list_list_a @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_58_ring_Onormalize_Ocong,axiom,
normalize_a_b = normalize_a_b ).
% ring.normalize.cong
thf(fact_59_ring_Onormalize_Ocong,axiom,
normal637505603836502915t_unit = normal637505603836502915t_unit ).
% ring.normalize.cong
thf(fact_60_ring_Ocoeff_Ocong,axiom,
coeff_a_b = coeff_a_b ).
% ring.coeff.cong
thf(fact_61_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_62_coeff__length,axiom,
! [P: list_a,I: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ P ) @ I )
=> ( ( coeff_a_b @ r @ P @ I )
= ( zero_a_b @ r ) ) ) ).
% coeff_length
thf(fact_63_associated__iff__same__ideal,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ( cgenid547466209912283029xt_a_b @ r @ A )
= ( cgenid547466209912283029xt_a_b @ r @ B ) ) ) ) ) ).
% associated_iff_same_ideal
thf(fact_64_domain_OsubdomainI_H,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( subdom7821232466298058046t_unit @ H @ R ) ) ) ).
% domain.subdomainI'
thf(fact_65_domain_OsubdomainI_H,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( subdomain_a_b @ H @ R ) ) ) ).
% domain.subdomainI'
thf(fact_66_subcringI,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( mult_a_ring_ext_a_b @ r @ H1 @ H2 )
= ( mult_a_ring_ext_a_b @ r @ H2 @ H1 ) ) ) )
=> ( subcring_a_b @ H @ r ) ) ) ).
% subcringI
thf(fact_67_list__eq__iff__nth__eq,axiom,
( ( ^ [Y: list_a,Z: list_a] : ( Y = Z ) )
= ( ^ [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_68_Skolem__list__nth,axiom,
! [K2: nat,P3: nat > a > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ? [X2: a] : ( P3 @ I2 @ X2 ) ) )
= ( ? [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= K2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( P3 @ I2 @ ( nth_a @ Xs @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_69_nth__equalityI,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
=> ( ( nth_a @ Xs2 @ I3 )
= ( nth_a @ Ys2 @ I3 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_70_ring_Ocarrier__is__subring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( subring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_71_ring_Ocarrier__is__subring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_72_ring_Ocarrier__is__subring,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( subrin3541368690557094692t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_73_eval__poly__in__carrier,axiom,
! [K: set_a,P: list_a,X3: a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% eval_poly_in_carrier
thf(fact_74_m__lcomm,axiom,
! [X3: a,Y2: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z2 ) )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z2 ) ) ) ) ) ) ).
% m_lcomm
thf(fact_75_m__comm,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X3 ) ) ) ) ).
% m_comm
thf(fact_76_m__assoc,axiom,
! [X3: a,Y2: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ r @ X3 @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% m_assoc
thf(fact_77_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_78_m__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_79_m__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_80_integral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_81_local_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_82_mult__cong__r,axiom,
! [B: a,B3: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B3 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_83_mult__cong__l,axiom,
! [A: a,A4: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ A4 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A4 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_84_eval__var,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X3 )
= X3 ) ) ).
% eval_var
thf(fact_85_cgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_86_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_87_m__closed,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_88_r__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_89_l__null,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_90_assms_I2_J,axiom,
member_list_a @ f @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ k ) ) ).
% assms(2)
thf(fact_91_subdomain_Osubintegral,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit,H12: set_list_a,H22: set_list_a] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( ( member_set_list_a @ H12 @ H )
=> ( ( member_set_list_a @ H22 @ H )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ H12 @ H22 )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( H12
= ( zero_s2910681146719230829t_unit @ R ) )
| ( H22
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_92_subdomain_Osubintegral,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H22 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( H12
= ( zero_l4142658623432671053t_unit @ R ) )
| ( H22
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_93_subdomain_Osubintegral,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H12 @ H22 )
= ( zero_a_b @ R ) )
=> ( ( H12
= ( zero_a_b @ R ) )
| ( H22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_94_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_95_ring_Oeval_Ocong,axiom,
eval_l34571156754992824t_unit = eval_l34571156754992824t_unit ).
% ring.eval.cong
thf(fact_96_subringE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subringE(2)
thf(fact_97_subringE_I2_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subrin5643252653130547402t_unit @ H @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ H ) ) ).
% subringE(2)
thf(fact_98_subringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subringE(2)
thf(fact_99_subringE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H22: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_100_subringE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H22: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_101_ring_Opoly__of__dense_Ocong,axiom,
poly_of_dense_a_b = poly_of_dense_a_b ).
% ring.poly_of_dense.cong
thf(fact_102_subcringE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_103_subcringE_I2_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subcri7783154434480317835t_unit @ H @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_104_subcringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_105_subcringE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H22: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_106_subcringE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H22: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_107_subcring_Osub__m__comm,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H22: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H22 )
= ( mult_l7073676228092353617t_unit @ R @ H22 @ H12 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_108_subcring_Osub__m__comm,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H22: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H12 @ H22 )
= ( mult_a_ring_ext_a_b @ R @ H22 @ H12 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_109_subdomainE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_110_subdomainE_I2_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_111_subdomainE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_112_subdomainE_I8_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H22 )
= ( mult_l7073676228092353617t_unit @ R @ H22 @ H12 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_113_subdomainE_I8_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H12 @ H22 )
= ( mult_a_ring_ext_a_b @ R @ H22 @ H12 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_114_subdomainE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_115_subdomainE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_116_ring_OsubcringI,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ! [H1: list_a,H2: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 )
= ( mult_l7073676228092353617t_unit @ R @ H2 @ H1 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H @ R ) ) ) ) ).
% ring.subcringI
thf(fact_117_ring_OsubcringI,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H2 )
= ( mult_a_ring_ext_a_b @ R @ H2 @ H1 ) ) ) )
=> ( subcring_a_b @ H @ R ) ) ) ) ).
% ring.subcringI
thf(fact_118_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_a] :
( ( size_size_list_a @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_119_neq__if__length__neq,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
!= ( size_size_list_a @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_120_ring_Oeval__var,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( var_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% ring.eval_var
thf(fact_121_ring_Oeval__var,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% ring.eval_var
thf(fact_122_ring_Oeval__var,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ ( var_li3532061862469730199t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% ring.eval_var
thf(fact_123_ring_Ocoeff__length,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ P ) @ I )
=> ( ( coeff_6360649920519955023t_unit @ R @ P @ I )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.coeff_length
thf(fact_124_ring_Ocoeff__length,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,I: nat] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( ord_less_eq_nat @ ( size_s1991367317912710102list_a @ P ) @ I )
=> ( ( coeff_5603115904260830831t_unit @ R @ P @ I )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% ring.coeff_length
thf(fact_125_ring_Ocoeff__length,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_nat @ ( size_size_list_a @ P ) @ I )
=> ( ( coeff_a_b @ R @ P @ I )
= ( zero_a_b @ R ) ) ) ) ).
% ring.coeff_length
thf(fact_126_ring_Oeval__poly__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,X3: a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( polynomial_a_b @ R @ K @ P )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ P @ X3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.eval_poly_in_carrier
thf(fact_127_ring_Oeval__poly__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K @ P )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ R @ P @ X3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% ring.eval_poly_in_carrier
thf(fact_128_ring_Oeval__poly__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( polyno42001669949040266t_unit @ R @ K @ P )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( eval_l1088911609197519410t_unit @ R @ P @ X3 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ).
% ring.eval_poly_in_carrier
thf(fact_129_length__induct,axiom,
! [P3: list_a > $o,Xs2: list_a] :
( ! [Xs3: list_a] :
( ! [Ys3: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs3 ) )
=> ( P3 @ Ys3 ) )
=> ( P3 @ Xs3 ) )
=> ( P3 @ Xs2 ) ) ).
% length_induct
thf(fact_130_subcring_Oaxioms_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( subring_a_b @ H @ R ) ) ).
% subcring.axioms(1)
thf(fact_131_subdomain_Oaxioms_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( subcring_a_b @ H @ R ) ) ).
% subdomain.axioms(1)
thf(fact_132_ring_Omonom__decomp,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K @ P )
=> ( P
= ( poly_o2635896782027652242t_unit @ R @ ( dense_5814815041220002634t_unit @ R @ P ) ) ) ) ) ) ).
% ring.monom_decomp
thf(fact_133_ring_Omonom__decomp,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( polynomial_a_b @ R @ K @ P )
=> ( P
= ( poly_of_dense_a_b @ R @ ( dense_repr_a_b @ R @ P ) ) ) ) ) ) ).
% ring.monom_decomp
thf(fact_134_boundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_135_ring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( P
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_136_ring__irreducibleE_I1_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( R2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_137_monoid__cancelI,axiom,
( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A2 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) )
=> ( ! [A2: a,B2: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A2 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A2 = B2 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_138_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_139_subdomainI,axiom,
! [H: set_a] :
( ( subcring_a_b @ H @ r )
=> ( ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ r @ H1 @ H2 )
= ( zero_a_b @ r ) )
=> ( ( H1
= ( zero_a_b @ r ) )
| ( H2
= ( zero_a_b @ r ) ) ) ) ) )
=> ( subdomain_a_b @ H @ r ) ) ) ) ).
% subdomainI
thf(fact_140_monom__coeff,axiom,
! [A: a,N: nat] :
( ( coeff_a_b @ r @ ( monom_a_b @ r @ A @ N ) )
= ( ^ [I2: nat] : ( if_a @ ( I2 = N ) @ A @ ( zero_a_b @ r ) ) ) ) ).
% monom_coeff
thf(fact_141_semiring_Or__null,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ X3 @ ( zero_s2910681146719230829t_unit @ R ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_142_semiring_Or__null,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.r_null
thf(fact_143_semiring_Or__null,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X3 @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_144_semiring_Or__null,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ X3 @ ( zero_l347298301471573063t_unit @ R ) )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.r_null
thf(fact_145_semiring_Ol__null,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X3 )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_146_semiring_Ol__null,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.l_null
thf(fact_147_semiring_Ol__null,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X3 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_148_semiring_Ol__null,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X3 )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% semiring.l_null
thf(fact_149_domain_Ointegral__iff,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
= ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_150_domain_Ointegral__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
= ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_151_domain_Ointegral__iff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_152_domain_Ointegral__iff,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( zero_l347298301471573063t_unit @ R ) )
= ( ( A
= ( zero_l347298301471573063t_unit @ R ) )
| ( B
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_153_domain_Om__rcancel,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ B @ A )
= ( mult_s7802724872828879953t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_154_domain_Om__rcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ B @ A )
= ( mult_a_ring_ext_a_b @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_155_domain_Om__rcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ B @ A )
= ( mult_l7073676228092353617t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_156_domain_Om__rcancel,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( A
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ B @ A )
= ( mult_l4853965630390486993t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_157_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_158_univ__poly__is__ring,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_ring
thf(fact_159_univ__poly__is__domain,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_domain
thf(fact_160_inv__unique,axiom,
! [Y2: a,X3: a,Y3: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y2 = Y3 ) ) ) ) ) ) ).
% inv_unique
thf(fact_161_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X4 )
= X4 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_162_ring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).
% ring_primeE(3)
thf(fact_163_ring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI
thf(fact_164_coeff__range,axiom,
! [K: set_a,F: list_a,I: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_a @ ( coeff_a_b @ r @ F @ I ) @ K ) ) ) ).
% coeff_range
thf(fact_165_var__closed_I1_J,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% var_closed(1)
thf(fact_166_pderiv__carr,axiom,
! [K: set_a,F: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( formal4452980811800949548iv_a_b @ r @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ).
% pderiv_carr
thf(fact_167_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_168_l__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_one
thf(fact_169_r__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( one_a_ring_ext_a_b @ r ) )
= X3 ) ) ).
% r_one
thf(fact_170_carrier__polynomial__shell,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_171_ring_Omonom_Ocong,axiom,
monom_a_b = monom_a_b ).
% ring.monom.cong
thf(fact_172_subringE_I3_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).
% subringE(3)
thf(fact_173_subringE_I3_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subrin5643252653130547402t_unit @ H @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ H ) ) ).
% subringE(3)
thf(fact_174_subringE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subringE(3)
thf(fact_175_univ__poly__carrier,axiom,
( polynomial_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,K3: set_a,P4: list_a] : ( member_list_a @ P4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K3 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_176_univ__poly__carrier,axiom,
( polyno1315193887021588240t_unit
= ( ^ [R3: partia2670972154091845814t_unit,K3: set_list_a,P4: list_list_a] : ( member_list_list_a @ P4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K3 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_177_subcringE_I3_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).
% subcringE(3)
thf(fact_178_subcringE_I3_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subcri7783154434480317835t_unit @ H @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ H ) ) ).
% subcringE(3)
thf(fact_179_subcringE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subcringE(3)
thf(fact_180_ring_Oring__simprules_I6_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_181_ring_Oring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_182_ring_Oring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_183_ring_Oring__simprules_I6_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% ring.ring_simprules(6)
thf(fact_184_domain_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% domain.one_not_zero
thf(fact_185_domain_Oone__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_186_domain_Oone__not__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_187_domain_Ozero__not__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( zero_a_b @ R )
!= ( one_a_ring_ext_a_b @ R ) ) ) ).
% domain.zero_not_one
thf(fact_188_domain_Ozero__not__one,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( zero_l4142658623432671053t_unit @ R )
!= ( one_li8328186300101108157t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_189_domain_Ozero__not__one,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( zero_s2910681146719230829t_unit @ R )
!= ( one_se1127990129394575805t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_190_subdomainE_I3_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H ) ) ).
% subdomainE(3)
thf(fact_191_subdomainE_I3_J,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ H ) ) ).
% subdomainE(3)
thf(fact_192_subdomainE_I3_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H ) ) ).
% subdomainE(3)
thf(fact_193_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R )
=> ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_194_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( one_a_ring_ext_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_195_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_196_semiring_Osemiring__simprules_I4_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% semiring.semiring_simprules(4)
thf(fact_197_ring_Oring__simprules_I12_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_198_ring_Oring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_199_ring_Oring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_200_ring_Oring__simprules_I12_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(12)
thf(fact_201_domain_Ouniv__poly__is__ring,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_ring
thf(fact_202_domain_Ouniv__poly__is__ring,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ring_l1939023646219158831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_ring
thf(fact_203_domain_Ouniv__poly__is__domain,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_204_domain_Ouniv__poly__is__domain,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_205_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_206_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_207_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_208_semiring_Osemiring__simprules_I9_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(9)
thf(fact_209_subdomain_Osub__one__not__zero,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_210_subdomain_Osub__one__not__zero,axiom,
! [H: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subdom3220114454046903646t_unit @ H @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_211_subdomain_Osub__one__not__zero,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subdomain.sub_one_not_zero
thf(fact_212_domain_Ocoeff__range,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,F: list_a,I: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_a @ ( coeff_a_b @ R @ F @ I ) @ K ) ) ) ) ).
% domain.coeff_range
thf(fact_213_domain_Ocoeff__range,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a,I: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_a @ ( coeff_6360649920519955023t_unit @ R @ F @ I ) @ K ) ) ) ) ).
% domain.coeff_range
thf(fact_214_domain_Ovar__closed_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( member_list_a @ ( var_a_b @ R ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_215_domain_Ovar__closed_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ R ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_216_domain_Opderiv__carr,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_list_a @ ( formal6075833236969493044t_unit @ R @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ).
% domain.pderiv_carr
thf(fact_217_domain_Opderiv__carr,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,F: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_a @ ( formal4452980811800949548iv_a_b @ R @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ).
% domain.pderiv_carr
thf(fact_218_ring_Omonom__coeff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ A @ N ) )
= ( ^ [I2: nat] : ( if_list_a @ ( I2 = N ) @ A @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_219_ring_Omonom__coeff,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,N: nat] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( coeff_5603115904260830831t_unit @ R @ ( monom_317758005976320064t_unit @ R @ A @ N ) )
= ( ^ [I2: nat] : ( if_set_list_a @ ( I2 = N ) @ A @ ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_220_ring_Omonom__coeff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ ( monom_a_b @ R @ A @ N ) )
= ( ^ [I2: nat] : ( if_a @ ( I2 = N ) @ A @ ( zero_a_b @ R ) ) ) ) ) ).
% ring.monom_coeff
thf(fact_221_ring_OsubdomainI,axiom,
! [R: partia7496981018696276118t_unit,H: set_set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( subcri7783154434480317835t_unit @ H @ R )
=> ( ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ! [H1: set_list_a,H2: set_list_a] :
( ( member_set_list_a @ H1 @ H )
=> ( ( member_set_list_a @ H2 @ H )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ H1 @ H2 )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( H1
= ( zero_s2910681146719230829t_unit @ R ) )
| ( H2
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) )
=> ( subdom3220114454046903646t_unit @ H @ R ) ) ) ) ) ).
% ring.subdomainI
thf(fact_222_ring_OsubdomainI,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ! [H1: list_a,H2: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( H1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( H2
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) )
=> ( subdom7821232466298058046t_unit @ H @ R ) ) ) ) ) ).
% ring.subdomainI
thf(fact_223_ring_OsubdomainI,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( subcring_a_b @ H @ R )
=> ( ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H1 @ H2 )
= ( zero_a_b @ R ) )
=> ( ( H1
= ( zero_a_b @ R ) )
| ( H2
= ( zero_a_b @ R ) ) ) ) ) )
=> ( subdomain_a_b @ H @ R ) ) ) ) ) ).
% ring.subdomainI
thf(fact_224_ring_Ois__abelian__group,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( abelian_group_a_b @ R ) ) ).
% ring.is_abelian_group
thf(fact_225_ring_Ois__abelian__group,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( abelia3891852623213500406t_unit @ R ) ) ).
% ring.is_abelian_group
thf(fact_226_abelian__group_Oaxioms_I1_J,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ G )
=> ( abelian_monoid_a_b @ G ) ) ).
% abelian_group.axioms(1)
thf(fact_227_abelian__group_Oaxioms_I1_J,axiom,
! [G: partia2670972154091845814t_unit] :
( ( abelia3891852623213500406t_unit @ G )
=> ( abelia226231641709521465t_unit @ G ) ) ).
% abelian_group.axioms(1)
thf(fact_228_semiring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( abelian_monoid_a_b @ R ) ) ).
% semiring.axioms(1)
thf(fact_229_semiring_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( abelia226231641709521465t_unit @ R ) ) ).
% semiring.axioms(1)
thf(fact_230_ring_Oring__simprules_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_231_ring_Oring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_232_ring_Oring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_233_ring_Oring__simprules_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% ring.ring_simprules(2)
thf(fact_234_ring_Oring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y2 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ R @ X3 @ ( mult_a_ring_ext_a_b @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_235_ring_Oring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ R @ X3 @ ( mult_l7073676228092353617t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_236_ring_Oring__simprules_I11_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( mult_l4853965630390486993t_unit @ R @ X3 @ ( mult_l4853965630390486993t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(11)
thf(fact_237_ring_Oring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_238_ring_Oring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_239_ring_Oring__simprules_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Y2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(5)
thf(fact_240_abelian__groupE_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( abelia5304159692179083286t_unit @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_241_abelian__groupE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_group_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_242_abelian__groupE_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( abelia3891852623213500406t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_243_abelian__groupE_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( abelia2778853791629620336t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% abelian_groupE(2)
thf(fact_244_abelian__monoidE_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( abelia3322010900105369177t_unit @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_245_abelian__monoidE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_246_abelian__monoidE_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_247_abelian__monoidE_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( abelia3641329199688042803t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% abelian_monoidE(2)
thf(fact_248_abelian__monoid_Ozero__closed,axiom,
! [G: partia7496981018696276118t_unit] :
( ( abelia3322010900105369177t_unit @ G )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ G ) @ ( partia141011252114345353t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_249_abelian__monoid_Ozero__closed,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( abelian_monoid_a_b @ G )
=> ( member_a @ ( zero_a_b @ G ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_250_abelian__monoid_Ozero__closed,axiom,
! [G: partia2670972154091845814t_unit] :
( ( abelia226231641709521465t_unit @ G )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ G ) @ ( partia5361259788508890537t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_251_abelian__monoid_Ozero__closed,axiom,
! [G: partia2956882679547061052t_unit] :
( ( abelia3641329199688042803t_unit @ G )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ G ) @ ( partia2464479390973590831t_unit @ G ) ) ) ).
% abelian_monoid.zero_closed
thf(fact_252_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( semiri4000464634269493571t_unit @ R )
=> ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_253_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( semiring_a_b @ R )
=> ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_254_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( semiri2871908745932252451t_unit @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_255_semiring_Osemiring__simprules_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( semiri2265994252334843677t_unit @ R )
=> ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% semiring.semiring_simprules(2)
thf(fact_256_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y2 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ R @ X3 @ ( mult_a_ring_ext_a_b @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_257_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ R @ X3 @ ( mult_l7073676228092353617t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_258_semiring_Osemiring__simprules_I8_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( mult_l4853965630390486993t_unit @ R @ X3 @ ( mult_l4853965630390486993t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(8)
thf(fact_259_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_260_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_261_semiring_Osemiring__simprules_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Y2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(3)
thf(fact_262_ring_Oring__simprules_I25_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ X3 @ ( zero_s2910681146719230829t_unit @ R ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_263_ring_Oring__simprules_I25_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_264_ring_Oring__simprules_I25_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X3 @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_265_ring_Oring__simprules_I25_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ X3 @ ( zero_l347298301471573063t_unit @ R ) )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% ring.ring_simprules(25)
thf(fact_266_ring_Oring__simprules_I24_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( mult_s7802724872828879953t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X3 )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_267_ring_Oring__simprules_I24_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_268_ring_Oring__simprules_I24_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X3 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_269_ring_Oring__simprules_I24_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X3 )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% ring.ring_simprules(24)
thf(fact_270_domain_Ointegral,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_271_domain_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_272_domain_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_273_domain_Ointegral,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( A
= ( zero_l347298301471573063t_unit @ R ) )
| ( B
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_274_domain_Om__lcancel,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( mult_s7802724872828879953t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_275_domain_Om__lcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( mult_a_ring_ext_a_b @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_276_domain_Om__lcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( mult_l7073676228092353617t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_277_domain_Om__lcancel,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( A
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( mult_l4853965630390486993t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_278_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q: list_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_279_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X3 )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X3 ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_280_bound_Ointro,axiom,
! [N: nat,F: nat > a,Z2: a] :
( ! [M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( F @ M2 )
= Z2 ) )
=> ( bound_a @ Z2 @ N @ F ) ) ).
% bound.intro
thf(fact_281_abelian__monoid_OboundD__carrier,axiom,
! [G: partia7496981018696276118t_unit,N: nat,F: nat > set_list_a,M: nat] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_set_list_a @ ( F @ M ) @ ( partia141011252114345353t_unit @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_282_abelian__monoid_OboundD__carrier,axiom,
! [G: partia2175431115845679010xt_a_b,N: nat,F: nat > a,M: nat] :
( ( abelian_monoid_a_b @ G )
=> ( ( bound_a @ ( zero_a_b @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_283_abelian__monoid_OboundD__carrier,axiom,
! [G: partia2670972154091845814t_unit,N: nat,F: nat > list_a,M: nat] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F @ M ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_284_abelian__monoid_OboundD__carrier,axiom,
! [G: partia2956882679547061052t_unit,N: nat,F: nat > list_list_a,M: nat] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ G ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_list_a @ ( F @ M ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ).
% abelian_monoid.boundD_carrier
thf(fact_285_pdivides__imp__root__sharing,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( eval_a_b @ r @ P @ A )
= ( zero_a_b @ r ) )
=> ( ( eval_a_b @ r @ Q @ A )
= ( zero_a_b @ r ) ) ) ) ) ) ).
% pdivides_imp_root_sharing
thf(fact_286_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_287_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_288_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_289_const__term__simprules__shell_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(2)
thf(fact_290_domain_Oring__primeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( prime_a_ring_ext_a_b @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_291_domain_Oring__primeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P )
=> ( prime_2011924034616061926t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_292_domain_Oring__primeE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P )
=> ( prime_1232919612140715622t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_293_ring_Oring__primeI,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( P
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( prime_5738381090551951334t_unit @ R @ P )
=> ( ring_r1091214237498979717t_unit @ R @ P ) ) ) ) ).
% ring.ring_primeI
thf(fact_294_ring_Oring__primeI,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_a_b @ R )
=> ( ( P
!= ( zero_a_b @ R ) )
=> ( ( prime_a_ring_ext_a_b @ R @ P )
=> ( ring_ring_prime_a_b @ R @ P ) ) ) ) ).
% ring.ring_primeI
thf(fact_295_ring_Oring__primeI,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( P
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( prime_2011924034616061926t_unit @ R @ P )
=> ( ring_r6430282645014804837t_unit @ R @ P ) ) ) ) ).
% ring.ring_primeI
thf(fact_296_univ__poly__is__abelian__group,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( abelia3891852623213500406t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_abelian_group
thf(fact_297_univ__poly__is__abelian__monoid,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_abelian_monoid
thf(fact_298_const__term__def,axiom,
! [P: list_a] :
( ( const_term_a_b @ r @ P )
= ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_299_const__term__simprules__shell_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ K ) ) ) ).
% const_term_simprules_shell(1)
thf(fact_300_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_301_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_302_ring_Oring__simprules_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_minus_a_b @ R @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_303_ring_Oring__simprules_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ R @ X3 @ Y2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_304_ring_Oring__simprules_I4_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( a_minu2241224857956505934t_unit @ R @ X3 @ Y2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(4)
thf(fact_305_domain_Ouniv__poly__is__abelian__group,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( abelia3891852623213500406t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_group
thf(fact_306_domain_Ouniv__poly__is__abelian__group,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( abelia2778853791629620336t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_group
thf(fact_307_abelian__group_Ominus__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( a_minus_a_b @ G @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_308_abelian__group_Ominus__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ G @ X3 @ Y2 ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_309_abelian__group_Ominus__closed,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( member_list_list_a @ ( a_minu2241224857956505934t_unit @ G @ X3 @ Y2 ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ).
% abelian_group.minus_closed
thf(fact_310_domain_Ouniv__poly__is__abelian__monoid,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_monoid
thf(fact_311_domain_Ouniv__poly__is__abelian__monoid,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( abelia3641329199688042803t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_monoid
thf(fact_312_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,X3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( polyno5142720416380192742t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q ) @ X3 )
=> ( ( polyno5142720416380192742t_unit @ R @ P @ X3 )
| ( polyno5142720416380192742t_unit @ R @ Q @ X3 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_313_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q ) @ X3 )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X3 )
| ( polyno6951661231331188332t_unit @ R @ Q @ X3 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_314_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X3: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q ) @ X3 )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X3 )
| ( polyno4133073214067823460ot_a_b @ R @ Q @ X3 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_315_ring_Oconst__term__def,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( const_3308765751713425893t_unit @ R @ P )
= ( eval_s5133945360527818456t_unit @ R @ P @ ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_316_ring_Oconst__term__def,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( const_6738166269504826821t_unit @ R @ P )
= ( eval_l34571156754992824t_unit @ R @ P @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_317_ring_Oconst__term__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( const_term_a_b @ R @ P )
= ( eval_a_b @ R @ P @ ( zero_a_b @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_318_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,Q: list_set_list_a,A: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( polyno9075941895896075626t_unit @ R @ P @ Q )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( eval_s5133945360527818456t_unit @ R @ P @ A )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( eval_s5133945360527818456t_unit @ R @ Q @ A )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_319_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,A: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( polyno4453881341673752516t_unit @ R @ P @ Q )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( eval_l1088911609197519410t_unit @ R @ P @ A )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ Q @ A )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_320_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,A: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ Q )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( eval_a_b @ R @ P @ A )
= ( zero_a_b @ R ) )
=> ( ( eval_a_b @ R @ Q @ A )
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_321_domain_Opdivides__imp__root__sharing,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,A: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ Q )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( eval_l34571156754992824t_unit @ R @ P @ A )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ Q @ A )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.pdivides_imp_root_sharing
thf(fact_322_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_a @ ( const_term_a_b @ R @ P ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_323_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ R @ P ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_324_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ R @ ( const_term_a_b @ R @ P ) @ ( const_term_a_b @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_325_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) @ ( const_6738166269504826821t_unit @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_326_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P @ Q )
= ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_327_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_328_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_329_bound_Obound,axiom,
! [Z2: a,N: nat,F: nat > a,M: nat] :
( ( bound_a @ Z2 @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( ( F @ M )
= Z2 ) ) ) ).
% bound.bound
thf(fact_330_bound__def,axiom,
( bound_a
= ( ^ [Z3: a,N2: nat,F2: nat > a] :
! [M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( ( F2 @ M3 )
= Z3 ) ) ) ) ).
% bound_def
thf(fact_331_bound__below,axiom,
! [Z2: a,M: nat,F: nat > a,N: nat] :
( ( bound_a @ Z2 @ M @ F )
=> ( ( ( F @ N )
!= Z2 )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% bound_below
thf(fact_332_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( prime_2011924034616061926t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_333_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( prime_5738381090551951334t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_334_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_335_ring__prime__def,axiom,
( ring_r1091214237498979717t_unit
= ( ^ [R3: partia7496981018696276118t_unit,A5: set_list_a] :
( ( A5
!= ( zero_s2910681146719230829t_unit @ R3 ) )
& ( prime_5738381090551951334t_unit @ R3 @ A5 ) ) ) ) ).
% ring_prime_def
thf(fact_336_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,A5: a] :
( ( A5
!= ( zero_a_b @ R3 ) )
& ( prime_a_ring_ext_a_b @ R3 @ A5 ) ) ) ) ).
% ring_prime_def
thf(fact_337_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R3: partia2670972154091845814t_unit,A5: list_a] :
( ( A5
!= ( zero_l4142658623432671053t_unit @ R3 ) )
& ( prime_2011924034616061926t_unit @ R3 @ A5 ) ) ) ) ).
% ring_prime_def
thf(fact_338_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,R2: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R2 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ R2 )
=> ( R2
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_339_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,R2: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R2 )
=> ( R2
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_340_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( R2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_341_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ( R2
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_342_domain_Oring__primeE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r1091214237498979717t_unit @ R @ P )
=> ( P
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_343_domain_Oring__primeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( P
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_344_domain_Oring__primeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_345_domain_Oring__primeE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P )
=> ( P
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_346_is__root__def,axiom,
! [P: list_a,X3: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
= ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X3 )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_347_pdivides__zero,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_348_unitary__monom__eq__var__pow,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( monom_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ).
% unitary_monom_eq_var_pow
thf(fact_349_bound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_b @ r ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ r ) @ N3 @ F ) ) ).
% bound_upD
thf(fact_350_monoid__cancel_Oprime__cong,axiom,
! [G: partia2175431115845679010xt_a_b,P: a,P5: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( prime_a_ring_ext_a_b @ G @ P )
=> ( ( associ5860276527279195403xt_a_b @ G @ P @ P5 )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ P5 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( prime_a_ring_ext_a_b @ G @ P5 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_351_monoid__cancel_Oprime__cong,axiom,
! [G: partia2670972154091845814t_unit,P: list_a,P5: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( prime_2011924034616061926t_unit @ G @ P )
=> ( ( associ8407585678920448409t_unit @ G @ P @ P5 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ P5 @ ( partia5361259788508890537t_unit @ G ) )
=> ( prime_2011924034616061926t_unit @ G @ P5 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_352_monoid__cancel_Oprime__cong,axiom,
! [G: partia2956882679547061052t_unit,P: list_list_a,P5: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( prime_1232919612140715622t_unit @ G @ P )
=> ( ( associ5603075271488036121t_unit @ G @ P @ P5 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ P5 @ ( partia2464479390973590831t_unit @ G ) )
=> ( prime_1232919612140715622t_unit @ G @ P5 ) ) ) ) ) ) ).
% monoid_cancel.prime_cong
thf(fact_353_polynomial__pow__division,axiom,
! [P: list_a,N: nat,M: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ M ) ) ) ) ).
% polynomial_pow_division
thf(fact_354_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a,B3: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ ( mult_a_ring_ext_a_b @ G @ A @ B ) @ ( mult_a_ring_ext_a_b @ G @ A @ B3 ) )
=> ( associ5860276527279195403xt_a_b @ G @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_355_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a,B3: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ ( mult_l7073676228092353617t_unit @ G @ A @ B ) @ ( mult_l7073676228092353617t_unit @ G @ A @ B3 ) )
=> ( associ8407585678920448409t_unit @ G @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_356_monoid__cancel_Oassoc__l__cancel,axiom,
! [G: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,B3: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( associ5603075271488036121t_unit @ G @ ( mult_l4853965630390486993t_unit @ G @ A @ B ) @ ( mult_l4853965630390486993t_unit @ G @ A @ B3 ) )
=> ( associ5603075271488036121t_unit @ G @ B @ B3 ) ) ) ) ) ) ).
% monoid_cancel.assoc_l_cancel
thf(fact_357_pprimeE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
| ( polyno5814909790663948098es_a_b @ r @ P @ R2 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_358_subring__props_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K ) ) ).
% subring_props(2)
thf(fact_359_subring__props_I6_J,axiom,
! [K: set_a,H12: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H12 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H12 @ H22 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_360_subring__props_I3_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K ) ) ).
% subring_props(3)
thf(fact_361_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_362_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_363_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_364_coeff_Osimps_I1_J,axiom,
( ( coeff_a_b @ r @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) ).
% coeff.simps(1)
thf(fact_365_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_366_const__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_b @ r @ P )
!= ( zero_a_b @ r ) )
=> ( P != nil_a ) ) ).
% const_term_not_zero
thf(fact_367_associated__polynomials__imp__same__length,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
=> ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) ) ) ) ) ) ).
% associated_polynomials_imp_same_length
thf(fact_368_pirreducibleE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_369_associated__polynomials__imp__same__is__root,axiom,
! [P: list_a,Q: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
= ( polyno4133073214067823460ot_a_b @ r @ Q @ X3 ) ) ) ) ) ).
% associated_polynomials_imp_same_is_root
thf(fact_370_pprime__iff__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_371_polynomial__pow__not__zero,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_372_subring__polynomial__pow__not__zero,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N )
!= nil_a ) ) ) ) ).
% subring_polynomial_pow_not_zero
thf(fact_373_var__pow__closed,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% var_pow_closed
thf(fact_374_pprimeE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_375_pirreducible__pow__pdivides__iff,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a,N: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R2 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_376_minus__closed,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_minus_a_b @ r @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% minus_closed
thf(fact_377_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_378_univ__poly__zero__closed,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_379_r__right__minus__eq,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_minus_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( A = B ) ) ) ) ).
% r_right_minus_eq
thf(fact_380_zero__is__polynomial,axiom,
! [K: set_a] : ( polynomial_a_b @ r @ K @ nil_a ) ).
% zero_is_polynomial
thf(fact_381_mem__upI,axiom,
! [F: nat > set_list_a,R: partia7496981018696276118t_unit] :
( ! [N3: nat] : ( member_set_list_a @ ( F @ N3 ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ? [N4: nat] : ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ N4 @ F )
=> ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_382_mem__upI,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b] :
( ! [N3: nat] : ( member_a @ ( F @ N3 ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ? [N4: nat] : ( bound_a @ ( zero_a_b @ R ) @ N4 @ F )
=> ( member_nat_a @ F @ ( up_a_b @ R ) ) ) ) ).
% mem_upI
thf(fact_383_mem__upI,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit] :
( ! [N3: nat] : ( member_list_a @ ( F @ N3 ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ? [N4: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ N4 @ F )
=> ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_384_mem__upI,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit] :
( ! [N3: nat] : ( member_list_list_a @ ( F @ N3 ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ? [N4: nat] : ( bound_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ N4 @ F )
=> ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) ) ) ) ).
% mem_upI
thf(fact_385_subfieldE_I4_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_a @ K1 @ K )
=> ( ( member_list_a @ K22 @ K )
=> ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
= ( mult_l7073676228092353617t_unit @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_386_subfieldE_I4_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K @ R )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K22 @ K )
=> ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( mult_a_ring_ext_a_b @ R @ K22 @ K1 ) ) ) ) ) ).
% subfieldE(4)
thf(fact_387_ring_Odense__repr_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( dense_5814815041220002634t_unit @ R @ nil_list_a )
= nil_Pr6246850598307483389_a_nat ) ) ).
% ring.dense_repr.simps(1)
thf(fact_388_ring_Odense__repr_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( dense_repr_a_b @ R @ nil_a )
= nil_Pr7402525243500994295_a_nat ) ) ).
% ring.dense_repr.simps(1)
thf(fact_389_subfieldE_I1_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subring_a_b @ K @ R ) ) ).
% subfieldE(1)
thf(fact_390_subfieldE_I2_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subcring_a_b @ K @ R ) ) ).
% subfieldE(2)
thf(fact_391_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_392_univ__poly__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_393_subfield_Oaxioms_I1_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( subdomain_a_b @ K @ R ) ) ).
% subfield.axioms(1)
thf(fact_394_mem__upD,axiom,
! [F: nat > a,R: partia2175431115845679010xt_a_b,N: nat] :
( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ( member_a @ ( F @ N ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% mem_upD
thf(fact_395_mem__upD,axiom,
! [F: nat > list_a,R: partia2670972154091845814t_unit,N: nat] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) )
=> ( member_list_a @ ( F @ N ) @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% mem_upD
thf(fact_396_mem__upD,axiom,
! [F: nat > list_list_a,R: partia2956882679547061052t_unit,N: nat] :
( ( member8650753269014980122list_a @ F @ ( up_lis8963924889346801084t_unit @ R ) )
=> ( member_list_list_a @ ( F @ N ) @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% mem_upD
thf(fact_397_domain_OpprimeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( P != nil_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_398_domain_OpprimeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ) ).
% domain.pprimeE(1)
thf(fact_399_domain_Opprime__iff__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_400_domain_Opprime__iff__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) ) ) ) ).
% domain.pprime_iff_pirreducible
thf(fact_401_domain_Opirreducible__pow__pdivides__iff,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R2: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( ~ ( polyno8016796738000020810t_unit @ R @ P @ Q )
=> ( ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ R2 ) )
= ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pirreducible_pow_pdivides_iff
thf(fact_402_domain_Opirreducible__pow__pdivides__iff,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R2: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ R @ P @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ R2 ) )
= ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pirreducible_pow_pdivides_iff
thf(fact_403_domain_Opolynomial__pow__not__zero,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ N )
!= nil_list_list_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_404_domain_Opolynomial__pow__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ N )
!= nil_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_405_domain_Opolynomial__pow__not__zero,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ N )
!= nil_list_a ) ) ) ) ).
% domain.polynomial_pow_not_zero
thf(fact_406_subfieldE_I5_J,axiom,
! [K: set_set_list_a,R: partia7496981018696276118t_unit,K1: set_list_a,K22: set_list_a] :
( ( subfie4339374749748326226t_unit @ K @ R )
=> ( ( member_set_list_a @ K1 @ K )
=> ( ( member_set_list_a @ K22 @ K )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ K1 @ K22 )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( K1
= ( zero_s2910681146719230829t_unit @ R ) )
| ( K22
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_407_subfieldE_I5_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit,K1: list_a,K22: list_a] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_a @ K1 @ K )
=> ( ( member_list_a @ K22 @ K )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ K1 @ K22 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( K1
= ( zero_l4142658623432671053t_unit @ R ) )
| ( K22
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_408_subfieldE_I5_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b,K1: a,K22: a] :
( ( subfield_a_b @ K @ R )
=> ( ( member_a @ K1 @ K )
=> ( ( member_a @ K22 @ K )
=> ( ( ( mult_a_ring_ext_a_b @ R @ K1 @ K22 )
= ( zero_a_b @ R ) )
=> ( ( K1
= ( zero_a_b @ R ) )
| ( K22
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subfieldE(5)
thf(fact_409_domain_Osubring__polynomial__pow__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ P @ N )
!= nil_a ) ) ) ) ) ).
% domain.subring_polynomial_pow_not_zero
thf(fact_410_domain_Osubring__polynomial__pow__not__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ N )
!= nil_list_a ) ) ) ) ) ).
% domain.subring_polynomial_pow_not_zero
thf(fact_411_subfieldE_I6_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% subfieldE(6)
thf(fact_412_subfieldE_I6_J,axiom,
! [K: set_set_list_a,R: partia7496981018696276118t_unit] :
( ( subfie4339374749748326226t_unit @ K @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% subfieldE(6)
thf(fact_413_subfieldE_I6_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% subfieldE(6)
thf(fact_414_domain_OpirreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_415_domain_OpirreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( P != nil_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_416_ring_Ozero__is__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( polyno1315193887021588240t_unit @ R @ K @ nil_list_a ) ) ).
% ring.zero_is_polynomial
thf(fact_417_ring_Ozero__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( ring_a_b @ R )
=> ( polynomial_a_b @ R @ K @ nil_a ) ) ).
% ring.zero_is_polynomial
thf(fact_418_ring_Onormalize_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ nil_a )
= nil_a ) ) ).
% ring.normalize.simps(1)
thf(fact_419_ring_Onormalize_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( normal637505603836502915t_unit @ R @ nil_list_a )
= nil_list_a ) ) ).
% ring.normalize.simps(1)
thf(fact_420_domain_Ozero__pdivides__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( polyno8016796738000020810t_unit @ R @ nil_list_a @ nil_list_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_421_domain_Ozero__pdivides__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( polyno5814909790663948098es_a_b @ R @ nil_a @ nil_a ) ) ).
% domain.zero_pdivides_zero
thf(fact_422_domain_Ozero__pdivides,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( polyno8016796738000020810t_unit @ R @ nil_list_a @ P )
= ( P = nil_list_a ) ) ) ).
% domain.zero_pdivides
thf(fact_423_domain_Ozero__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( domain_a_b @ R )
=> ( ( polyno5814909790663948098es_a_b @ R @ nil_a @ P )
= ( P = nil_a ) ) ) ).
% domain.zero_pdivides
thf(fact_424_ring_Ocoeff_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( coeff_6360649920519955023t_unit @ R @ nil_list_a )
= ( ^ [Uu: nat] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.coeff.simps(1)
thf(fact_425_ring_Ocoeff_Osimps_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( coeff_5603115904260830831t_unit @ R @ nil_set_list_a )
= ( ^ [Uu: nat] : ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% ring.coeff.simps(1)
thf(fact_426_ring_Ocoeff_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( coeff_a_b @ R @ nil_a )
= ( ^ [Uu: nat] : ( zero_a_b @ R ) ) ) ) ).
% ring.coeff.simps(1)
thf(fact_427_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( eval_s5133945360527818456t_unit @ R @ nil_set_list_a )
= ( ^ [Uu: set_list_a] : ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_428_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( eval_a_b @ R @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_429_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( eval_l34571156754992824t_unit @ R @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_430_ring_Oconst__term__not__zero,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( const_6738166269504826821t_unit @ R @ P )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( P != nil_list_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_431_ring_Oconst__term__not__zero,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( ( const_3308765751713425893t_unit @ R @ P )
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( P != nil_set_list_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_432_ring_Oconst__term__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ( const_term_a_b @ R @ P )
!= ( zero_a_b @ R ) )
=> ( P != nil_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_433_monoid__cancel_Ois__monoid__cancel,axiom,
! [G: partia2175431115845679010xt_a_b] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( monoid5798828371819920185xt_a_b @ G ) ) ).
% monoid_cancel.is_monoid_cancel
thf(fact_434_ring_Obound__upD,axiom,
! [R: partia2670972154091845814t_unit,F: nat > list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ R ) )
=> ? [N3: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ N3 @ F ) ) ) ).
% ring.bound_upD
thf(fact_435_ring_Obound__upD,axiom,
! [R: partia7496981018696276118t_unit,F: nat > set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( member491565700723299188list_a @ F @ ( up_set529185716248919906t_unit @ R ) )
=> ? [N3: nat] : ( bound_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ N3 @ F ) ) ) ).
% ring.bound_upD
thf(fact_436_ring_Obound__upD,axiom,
! [R: partia2175431115845679010xt_a_b,F: nat > a] :
( ( ring_a_b @ R )
=> ( ( member_nat_a @ F @ ( up_a_b @ R ) )
=> ? [N3: nat] : ( bound_a @ ( zero_a_b @ R ) @ N3 @ F ) ) ) ).
% ring.bound_upD
thf(fact_437_domain_Oassociated__polynomials__imp__same__length,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q )
=> ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_length
thf(fact_438_domain_Oassociated__polynomials__imp__same__length,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q )
=> ( ( size_s349497388124573686list_a @ P )
= ( size_s349497388124573686list_a @ Q ) ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_length
thf(fact_439_ring_Oeval__monom,axiom,
! [R: partia2175431115845679010xt_a_b,B: a,A: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( monom_a_b @ R @ B @ N ) @ A )
= ( mult_a_ring_ext_a_b @ R @ B @ ( pow_a_1026414303147256608_b_nat @ R @ A @ N ) ) ) ) ) ) ).
% ring.eval_monom
thf(fact_440_ring_Oeval__monom,axiom,
! [R: partia2670972154091845814t_unit,B: list_a,A: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( monom_7446464087056152608t_unit @ R @ B @ N ) @ A )
= ( mult_l7073676228092353617t_unit @ R @ B @ ( pow_li1142815632869257134it_nat @ R @ A @ N ) ) ) ) ) ) ).
% ring.eval_monom
thf(fact_441_ring_Oeval__monom,axiom,
! [R: partia2956882679547061052t_unit,B: list_list_a,A: list_list_a,N: nat] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ ( monom_4043874212805408666t_unit @ R @ B @ N ) @ A )
= ( mult_l4853965630390486993t_unit @ R @ B @ ( pow_li488931774710091566it_nat @ R @ A @ N ) ) ) ) ) ) ).
% ring.eval_monom
thf(fact_442_domain_Oassociated__polynomials__imp__same__is__root,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,X3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( associ9038253669175192217t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q )
=> ( ( polyno5142720416380192742t_unit @ R @ P @ X3 )
= ( polyno5142720416380192742t_unit @ R @ Q @ X3 ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_is_root
thf(fact_443_domain_Oassociated__polynomials__imp__same__is__root,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X3 )
= ( polyno6951661231331188332t_unit @ R @ Q @ X3 ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_is_root
thf(fact_444_domain_Oassociated__polynomials__imp__same__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X3: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X3 )
= ( polyno4133073214067823460ot_a_b @ R @ Q @ X3 ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_is_root
thf(fact_445_domain_Ovar__pow__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ ( var_a_b @ R ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ).
% domain.var_pow_closed
thf(fact_446_domain_Ovar__pow__closed,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( var_li8453953174693405341t_unit @ R ) @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ).
% domain.var_pow_closed
thf(fact_447_domain_OpprimeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ Q )
| ( polyno5814909790663948098es_a_b @ R @ P @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_448_domain_OpprimeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ R2 ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ Q )
| ( polyno8016796738000020810t_unit @ R @ P @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.pprimeE(3)
thf(fact_449_ring_Ois__root__def,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( polyno4320237611291262604t_unit @ R @ P @ X3 )
= ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
& ( ( eval_s5133945360527818456t_unit @ R @ P @ X3 )
= ( zero_s2910681146719230829t_unit @ R ) )
& ( P != nil_set_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_450_ring_Ois__root__def,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X3 )
= ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
& ( ( eval_l34571156754992824t_unit @ R @ P @ X3 )
= ( zero_l4142658623432671053t_unit @ R ) )
& ( P != nil_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_451_ring_Ois__root__def,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( polyno5142720416380192742t_unit @ R @ P @ X3 )
= ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
& ( ( eval_l1088911609197519410t_unit @ R @ P @ X3 )
= ( zero_l347298301471573063t_unit @ R ) )
& ( P != nil_list_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_452_ring_Ois__root__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X3: a] :
( ( ring_a_b @ R )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X3 )
= ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( eval_a_b @ R @ P @ X3 )
= ( zero_a_b @ R ) )
& ( P != nil_a ) ) ) ) ).
% ring.is_root_def
thf(fact_453_domain_Opdivides__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ R @ P @ nil_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_454_domain_Opdivides__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( polyno8016796738000020810t_unit @ R @ P @ nil_list_a ) ) ) ) ).
% domain.pdivides_zero
thf(fact_455_domain_Opolynomial__pow__division,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,N: nat,M: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ N ) @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_456_domain_Opolynomial__pow__division,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,N: nat,M: nat] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_457_domain_Opolynomial__pow__division,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,N: nat,M: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ M ) ) ) ) ) ).
% domain.polynomial_pow_division
thf(fact_458_domain_Ounitary__monom__eq__var__pow,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( ( monom_317758005976320064t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) @ N )
= ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( var_se6008125447796440765t_unit @ R ) @ N ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_459_domain_Ounitary__monom__eq__var__pow,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( monom_7446464087056152608t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) @ N )
= ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( var_li8453953174693405341t_unit @ R ) @ N ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_460_domain_Ounitary__monom__eq__var__pow,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( monom_a_b @ R @ ( one_a_ring_ext_a_b @ R ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ K ) @ ( var_a_b @ R ) @ N ) ) ) ) ).
% domain.unitary_monom_eq_var_pow
thf(fact_461_monoid__cancel_Ol__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,C: a,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ C @ A )
= ( mult_a_ring_ext_a_b @ G @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_462_monoid__cancel_Ol__cancel,axiom,
! [G: partia2670972154091845814t_unit,C: list_a,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ C @ A )
= ( mult_l7073676228092353617t_unit @ G @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_463_monoid__cancel_Ol__cancel,axiom,
! [G: partia2956882679547061052t_unit,C: list_list_a,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( ( mult_l4853965630390486993t_unit @ G @ C @ A )
= ( mult_l4853965630390486993t_unit @ G @ C @ B ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.l_cancel
thf(fact_464_monoid__cancel_Or__cancel,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,C: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( ( mult_a_ring_ext_a_b @ G @ A @ C )
= ( mult_a_ring_ext_a_b @ G @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_465_monoid__cancel_Or__cancel,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,C: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( ( mult_l7073676228092353617t_unit @ G @ A @ C )
= ( mult_l7073676228092353617t_unit @ G @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_466_monoid__cancel_Or__cancel,axiom,
! [G: partia2956882679547061052t_unit,A: list_list_a,C: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( ( mult_l4853965630390486993t_unit @ G @ A @ C )
= ( mult_l4853965630390486993t_unit @ G @ B @ C ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ G ) )
=> ( A = B ) ) ) ) ) ) ).
% monoid_cancel.r_cancel
thf(fact_467_univ__poly__is__principal,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_principal
thf(fact_468_exists__unique__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X4: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ X4 )
& ! [Y4: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ Y4 )
=> ( Y4 = X4 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_469_long__division__zero_I1_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_470_long__division__closed_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_471_same__pmod__iff__pdivides,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( polyno5814909790663948098es_a_b @ r @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_472_pprimeI,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
| ( polyno5814909790663948098es_a_b @ r @ P @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).
% pprimeI
thf(fact_473_pmod__zero__iff__pdivides,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q @ P ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_474_subring__degree__one__associatedI,axiom,
! [K: set_a,A: a,A4: a,B: a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ K )
=> ( ( member_a @ A4 @ K )
=> ( ( member_a @ B @ K )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ r @ A4 @ B ) @ nil_a ) ) ) ) ) ) ) ) ).
% subring_degree_one_associatedI
thf(fact_475_normalize_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X3
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_476_pow__non__zero,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
!= ( zero_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N )
!= ( zero_a_b @ r ) ) ) ) ).
% pow_non_zero
thf(fact_477_pow__mult__distrib,axiom,
! [X3: a,Y2: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_478_nat__pow__distrib,axiom,
! [X3: a,Y2: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_479_nat__pow__comm,axiom,
! [X3: a,N: nat,M: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_480_group__commutes__pow,axiom,
! [X3: a,Y2: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X3 ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_481_dense__repr_Osimps_I1_J,axiom,
( ( dense_repr_a_b @ r @ nil_a )
= nil_Pr7402525243500994295_a_nat ) ).
% dense_repr.simps(1)
thf(fact_482_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_483_long__division__closed_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_484_eval__monom,axiom,
! [B: a,A: a,N: nat] :
( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( monom_a_b @ r @ B @ N ) @ A )
= ( mult_a_ring_ext_a_b @ r @ B @ ( pow_a_1026414303147256608_b_nat @ r @ A @ N ) ) ) ) ) ).
% eval_monom
thf(fact_485_long__division__zero_I2_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_486_pirreducibleE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_487_pprimeE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_488_pirreducibleE_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R2 ) )
=> ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ) ) ) ).
% pirreducibleE(3)
thf(fact_489_pirreducibleI,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R4 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_490_nat__pow__closed,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_491_nat__pow__one,axiom,
! [N: nat] :
( ( pow_a_1026414303147256608_b_nat @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( one_a_ring_ext_a_b @ r ) ) ).
% nat_pow_one
thf(fact_492_one__is__polynomial,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( polynomial_a_b @ r @ K @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) ) ) ).
% one_is_polynomial
thf(fact_493_transpose_Ocases,axiom,
! [X3: list_l2471972001652375325_a_nat] :
( ( X3 != nil_li191968740515856775_a_nat )
=> ( ! [Xss: list_l2471972001652375325_a_nat] :
( X3
!= ( cons_l2046435710214046167_a_nat @ nil_Pr7402525243500994295_a_nat @ Xss ) )
=> ~ ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Xss: list_l2471972001652375325_a_nat] :
( X3
!= ( cons_l2046435710214046167_a_nat @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_494_transpose_Ocases,axiom,
! [X3: list_list_a] :
( ( X3 != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X3
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X4: a,Xs3: list_a,Xss: list_list_a] :
( X3
!= ( cons_list_a @ ( cons_a @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_495_not__Cons__self2,axiom,
! [X3: a,Xs2: list_a] :
( ( cons_a @ X3 @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_496_ring_Olong__divides_Ocong,axiom,
polyno2806191415236617128es_a_b = polyno2806191415236617128es_a_b ).
% ring.long_divides.cong
thf(fact_497_ring_Opmod_Ocong,axiom,
polynomial_pmod_a_b = polynomial_pmod_a_b ).
% ring.pmod.cong
thf(fact_498_ring_Opdiv_Ocong,axiom,
polynomial_pdiv_a_b = polynomial_pdiv_a_b ).
% ring.pdiv.cong
thf(fact_499_list__nonempty__induct,axiom,
! [Xs2: list_P3592885314253461005_a_nat,P3: list_P3592885314253461005_a_nat > $o] :
( ( Xs2 != nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: product_prod_a_nat] : ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ nil_Pr7402525243500994295_a_nat ) )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] :
( ( Xs3 != nil_Pr7402525243500994295_a_nat )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) ) ) )
=> ( P3 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_500_list__nonempty__induct,axiom,
! [Xs2: list_a,P3: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X4: a] : ( P3 @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) ) ) )
=> ( P3 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_501_list__induct2_H,axiom,
! [P3: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o,Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] : ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ nil_Pr7402525243500994295_a_nat )
=> ( ! [Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] : ( P3 @ nil_Pr7402525243500994295_a_nat @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( P3 @ Xs3 @ Ys4 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) ) )
=> ( P3 @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_502_list__induct2_H,axiom,
! [P3: list_P3592885314253461005_a_nat > list_a > $o,Xs2: list_P3592885314253461005_a_nat,Ys2: list_a] :
( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] : ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ nil_a )
=> ( ! [Y5: a,Ys4: list_a] : ( P3 @ nil_Pr7402525243500994295_a_nat @ ( cons_a @ Y5 @ Ys4 ) )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: a,Ys4: list_a] :
( ( P3 @ Xs3 @ Ys4 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) ) )
=> ( P3 @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_503_list__induct2_H,axiom,
! [P3: list_a > list_P3592885314253461005_a_nat > $o,Xs2: list_a,Ys2: list_P3592885314253461005_a_nat] :
( ( P3 @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: a,Xs3: list_a] : ( P3 @ ( cons_a @ X4 @ Xs3 ) @ nil_Pr7402525243500994295_a_nat )
=> ( ! [Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] : ( P3 @ nil_a @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) )
=> ( ! [X4: a,Xs3: list_a,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( P3 @ Xs3 @ Ys4 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) ) )
=> ( P3 @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_504_list__induct2_H,axiom,
! [P3: list_a > list_a > $o,Xs2: list_a,Ys2: list_a] :
( ( P3 @ nil_a @ nil_a )
=> ( ! [X4: a,Xs3: list_a] : ( P3 @ ( cons_a @ X4 @ Xs3 ) @ nil_a )
=> ( ! [Y5: a,Ys4: list_a] : ( P3 @ nil_a @ ( cons_a @ Y5 @ Ys4 ) )
=> ( ! [X4: a,Xs3: list_a,Y5: a,Ys4: list_a] :
( ( P3 @ Xs3 @ Ys4 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) ) )
=> ( P3 @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_505_neq__Nil__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( Xs2 != nil_Pr7402525243500994295_a_nat )
= ( ? [Y6: product_prod_a_nat,Ys: list_P3592885314253461005_a_nat] :
( Xs2
= ( cons_P5205166803686508359_a_nat @ Y6 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_506_neq__Nil__conv,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
= ( ? [Y6: a,Ys: list_a] :
( Xs2
= ( cons_a @ Y6 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_507_remdups__adj_Ocases,axiom,
! [X3: list_P3592885314253461005_a_nat] :
( ( X3 != nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: product_prod_a_nat] :
( X3
!= ( cons_P5205166803686508359_a_nat @ X4 @ nil_Pr7402525243500994295_a_nat ) )
=> ~ ! [X4: product_prod_a_nat,Y5: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] :
( X3
!= ( cons_P5205166803686508359_a_nat @ X4 @ ( cons_P5205166803686508359_a_nat @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_508_remdups__adj_Ocases,axiom,
! [X3: list_a] :
( ( X3 != nil_a )
=> ( ! [X4: a] :
( X3
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,Y5: a,Xs3: list_a] :
( X3
!= ( cons_a @ X4 @ ( cons_a @ Y5 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_509_list_Oexhaust,axiom,
! [Y2: list_P3592885314253461005_a_nat] :
( ( Y2 != nil_Pr7402525243500994295_a_nat )
=> ~ ! [X212: product_prod_a_nat,X222: list_P3592885314253461005_a_nat] :
( Y2
!= ( cons_P5205166803686508359_a_nat @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_510_list_Oexhaust,axiom,
! [Y2: list_a] :
( ( Y2 != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y2
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_511_list_OdiscI,axiom,
! [List: list_P3592885314253461005_a_nat,X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat] :
( ( List
= ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) )
=> ( List != nil_Pr7402525243500994295_a_nat ) ) ).
% list.discI
thf(fact_512_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_513_list_Odistinct_I1_J,axiom,
! [X21: product_prod_a_nat,X22: list_P3592885314253461005_a_nat] :
( nil_Pr7402525243500994295_a_nat
!= ( cons_P5205166803686508359_a_nat @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_514_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_515_list__induct4,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_a,Ws: list_a,P3: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs3: list_a,Y5: a,Ys4: list_a,Z4: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_516_list__induct4,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_a,Zs: list_a,Ws: list_a,P3: list_P3592885314253461005_a_nat > list_a > list_a > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: a,Ys4: list_a,Z4: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_517_list__induct4,axiom,
! [Xs2: list_a,Ys2: list_P3592885314253461005_a_nat,Zs: list_a,Ws: list_a,P3: list_a > list_P3592885314253461005_a_nat > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a )
=> ( ! [X4: a,Xs3: list_a,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z4: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_518_list__induct4,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_P3592885314253461005_a_nat,Ws: list_a,P3: list_a > list_a > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X4: a,Xs3: list_a,Y5: a,Ys4: list_a,Z4: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z4 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_519_list__induct4,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_a,Ws: list_P3592885314253461005_a_nat,P3: list_a > list_a > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s984997627204368545_a_nat @ Ws ) )
=> ( ( P3 @ nil_a @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: a,Xs3: list_a,Y5: a,Ys4: list_a,Z4: a,Zs2: list_a,W: product_prod_a_nat,Ws2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s984997627204368545_a_nat @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) @ ( cons_P5205166803686508359_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_520_list__induct4,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat,Zs: list_a,Ws: list_a,P3: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_a > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z4: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_521_list__induct4,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_a,Zs: list_P3592885314253461005_a_nat,Ws: list_a,P3: list_P3592885314253461005_a_nat > list_a > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: a,Ys4: list_a,Z4: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat,W: a,Ws2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z4 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_522_list__induct4,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_a,Zs: list_a,Ws: list_P3592885314253461005_a_nat,P3: list_P3592885314253461005_a_nat > list_a > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s984997627204368545_a_nat @ Ws ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: a,Ys4: list_a,Z4: a,Zs2: list_a,W: product_prod_a_nat,Ws2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s984997627204368545_a_nat @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) @ ( cons_P5205166803686508359_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_523_list__induct4,axiom,
! [Xs2: list_a,Ys2: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,Ws: list_a,P3: list_a > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P3 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X4: a,Xs3: list_a,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z4: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z4 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_524_list__induct4,axiom,
! [Xs2: list_a,Ys2: list_P3592885314253461005_a_nat,Zs: list_a,Ws: list_P3592885314253461005_a_nat,P3: list_a > list_P3592885314253461005_a_nat > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s984997627204368545_a_nat @ Ws ) )
=> ( ( P3 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: a,Xs3: list_a,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z4: a,Zs2: list_a,W: product_prod_a_nat,Ws2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s984997627204368545_a_nat @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) @ ( cons_P5205166803686508359_a_nat @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_525_list__induct3,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,P3: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z4: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_526_list__induct3,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat,Zs: list_a,P3: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z4: a,Zs2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_527_list__induct3,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_a,Zs: list_P3592885314253461005_a_nat,P3: list_P3592885314253461005_a_nat > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: a,Ys4: list_a,Z4: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_528_list__induct3,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_a,Zs: list_a,P3: list_P3592885314253461005_a_nat > list_a > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_a @ nil_a )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: a,Ys4: list_a,Z4: a,Zs2: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_529_list__induct3,axiom,
! [Xs2: list_a,Ys2: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,P3: list_a > list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P3 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: a,Xs3: list_a,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z4: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_530_list__induct3,axiom,
! [Xs2: list_a,Ys2: list_P3592885314253461005_a_nat,Zs: list_a,P3: list_a > list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P3 @ nil_a @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X4: a,Xs3: list_a,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat,Z4: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( ( size_s984997627204368545_a_nat @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_531_list__induct3,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_P3592885314253461005_a_nat,P3: list_a > list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s984997627204368545_a_nat @ Zs ) )
=> ( ( P3 @ nil_a @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: a,Xs3: list_a,Y5: a,Ys4: list_a,Z4: product_prod_a_nat,Zs2: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_s984997627204368545_a_nat @ Zs2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) @ ( cons_P5205166803686508359_a_nat @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_532_list__induct3,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_a,P3: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs ) )
=> ( ( P3 @ nil_a @ nil_a @ nil_a )
=> ( ! [X4: a,Xs3: list_a,Y5: a,Ys4: list_a,Z4: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( ( size_size_list_a @ Ys4 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) @ ( cons_a @ Z4 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_533_list__induct2,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat,P3: list_P3592885314253461005_a_nat > list_P3592885314253461005_a_nat > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( P3 @ Xs3 @ Ys4 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) ) ) )
=> ( P3 @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_534_list__induct2,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_a,P3: list_P3592885314253461005_a_nat > list_a > $o] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P3 @ nil_Pr7402525243500994295_a_nat @ nil_a )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: a,Ys4: list_a] :
( ( ( size_s984997627204368545_a_nat @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P3 @ Xs3 @ Ys4 )
=> ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) ) ) )
=> ( P3 @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_535_list__induct2,axiom,
! [Xs2: list_a,Ys2: list_P3592885314253461005_a_nat,P3: list_a > list_P3592885314253461005_a_nat > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ( ( P3 @ nil_a @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: a,Xs3: list_a,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( ( ( size_size_list_a @ Xs3 )
= ( size_s984997627204368545_a_nat @ Ys4 ) )
=> ( ( P3 @ Xs3 @ Ys4 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) ) ) )
=> ( P3 @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_536_list__induct2,axiom,
! [Xs2: list_a,Ys2: list_a,P3: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P3 @ nil_a @ nil_a )
=> ( ! [X4: a,Xs3: list_a,Y5: a,Ys4: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys4 ) )
=> ( ( P3 @ Xs3 @ Ys4 )
=> ( P3 @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) ) ) )
=> ( P3 @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_537_impossible__Cons,axiom,
! [Xs2: list_a,Ys2: list_a,X3: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs2 ) @ ( size_size_list_a @ Ys2 ) )
=> ( Xs2
!= ( cons_a @ X3 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_538_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_539_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_540_ring_Odense__repr_Ocases,axiom,
! [R: partia2670972154091845814t_unit,X3: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( X3 != nil_list_a )
=> ~ ! [V: list_a,Va: list_list_a] :
( X3
!= ( cons_list_a @ V @ Va ) ) ) ) ).
% ring.dense_repr.cases
thf(fact_541_ring_Odense__repr_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,X3: list_a] :
( ( ring_a_b @ R )
=> ( ( X3 != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X3
!= ( cons_a @ V @ Va ) ) ) ) ).
% ring.dense_repr.cases
thf(fact_542_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2956882679547061052t_unit,R2: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R2 )
=> ~ ( member_list_list_a @ R2 @ ( units_4903515905731149798t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_543_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_544_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ~ ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_545_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_546_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_547_monoid__cancel_Oassoc__unit__r,axiom,
! [G: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ G ) )
=> ( ( associ5603075271488036121t_unit @ G @ A @ B )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_r
thf(fact_548_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ G ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_549_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ G ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_550_monoid__cancel_Oassoc__unit__l,axiom,
! [G: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( associ5603075271488036121t_unit @ G @ A @ B )
=> ( ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ G ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
=> ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.assoc_unit_l
thf(fact_551_univ__poly__one,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
( ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) )
= ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) ) ).
% univ_poly_one
thf(fact_552_univ__poly__one,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ K ) )
= ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_553_univ__poly__one,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
= ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) ) ).
% univ_poly_one
thf(fact_554_domain_Oring__associated__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( associ5860276527279195403xt_a_b @ R @ A @ B )
= ( ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ R ) )
& ( A
= ( mult_a_ring_ext_a_b @ R @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_555_domain_Oring__associated__iff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( associ8407585678920448409t_unit @ R @ A @ B )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ R ) )
& ( A
= ( mult_l7073676228092353617t_unit @ R @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_556_domain_Oring__associated__iff,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( associ5603075271488036121t_unit @ R @ A @ B )
= ( ? [X: list_list_a] :
( ( member_list_list_a @ X @ ( units_4903515905731149798t_unit @ R ) )
& ( A
= ( mult_l4853965630390486993t_unit @ R @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_557_domain_OpirreducibleE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_558_domain_OpirreducibleE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.pirreducibleE(2)
thf(fact_559_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2956882679547061052t_unit,R2: list_list_a,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R2 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( R2
= ( mult_l4853965630390486993t_unit @ R @ A @ B ) )
=> ( ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ R ) )
| ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_560_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,R2: a,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ R @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_561_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,R2: list_a,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( R2
= ( mult_l7073676228092353617t_unit @ R @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_562_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
= ( ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ G ) )
& ( A
= ( mult_a_ring_ext_a_b @ G @ B @ X ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_563_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ G ) )
& ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ X ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_564_monoid__cancel_Oassociated__iff,axiom,
! [G: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( associ5603075271488036121t_unit @ G @ A @ B )
= ( ? [X: list_list_a] :
( ( member_list_list_a @ X @ ( units_4903515905731149798t_unit @ G ) )
& ( A
= ( mult_l4853965630390486993t_unit @ G @ B @ X ) ) ) ) ) ) ) ) ).
% monoid_cancel.associated_iff
thf(fact_565_monoid__cancel_OassociatedE2,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_ring_ext_a_b @ G @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_ring_ext_a_b @ G ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ~ ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_566_monoid__cancel_OassociatedE2,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ! [U2: list_a] :
( ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ U2 ) )
=> ~ ( member_list_a @ U2 @ ( units_2932844235741507942t_unit @ G ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ~ ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_567_monoid__cancel_OassociatedE2,axiom,
! [G: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( associ5603075271488036121t_unit @ G @ A @ B )
=> ( ! [U2: list_list_a] :
( ( A
= ( mult_l4853965630390486993t_unit @ G @ B @ U2 ) )
=> ~ ( member_list_list_a @ U2 @ ( units_4903515905731149798t_unit @ G ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
=> ~ ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ) ).
% monoid_cancel.associatedE2
thf(fact_568_monoid__cancel_OassociatedD2,axiom,
! [G: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( monoid5798828371819920185xt_a_b @ G )
=> ( ( associ5860276527279195403xt_a_b @ G @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ G ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( units_a_ring_ext_a_b @ G ) )
& ( A
= ( mult_a_ring_ext_a_b @ G @ B @ X4 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_569_monoid__cancel_OassociatedD2,axiom,
! [G: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( monoid4303264861975686087t_unit @ G )
=> ( ( associ8407585678920448409t_unit @ G @ A @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ G ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( units_2932844235741507942t_unit @ G ) )
& ( A
= ( mult_l7073676228092353617t_unit @ G @ B @ X4 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_570_monoid__cancel_OassociatedD2,axiom,
! [G: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( monoid576229335242748231t_unit @ G )
=> ( ( associ5603075271488036121t_unit @ G @ A @ B )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ G ) )
=> ? [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( units_4903515905731149798t_unit @ G ) )
& ( A
= ( mult_l4853965630390486993t_unit @ G @ B @ X4 ) ) ) ) ) ) ) ).
% monoid_cancel.associatedD2
thf(fact_571_var__def,axiom,
( var_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b] : ( cons_a @ ( one_a_ring_ext_a_b @ R3 ) @ ( cons_a @ ( zero_a_b @ R3 ) @ nil_a ) ) ) ) ).
% var_def
thf(fact_572_var__def,axiom,
( var_li8453953174693405341t_unit
= ( ^ [R3: partia2670972154091845814t_unit] : ( cons_list_a @ ( one_li8328186300101108157t_unit @ R3 ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R3 ) @ nil_list_a ) ) ) ) ).
% var_def
thf(fact_573_var__def,axiom,
( var_se6008125447796440765t_unit
= ( ^ [R3: partia7496981018696276118t_unit] : ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R3 ) @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R3 ) @ nil_set_list_a ) ) ) ) ).
% var_def
thf(fact_574_domain_OpirreducibleE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ R2 ) )
=> ( ( member_list_list_a @ Q @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
| ( member_list_list_a @ R2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_575_domain_OpirreducibleE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ R2 ) )
=> ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
| ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ) ) ) ) ).
% domain.pirreducibleE(3)
thf(fact_576_domain_OpprimeE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_577_domain_OpprimeE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.pprimeE(2)
thf(fact_578_domain_Oone__is__polynomial,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( polyno1315193887021588240t_unit @ R @ K @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_579_domain_Oone__is__polynomial,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( polyno3115169382166032176t_unit @ R @ K @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ nil_set_list_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_580_domain_Oone__is__polynomial,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( polynomial_a_b @ R @ K @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ) ) ).
% domain.one_is_polynomial
thf(fact_581_domain_Olong__division__closed_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_582_domain_Olong__division__closed_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ R @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(2)
thf(fact_583_domain_Ouniv__poly__is__principal,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_584_domain_Ouniv__poly__is__principal,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_principal
thf(fact_585_domain_OpirreducibleI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ! [Q2: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q2 @ R4 ) )
=> ( ( member_list_list_a @ Q2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
| ( member_list_list_a @ R4 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_586_domain_OpirreducibleI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q2 @ R4 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
| ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pirreducibleI
thf(fact_587_principal__domain_Oprimeness__condition,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ P )
= ( ring_r5437400583859147359t_unit @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_588_principal__domain_Oprimeness__condition,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P )
= ( ring_ring_prime_a_b @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_589_principal__domain_Oprimeness__condition,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P )
= ( ring_r6430282645014804837t_unit @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_590_domain_Osubring__degree__one__associatedI,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a,A: set_list_a,A4: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( ( member_set_list_a @ A @ K )
=> ( ( member_set_list_a @ A4 @ K )
=> ( ( member_set_list_a @ B @ K )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ A4 )
= ( one_se1127990129394575805t_unit @ R ) )
=> ( associ8249012953061539097t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ ( cons_set_list_a @ A @ ( cons_set_list_a @ B @ nil_set_list_a ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( mult_s7802724872828879953t_unit @ R @ A4 @ B ) @ nil_set_list_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_591_domain_Osubring__degree__one__associatedI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_a,A4: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_a @ A @ K )
=> ( ( member_list_a @ A4 @ K )
=> ( ( member_list_a @ B @ K )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ A4 )
= ( one_li8328186300101108157t_unit @ R ) )
=> ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( mult_l7073676228092353617t_unit @ R @ A4 @ B ) @ nil_list_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_592_domain_Osubring__degree__one__associatedI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: a,A4: a,B: a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_a @ A @ K )
=> ( ( member_a @ A4 @ K )
=> ( ( member_a @ B @ K )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ A4 )
= ( one_a_ring_ext_a_b @ R ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ K ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ R @ A4 @ B ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% domain.subring_degree_one_associatedI
thf(fact_593_domain_Olong__division__zero_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ nil_list_a @ Q )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_594_domain_Olong__division__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pmod_a_b @ R @ nil_a @ Q )
= nil_a ) ) ) ) ).
% domain.long_division_zero(2)
thf(fact_595_domain_Olong__division__closed_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_596_domain_Olong__division__closed_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ R @ P @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ).
% domain.long_division_closed(1)
thf(fact_597_domain_Opmod__zero__iff__pdivides,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ P @ Q )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ R @ Q @ P ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_598_domain_Opmod__zero__iff__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ R @ P @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ R @ Q @ P ) ) ) ) ) ) ).
% domain.pmod_zero_iff_pdivides
thf(fact_599_domain_Olong__division__zero_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ nil_list_a @ Q )
= nil_list_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_600_domain_Olong__division__zero_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ R @ nil_a @ Q )
= nil_a ) ) ) ) ).
% domain.long_division_zero(1)
thf(fact_601_domain_OpprimeI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ! [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q2 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ Q2 )
| ( polyno5814909790663948098es_a_b @ R @ P @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_602_domain_OpprimeI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ! [Q2: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q2 @ R4 ) )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ Q2 )
| ( polyno8016796738000020810t_unit @ R @ P @ R4 ) ) ) ) )
=> ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) ) ) ) ) ) ) ).
% domain.pprimeI
thf(fact_603_domain_Osame__pmod__iff__pdivides,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ A @ Q )
= ( polyno1727750685288865234t_unit @ R @ B @ Q ) )
= ( polyno8016796738000020810t_unit @ R @ Q @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_604_domain_Osame__pmod__iff__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ R @ A @ Q )
= ( polynomial_pmod_a_b @ R @ B @ Q ) )
= ( polyno5814909790663948098es_a_b @ R @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ B ) ) ) ) ) ) ) ) ).
% domain.same_pmod_iff_pdivides
thf(fact_605_domain_Oexists__unique__long__division,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q != nil_list_a )
=> ? [X4: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R @ P @ Q @ X4 )
& ! [Y4: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ R @ P @ Q @ Y4 )
=> ( Y4 = X4 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_606_domain_Oexists__unique__long__division,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X4: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R @ P @ Q @ X4 )
& ! [Y4: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ R @ P @ Q @ Y4 )
=> ( Y4 = X4 ) ) ) ) ) ) ) ) ).
% domain.exists_unique_long_division
thf(fact_607_long__divisionI,axiom,
! [K: set_a,P: list_a,Q: list_a,B: list_a,R2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_608_long__divisionE,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_609_poly__mult__var,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( P = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
= ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).
% poly_mult_var
thf(fact_610_is__root__imp__pdivides,axiom,
! [P: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X3 )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ P ) ) ) ).
% is_root_imp_pdivides
thf(fact_611_eval__as__unique__hom,axiom,
! [K: set_a,X3: a,H3: list_a > a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ r @ H3 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K )
=> ( ( H3 @ ( cons_a @ K4 @ nil_a ) )
= K4 ) )
=> ( ( ( H3 @ ( var_a_b @ r ) )
= X3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( H3 @ P )
= ( eval_a_b @ r @ P @ X3 ) ) ) ) ) ) ) ) ).
% eval_as_unique_hom
thf(fact_612_pdiv__pmod,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_613_const__term__zero,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= ( zero_a_b @ r ) )
=> ~ ! [P6: list_a] :
( ( polynomial_a_b @ r @ K @ P6 )
=> ( ( P6 != nil_a )
=> ( P
!= ( append_a @ P6 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ) ) ).
% const_term_zero
thf(fact_614_poly__add_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
~ ! [P12: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ P12 @ P22 ) ) ).
% poly_add.cases
thf(fact_615_Units__closed,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_616_Units__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ).
% Units_assoc
thf(fact_617_Units__pow__closed,axiom,
! [X3: a,D: nat] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ D ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_618_poly__mult_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V: a,Va: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_619_combine_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
( ! [K4: a,Ks: list_a,U2: a,Us: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ K4 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X3
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_620_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_621_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_622_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_623_Units__inv__comm,axiom,
! [X3: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_624_Units__cong,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_cong
thf(fact_625_subring__props_I5_J,axiom,
! [K: set_a,H3: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H3 @ K )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ K ) ) ) ).
% subring_props(5)
thf(fact_626_local_Onormalize__idem,axiom,
! [P: list_a,Q: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P ) @ Q ) )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ).
% local.normalize_idem
thf(fact_627_ideal__eq__carrier__iff,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_628_ring__irreducibleE_I4_J,axiom,
! [R2: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ~ ( member_a @ R2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_629_Units__r__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_630_Units__l__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X4 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_631_l__minus,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Y2 )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) ) ) ) ) ).
% l_minus
thf(fact_632_r__minus,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X3 @ ( a_inv_a_b @ r @ Y2 ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) ) ) ) ) ).
% r_minus
thf(fact_633_ring__associated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ X @ B ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_634_associatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_635_associatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_636_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_637_append__assoc,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs2 @ Ys2 ) @ Zs )
= ( append_a @ Xs2 @ ( append_a @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_638_append__same__eq,axiom,
! [Ys2: list_a,Xs2: list_a,Zs: list_a] :
( ( ( append_a @ Ys2 @ Xs2 )
= ( append_a @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_639_same__append__eq,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_a] :
( ( ( append_a @ Xs2 @ Ys2 )
= ( append_a @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_640_ring__irreducibleE_I5_J,axiom,
! [R2: a,A: a,B: a] :
( ( member_a @ R2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R2
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_641_square__eq__one,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ X3 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( X3
= ( one_a_ring_ext_a_b @ r ) )
| ( X3
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% square_eq_one
thf(fact_642_long__division__add__iff,axiom,
! [K: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ C ) @ Q )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_643_long__division__add_I2_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ A @ Q ) @ ( polynomial_pmod_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_644_long__division__add_I1_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ A @ Q ) @ ( polynomial_pdiv_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_645_monic__degree__one__root__condition,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) @ B )
= ( A = B ) ) ) ).
% monic_degree_one_root_condition
thf(fact_646_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_647_append_Oright__neutral,axiom,
! [A: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ A @ nil_Pr7402525243500994295_a_nat )
= A ) ).
% append.right_neutral
thf(fact_648_append__Nil2,axiom,
! [Xs2: list_a] :
( ( append_a @ Xs2 @ nil_a )
= Xs2 ) ).
% append_Nil2
thf(fact_649_append__Nil2,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ Xs2 @ nil_Pr7402525243500994295_a_nat )
= Xs2 ) ).
% append_Nil2
thf(fact_650_append__self__conv,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( ( append_a @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_a ) ) ).
% append_self_conv
thf(fact_651_append__self__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_Pr7402525243500994295_a_nat ) ) ).
% append_self_conv
thf(fact_652_self__append__conv,axiom,
! [Y2: list_a,Ys2: list_a] :
( ( Y2
= ( append_a @ Y2 @ Ys2 ) )
= ( Ys2 = nil_a ) ) ).
% self_append_conv
thf(fact_653_self__append__conv,axiom,
! [Y2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( Y2
= ( append7679239579558125090_a_nat @ Y2 @ Ys2 ) )
= ( Ys2 = nil_Pr7402525243500994295_a_nat ) ) ).
% self_append_conv
thf(fact_654_append__self__conv2,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( ( append_a @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_a ) ) ).
% append_self_conv2
thf(fact_655_append__self__conv2,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% append_self_conv2
thf(fact_656_self__append__conv2,axiom,
! [Y2: list_a,Xs2: list_a] :
( ( Y2
= ( append_a @ Xs2 @ Y2 ) )
= ( Xs2 = nil_a ) ) ).
% self_append_conv2
thf(fact_657_self__append__conv2,axiom,
! [Y2: list_P3592885314253461005_a_nat,Xs2: list_P3592885314253461005_a_nat] :
( ( Y2
= ( append7679239579558125090_a_nat @ Xs2 @ Y2 ) )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% self_append_conv2
thf(fact_658_Nil__is__append__conv,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( nil_a
= ( append_a @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_a )
& ( Ys2 = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_659_Nil__is__append__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( nil_Pr7402525243500994295_a_nat
= ( append7679239579558125090_a_nat @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_Pr7402525243500994295_a_nat )
& ( Ys2 = nil_Pr7402525243500994295_a_nat ) ) ) ).
% Nil_is_append_conv
thf(fact_660_append__is__Nil__conv,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( ( append_a @ Xs2 @ Ys2 )
= nil_a )
= ( ( Xs2 = nil_a )
& ( Ys2 = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_661_append__is__Nil__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs2 @ Ys2 )
= nil_Pr7402525243500994295_a_nat )
= ( ( Xs2 = nil_Pr7402525243500994295_a_nat )
& ( Ys2 = nil_Pr7402525243500994295_a_nat ) ) ) ).
% append_is_Nil_conv
thf(fact_662_append__eq__append__conv,axiom,
! [Xs2: list_a,Ys2: list_a,Us2: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
| ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs2 @ Us2 )
= ( append_a @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_663_exists__long__division,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B2: list_a] :
( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_664_pdivides__imp__is__root,axiom,
! [P: list_a,X3: a] :
( ( P != nil_a )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ r @ P @ X3 ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_665_determination__of__hom,axiom,
! [K: set_a,A3: partia2175431115845679010xt_a_b,H3: list_a > a,G2: list_a > a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ A3 @ H3 )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ r @ K ) @ A3 @ G2 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K )
=> ( ( H3 @ ( cons_a @ K4 @ nil_a ) )
= ( G2 @ ( cons_a @ K4 @ nil_a ) ) ) )
=> ( ( ( H3 @ ( var_a_b @ r ) )
= ( G2 @ ( var_a_b @ r ) ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( H3 @ P )
= ( G2 @ P ) ) ) ) ) ) ) ) ).
% determination_of_hom
thf(fact_666_Units__m__closed,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_667_Units__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_one_closed
thf(fact_668_append1__eq__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat,X3: product_prod_a_nat,Ys2: list_P3592885314253461005_a_nat,Y2: product_prod_a_nat] :
( ( ( append7679239579558125090_a_nat @ Xs2 @ ( cons_P5205166803686508359_a_nat @ X3 @ nil_Pr7402525243500994295_a_nat ) )
= ( append7679239579558125090_a_nat @ Ys2 @ ( cons_P5205166803686508359_a_nat @ Y2 @ nil_Pr7402525243500994295_a_nat ) ) )
= ( ( Xs2 = Ys2 )
& ( X3 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_669_append1__eq__conv,axiom,
! [Xs2: list_a,X3: a,Ys2: list_a,Y2: a] :
( ( ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) )
= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) )
= ( ( Xs2 = Ys2 )
& ( X3 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_670_Units__l__cancel,axiom,
! [X3: a,Y2: a,Z2: a] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X3 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ X3 @ Z2 ) )
= ( Y2 = Z2 ) ) ) ) ) ).
% Units_l_cancel
thf(fact_671_a__inv__closed,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_672_local_Ominus__minus,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X3 ) )
= X3 ) ) ).
% local.minus_minus
thf(fact_673_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_674_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_675_nth__append__length,axiom,
! [Xs2: list_a,X3: a,Ys2: list_a] :
( ( nth_a @ ( append_a @ Xs2 @ ( cons_a @ X3 @ Ys2 ) ) @ ( size_size_list_a @ Xs2 ) )
= X3 ) ).
% nth_append_length
thf(fact_676_add_Oinv__eq__1__iff,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X3 )
= ( zero_a_b @ r ) )
= ( X3
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_677_a__minus__def,axiom,
( a_minu3984020753470702548t_unit
= ( ^ [R3: partia2670972154091845814t_unit,X: list_a,Y6: list_a] : ( add_li7652885771158616974t_unit @ R3 @ X @ ( a_inv_8944721093294617173t_unit @ R3 @ Y6 ) ) ) ) ).
% a_minus_def
thf(fact_678_a__minus__def,axiom,
( a_minus_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,X: a,Y6: a] : ( add_a_b @ R3 @ X @ ( a_inv_a_b @ R3 @ Y6 ) ) ) ) ).
% a_minus_def
thf(fact_679_append__eq__appendI,axiom,
! [Xs2: list_a,Xs1: list_a,Zs: list_a,Ys2: list_a,Us2: list_a] :
( ( ( append_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_a @ Xs1 @ Us2 ) )
=> ( ( append_a @ Xs2 @ Ys2 )
= ( append_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_680_append__eq__append__conv2,axiom,
! [Xs2: list_a,Ys2: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs2 @ Ys2 )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us3: list_a] :
( ( ( Xs2
= ( append_a @ Zs @ Us3 ) )
& ( ( append_a @ Us3 @ Ys2 )
= Ts ) )
| ( ( ( append_a @ Xs2 @ Us3 )
= Zs )
& ( Ys2
= ( append_a @ Us3 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_681_ring_Oring__simprules_I19_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( add_a_b @ R @ X3 @ Y2 ) )
= ( add_a_b @ R @ ( a_inv_a_b @ R @ X3 ) @ ( a_inv_a_b @ R @ Y2 ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_682_ring_Oring__simprules_I19_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) )
= ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ ( a_inv_8944721093294617173t_unit @ R @ Y2 ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_683_ring_Oring__simprules_I19_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( a_inv_7033018035630854991t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) )
= ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ ( a_inv_7033018035630854991t_unit @ R @ Y2 ) ) ) ) ) ) ).
% ring.ring_simprules(19)
thf(fact_684_ring_Oring__simprules_I18_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X3 ) @ ( add_a_b @ R @ X3 @ Y2 ) )
= Y2 ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_685_ring_Oring__simprules_I18_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) )
= Y2 ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_686_ring_Oring__simprules_I18_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) )
= Y2 ) ) ) ) ).
% ring.ring_simprules(18)
thf(fact_687_ring_Oring__simprules_I17_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( add_a_b @ R @ ( a_inv_a_b @ R @ X3 ) @ Y2 ) )
= Y2 ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_688_ring_Oring__simprules_I17_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ Y2 ) )
= Y2 ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_689_ring_Oring__simprules_I17_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ Y2 ) )
= Y2 ) ) ) ) ).
% ring.ring_simprules(17)
thf(fact_690_abelian__group_Or__neg1,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X3 ) @ ( add_a_b @ G @ X3 @ Y2 ) )
= Y2 ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_691_abelian__group_Or__neg1,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X3 ) @ ( add_li7652885771158616974t_unit @ G @ X3 @ Y2 ) )
= Y2 ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_692_abelian__group_Or__neg1,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ ( a_inv_7033018035630854991t_unit @ G @ X3 ) @ ( add_li174743652000525320t_unit @ G @ X3 @ Y2 ) )
= Y2 ) ) ) ) ).
% abelian_group.r_neg1
thf(fact_693_abelian__group_Or__neg2,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( add_a_b @ G @ ( a_inv_a_b @ G @ X3 ) @ Y2 ) )
= Y2 ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_694_abelian__group_Or__neg2,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X3 ) @ Y2 ) )
= Y2 ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_695_abelian__group_Or__neg2,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X3 @ ( add_li174743652000525320t_unit @ G @ ( a_inv_7033018035630854991t_unit @ G @ X3 ) @ Y2 ) )
= Y2 ) ) ) ) ).
% abelian_group.r_neg2
thf(fact_696_abelian__group_Ominus__add,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ ( add_a_b @ G @ X3 @ Y2 ) )
= ( add_a_b @ G @ ( a_inv_a_b @ G @ X3 ) @ ( a_inv_a_b @ G @ Y2 ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_697_abelian__group_Ominus__add,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_inv_8944721093294617173t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ X3 @ Y2 ) )
= ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X3 ) @ ( a_inv_8944721093294617173t_unit @ G @ Y2 ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_698_abelian__group_Ominus__add,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( a_inv_7033018035630854991t_unit @ G @ ( add_li174743652000525320t_unit @ G @ X3 @ Y2 ) )
= ( add_li174743652000525320t_unit @ G @ ( a_inv_7033018035630854991t_unit @ G @ X3 ) @ ( a_inv_7033018035630854991t_unit @ G @ Y2 ) ) ) ) ) ) ).
% abelian_group.minus_add
thf(fact_699_ring_Oring__simprules_I14_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( a_minu3984020753470702548t_unit @ R @ X3 @ Y2 )
= ( add_li7652885771158616974t_unit @ R @ X3 @ ( a_inv_8944721093294617173t_unit @ R @ Y2 ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_700_ring_Oring__simprules_I14_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( a_minus_a_b @ R @ X3 @ Y2 )
= ( add_a_b @ R @ X3 @ ( a_inv_a_b @ R @ Y2 ) ) ) ) ).
% ring.ring_simprules(14)
thf(fact_701_abelian__group_Ominus__eq,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_group_a_b @ G )
=> ( ( a_minus_a_b @ G @ X3 @ Y2 )
= ( add_a_b @ G @ X3 @ ( a_inv_a_b @ G @ Y2 ) ) ) ) ).
% abelian_group.minus_eq
thf(fact_702_abelian__group_Ominus__eq,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( a_minu3984020753470702548t_unit @ G @ X3 @ Y2 )
= ( add_li7652885771158616974t_unit @ G @ X3 @ ( a_inv_8944721093294617173t_unit @ G @ Y2 ) ) ) ) ).
% abelian_group.minus_eq
thf(fact_703_subringE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H3: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H3 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ H ) ) ) ).
% subringE(5)
thf(fact_704_subringE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H3: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H ) ) ) ).
% subringE(5)
thf(fact_705_append__Cons,axiom,
! [X3: a,Xs2: list_a,Ys2: list_a] :
( ( append_a @ ( cons_a @ X3 @ Xs2 ) @ Ys2 )
= ( cons_a @ X3 @ ( append_a @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_706_Cons__eq__appendI,axiom,
! [X3: a,Xs1: list_a,Ys2: list_a,Xs2: list_a,Zs: list_a] :
( ( ( cons_a @ X3 @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X3 @ Xs2 )
= ( append_a @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_707_append__Nil,axiom,
! [Ys2: list_a] :
( ( append_a @ nil_a @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_708_append__Nil,axiom,
! [Ys2: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_709_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_710_append_Oleft__neutral,axiom,
! [A: list_P3592885314253461005_a_nat] :
( ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ A )
= A ) ).
% append.left_neutral
thf(fact_711_eq__Nil__appendI,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append_a @ nil_a @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_712_eq__Nil__appendI,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append7679239579558125090_a_nat @ nil_Pr7402525243500994295_a_nat @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_713_subringE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H22: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_714_subringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H22: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_715_subcringE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H3: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H3 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ H ) ) ) ).
% subcringE(5)
thf(fact_716_subcringE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H3: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H ) ) ) ).
% subcringE(5)
thf(fact_717_ring_Opoly__add_Ocases,axiom,
! [R: partia2670972154091845814t_unit,X3: produc7709606177366032167list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ~ ! [P12: list_list_a,P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ P12 @ P22 ) ) ) ).
% ring.poly_add.cases
thf(fact_718_ring_Opoly__add_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,X3: produc9164743771328383783list_a] :
( ( ring_a_b @ R )
=> ~ ! [P12: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ P12 @ P22 ) ) ) ).
% ring.poly_add.cases
thf(fact_719_subdomainE_I5_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H3: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H3 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H3 ) @ H ) ) ) ).
% subdomainE(5)
thf(fact_720_subdomainE_I5_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H3: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H3 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H3 ) @ H ) ) ) ).
% subdomainE(5)
thf(fact_721_subcringE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H22: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_722_subcringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H22: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_723_subdomainE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H22: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_724_subdomainE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H22: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H22 @ H )
=> ( member_a @ ( add_a_b @ R @ H12 @ H22 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_725_ring_Oring__simprules_I9_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( a_inv_5715216516650856053t_unit @ R @ X3 ) @ X3 )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_726_ring_Oring__simprules_I9_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( a_inv_a_b @ R @ X3 ) @ X3 )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_727_ring_Oring__simprules_I9_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ X3 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_728_ring_Oring__simprules_I9_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ X3 )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% ring.ring_simprules(9)
thf(fact_729_ring_Oring__simprules_I16_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X3 @ ( a_inv_5715216516650856053t_unit @ R @ X3 ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_730_ring_Oring__simprules_I16_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( a_inv_a_b @ R @ X3 ) )
= ( zero_a_b @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_731_ring_Oring__simprules_I16_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_732_ring_Oring__simprules_I16_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ).
% ring.ring_simprules(16)
thf(fact_733_abelian__group_Ominus__equality,axiom,
! [G: partia7496981018696276118t_unit,Y2: set_list_a,X3: set_list_a] :
( ( abelia5304159692179083286t_unit @ G )
=> ( ( ( add_se2486902527185523630t_unit @ G @ Y2 @ X3 )
= ( zero_s2910681146719230829t_unit @ G ) )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( a_inv_5715216516650856053t_unit @ G @ X3 )
= Y2 ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_734_abelian__group_Ominus__equality,axiom,
! [G: partia2175431115845679010xt_a_b,Y2: a,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( ( add_a_b @ G @ Y2 @ X3 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ X3 )
= Y2 ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_735_abelian__group_Ominus__equality,axiom,
! [G: partia2670972154091845814t_unit,Y2: list_a,X3: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( ( add_li7652885771158616974t_unit @ G @ Y2 @ X3 )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_inv_8944721093294617173t_unit @ G @ X3 )
= Y2 ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_736_abelian__group_Ominus__equality,axiom,
! [G: partia2956882679547061052t_unit,Y2: list_list_a,X3: list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( ( add_li174743652000525320t_unit @ G @ Y2 @ X3 )
= ( zero_l347298301471573063t_unit @ G ) )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( a_inv_7033018035630854991t_unit @ G @ X3 )
= Y2 ) ) ) ) ) ).
% abelian_group.minus_equality
thf(fact_737_abelian__group_Or__neg,axiom,
! [G: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia5304159692179083286t_unit @ G )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( add_se2486902527185523630t_unit @ G @ X3 @ ( a_inv_5715216516650856053t_unit @ G @ X3 ) )
= ( zero_s2910681146719230829t_unit @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_738_abelian__group_Or__neg,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( a_inv_a_b @ G @ X3 ) )
= ( zero_a_b @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_739_abelian__group_Or__neg,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ ( a_inv_8944721093294617173t_unit @ G @ X3 ) )
= ( zero_l4142658623432671053t_unit @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_740_abelian__group_Or__neg,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X3 @ ( a_inv_7033018035630854991t_unit @ G @ X3 ) )
= ( zero_l347298301471573063t_unit @ G ) ) ) ) ).
% abelian_group.r_neg
thf(fact_741_abelian__group_Ol__neg,axiom,
! [G: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia5304159692179083286t_unit @ G )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( add_se2486902527185523630t_unit @ G @ ( a_inv_5715216516650856053t_unit @ G @ X3 ) @ X3 )
= ( zero_s2910681146719230829t_unit @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_742_abelian__group_Ol__neg,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( a_inv_a_b @ G @ X3 ) @ X3 )
= ( zero_a_b @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_743_abelian__group_Ol__neg,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X3 ) @ X3 )
= ( zero_l4142658623432671053t_unit @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_744_abelian__group_Ol__neg,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ ( a_inv_7033018035630854991t_unit @ G @ X3 ) @ X3 )
= ( zero_l347298301471573063t_unit @ G ) ) ) ) ).
% abelian_group.l_neg
thf(fact_745_ring_Oring__simprules_I20_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( a_inv_a_b @ R @ ( a_inv_a_b @ R @ X3 ) )
= X3 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_746_ring_Oring__simprules_I20_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) )
= X3 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_747_ring_Oring__simprules_I20_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( a_inv_7033018035630854991t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) )
= X3 ) ) ) ).
% ring.ring_simprules(20)
thf(fact_748_ring_Oring__simprules_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( a_inv_a_b @ R @ X3 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_749_ring_Oring__simprules_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_750_ring_Oring__simprules_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% ring.ring_simprules(3)
thf(fact_751_ring_Ominus__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( a_inv_a_b @ R @ ( zero_a_b @ R ) )
= ( zero_a_b @ R ) ) ) ).
% ring.minus_zero
thf(fact_752_ring_Ominus__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( a_inv_8944721093294617173t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% ring.minus_zero
thf(fact_753_ring_Ominus__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( a_inv_5715216516650856053t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% ring.minus_zero
thf(fact_754_abelian__group_Ominus__minus,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( a_inv_a_b @ G @ ( a_inv_a_b @ G @ X3 ) )
= X3 ) ) ) ).
% abelian_group.minus_minus
thf(fact_755_abelian__group_Ominus__minus,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( a_inv_8944721093294617173t_unit @ G @ ( a_inv_8944721093294617173t_unit @ G @ X3 ) )
= X3 ) ) ) ).
% abelian_group.minus_minus
thf(fact_756_abelian__group_Ominus__minus,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( a_inv_7033018035630854991t_unit @ G @ ( a_inv_7033018035630854991t_unit @ G @ X3 ) )
= X3 ) ) ) ).
% abelian_group.minus_minus
thf(fact_757_abelian__group_Oa__inv__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( a_inv_a_b @ G @ X3 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_758_abelian__group_Oa__inv__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia3891852623213500406t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ G @ X3 ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_759_abelian__group_Oa__inv__closed,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia2778853791629620336t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( member_list_list_a @ ( a_inv_7033018035630854991t_unit @ G @ X3 ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ).
% abelian_group.a_inv_closed
thf(fact_760_ring_Oring__simprules_I22_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y2 @ Z2 ) )
= ( add_a_b @ R @ Y2 @ ( add_a_b @ R @ X3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_761_ring_Oring__simprules_I22_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ Y2 @ Z2 ) )
= ( add_li7652885771158616974t_unit @ R @ Y2 @ ( add_li7652885771158616974t_unit @ R @ X3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_762_ring_Oring__simprules_I22_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ Y2 @ Z2 ) )
= ( add_li174743652000525320t_unit @ R @ Y2 @ ( add_li174743652000525320t_unit @ R @ X3 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(22)
thf(fact_763_ring_Oring__simprules_I10_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y2 )
= ( add_a_b @ R @ Y2 @ X3 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_764_ring_Oring__simprules_I10_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 )
= ( add_li7652885771158616974t_unit @ R @ Y2 @ X3 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_765_ring_Oring__simprules_I10_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ Y2 )
= ( add_li174743652000525320t_unit @ R @ Y2 @ X3 ) ) ) ) ) ).
% ring.ring_simprules(10)
thf(fact_766_ring_Oring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y2 ) @ Z2 )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_767_ring_Oring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_768_ring_Oring__simprules_I7_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(7)
thf(fact_769_ring_Oring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_770_ring_Oring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_771_ring_Oring__simprules_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% ring.ring_simprules(1)
thf(fact_772_rev__nonempty__induct,axiom,
! [Xs2: list_P3592885314253461005_a_nat,P3: list_P3592885314253461005_a_nat > $o] :
( ( Xs2 != nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: product_prod_a_nat] : ( P3 @ ( cons_P5205166803686508359_a_nat @ X4 @ nil_Pr7402525243500994295_a_nat ) )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] :
( ( Xs3 != nil_Pr7402525243500994295_a_nat )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( append7679239579558125090_a_nat @ Xs3 @ ( cons_P5205166803686508359_a_nat @ X4 @ nil_Pr7402525243500994295_a_nat ) ) ) ) )
=> ( P3 @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_773_rev__nonempty__induct,axiom,
! [Xs2: list_a,P3: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X4: a] : ( P3 @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs3: list_a] :
( ( Xs3 != nil_a )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( append_a @ Xs3 @ ( cons_a @ X4 @ nil_a ) ) ) ) )
=> ( P3 @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_774_append__eq__Cons__conv,axiom,
! [Ys2: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat,X3: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat] :
( ( ( append7679239579558125090_a_nat @ Ys2 @ Zs )
= ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 ) )
= ( ( ( Ys2 = nil_Pr7402525243500994295_a_nat )
& ( Zs
= ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 ) ) )
| ? [Ys5: list_P3592885314253461005_a_nat] :
( ( Ys2
= ( cons_P5205166803686508359_a_nat @ X3 @ Ys5 ) )
& ( ( append7679239579558125090_a_nat @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_775_append__eq__Cons__conv,axiom,
! [Ys2: list_a,Zs: list_a,X3: a,Xs2: list_a] :
( ( ( append_a @ Ys2 @ Zs )
= ( cons_a @ X3 @ Xs2 ) )
= ( ( ( Ys2 = nil_a )
& ( Zs
= ( cons_a @ X3 @ Xs2 ) ) )
| ? [Ys5: list_a] :
( ( Ys2
= ( cons_a @ X3 @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_776_Cons__eq__append__conv,axiom,
! [X3: product_prod_a_nat,Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat,Zs: list_P3592885314253461005_a_nat] :
( ( ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 )
= ( append7679239579558125090_a_nat @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_Pr7402525243500994295_a_nat )
& ( ( cons_P5205166803686508359_a_nat @ X3 @ Xs2 )
= Zs ) )
| ? [Ys5: list_P3592885314253461005_a_nat] :
( ( ( cons_P5205166803686508359_a_nat @ X3 @ Ys5 )
= Ys2 )
& ( Xs2
= ( append7679239579558125090_a_nat @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_777_Cons__eq__append__conv,axiom,
! [X3: a,Xs2: list_a,Ys2: list_a,Zs: list_a] :
( ( ( cons_a @ X3 @ Xs2 )
= ( append_a @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_a )
& ( ( cons_a @ X3 @ Xs2 )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X3 @ Ys5 )
= Ys2 )
& ( Xs2
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_778_rev__exhaust,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( Xs2 != nil_Pr7402525243500994295_a_nat )
=> ~ ! [Ys4: list_P3592885314253461005_a_nat,Y5: product_prod_a_nat] :
( Xs2
!= ( append7679239579558125090_a_nat @ Ys4 @ ( cons_P5205166803686508359_a_nat @ Y5 @ nil_Pr7402525243500994295_a_nat ) ) ) ) ).
% rev_exhaust
thf(fact_779_rev__exhaust,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ~ ! [Ys4: list_a,Y5: a] :
( Xs2
!= ( append_a @ Ys4 @ ( cons_a @ Y5 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_780_rev__induct,axiom,
! [P3: list_P3592885314253461005_a_nat > $o,Xs2: list_P3592885314253461005_a_nat] :
( ( P3 @ nil_Pr7402525243500994295_a_nat )
=> ( ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat] :
( ( P3 @ Xs3 )
=> ( P3 @ ( append7679239579558125090_a_nat @ Xs3 @ ( cons_P5205166803686508359_a_nat @ X4 @ nil_Pr7402525243500994295_a_nat ) ) ) )
=> ( P3 @ Xs2 ) ) ) ).
% rev_induct
thf(fact_781_rev__induct,axiom,
! [P3: list_a > $o,Xs2: list_a] :
( ( P3 @ nil_a )
=> ( ! [X4: a,Xs3: list_a] :
( ( P3 @ Xs3 )
=> ( P3 @ ( append_a @ Xs3 @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ( P3 @ Xs2 ) ) ) ).
% rev_induct
thf(fact_782_abelian__groupE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y2 )
= ( add_a_b @ R @ Y2 @ X3 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_783_abelian__groupE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 )
= ( add_li7652885771158616974t_unit @ R @ Y2 @ X3 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_784_abelian__groupE_I4_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( abelia2778853791629620336t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ Y2 )
= ( add_li174743652000525320t_unit @ R @ Y2 @ X3 ) ) ) ) ) ).
% abelian_groupE(4)
thf(fact_785_abelian__groupE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y2 ) @ Z2 )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_786_abelian__groupE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_787_abelian__groupE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( abelia2778853791629620336t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% abelian_groupE(3)
thf(fact_788_abelian__groupE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_789_abelian__groupE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_790_abelian__groupE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( abelia2778853791629620336t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% abelian_groupE(1)
thf(fact_791_shuffles_Ocases,axiom,
! [X3: produc417292134775302395_a_nat] :
( ! [Ys4: list_P3592885314253461005_a_nat] :
( X3
!= ( produc5384655689722402227_a_nat @ nil_Pr7402525243500994295_a_nat @ Ys4 ) )
=> ( ! [Xs3: list_P3592885314253461005_a_nat] :
( X3
!= ( produc5384655689722402227_a_nat @ Xs3 @ nil_Pr7402525243500994295_a_nat ) )
=> ~ ! [X4: product_prod_a_nat,Xs3: list_P3592885314253461005_a_nat,Y5: product_prod_a_nat,Ys4: list_P3592885314253461005_a_nat] :
( X3
!= ( produc5384655689722402227_a_nat @ ( cons_P5205166803686508359_a_nat @ X4 @ Xs3 ) @ ( cons_P5205166803686508359_a_nat @ Y5 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_792_shuffles_Ocases,axiom,
! [X3: produc9164743771328383783list_a] :
( ! [Ys4: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ Ys4 ) )
=> ( ! [Xs3: list_a] :
( X3
!= ( produc6837034575241423639list_a @ Xs3 @ nil_a ) )
=> ~ ! [X4: a,Xs3: list_a,Y5: a,Ys4: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ X4 @ Xs3 ) @ ( cons_a @ Y5 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_793_ring_Onormalize__idem,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ ( append_a @ ( normalize_a_b @ R @ P ) @ Q ) )
= ( normalize_a_b @ R @ ( append_a @ P @ Q ) ) ) ) ).
% ring.normalize_idem
thf(fact_794_ring_Onormalize__idem,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( normal637505603836502915t_unit @ R @ ( append_list_a @ ( normal637505603836502915t_unit @ R @ P ) @ Q ) )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ P @ Q ) ) ) ) ).
% ring.normalize_idem
thf(fact_795_abelian__monoidE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y2 )
= ( add_a_b @ R @ Y2 @ X3 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_796_abelian__monoidE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 )
= ( add_li7652885771158616974t_unit @ R @ Y2 @ X3 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_797_abelian__monoidE_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ Y2 )
= ( add_li174743652000525320t_unit @ R @ Y2 @ X3 ) ) ) ) ) ).
% abelian_monoidE(5)
thf(fact_798_abelian__monoidE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y2 ) @ Z2 )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_799_abelian__monoidE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_800_abelian__monoidE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoidE(3)
thf(fact_801_abelian__monoidE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_802_abelian__monoidE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_803_abelian__monoidE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% abelian_monoidE(1)
thf(fact_804_abelian__monoid_Oa__comm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ Y2 )
= ( add_a_b @ G @ Y2 @ X3 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_805_abelian__monoid_Oa__comm,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ Y2 )
= ( add_li7652885771158616974t_unit @ G @ Y2 @ X3 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_806_abelian__monoid_Oa__comm,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X3 @ Y2 )
= ( add_li174743652000525320t_unit @ G @ Y2 @ X3 ) ) ) ) ) ).
% abelian_monoid.a_comm
thf(fact_807_abelian__monoid_Oa__assoc,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( add_a_b @ G @ X3 @ Y2 ) @ Z2 )
= ( add_a_b @ G @ X3 @ ( add_a_b @ G @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_808_abelian__monoid_Oa__assoc,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( add_li7652885771158616974t_unit @ G @ X3 @ Y2 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ G @ X3 @ ( add_li7652885771158616974t_unit @ G @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_809_abelian__monoid_Oa__assoc,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ ( add_li174743652000525320t_unit @ G @ X3 @ Y2 ) @ Z2 )
= ( add_li174743652000525320t_unit @ G @ X3 @ ( add_li174743652000525320t_unit @ G @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_assoc
thf(fact_810_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( add_a_b @ G @ Y2 @ Z2 ) )
= ( add_a_b @ G @ Y2 @ ( add_a_b @ G @ X3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_811_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ ( add_li7652885771158616974t_unit @ G @ Y2 @ Z2 ) )
= ( add_li7652885771158616974t_unit @ G @ Y2 @ ( add_li7652885771158616974t_unit @ G @ X3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_812_abelian__monoid_Oa__lcomm,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X3 @ ( add_li174743652000525320t_unit @ G @ Y2 @ Z2 ) )
= ( add_li174743652000525320t_unit @ G @ Y2 @ ( add_li174743652000525320t_unit @ G @ X3 @ Z2 ) ) ) ) ) ) ) ).
% abelian_monoid.a_lcomm
thf(fact_813_abelian__monoid_Oa__closed,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( member_a @ ( add_a_b @ G @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_814_abelian__monoid_Oa__closed,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ G @ X3 @ Y2 ) @ ( partia5361259788508890537t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_815_abelian__monoid_Oa__closed,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ G @ X3 @ Y2 ) @ ( partia2464479390973590831t_unit @ G ) ) ) ) ) ).
% abelian_monoid.a_closed
thf(fact_816_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y2 @ Z2 ) )
= ( add_a_b @ R @ Y2 @ ( add_a_b @ R @ X3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_817_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ Y2 @ Z2 ) )
= ( add_li7652885771158616974t_unit @ R @ Y2 @ ( add_li7652885771158616974t_unit @ R @ X3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_818_semiring_Osemiring__simprules_I12_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ Y2 @ Z2 ) )
= ( add_li174743652000525320t_unit @ R @ Y2 @ ( add_li174743652000525320t_unit @ R @ X3 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(12)
thf(fact_819_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ Y2 )
= ( add_a_b @ R @ Y2 @ X3 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_820_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 )
= ( add_li7652885771158616974t_unit @ R @ Y2 @ X3 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_821_semiring_Osemiring__simprules_I7_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ Y2 )
= ( add_li174743652000525320t_unit @ R @ Y2 @ X3 ) ) ) ) ) ).
% semiring.semiring_simprules(7)
thf(fact_822_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X3 @ Y2 ) @ Z2 )
= ( add_a_b @ R @ X3 @ ( add_a_b @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_823_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ X3 @ ( add_li7652885771158616974t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_824_semiring_Osemiring__simprules_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li174743652000525320t_unit @ R @ X3 @ ( add_li174743652000525320t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% semiring.semiring_simprules(5)
thf(fact_825_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_826_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_827_semiring_Osemiring__simprules_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% semiring.semiring_simprules(1)
thf(fact_828_domain_Oconst__term__simprules__shell_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) )
= ( a_inv_a_b @ R @ ( const_term_a_b @ R @ P ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(4)
thf(fact_829_domain_Oconst__term__simprules__shell_I4_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) )
= ( a_inv_8944721093294617173t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(4)
thf(fact_830_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q ) )
= ( add_a_b @ R @ ( const_term_a_b @ R @ P ) @ ( const_term_a_b @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_831_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) @ ( const_6738166269504826821t_unit @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_832_ring_Ol__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( a_inv_a_b @ R @ X3 ) @ Y2 )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y2 ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_833_ring_Ol__minus,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ Y2 )
= ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y2 ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_834_ring_Ol__minus,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ Y2 )
= ( a_inv_7033018035630854991t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Y2 ) ) ) ) ) ) ).
% ring.l_minus
thf(fact_835_ring_Or__minus,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ X3 @ ( a_inv_a_b @ R @ Y2 ) )
= ( a_inv_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Y2 ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_836_ring_Or__minus,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ X3 @ ( a_inv_8944721093294617173t_unit @ R @ Y2 ) )
= ( a_inv_8944721093294617173t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Y2 ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_837_ring_Or__minus,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ X3 @ ( a_inv_7033018035630854991t_unit @ R @ Y2 ) )
= ( a_inv_7033018035630854991t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Y2 ) ) ) ) ) ) ).
% ring.r_minus
thf(fact_838_ring_OUnits__minus__one__closed,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( member_a @ ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) @ ( units_a_ring_ext_a_b @ R ) ) ) ).
% ring.Units_minus_one_closed
thf(fact_839_ring_OUnits__minus__one__closed,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) ) @ ( units_2932844235741507942t_unit @ R ) ) ) ).
% ring.Units_minus_one_closed
thf(fact_840_ring_OUnits__minus__one__closed,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( member_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) ) @ ( units_5837875185506529638t_unit @ R ) ) ) ).
% ring.Units_minus_one_closed
thf(fact_841_ring_Oring__simprules_I15_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X3 @ ( zero_s2910681146719230829t_unit @ R ) )
= X3 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_842_ring_Oring__simprules_I15_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= X3 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_843_ring_Oring__simprules_I15_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ ( zero_l4142658623432671053t_unit @ R ) )
= X3 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_844_ring_Oring__simprules_I15_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ ( zero_l347298301471573063t_unit @ R ) )
= X3 ) ) ) ).
% ring.ring_simprules(15)
thf(fact_845_ring_Oring__simprules_I8_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_846_ring_Oring__simprules_I8_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_847_ring_Oring__simprules_I8_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_848_ring_Oring__simprules_I8_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% ring.ring_simprules(8)
thf(fact_849_ring_Oring__simprules_I23_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z2 @ ( add_a_b @ R @ X3 @ Y2 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z2 @ X3 ) @ ( mult_a_ring_ext_a_b @ R @ Z2 @ Y2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_850_ring_Oring__simprules_I23_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z2 @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ X3 ) @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ Y2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_851_ring_Oring__simprules_I23_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ Z2 @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) )
= ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ Z2 @ X3 ) @ ( mult_l4853965630390486993t_unit @ R @ Z2 @ Y2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(23)
thf(fact_852_ring_Oring__simprules_I13_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X3 @ Y2 ) @ Z2 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_853_ring_Oring__simprules_I13_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_854_ring_Oring__simprules_I13_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Z2 ) @ ( mult_l4853965630390486993t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% ring.ring_simprules(13)
thf(fact_855_same__length__different,axiom,
! [Xs2: list_P3592885314253461005_a_nat,Ys2: list_P3592885314253461005_a_nat] :
( ( Xs2 != Ys2 )
=> ( ( ( size_s984997627204368545_a_nat @ Xs2 )
= ( size_s984997627204368545_a_nat @ Ys2 ) )
=> ? [Pre: list_P3592885314253461005_a_nat,X4: product_prod_a_nat,Xs4: list_P3592885314253461005_a_nat,Y5: product_prod_a_nat,Ys6: list_P3592885314253461005_a_nat] :
( ( X4 != Y5 )
& ( Xs2
= ( append7679239579558125090_a_nat @ Pre @ ( append7679239579558125090_a_nat @ ( cons_P5205166803686508359_a_nat @ X4 @ nil_Pr7402525243500994295_a_nat ) @ Xs4 ) ) )
& ( Ys2
= ( append7679239579558125090_a_nat @ Pre @ ( append7679239579558125090_a_nat @ ( cons_P5205166803686508359_a_nat @ Y5 @ nil_Pr7402525243500994295_a_nat ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_856_same__length__different,axiom,
! [Xs2: list_a,Ys2: list_a] :
( ( Xs2 != Ys2 )
=> ( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ? [Pre: list_a,X4: a,Xs4: list_a,Y5: a,Ys6: list_a] :
( ( X4 != Y5 )
& ( Xs2
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X4 @ nil_a ) @ Xs4 ) ) )
& ( Ys2
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y5 @ nil_a ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_857_abelian__groupE_I6_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia5304159692179083286t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ? [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
& ( ( add_se2486902527185523630t_unit @ R @ X4 @ X3 )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_858_abelian__groupE_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ X4 @ X3 )
= ( zero_a_b @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_859_abelian__groupE_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
& ( ( add_li7652885771158616974t_unit @ R @ X4 @ X3 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_860_abelian__groupE_I6_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia2778853791629620336t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ? [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
& ( ( add_li174743652000525320t_unit @ R @ X4 @ X3 )
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% abelian_groupE(6)
thf(fact_861_abelian__groupE_I5_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia5304159692179083286t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_groupE(5)
thf(fact_862_abelian__groupE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_group_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_groupE(5)
thf(fact_863_abelian__groupE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia3891852623213500406t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_groupE(5)
thf(fact_864_abelian__groupE_I5_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia2778853791629620336t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_groupE(5)
thf(fact_865_abelian__groupI,axiom,
! [R: partia7496981018696276118t_unit] :
( ! [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ! [Y5: set_list_a] :
( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ X4 @ Y5 ) @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ! [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ! [Y5: set_list_a] :
( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R ) )
=> ! [Z4: set_list_a] :
( ( member_set_list_a @ Z4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X4 @ Y5 ) @ Z4 )
= ( add_se2486902527185523630t_unit @ R @ X4 @ ( add_se2486902527185523630t_unit @ R @ Y5 @ Z4 ) ) ) ) ) )
=> ( ! [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ! [Y5: set_list_a] :
( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X4 @ Y5 )
= ( add_se2486902527185523630t_unit @ R @ Y5 @ X4 ) ) ) )
=> ( ! [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ? [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ ( partia141011252114345353t_unit @ R ) )
& ( ( add_se2486902527185523630t_unit @ R @ Xa @ X4 )
= ( zero_s2910681146719230829t_unit @ R ) ) ) )
=> ( abelia5304159692179083286t_unit @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_866_abelian__groupI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X4 @ Y5 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Z4: a] :
( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X4 @ Y5 ) @ Z4 )
= ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y5 @ Z4 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ! [Y5: a] :
( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ Y5 )
= ( add_a_b @ R @ Y5 @ X4 ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( add_a_b @ R @ Xa @ X4 )
= ( zero_a_b @ R ) ) ) )
=> ( abelian_group_a_b @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_867_abelian__groupI,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Y5: list_a] :
( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y5 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Y5: list_a] :
( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Z4: list_a] :
( ( member_list_a @ Z4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y5 ) @ Z4 )
= ( add_li7652885771158616974t_unit @ R @ X4 @ ( add_li7652885771158616974t_unit @ R @ Y5 @ Z4 ) ) ) ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ! [Y5: list_a] :
( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X4 @ Y5 )
= ( add_li7652885771158616974t_unit @ R @ Y5 @ X4 ) ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ? [Xa: list_a] :
( ( member_list_a @ Xa @ ( partia5361259788508890537t_unit @ R ) )
& ( ( add_li7652885771158616974t_unit @ R @ Xa @ X4 )
= ( zero_l4142658623432671053t_unit @ R ) ) ) )
=> ( abelia3891852623213500406t_unit @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_868_abelian__groupI,axiom,
! [R: partia2956882679547061052t_unit] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ! [Y5: list_list_a] :
( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X4 @ Y5 ) @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ! [Y5: list_list_a] :
( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ! [Z4: list_list_a] :
( ( member_list_list_a @ Z4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X4 @ Y5 ) @ Z4 )
= ( add_li174743652000525320t_unit @ R @ X4 @ ( add_li174743652000525320t_unit @ R @ Y5 @ Z4 ) ) ) ) ) )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ! [Y5: list_list_a] :
( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X4 @ Y5 )
= ( add_li174743652000525320t_unit @ R @ Y5 @ X4 ) ) ) )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ? [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ ( partia2464479390973590831t_unit @ R ) )
& ( ( add_li174743652000525320t_unit @ R @ Xa @ X4 )
= ( zero_l347298301471573063t_unit @ R ) ) ) )
=> ( abelia2778853791629620336t_unit @ R ) ) ) ) ) ) ) ).
% abelian_groupI
thf(fact_869_abelian__monoidE_I4_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia3322010900105369177t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_870_abelian__monoidE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_871_abelian__monoidE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia226231641709521465t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_872_abelian__monoidE_I4_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia3641329199688042803t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% abelian_monoidE(4)
thf(fact_873_abelian__monoid_Ol__zero,axiom,
! [G: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( add_se2486902527185523630t_unit @ G @ ( zero_s2910681146719230829t_unit @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_874_abelian__monoid_Ol__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ ( zero_a_b @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_875_abelian__monoid_Ol__zero,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ ( zero_l4142658623432671053t_unit @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_876_abelian__monoid_Ol__zero,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ ( zero_l347298301471573063t_unit @ G ) @ X3 )
= X3 ) ) ) ).
% abelian_monoid.l_zero
thf(fact_877_abelian__monoid_Or__zero,axiom,
! [G: partia7496981018696276118t_unit,X3: set_list_a] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( add_se2486902527185523630t_unit @ G @ X3 @ ( zero_s2910681146719230829t_unit @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_878_abelian__monoid_Or__zero,axiom,
! [G: partia2175431115845679010xt_a_b,X3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( add_a_b @ G @ X3 @ ( zero_a_b @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_879_abelian__monoid_Or__zero,axiom,
! [G: partia2670972154091845814t_unit,X3: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( add_li7652885771158616974t_unit @ G @ X3 @ ( zero_l4142658623432671053t_unit @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_880_abelian__monoid_Or__zero,axiom,
! [G: partia2956882679547061052t_unit,X3: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( add_li174743652000525320t_unit @ G @ X3 @ ( zero_l347298301471573063t_unit @ G ) )
= X3 ) ) ) ).
% abelian_monoid.r_zero
thf(fact_881_abelian__monoid_Ominus__unique,axiom,
! [G: partia7496981018696276118t_unit,Y2: set_list_a,X3: set_list_a,Y3: set_list_a] :
( ( abelia3322010900105369177t_unit @ G )
=> ( ( ( add_se2486902527185523630t_unit @ G @ Y2 @ X3 )
= ( zero_s2910681146719230829t_unit @ G ) )
=> ( ( ( add_se2486902527185523630t_unit @ G @ X3 @ Y3 )
= ( zero_s2910681146719230829t_unit @ G ) )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( member_set_list_a @ Y2 @ ( partia141011252114345353t_unit @ G ) )
=> ( ( member_set_list_a @ Y3 @ ( partia141011252114345353t_unit @ G ) )
=> ( Y2 = Y3 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_882_abelian__monoid_Ominus__unique,axiom,
! [G: partia2175431115845679010xt_a_b,Y2: a,X3: a,Y3: a] :
( ( abelian_monoid_a_b @ G )
=> ( ( ( add_a_b @ G @ Y2 @ X3 )
= ( zero_a_b @ G ) )
=> ( ( ( add_a_b @ G @ X3 @ Y3 )
= ( zero_a_b @ G ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ G ) )
=> ( Y2 = Y3 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_883_abelian__monoid_Ominus__unique,axiom,
! [G: partia2670972154091845814t_unit,Y2: list_a,X3: list_a,Y3: list_a] :
( ( abelia226231641709521465t_unit @ G )
=> ( ( ( add_li7652885771158616974t_unit @ G @ Y2 @ X3 )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( ( add_li7652885771158616974t_unit @ G @ X3 @ Y3 )
= ( zero_l4142658623432671053t_unit @ G ) )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ G ) )
=> ( ( member_list_a @ Y3 @ ( partia5361259788508890537t_unit @ G ) )
=> ( Y2 = Y3 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_884_abelian__monoid_Ominus__unique,axiom,
! [G: partia2956882679547061052t_unit,Y2: list_list_a,X3: list_list_a,Y3: list_list_a] :
( ( abelia3641329199688042803t_unit @ G )
=> ( ( ( add_li174743652000525320t_unit @ G @ Y2 @ X3 )
= ( zero_l347298301471573063t_unit @ G ) )
=> ( ( ( add_li174743652000525320t_unit @ G @ X3 @ Y3 )
= ( zero_l347298301471573063t_unit @ G ) )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ G ) )
=> ( ( member_list_list_a @ Y3 @ ( partia2464479390973590831t_unit @ G ) )
=> ( Y2 = Y3 ) ) ) ) ) ) ) ).
% abelian_monoid.minus_unique
thf(fact_885_abelian__monoidI,axiom,
! [R: partia7496981018696276118t_unit] :
( ! [X4: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ X4 @ Y5 ) @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( member_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ ( partia141011252114345353t_unit @ R ) )
=> ( ! [X4: set_list_a,Y5: set_list_a,Z4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Z4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( add_se2486902527185523630t_unit @ R @ X4 @ Y5 ) @ Z4 )
= ( add_se2486902527185523630t_unit @ R @ X4 @ ( add_se2486902527185523630t_unit @ R @ Y5 @ Z4 ) ) ) ) ) )
=> ( ! [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X4 @ Y5 )
= ( add_se2486902527185523630t_unit @ R @ Y5 @ X4 ) ) ) )
=> ( abelia3322010900105369177t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_886_abelian__monoidI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( add_a_b @ R @ X4 @ Y5 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ ( zero_a_b @ R ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ! [X4: a,Y5: a,Z4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( add_a_b @ R @ X4 @ Y5 ) @ Z4 )
= ( add_a_b @ R @ X4 @ ( add_a_b @ R @ Y5 @ Z4 ) ) ) ) ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X4 @ Y5 )
= ( add_a_b @ R @ Y5 @ X4 ) ) ) )
=> ( abelian_monoid_a_b @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_887_abelian__monoidI,axiom,
! [R: partia2670972154091845814t_unit] :
( ! [X4: list_a,Y5: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y5 ) @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ! [X4: list_a,Y5: list_a,Z4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y5 ) @ Z4 )
= ( add_li7652885771158616974t_unit @ R @ X4 @ ( add_li7652885771158616974t_unit @ R @ Y5 @ Z4 ) ) ) ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: list_a,Y5: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X4 @ Y5 )
= ( add_li7652885771158616974t_unit @ R @ Y5 @ X4 ) ) ) )
=> ( abelia226231641709521465t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_888_abelian__monoidI,axiom,
! [R: partia2956882679547061052t_unit] :
( ! [X4: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ X4 @ Y5 ) @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ! [X4: list_list_a,Y5: list_list_a,Z4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X4 @ Y5 ) @ Z4 )
= ( add_li174743652000525320t_unit @ R @ X4 @ ( add_li174743652000525320t_unit @ R @ Y5 @ Z4 ) ) ) ) ) )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X4 )
= X4 ) )
=> ( ! [X4: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X4 @ Y5 )
= ( add_li174743652000525320t_unit @ R @ Y5 @ X4 ) ) ) )
=> ( abelia3641329199688042803t_unit @ R ) ) ) ) ) ) ).
% abelian_monoidI
thf(fact_889_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ X3 @ ( zero_s2910681146719230829t_unit @ R ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_890_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ X3 @ ( zero_a_b @ R ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_891_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ X3 @ ( zero_l4142658623432671053t_unit @ R ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_892_semiring_Osemiring__simprules_I11_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ X3 @ ( zero_l347298301471573063t_unit @ R ) )
= X3 ) ) ) ).
% semiring.semiring_simprules(11)
thf(fact_893_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( add_se2486902527185523630t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_894_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_895_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( add_li7652885771158616974t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_896_semiring_Osemiring__simprules_I6_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( add_li174743652000525320t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) @ X3 )
= X3 ) ) ) ).
% semiring.semiring_simprules(6)
thf(fact_897_ring_Opoly__mult_Ocases,axiom,
! [R: partia2670972154091845814t_unit,X3: produc7709606177366032167list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ! [P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
=> ~ ! [V: list_a,Va: list_list_a,P22: list_list_a] :
( X3
!= ( produc8696003437204565271list_a @ ( cons_list_a @ V @ Va ) @ P22 ) ) ) ) ).
% ring.poly_mult.cases
thf(fact_898_ring_Opoly__mult_Ocases,axiom,
! [R: partia2175431115845679010xt_a_b,X3: produc9164743771328383783list_a] :
( ( ring_a_b @ R )
=> ( ! [P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V: a,Va: list_a,P22: list_a] :
( X3
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P22 ) ) ) ) ).
% ring.poly_mult.cases
thf(fact_899_semiring_Ol__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X3 @ Y2 ) @ Z2 )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X3 @ Z2 ) @ ( mult_a_ring_ext_a_b @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_900_semiring_Ol__distr,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ X3 @ Z2 ) @ ( mult_l7073676228092353617t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_901_semiring_Ol__distr,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) @ Z2 )
= ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ X3 @ Z2 ) @ ( mult_l4853965630390486993t_unit @ R @ Y2 @ Z2 ) ) ) ) ) ) ) ).
% semiring.l_distr
thf(fact_902_semiring_Or__distr,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,Y2: a,Z2: a] :
( ( semiring_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( mult_a_ring_ext_a_b @ R @ Z2 @ ( add_a_b @ R @ X3 @ Y2 ) )
= ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z2 @ X3 ) @ ( mult_a_ring_ext_a_b @ R @ Z2 @ Y2 ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_903_semiring_Or__distr,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,Y2: list_a,Z2: list_a] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( mult_l7073676228092353617t_unit @ R @ Z2 @ ( add_li7652885771158616974t_unit @ R @ X3 @ Y2 ) )
= ( add_li7652885771158616974t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ X3 ) @ ( mult_l7073676228092353617t_unit @ R @ Z2 @ Y2 ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_904_semiring_Or__distr,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,Y2: list_list_a,Z2: list_list_a] :
( ( semiri2265994252334843677t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Z2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( mult_l4853965630390486993t_unit @ R @ Z2 @ ( add_li174743652000525320t_unit @ R @ X3 @ Y2 ) )
= ( add_li174743652000525320t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ Z2 @ X3 ) @ ( mult_l4853965630390486993t_unit @ R @ Z2 @ Y2 ) ) ) ) ) ) ) ).
% semiring.r_distr
thf(fact_905_domain_Osquare__eq__one,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ X3 @ X3 )
= ( one_se1127990129394575805t_unit @ R ) )
=> ( ( X3
= ( one_se1127990129394575805t_unit @ R ) )
| ( X3
= ( a_inv_5715216516650856053t_unit @ R @ ( one_se1127990129394575805t_unit @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_906_domain_Osquare__eq__one,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ X3 @ X3 )
= ( one_a_ring_ext_a_b @ R ) )
=> ( ( X3
= ( one_a_ring_ext_a_b @ R ) )
| ( X3
= ( a_inv_a_b @ R @ ( one_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_907_domain_Osquare__eq__one,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ X3 @ X3 )
= ( one_li8328186300101108157t_unit @ R ) )
=> ( ( X3
= ( one_li8328186300101108157t_unit @ R ) )
| ( X3
= ( a_inv_8944721093294617173t_unit @ R @ ( one_li8328186300101108157t_unit @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_908_domain_Osquare__eq__one,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ X3 @ X3 )
= ( one_li8234411390022467901t_unit @ R ) )
=> ( ( X3
= ( one_li8234411390022467901t_unit @ R ) )
| ( X3
= ( a_inv_7033018035630854991t_unit @ R @ ( one_li8234411390022467901t_unit @ R ) ) ) ) ) ) ) ).
% domain.square_eq_one
thf(fact_909_domain_Olong__division__add_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno1727750685288865234t_unit @ R @ A @ Q ) @ ( polyno1727750685288865234t_unit @ R @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_910_domain_Olong__division__add_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pmod_a_b @ R @ A @ Q ) @ ( polynomial_pmod_a_b @ R @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(2)
thf(fact_911_domain_Olong__division__add__iff,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ R @ A @ Q )
= ( polyno1727750685288865234t_unit @ R @ B @ Q ) )
= ( ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ C ) @ Q )
= ( polyno1727750685288865234t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_912_domain_Olong__division__add__iff,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ R @ A @ Q )
= ( polynomial_pmod_a_b @ R @ B @ Q ) )
= ( ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ C ) @ Q )
= ( polynomial_pmod_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_division_add_iff
thf(fact_913_domain_Olong__division__add_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno5893782122288709345t_unit @ R @ A @ Q ) @ ( polyno5893782122288709345t_unit @ R @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_914_domain_Olong__division__add_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pdiv_a_b @ R @ A @ Q ) @ ( polynomial_pdiv_a_b @ R @ B @ Q ) ) ) ) ) ) ) ) ).
% domain.long_division_add(1)
thf(fact_915_ring_Omonic__degree__one__root__condition,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( polyno4320237611291262604t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ A ) @ nil_set_list_a ) ) @ B )
= ( A = B ) ) ) ) ).
% ring.monic_degree_one_root_condition
thf(fact_916_ring_Omonic__degree__one__root__condition,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( polyno6951661231331188332t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ A ) @ nil_list_a ) ) @ B )
= ( A = B ) ) ) ) ).
% ring.monic_degree_one_root_condition
thf(fact_917_ring_Omonic__degree__one__root__condition,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( polyno5142720416380192742t_unit @ R @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ A ) @ nil_list_list_a ) ) @ B )
= ( A = B ) ) ) ) ).
% ring.monic_degree_one_root_condition
thf(fact_918_ring_Omonic__degree__one__root__condition,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ A ) @ nil_a ) ) @ B )
= ( A = B ) ) ) ) ).
% ring.monic_degree_one_root_condition
thf(fact_919_domain_Oexists__long__division,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q != nil_list_a )
=> ~ ! [B2: list_list_a] :
( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ! [R4: list_list_a] :
( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ~ ( polyno6947042923167803568t_unit @ R @ P @ Q @ ( produc8696003437204565271list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_920_domain_Oexists__long__division,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B2: list_a] :
( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ R @ P @ Q @ ( produc6837034575241423639list_a @ B2 @ R4 ) ) ) ) ) ) ) ) ) ).
% domain.exists_long_division
thf(fact_921_domain_Opdivides__imp__is__root,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,X3: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( P != nil_set_list_a )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( polyno9075941895896075626t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X3 ) @ nil_set_list_a ) ) @ P )
=> ( polyno4320237611291262604t_unit @ R @ P @ X3 ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_922_domain_Opdivides__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( P != nil_list_a )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ nil_list_a ) ) @ P )
=> ( polyno6951661231331188332t_unit @ R @ P @ X3 ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_923_domain_Opdivides__imp__is__root,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( P != nil_list_list_a )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( polyno4453881341673752516t_unit @ R @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ nil_list_list_a ) ) @ P )
=> ( polyno5142720416380192742t_unit @ R @ P @ X3 ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_924_domain_Opdivides__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X3: a] :
( ( domain_a_b @ R )
=> ( ( P != nil_a )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( polyno5814909790663948098es_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X3 ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ R @ P @ X3 ) ) ) ) ) ).
% domain.pdivides_imp_is_root
thf(fact_925_domain_Odetermination__of__hom,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,A3: partia2175431115845679010xt_a_b,H3: list_a > a,G2: list_a > a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ A3 @ H3 )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ A3 @ G2 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K )
=> ( ( H3 @ ( cons_a @ K4 @ nil_a ) )
= ( G2 @ ( cons_a @ K4 @ nil_a ) ) ) )
=> ( ( ( H3 @ ( var_a_b @ R ) )
= ( G2 @ ( var_a_b @ R ) ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( H3 @ P )
= ( G2 @ P ) ) ) ) ) ) ) ) ) ).
% domain.determination_of_hom
thf(fact_926_ring_Oconst__term__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( polyno1315193887021588240t_unit @ R @ K @ P )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ R @ P )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ~ ! [P6: list_list_a] :
( ( polyno1315193887021588240t_unit @ R @ K @ P6 )
=> ( ( P6 != nil_list_a )
=> ( P
!= ( append_list_a @ P6 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ) ) ) ) ).
% ring.const_term_zero
thf(fact_927_ring_Oconst__term__zero,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( ( polyno3115169382166032176t_unit @ R @ K @ P )
=> ( ( P != nil_set_list_a )
=> ( ( ( const_3308765751713425893t_unit @ R @ P )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ~ ! [P6: list_set_list_a] :
( ( polyno3115169382166032176t_unit @ R @ K @ P6 )
=> ( ( P6 != nil_set_list_a )
=> ( P
!= ( append_set_list_a @ P6 @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) ) ) ) ) ) ) ) ) ).
% ring.const_term_zero
thf(fact_928_ring_Oconst__term__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( polynomial_a_b @ R @ K @ P )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ R @ P )
= ( zero_a_b @ R ) )
=> ~ ! [P6: list_a] :
( ( polynomial_a_b @ R @ K @ P6 )
=> ( ( P6 != nil_a )
=> ( P
!= ( append_a @ P6 @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ) ) ) ).
% ring.const_term_zero
thf(fact_929_domain_Opdiv__pmod,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_930_domain_Opdiv__pmod,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ Q @ ( polynomial_pdiv_a_b @ R @ P @ Q ) ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.pdiv_pmod
thf(fact_931_domain_Ois__root__imp__pdivides,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,X3: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( polyno4320237611291262604t_unit @ R @ P @ X3 )
=> ( polyno9075941895896075626t_unit @ R @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X3 ) @ nil_set_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_932_domain_Ois__root__imp__pdivides,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( polyno5142720416380192742t_unit @ R @ P @ X3 )
=> ( polyno4453881341673752516t_unit @ R @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ nil_list_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_933_domain_Ois__root__imp__pdivides,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X3 )
=> ( polyno8016796738000020810t_unit @ R @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ nil_list_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_934_domain_Ois__root__imp__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X3: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X3 )
=> ( polyno5814909790663948098es_a_b @ R @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X3 ) @ nil_a ) ) @ P ) ) ) ) ).
% domain.is_root_imp_pdivides
thf(fact_935_domain_Oeval__as__unique__hom,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,X3: a,H3: list_a > a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_h7848885096329822662it_a_b @ ( univ_poly_a_b @ R @ K ) @ R @ H3 )
=> ( ! [K4: a] :
( ( member_a @ K4 @ K )
=> ( ( H3 @ ( cons_a @ K4 @ nil_a ) )
= K4 ) )
=> ( ( ( H3 @ ( var_a_b @ R ) )
= X3 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( H3 @ P )
= ( eval_a_b @ R @ P @ X3 ) ) ) ) ) ) ) ) ) ).
% domain.eval_as_unique_hom
thf(fact_936_domain_Oeval__as__unique__hom,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,X3: list_a,H3: list_list_a > list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_h4589914651911841480t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ R @ H3 )
=> ( ! [K4: list_a] :
( ( member_list_a @ K4 @ K )
=> ( ( H3 @ ( cons_list_a @ K4 @ nil_list_a ) )
= K4 ) )
=> ( ( ( H3 @ ( var_li8453953174693405341t_unit @ R ) )
= X3 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( H3 @ P )
= ( eval_l34571156754992824t_unit @ R @ P @ X3 ) ) ) ) ) ) ) ) ) ).
% domain.eval_as_unique_hom
thf(fact_937_domain_Oeval__as__unique__hom,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,X3: list_list_a,H3: list_list_list_a > list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_h7694777735462631100t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ R @ H3 )
=> ( ! [K4: list_list_a] :
( ( member_list_list_a @ K4 @ K )
=> ( ( H3 @ ( cons_list_list_a @ K4 @ nil_list_list_a ) )
= K4 ) )
=> ( ( ( H3 @ ( var_li3532061862469730199t_unit @ R ) )
= X3 )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( H3 @ P )
= ( eval_l1088911609197519410t_unit @ R @ P @ X3 ) ) ) ) ) ) ) ) ) ).
% domain.eval_as_unique_hom
thf(fact_938_domain_Olong__divisionE,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q != nil_list_a )
=> ( polyno6947042923167803568t_unit @ R @ P @ Q @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_939_domain_Olong__divisionE,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ R @ P @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P @ Q ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ) ).
% domain.long_divisionE
thf(fact_940_domain_Olong__divisionI,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a,B: list_list_a,R2: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ R @ P @ Q @ ( produc8696003437204565271list_a @ B @ R2 ) )
=> ( ( produc8696003437204565271list_a @ B @ R2 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_941_domain_Olong__divisionI,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a,B: list_a,R2: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ R @ P @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) )
=> ( ( produc6837034575241423639list_a @ B @ R2 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ R @ P @ Q ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ) ) ).
% domain.long_divisionI
thf(fact_942_domain_Opoly__mult__var,axiom,
! [R: partia7496981018696276118t_unit,K: set_set_list_a,P: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( subrin5643252653130547402t_unit @ K @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) ) )
=> ( ( ( P = nil_set_list_a )
=> ( ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P @ ( var_se6008125447796440765t_unit @ R ) )
= nil_set_list_a ) )
& ( ( P != nil_set_list_a )
=> ( ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ K ) @ P @ ( var_se6008125447796440765t_unit @ R ) )
= ( append_set_list_a @ P @ ( cons_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ nil_set_list_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_943_domain_Opoly__mult__var,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ( P = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ ( var_a_b @ R ) )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ ( var_a_b @ R ) )
= ( append_a @ P @ ( cons_a @ ( zero_a_b @ R ) @ nil_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_944_domain_Opoly__mult__var,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ( P = nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ ( var_li8453953174693405341t_unit @ R ) )
= nil_list_a ) )
& ( ( P != nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ ( var_li8453953174693405341t_unit @ R ) )
= ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ nil_list_a ) ) ) ) ) ) ) ) ).
% domain.poly_mult_var
thf(fact_945_le__alg__mult__imp__pdivides,axiom,
! [X3: a,P: list_a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X3 ) )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ N ) @ P ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_946_alg__multE_I2_J,axiom,
! [X3: a,P: list_a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X3 ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_947_alg__multE_I1_J,axiom,
! [X3: a,P: list_a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X3 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ X3 ) ) @ P ) ) ) ) ).
% alg_multE(1)
thf(fact_948_ring_Ocoeff__add,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,F: list_a,G2: list_a,I: nat] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ G2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( coeff_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ F @ G2 ) @ I )
= ( add_a_b @ R @ ( coeff_a_b @ R @ F @ I ) @ ( coeff_a_b @ R @ G2 @ I ) ) ) ) ) ) ) ).
% ring.coeff_add
thf(fact_949_ring_Ocoeff__add,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,F: list_list_a,G2: list_list_a,I: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ G2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( coeff_6360649920519955023t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ F @ G2 ) @ I )
= ( add_li7652885771158616974t_unit @ R @ ( coeff_6360649920519955023t_unit @ R @ F @ I ) @ ( coeff_6360649920519955023t_unit @ R @ G2 @ I ) ) ) ) ) ) ) ).
% ring.coeff_add
thf(fact_950_ring__hom__ringI,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H3: a > a] :
( ( ring_a_b @ R )
=> ( ( ring_a_b @ S )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H3 @ X4 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X4 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( ring_h4024360765257340990_b_a_b @ R @ S @ H3 ) ) ) ) ) ) ) ).
% ring_hom_ringI
thf(fact_951_ring__hom__ringI,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H3: a > list_a] :
( ( ring_a_b @ R )
=> ( ( ring_l6212528067271185461t_unit @ S )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H3 @ X4 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y5 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X4 @ Y5 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( ring_h5357930050666032198t_unit @ R @ S @ H3 ) ) ) ) ) ) ) ).
% ring_hom_ringI
thf(fact_952_ring__hom__ringI,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H3: list_a > a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ring_a_b @ S )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H3 @ X4 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X4: list_a,Y5: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X4 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X4: list_a,Y5: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( ring_h7848885096329822662it_a_b @ R @ S @ H3 ) ) ) ) ) ) ) ).
% ring_hom_ringI
thf(fact_953_ring__hom__ringI,axiom,
! [R: partia7496981018696276118t_unit,S: partia2175431115845679010xt_a_b,H3: set_list_a > a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( ring_a_b @ S )
=> ( ! [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_a @ ( H3 @ X4 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X4: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H3 @ ( mult_s7802724872828879953t_unit @ R @ X4 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X4: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H3 @ ( add_se2486902527185523630t_unit @ R @ X4 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( ring_h1355656666774236646it_a_b @ R @ S @ H3 ) ) ) ) ) ) ) ).
% ring_hom_ringI
thf(fact_954_ring__hom__ringI,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia7496981018696276118t_unit,H3: a > set_list_a] :
( ( ring_a_b @ R )
=> ( ( ring_s8247141995668492373t_unit @ S )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_set_list_a @ ( H3 @ X4 ) @ ( partia141011252114345353t_unit @ S ) ) )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y5 ) )
= ( mult_s7802724872828879953t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X4 @ Y5 ) )
= ( add_se2486902527185523630t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_se1127990129394575805t_unit @ S ) )
=> ( ring_h7781647138149442662t_unit @ R @ S @ H3 ) ) ) ) ) ) ) ).
% ring_hom_ringI
thf(fact_955_ring__hom__ringI,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,H3: a > list_list_a] :
( ( ring_a_b @ R )
=> ( ( ring_l1939023646219158831t_unit @ S )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H3 @ X4 ) @ ( partia2464479390973590831t_unit @ S ) ) )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( mult_a_ring_ext_a_b @ R @ X4 @ Y5 ) )
= ( mult_l4853965630390486993t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X4: a,Y5: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( add_a_b @ R @ X4 @ Y5 ) )
= ( add_li174743652000525320t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_a_ring_ext_a_b @ R ) )
= ( one_li8234411390022467901t_unit @ S ) )
=> ( ring_h2621993495375454400t_unit @ R @ S @ H3 ) ) ) ) ) ) ) ).
% ring_hom_ringI
thf(fact_956_ring__hom__ringI,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H3: list_a > list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ring_l6212528067271185461t_unit @ S )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H3 @ X4 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X4: list_a,Y5: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X4 @ Y5 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X4: list_a,Y5: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y5 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( ring_h1334922693953046990t_unit @ R @ S @ H3 ) ) ) ) ) ) ) ).
% ring_hom_ringI
thf(fact_957_ring__hom__ringI,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H3: list_list_a > a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ring_a_b @ S )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H3 @ X4 ) @ ( partia707051561876973205xt_a_b @ S ) ) )
=> ( ! [X4: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( mult_l4853965630390486993t_unit @ R @ X4 @ Y5 ) )
= ( mult_a_ring_ext_a_b @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X4: list_list_a,Y5: list_list_a] :
( ( member_list_list_a @ X4 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( add_li174743652000525320t_unit @ R @ X4 @ Y5 ) )
= ( add_a_b @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8234411390022467901t_unit @ R ) )
= ( one_a_ring_ext_a_b @ S ) )
=> ( ring_h3841606220870141376it_a_b @ R @ S @ H3 ) ) ) ) ) ) ) ).
% ring_hom_ringI
thf(fact_958_ring__hom__ringI,axiom,
! [R: partia7496981018696276118t_unit,S: partia2670972154091845814t_unit,H3: set_list_a > list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( ring_l6212528067271185461t_unit @ S )
=> ( ! [X4: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( member_list_a @ ( H3 @ X4 ) @ ( partia5361259788508890537t_unit @ S ) ) )
=> ( ! [X4: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H3 @ ( mult_s7802724872828879953t_unit @ R @ X4 @ Y5 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X4: set_list_a,Y5: set_list_a] :
( ( member_set_list_a @ X4 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ Y5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( H3 @ ( add_se2486902527185523630t_unit @ R @ X4 @ Y5 ) )
= ( add_li7652885771158616974t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_se1127990129394575805t_unit @ R ) )
= ( one_li8328186300101108157t_unit @ S ) )
=> ( ring_h8569328188485256686t_unit @ R @ S @ H3 ) ) ) ) ) ) ) ).
% ring_hom_ringI
thf(fact_959_ring__hom__ringI,axiom,
! [R: partia2670972154091845814t_unit,S: partia7496981018696276118t_unit,H3: list_a > set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ring_s8247141995668492373t_unit @ S )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_set_list_a @ ( H3 @ X4 ) @ ( partia141011252114345353t_unit @ S ) ) )
=> ( ! [X4: list_a,Y5: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( mult_l7073676228092353617t_unit @ R @ X4 @ Y5 ) )
= ( mult_s7802724872828879953t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ! [X4: list_a,Y5: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( add_li7652885771158616974t_unit @ R @ X4 @ Y5 ) )
= ( add_se2486902527185523630t_unit @ S @ ( H3 @ X4 ) @ ( H3 @ Y5 ) ) ) ) )
=> ( ( ( H3 @ ( one_li8328186300101108157t_unit @ R ) )
= ( one_se1127990129394575805t_unit @ S ) )
=> ( ring_h6719293541701509614t_unit @ R @ S @ H3 ) ) ) ) ) ) ) ).
% ring_hom_ringI
thf(fact_960_const__term__simprules__shell_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(3)
thf(fact_961_a__lcomm,axiom,
! [X3: a,Y2: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y2 @ Z2 ) )
= ( add_a_b @ r @ Y2 @ ( add_a_b @ r @ X3 @ Z2 ) ) ) ) ) ) ).
% a_lcomm
thf(fact_962_a__comm,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ Y2 )
= ( add_a_b @ r @ Y2 @ X3 ) ) ) ) ).
% a_comm
thf(fact_963_a__assoc,axiom,
! [X3: a,Y2: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X3 @ Y2 ) @ Z2 )
= ( add_a_b @ r @ X3 @ ( add_a_b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% a_assoc
thf(fact_964_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_965_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_966_subring__props_I7_J,axiom,
! [K: set_a,H12: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H12 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( add_a_b @ r @ H12 @ H22 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_967_local_Ominus__unique,axiom,
! [Y2: a,X3: a,Y3: a] :
( ( ( add_a_b @ r @ Y2 @ X3 )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ Y3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y2 = Y3 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_968_add_Or__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X4 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_969_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X4 )
= X4 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_970_add_Ol__inv__ex,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X4 @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_971_add_Oinv__comm,axiom,
! [X3: a,Y2: a] :
( ( ( add_a_b @ r @ X3 @ Y2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y2 @ X3 )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_972_r__distr,axiom,
! [X3: a,Y2: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z2 @ ( add_a_b @ r @ X3 @ Y2 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z2 @ X3 ) @ ( mult_a_ring_ext_a_b @ r @ Z2 @ Y2 ) ) ) ) ) ) ).
% r_distr
thf(fact_973_l__distr,axiom,
! [X3: a,Y2: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X3 @ Y2 ) @ Z2 )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X3 @ Z2 ) @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% l_distr
thf(fact_974_r__neg2,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Y2 ) )
= Y2 ) ) ) ).
% r_neg2
thf(fact_975_r__neg1,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ ( add_a_b @ r @ X3 @ Y2 ) )
= Y2 ) ) ) ).
% r_neg1
thf(fact_976_local_Ominus__add,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X3 @ Y2 ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ ( a_inv_a_b @ r @ Y2 ) ) ) ) ) ).
% local.minus_add
thf(fact_977_a__transpose__inv,axiom,
! [X3: a,Y2: a,Z2: a] :
( ( ( add_a_b @ r @ X3 @ Y2 )
= Z2 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ Z2 )
= Y2 ) ) ) ) ) ).
% a_transpose_inv
thf(fact_978_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_979_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_980_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_981_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_982_add_Oinv__mult__group,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X3 @ Y2 ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y2 ) @ ( a_inv_a_b @ r @ X3 ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_983_minus__eq,axiom,
! [X3: a,Y2: a] :
( ( a_minus_a_b @ r @ X3 @ Y2 )
= ( add_a_b @ r @ X3 @ ( a_inv_a_b @ r @ Y2 ) ) ) ).
% minus_eq
thf(fact_984_l__neg,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X3 ) @ X3 )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_985_minus__equality,axiom,
! [Y2: a,X3: a] :
( ( ( add_a_b @ r @ Y2 @ X3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X3 )
= Y2 ) ) ) ) ).
% minus_equality
thf(fact_986_r__neg,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( a_inv_a_b @ r @ X3 ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_987_sum__zero__eq__neg,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ Y2 )
= ( zero_a_b @ r ) )
=> ( X3
= ( a_inv_a_b @ r @ Y2 ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_988_univ__poly__a__inv__consistent,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_989_univ__poly__a__inv__length,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
= ( size_size_list_a @ P ) ) ) ) ).
% univ_poly_a_inv_length
thf(fact_990_long__division__a__inv_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_991_long__division__a__inv_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_992_const__term__simprules__shell_I4_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
= ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P ) ) ) ) ) ).
% const_term_simprules_shell(4)
thf(fact_993_coeff__add,axiom,
! [K: set_a,F: list_a,G2: list_a,I: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ G2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( coeff_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ F @ G2 ) @ I )
= ( add_a_b @ r @ ( coeff_a_b @ r @ F @ I ) @ ( coeff_a_b @ r @ G2 @ I ) ) ) ) ) ) ).
% coeff_add
thf(fact_994_a__closed,axiom,
! [X3: a,Y2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X3 @ Y2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_995_local_Oadd_Oright__cancel,axiom,
! [X3: a,Y2: a,Z2: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y2 @ X3 )
= ( add_a_b @ r @ Z2 @ X3 ) )
= ( Y2 = Z2 ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_996_r__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X3 @ ( zero_a_b @ r ) )
= X3 ) ) ).
% r_zero
thf(fact_997_l__zero,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X3 )
= X3 ) ) ).
% l_zero
thf(fact_998_add_Or__cancel__one_H,axiom,
! [X3: a,A: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ A @ X3 ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_999_add_Or__cancel__one,axiom,
! [X3: a,A: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X3 )
= X3 )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_1000_add_Ol__cancel__one_H,axiom,
! [X3: a,A: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X3
= ( add_a_b @ r @ X3 @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_1001_add_Ol__cancel__one,axiom,
! [X3: a,A: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X3 @ A )
= X3 )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_1002_ring_Oalg__mult_Ocong,axiom,
polyno4422430861927485590lt_a_b = polyno4422430861927485590lt_a_b ).
% ring.alg_mult.cong
thf(fact_1003_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P )
= ( a_inv_5142495083975434441t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_1004_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_1005_domain_Ouniv__poly__a__inv__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_consistent
thf(fact_1006_domain_Ouniv__poly__a__inv__length,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) )
= ( size_size_list_a @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_length
thf(fact_1007_domain_Ouniv__poly__a__inv__length,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) )
= ( size_s349497388124573686list_a @ P ) ) ) ) ) ).
% domain.univ_poly_a_inv_length
thf(fact_1008_domain_Olong__division__a__inv_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno1727750685288865234t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(2)
thf(fact_1009_domain_Olong__division__a__inv_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pmod_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pmod_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(2)
thf(fact_1010_domain_Olong__division__a__inv_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ R @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( polyno5893782122288709345t_unit @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_1011_domain_Olong__division__a__inv_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ R @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ R @ K ) @ ( polynomial_pdiv_a_b @ R @ P @ Q ) ) ) ) ) ) ) ).
% domain.long_division_a_inv(1)
thf(fact_1012_ring__hom__ring_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H3: list_a > a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H3 )
=> ( ring_l6212528067271185461t_unit @ R ) ) ).
% ring_hom_ring.axioms(1)
thf(fact_1013_ring__hom__ring_Oaxioms_I2_J,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H3: list_a > a] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H3 )
=> ( ring_a_b @ S ) ) ).
% ring_hom_ring.axioms(2)
thf(fact_1014_ring__hom__ring_Ohom__nat__pow,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,H3: a > list_a,X3: a,N: nat] :
( ( ring_h5357930050666032198t_unit @ R @ S @ H3 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( pow_a_1026414303147256608_b_nat @ R @ X3 @ N ) )
= ( pow_li1142815632869257134it_nat @ S @ ( H3 @ X3 ) @ N ) ) ) ) ).
% ring_hom_ring.hom_nat_pow
thf(fact_1015_ring__hom__ring_Ohom__nat__pow,axiom,
! [R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,H3: a > a,X3: a,N: nat] :
( ( ring_h4024360765257340990_b_a_b @ R @ S @ H3 )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H3 @ ( pow_a_1026414303147256608_b_nat @ R @ X3 @ N ) )
= ( pow_a_1026414303147256608_b_nat @ S @ ( H3 @ X3 ) @ N ) ) ) ) ).
% ring_hom_ring.hom_nat_pow
thf(fact_1016_ring__hom__ring_Ohom__nat__pow,axiom,
! [R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,H3: list_a > list_a,X3: list_a,N: nat] :
( ( ring_h1334922693953046990t_unit @ R @ S @ H3 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( pow_li1142815632869257134it_nat @ R @ X3 @ N ) )
= ( pow_li1142815632869257134it_nat @ S @ ( H3 @ X3 ) @ N ) ) ) ) ).
% ring_hom_ring.hom_nat_pow
thf(fact_1017_ring__hom__ring_Ohom__nat__pow,axiom,
! [R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,H3: list_a > a,X3: list_a,N: nat] :
( ( ring_h7848885096329822662it_a_b @ R @ S @ H3 )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H3 @ ( pow_li1142815632869257134it_nat @ R @ X3 @ N ) )
= ( pow_a_1026414303147256608_b_nat @ S @ ( H3 @ X3 ) @ N ) ) ) ) ).
% ring_hom_ring.hom_nat_pow
thf(fact_1018_ring__hom__ring_Ohom__nat__pow,axiom,
! [R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,H3: list_list_a > list_a,X3: list_list_a,N: nat] :
( ( ring_h4589914651911841480t_unit @ R @ S @ H3 )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( pow_li488931774710091566it_nat @ R @ X3 @ N ) )
= ( pow_li1142815632869257134it_nat @ S @ ( H3 @ X3 ) @ N ) ) ) ) ).
% ring_hom_ring.hom_nat_pow
thf(fact_1019_ring__hom__ring_Ohom__nat__pow,axiom,
! [R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,H3: list_list_a > a,X3: list_list_a,N: nat] :
( ( ring_h3841606220870141376it_a_b @ R @ S @ H3 )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H3 @ ( pow_li488931774710091566it_nat @ R @ X3 @ N ) )
= ( pow_a_1026414303147256608_b_nat @ S @ ( H3 @ X3 ) @ N ) ) ) ) ).
% ring_hom_ring.hom_nat_pow
thf(fact_1020_domain_Opow__non__zero,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( X3
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( pow_se8252319793075206062it_nat @ R @ X3 @ N )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_1021_domain_Opow__non__zero,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( X3
!= ( zero_a_b @ R ) )
=> ( ( pow_a_1026414303147256608_b_nat @ R @ X3 @ N )
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_1022_domain_Opow__non__zero,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( X3
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( pow_li1142815632869257134it_nat @ R @ X3 @ N )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_1023_domain_Opow__non__zero,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( X3
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( pow_li488931774710091566it_nat @ R @ X3 @ N )
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.pow_non_zero
thf(fact_1024_Units__hom,axiom,
! [H3: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X3: a] :
( ( member_a_a @ H3 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( domain_a_b @ R )
=> ( ( domain_a_b @ S )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ R ) )
= ( member_a @ ( H3 @ X3 ) @ ( units_a_ring_ext_a_b @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_1025_Units__hom,axiom,
! [H3: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X3: a] :
( ( member_a_list_a @ H3 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( domain_a_b @ R )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ R ) )
= ( member_list_a @ ( H3 @ X3 ) @ ( units_2932844235741507942t_unit @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_1026_Units__hom,axiom,
! [H3: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X3: list_a] :
( ( member_list_a_a @ H3 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( domain6553523120543210313t_unit @ R )
=> ( ( domain_a_b @ S )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ R ) )
= ( member_a @ ( H3 @ X3 ) @ ( units_a_ring_ext_a_b @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_1027_Units__hom,axiom,
! [H3: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X3: list_a] :
( ( member_list_a_list_a @ H3 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( domain6553523120543210313t_unit @ R )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ R ) )
= ( member_list_a @ ( H3 @ X3 ) @ ( units_2932844235741507942t_unit @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_1028_Units__hom,axiom,
! [H3: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X3: list_list_a] :
( ( member_list_list_a_a @ H3 @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( domain7810152921033798211t_unit @ R )
=> ( ( domain_a_b @ S )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ X3 @ ( units_4903515905731149798t_unit @ R ) )
= ( member_a @ ( H3 @ X3 ) @ ( units_a_ring_ext_a_b @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_1029_Units__hom,axiom,
! [H3: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X3: list_list_a] :
( ( member7168557129179038582list_a @ H3 @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( domain7810152921033798211t_unit @ R )
=> ( ( domain6553523120543210313t_unit @ S )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ X3 @ ( units_4903515905731149798t_unit @ R ) )
= ( member_list_a @ ( H3 @ X3 ) @ ( units_2932844235741507942t_unit @ S ) ) ) ) ) ) ) ).
% Units_hom
thf(fact_1030_domain_Oalg__multE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a,P: list_set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X3 ) @ nil_set_list_a ) ) @ ( polyno1088517687229135038t_unit @ R @ P @ X3 ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_1031_domain_Oalg__multE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P != nil_list_a )
=> ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ R @ P @ X3 ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_1032_domain_Oalg__multE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ nil_list_list_a ) ) @ ( polyno1672195411705137432t_unit @ R @ P @ X3 ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_1033_domain_Oalg__multE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X3 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ R @ P @ X3 ) ) @ P ) ) ) ) ) ).
% domain.alg_multE(1)
thf(fact_1034_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a,P: list_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno1088517687229135038t_unit @ R @ P @ X3 ) )
=> ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X3 ) @ nil_set_list_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_1035_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ R @ P @ X3 ) )
=> ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ nil_list_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_1036_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno1672195411705137432t_unit @ R @ P @ X3 ) )
=> ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ nil_list_list_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_1037_domain_Ole__alg__mult__imp__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,P: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ R @ P @ X3 ) )
=> ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X3 ) @ nil_a ) ) @ N ) @ P ) ) ) ) ) ).
% domain.le_alg_mult_imp_pdivides
thf(fact_1038_domain_Oalg__multE_I2_J,axiom,
! [R: partia7496981018696276118t_unit,X3: set_list_a,P: list_set_list_a,N: nat] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ X3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X3 ) @ nil_set_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno1088517687229135038t_unit @ R @ P @ X3 ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_1039_domain_Oalg__multE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,X3: list_a,P: list_list_a,N: nat] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X3 ) @ nil_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ R @ P @ X3 ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_1040_domain_Oalg__multE_I2_J,axiom,
! [R: partia2956882679547061052t_unit,X3: list_list_a,P: list_list_list_a,N: nat] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X3 ) @ nil_list_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno1672195411705137432t_unit @ R @ P @ X3 ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_1041_domain_Oalg__multE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,X3: a,P: list_a,N: nat] :
( ( domain_a_b @ R )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X3 ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ R @ P @ X3 ) ) ) ) ) ) ) ).
% domain.alg_multE(2)
thf(fact_1042_alg__mult__gt__zero__iff__is__root,axiom,
! [P: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ X3 ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X3 ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1043_subringI,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ r ) @ H )
=> ( ! [H4: a] :
( ( member_a @ H4 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ H4 ) @ H ) )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H2 ) @ H ) ) )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( add_a_b @ r @ H1 @ H2 ) @ H ) ) )
=> ( subring_a_b @ H @ r ) ) ) ) ) ) ).
% subringI
thf(fact_1044_cgenideal__pirreducible,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Q )
=> ( ( member_list_a @ Q @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) ) ) ) ) ) ).
% cgenideal_pirreducible
thf(fact_1045_associated__polynomials__imp__same__roots,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q )
=> ( ( polynomial_roots_a_b @ r @ P )
= ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ).
% associated_polynomials_imp_same_roots
thf(fact_1046_subring__props_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_1047_nat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_1048_length__0__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( ( size_s984997627204368545_a_nat @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_Pr7402525243500994295_a_nat ) ) ).
% length_0_conv
thf(fact_1049_length__0__conv,axiom,
! [Xs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_a ) ) ).
% length_0_conv
thf(fact_1050_nth__Cons__0,axiom,
! [X3: a,Xs2: list_a] :
( ( nth_a @ ( cons_a @ X3 @ Xs2 ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_1051_length__greater__0__conv,axiom,
! [Xs2: list_P3592885314253461005_a_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s984997627204368545_a_nat @ Xs2 ) )
= ( Xs2 != nil_Pr7402525243500994295_a_nat ) ) ).
% length_greater_0_conv
thf(fact_1052_length__greater__0__conv,axiom,
! [Xs2: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs2 ) )
= ( Xs2 != nil_a ) ) ).
% length_greater_0_conv
thf(fact_1053_local_Onat__pow__0,axiom,
! [X3: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X3 @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_1054_ring_Oroots_Ocong,axiom,
polynomial_roots_a_b = polynomial_roots_a_b ).
% ring.roots.cong
thf(fact_1055_subfieldE_I3_J,axiom,
! [K: set_a,R: partia2175431115845679010xt_a_b] :
( ( subfield_a_b @ K @ R )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subfieldE(3)
thf(fact_1056_subfieldE_I3_J,axiom,
! [K: set_list_a,R: partia2670972154091845814t_unit] :
( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_1057_subfieldE_I3_J,axiom,
! [K: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subfie4546268998243038636t_unit @ K @ R )
=> ( ord_le8488217952732425610list_a @ K @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subfieldE(3)
thf(fact_1058_subringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subringE(1)
thf(fact_1059_subringE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_1060_subringE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subrin3541368690557094692t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_1061_list_Osize_I3_J,axiom,
( ( size_s984997627204368545_a_nat @ nil_Pr7402525243500994295_a_nat )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1062_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_1063_subcringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subcringE(1)
thf(fact_1064_subcringE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_1065_subcringE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subcri8676831449680469861t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_1066_subdomainE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subdomainE(1)
thf(fact_1067_subdomainE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_1068_subdomainE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subdom561091866123308472t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_1069_semiring_Onat__pow__zero,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( semiring_a_b @ R )
=> ( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ R @ ( zero_a_b @ R ) @ N )
= ( zero_a_b @ R ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_1070_semiring_Onat__pow__zero,axiom,
! [R: partia2670972154091845814t_unit,N: nat] :
( ( semiri2871908745932252451t_unit @ R )
=> ( ( N != zero_zero_nat )
=> ( ( pow_li1142815632869257134it_nat @ R @ ( zero_l4142658623432671053t_unit @ R ) @ N )
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_1071_semiring_Onat__pow__zero,axiom,
! [R: partia7496981018696276118t_unit,N: nat] :
( ( semiri4000464634269493571t_unit @ R )
=> ( ( N != zero_zero_nat )
=> ( ( pow_se8252319793075206062it_nat @ R @ ( zero_s2910681146719230829t_unit @ R ) @ N )
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ).
% semiring.nat_pow_zero
thf(fact_1072_domain_Oassociated__polynomials__imp__same__roots,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( associ9038253669175192217t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q )
=> ( ( polyno3707469075594375645t_unit @ R @ P )
= ( polyno3707469075594375645t_unit @ R @ Q ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_roots
thf(fact_1073_domain_Oassociated__polynomials__imp__same__roots,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q )
=> ( ( polyno7858422826990252003t_unit @ R @ P )
= ( polyno7858422826990252003t_unit @ R @ Q ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_roots
thf(fact_1074_domain_Oassociated__polynomials__imp__same__roots,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q )
=> ( ( polynomial_roots_a_b @ R @ P )
= ( polynomial_roots_a_b @ R @ Q ) ) ) ) ) ) ).
% domain.associated_polynomials_imp_same_roots
thf(fact_1075_domain_Ocgenideal__pirreducible,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subfie1779122896746047282t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ Q )
=> ( ( member_list_list_a @ Q @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P ) )
=> ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q ) ) ) ) ) ) ) ).
% domain.cgenideal_pirreducible
thf(fact_1076_domain_Ocgenideal__pirreducible,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subfield_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ Q )
=> ( ( member_list_a @ Q @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ R @ K ) @ P ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q ) ) ) ) ) ) ) ).
% domain.cgenideal_pirreducible
thf(fact_1077_ring_OsubringI,axiom,
! [R: partia7496981018696276118t_unit,H: set_set_list_a] :
( ( ring_s8247141995668492373t_unit @ R )
=> ( ( ord_le8877086941679407844list_a @ H @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ H )
=> ( ! [H4: set_list_a] :
( ( member_set_list_a @ H4 @ H )
=> ( member_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ H4 ) @ H ) )
=> ( ! [H1: set_list_a,H2: set_list_a] :
( ( member_set_list_a @ H1 @ H )
=> ( ( member_set_list_a @ H2 @ H )
=> ( member_set_list_a @ ( mult_s7802724872828879953t_unit @ R @ H1 @ H2 ) @ H ) ) )
=> ( ! [H1: set_list_a,H2: set_list_a] :
( ( member_set_list_a @ H1 @ H )
=> ( ( member_set_list_a @ H2 @ H )
=> ( member_set_list_a @ ( add_se2486902527185523630t_unit @ R @ H1 @ H2 ) @ H ) ) )
=> ( subrin5643252653130547402t_unit @ H @ R ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_1078_ring_OsubringI,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( one_a_ring_ext_a_b @ R ) @ H )
=> ( ! [H4: a] :
( ( member_a @ H4 @ H )
=> ( member_a @ ( a_inv_a_b @ R @ H4 ) @ H ) )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H1 @ H2 ) @ H ) ) )
=> ( ! [H1: a,H2: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H2 @ H )
=> ( member_a @ ( add_a_b @ R @ H1 @ H2 ) @ H ) ) )
=> ( subring_a_b @ H @ R ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_1079_ring_OsubringI,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ R ) @ H )
=> ( ! [H4: list_a] :
( ( member_list_a @ H4 @ H )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ R @ H4 ) @ H ) )
=> ( ! [H1: list_a,H2: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H1 @ H2 ) @ H ) ) )
=> ( ! [H1: list_a,H2: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H2 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H1 @ H2 ) @ H ) ) )
=> ( subrin6918843898125473962t_unit @ H @ R ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_1080_ring_OsubringI,axiom,
! [R: partia2956882679547061052t_unit,H: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ H )
=> ( ! [H4: list_list_a] :
( ( member_list_list_a @ H4 @ H )
=> ( member_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ H4 ) @ H ) )
=> ( ! [H1: list_list_a,H2: list_list_a] :
( ( member_list_list_a @ H1 @ H )
=> ( ( member_list_list_a @ H2 @ H )
=> ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ H1 @ H2 ) @ H ) ) )
=> ( ! [H1: list_list_a,H2: list_list_a] :
( ( member_list_list_a @ H1 @ H )
=> ( ( member_list_list_a @ H2 @ H )
=> ( member_list_list_a @ ( add_li174743652000525320t_unit @ R @ H1 @ H2 ) @ H ) ) )
=> ( subrin3541368690557094692t_unit @ H @ R ) ) ) ) ) ) ) ).
% ring.subringI
thf(fact_1081_roots__inclI,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A2 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ A2 ) ) @ Q ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% roots_inclI
thf(fact_1082_eval__append,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ Q ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( pow_a_1026414303147256608_b_nat @ r @ A @ ( size_size_list_a @ Q ) ) ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_append
thf(fact_1083_eval__append__aux,axiom,
! [P: list_a,B: a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ ( cons_a @ B @ nil_a ) ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ A ) @ B ) ) ) ) ) ).
% eval_append_aux
thf(fact_1084_polynomial__incl,axiom,
! [K: set_a,P: list_a] :
( ( polynomial_a_b @ r @ K @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K ) ) ).
% polynomial_incl
thf(fact_1085_normalize__in__carrier,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ r @ P ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% normalize_in_carrier
thf(fact_1086_coeff__in__carrier,axiom,
! [P: list_a,I: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( coeff_a_b @ r @ P @ I ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% coeff_in_carrier
thf(fact_1087_eval__in__carrier,axiom,
! [P: list_a,X3: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_1088_const__term__simprules_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% const_term_simprules(1)
thf(fact_1089_normalize__gives__polynomial,axiom,
! [P: list_a,K: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ K )
=> ( polynomial_a_b @ r @ K @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_gives_polynomial
thf(fact_1090_ee__sym,axiom,
! [As: list_a,Bs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ Bs @ As ) ) ) ) ).
% ee_sym
thf(fact_1091_ee__trans,axiom,
! [As: list_a,Bs: list_a,Cs: list_a] :
( ( essent8953798148185448568xt_a_b @ r @ As @ Bs )
=> ( ( essent8953798148185448568xt_a_b @ r @ Bs @ Cs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Cs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ Cs ) ) ) ) ) ) ).
% ee_trans
thf(fact_1092_eval__normalize,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( normalize_a_b @ r @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_normalize
thf(fact_1093_const__term__explicit,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= A )
=> ~ ! [P6: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P6 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P6 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_1094_const__term__eq__last,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ).
% const_term_eq_last
thf(fact_1095_pdivides__imp__roots__incl,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% pdivides_imp_roots_incl
thf(fact_1096_monom__in__carrier,axiom,
! [A: a,N: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monom_in_carrier
thf(fact_1097_ee__refl,axiom,
! [As: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( essent8953798148185448568xt_a_b @ r @ As @ As ) ) ).
% ee_refl
thf(fact_1098_polynomial__in__carrier,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% polynomial_in_carrier
thf(fact_1099_units__of__pow,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( pow_a_1875594501834816709it_nat @ ( units_8174867845824275201xt_a_b @ r ) @ X3 @ N )
= ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) ) ) ).
% units_of_pow
thf(fact_1100_exp__base__closed,axiom,
! [X3: a,N: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X3 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_1101_poly__mult__var_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( var_a_b @ r ) @ P )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(1)
thf(fact_1102_poly__mult_Osimps_I1_J,axiom,
! [P2: list_a] :
( ( poly_mult_a_b @ r @ nil_a @ P2 )
= nil_a ) ).
% poly_mult.simps(1)
thf(fact_1103_poly__mult__closed,axiom,
! [K: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( polynomial_a_b @ r @ K @ ( poly_mult_a_b @ r @ P1 @ P2 ) ) ) ) ) ).
% poly_mult_closed
thf(fact_1104_poly__mult__comm,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P2 )
= ( poly_mult_a_b @ r @ P2 @ P1 ) ) ) ) ).
% poly_mult_comm
thf(fact_1105_poly__mult__in__carrier,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ r @ P1 @ P2 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_mult_in_carrier
thf(fact_1106_poly__mult__integral,axiom,
! [K: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ( ( poly_mult_a_b @ r @ P1 @ P2 )
= nil_a )
=> ( ( P1 = nil_a )
| ( P2 = nil_a ) ) ) ) ) ) ).
% poly_mult_integral
thf(fact_1107_poly__mult__zero_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ nil_a @ P )
= nil_a ) ) ).
% poly_mult_zero(1)
thf(fact_1108_poly__mult__zero_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ nil_a )
= nil_a ) ) ).
% poly_mult_zero(2)
thf(fact_1109_poly__mult__normalize,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P2 )
= ( poly_mult_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P2 ) ) ) ) ).
% poly_mult_normalize
thf(fact_1110_poly__mult__is__polynomial,axiom,
! [K: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ K )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K )
=> ( polynomial_a_b @ r @ K @ ( poly_mult_a_b @ r @ P1 @ P2 ) ) ) ) ) ).
% poly_mult_is_polynomial
thf(fact_1111_poly__mult__monom__assoc,axiom,
! [P: list_a,Q: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_monom_assoc
thf(fact_1112_poly__mult__semiassoc,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_semiassoc
thf(fact_1113_eval__poly__mult,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) @ A )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_mult
thf(fact_1114_const__term__simprules_I2_J,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(2)
thf(fact_1115_poly__mult__one_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_mult_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) @ P )
= P ) ) ) ).
% poly_mult_one(1)
thf(fact_1116_poly__mult__one_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_mult_a_b @ r @ P @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) )
= P ) ) ) ).
% poly_mult_one(2)
thf(fact_1117_poly__mult__append__zero__rcancel,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( polynomial_a_b @ r @ K @ Q )
=> ( ( ( poly_mult_a_b @ r @ P @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( append_a @ R2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
=> ( ( poly_mult_a_b @ r @ P @ Q )
= R2 ) ) ) ) ) ).
% poly_mult_append_zero_rcancel
thf(fact_1118_poly__mult__append__zero__lcancel,axiom,
! [K: set_a,P: list_a,Q: list_a,R2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( polynomial_a_b @ r @ K @ Q )
=> ( ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
= ( append_a @ R2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
=> ( ( poly_mult_a_b @ r @ P @ Q )
= R2 ) ) ) ) ) ).
% poly_mult_append_zero_lcancel
thf(fact_1119_poly__mult__one_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_mult_one'(2)
thf(fact_1120_poly__mult__one_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ nil_a ) @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_mult_one'(1)
thf(fact_1121_poly__mult__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_mult_append_zero
thf(fact_1122_poly__mult__var_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( var_a_b @ r ) )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(2)
thf(fact_1123_roots__inclI_H,axiom,
! [P: list_a,M: multiset_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ A2 ) @ ( count_a @ M @ A2 ) ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ M ) ) ) ).
% roots_inclI'
thf(fact_1124_factors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% factors_mult
thf(fact_1125_freshmans__dream,axiom,
! [X3: a,Y2: a] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( add_a_b @ r @ X3 @ Y2 ) @ ( ring_char_a_b @ r ) )
= ( add_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ ( ring_char_a_b @ r ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ ( ring_char_a_b @ r ) ) ) ) ) ) ) ).
% freshmans_dream
thf(fact_1126_factors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_1127_nunit__factors,axiom,
! [A: a,As: list_a] :
( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( factor5638265376665762323xt_a_b @ r @ As @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ As ) ) ) ) ).
% nunit_factors
thf(fact_1128_alg__mult__eq__count__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4422430861927485590lt_a_b @ r @ P )
= ( count_a @ ( polynomial_roots_a_b @ r @ P ) ) ) ) ).
% alg_mult_eq_count_roots
thf(fact_1129_add_Oone__in__subset,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H != bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H )
=> ( member_a @ ( a_inv_a_b @ r @ X4 ) @ H ) )
=> ( ! [X4: a] :
( ( member_a @ X4 @ H )
=> ! [Xa2: a] :
( ( member_a @ Xa2 @ H )
=> ( member_a @ ( add_a_b @ r @ X4 @ Xa2 ) @ H ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% add.one_in_subset
thf(fact_1130_freshmans__dream__ext,axiom,
! [X3: a,Y2: a,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_char_a_b @ r ) )
=> ( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( add_a_b @ r @ X3 @ Y2 ) @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M ) )
= ( add_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M ) ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ ( power_power_nat @ ( ring_char_a_b @ r ) @ M ) ) ) ) ) ) ) ).
% freshmans_dream_ext
thf(fact_1131_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_1132_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1133_nat__zero__less__power__iff,axiom,
! [X3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X3 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1134_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1135_not__empty__rootsE,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( polynomial_roots_a_b @ r @ P )
!= zero_zero_multiset_a )
=> ~ ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A2 @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P ) ) )
=> ( ( member_list_a @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A2 ) @ nil_a ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A2 ) @ nil_a ) ) @ P ) ) ) ) ) ) ).
% not_empty_rootsE
thf(fact_1136_degree__one__roots,axiom,
! [A: a,A4: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( polynomial_roots_a_b @ r @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) )
= ( add_mset_a @ ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A4 @ B ) ) @ zero_zero_multiset_a ) ) ) ) ) ) ).
% degree_one_roots
thf(fact_1137_poly__add__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_add_append_zero
thf(fact_1138_poly__add__closed,axiom,
! [K: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P1 @ P2 ) ) ) ) ) ).
% poly_add_closed
thf(fact_1139_poly__add__comm,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P2 )
= ( poly_add_a_b @ r @ P2 @ P1 ) ) ) ) ).
% poly_add_comm
thf(fact_1140_poly__add__in__carrier,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P1 @ P2 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_add_in_carrier
thf(fact_1141_poly__add__zero_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= P ) ) ) ).
% poly_add_zero(1)
thf(fact_1142_poly__add__zero_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= P ) ) ) ).
% poly_add_zero(2)
thf(fact_1143_poly__mult__r__distr,axiom,
! [K: set_a,P1: list_a,P2: list_a,P32: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ( polynomial_a_b @ r @ K @ P32 )
=> ( ( poly_mult_a_b @ r @ P1 @ ( poly_add_a_b @ r @ P2 @ P32 ) )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P2 ) @ ( poly_mult_a_b @ r @ P1 @ P32 ) ) ) ) ) ) ) ).
% poly_mult_r_distr
thf(fact_1144_poly__mult__l__distr,axiom,
! [K: set_a,P1: list_a,P2: list_a,P32: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ( polynomial_a_b @ r @ K @ P32 )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P2 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P32 ) @ ( poly_mult_a_b @ r @ P2 @ P32 ) ) ) ) ) ) ) ).
% poly_mult_l_distr
thf(fact_1145_poly__mult__l__distr_H,axiom,
! [P1: list_a,P2: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P2 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P32 ) @ ( poly_mult_a_b @ r @ P2 @ P32 ) ) ) ) ) ) ).
% poly_mult_l_distr'
thf(fact_1146_poly__mult__r__distr_H,axiom,
! [P1: list_a,P2: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ ( poly_add_a_b @ r @ P2 @ P32 ) )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P2 ) @ ( poly_mult_a_b @ r @ P1 @ P32 ) ) ) ) ) ) ).
% poly_mult_r_distr'
thf(fact_1147_poly__add__normalize_I3_J,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P2 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ ( normalize_a_b @ r @ P2 ) ) ) ) ) ).
% poly_add_normalize(3)
thf(fact_1148_poly__add__normalize_I2_J,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P2 )
= ( poly_add_a_b @ r @ P1 @ ( normalize_a_b @ r @ P2 ) ) ) ) ) ).
% poly_add_normalize(2)
thf(fact_1149_poly__add__normalize__aux,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P2 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P2 ) ) ) ) ).
% poly_add_normalize_aux
thf(fact_1150_poly__add__coeff__aux,axiom,
! [P2: list_a,P1: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ P2 ) @ ( size_size_list_a @ P1 ) )
=> ( ( coeff_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P2 ) )
= ( ^ [I2: nat] : ( add_a_b @ r @ ( coeff_a_b @ r @ P1 @ I2 ) @ ( coeff_a_b @ r @ P2 @ I2 ) ) ) ) ) ).
% poly_add_coeff_aux
thf(fact_1151_poly__add__is__polynomial,axiom,
! [K: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ K )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K )
=> ( polynomial_a_b @ r @ K @ ( poly_add_a_b @ r @ P1 @ P2 ) ) ) ) ) ).
% poly_add_is_polynomial
thf(fact_1152_poly__add__coeff,axiom,
! [P1: list_a,P2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( coeff_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P2 ) )
= ( ^ [I2: nat] : ( add_a_b @ r @ ( coeff_a_b @ r @ P1 @ I2 ) @ ( coeff_a_b @ r @ P2 @ I2 ) ) ) ) ) ) ).
% poly_add_coeff
thf(fact_1153_eval__poly__add,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_add
thf(fact_1154_poly__add__zero_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(1)
thf(fact_1155_poly__add__zero_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(2)
thf(fact_1156_const__term__simprules_I3_J,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(3)
thf(fact_1157_eval__poly__add__aux,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ) ).
% eval_poly_add_aux
thf(fact_1158_roots__mem__iff__is__root,axiom,
! [P: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ X3 @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P ) ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X3 ) ) ) ).
% roots_mem_iff_is_root
thf(fact_1159_monic__degree__one__roots,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polynomial_roots_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A ) @ nil_a ) ) )
= ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ).
% monic_degree_one_roots
thf(fact_1160_poly__add__append__replicate,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( replicate_a @ ( size_size_list_a @ Q ) @ ( zero_a_b @ r ) ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ) ) ).
% poly_add_append_replicate
thf(fact_1161_line__extension__smult__closed,axiom,
! [K: set_a,E: set_a,A: a,K2: a,U: a] :
( ( subfield_a_b @ K @ r )
=> ( ! [K4: a,V: a] :
( ( member_a @ K4 @ K )
=> ( ( member_a @ V @ E )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K4 @ V ) @ E ) ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ K )
=> ( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) ) ) ) ) ) ) ) ).
% line_extension_smult_closed
thf(fact_1162_line__extension__in__carrier,axiom,
! [K: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_1163_line__extension__mem__iff,axiom,
! [U: a,K: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
= ( ? [X: a] :
( ( member_a @ X @ K )
& ? [Y6: a] :
( ( member_a @ Y6 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ A ) @ Y6 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_1164_normalize__replicate__zero,axiom,
! [N: nat,P: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
= ( normalize_a_b @ r @ P ) ) ).
% normalize_replicate_zero
thf(fact_1165_prefix__replicate__zero__coeff,axiom,
! [P: list_a,N: nat] :
( ( coeff_a_b @ r @ P )
= ( coeff_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) ) ) ).
% prefix_replicate_zero_coeff
thf(fact_1166_local_Omonom__def,axiom,
! [A: a,N: nat] :
( ( monom_a_b @ r @ A @ N )
= ( cons_a @ A @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ).
% local.monom_def
thf(fact_1167_dense__repr__replicate__zero,axiom,
! [N: nat,P: list_a] :
( ( dense_repr_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
= ( dense_repr_a_b @ r @ P ) ) ).
% dense_repr_replicate_zero
thf(fact_1168_poly__add__replicate__zero_I2_J,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= P ) ) ) ).
% poly_add_replicate_zero(2)
thf(fact_1169_poly__add__replicate__zero_I1_J,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= P ) ) ) ).
% poly_add_replicate_zero(1)
thf(fact_1170_append__is__polynomial,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( P != nil_a )
=> ( polynomial_a_b @ r @ K @ ( append_a @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% append_is_polynomial
thf(fact_1171_poly__mult__replicate__zero_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= nil_a ) ) ).
% poly_mult_replicate_zero(1)
thf(fact_1172_poly__mult__replicate__zero_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= nil_a ) ) ).
% poly_mult_replicate_zero(2)
thf(fact_1173_poly__mult__prepend__replicate__zero,axiom,
! [P1: list_a,P2: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P2 )
= ( poly_mult_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P1 ) @ P2 ) ) ) ) ).
% poly_mult_prepend_replicate_zero
thf(fact_1174_poly__add__replicate__zero_H_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(2)
thf(fact_1175_poly__add__replicate__zero_H_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_replicate_zero'(1)
thf(fact_1176_eval__replicate,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_replicate
thf(fact_1177_replicate__zero__coeff,axiom,
! [N: nat] :
( ( coeff_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= ( ^ [Uu: nat] : ( zero_a_b @ r ) ) ) ).
% replicate_zero_coeff
thf(fact_1178_poly__mult__monom_H,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
= ( normalize_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ A ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% poly_mult_monom'
thf(fact_1179_normalize__trick,axiom,
! [P: list_a] :
( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ ( zero_a_b @ r ) ) @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_trick
thf(fact_1180_combine__append__zero,axiom,
! [Us4: list_a,Ks2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks2 @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Us4 )
= ( embedded_combine_a_b @ r @ Ks2 @ Us4 ) ) ) ).
% combine_append_zero
thf(fact_1181_norm__map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_a_list_a @ F @ P ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% norm_map_in_poly_ring_carrier
thf(fact_1182_combine_Osimps_I2_J,axiom,
! [Us4: list_a] :
( ( embedded_combine_a_b @ r @ nil_a @ Us4 )
= ( zero_a_b @ r ) ) ).
% combine.simps(2)
thf(fact_1183_combine_Osimps_I3_J,axiom,
! [Ks2: list_a] :
( ( embedded_combine_a_b @ r @ Ks2 @ nil_a )
= ( zero_a_b @ r ) ) ).
% combine.simps(3)
thf(fact_1184_combine_Osimps_I1_J,axiom,
! [K2: a,Ks2: list_a,U: a,Us4: list_a] :
( ( embedded_combine_a_b @ r @ ( cons_a @ K2 @ Ks2 ) @ ( cons_a @ U @ Us4 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedded_combine_a_b @ r @ Ks2 @ Us4 ) ) ) ).
% combine.simps(1)
thf(fact_1185_combine__eq__eval,axiom,
! [Ks2: list_a,X3: a] :
( ( embedded_combine_a_b @ r @ Ks2 @ ( polyno2922411391617481336se_a_b @ r @ X3 @ ( size_size_list_a @ Ks2 ) ) )
= ( eval_a_b @ r @ Ks2 @ X3 ) ) ).
% combine_eq_eval
thf(fact_1186_map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A2: a] :
( ( member_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [A2: a] :
( ( A2
!= ( zero_a_b @ r ) )
=> ( ( F @ A2 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% map_in_poly_ring_carrier
thf(fact_1187_append__coeff,axiom,
! [P: list_a,Q: list_a] :
( ( coeff_a_b @ r @ ( append_a @ P @ Q ) )
= ( ^ [I2: nat] : ( if_a @ ( ord_less_nat @ I2 @ ( size_size_list_a @ Q ) ) @ ( coeff_a_b @ r @ Q @ I2 ) @ ( coeff_a_b @ r @ P @ ( minus_minus_nat @ I2 @ ( size_size_list_a @ Q ) ) ) ) ) ) ).
% append_coeff
thf(fact_1188_combine_Oelims,axiom,
! [X3: list_a,Xa3: list_a,Y2: a] :
( ( ( embedded_combine_a_b @ r @ X3 @ Xa3 )
= Y2 )
=> ( ! [K4: a,Ks: list_a] :
( ( X3
= ( cons_a @ K4 @ Ks ) )
=> ! [U2: a,Us: list_a] :
( ( Xa3
= ( cons_a @ U2 @ Us ) )
=> ( Y2
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K4 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks @ Us ) ) ) ) )
=> ( ( ( X3 = nil_a )
=> ( Y2
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa3 = nil_a )
=> ( Y2
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_1189_combine__r__distr,axiom,
! [Ks2: list_a,Us4: list_a,K2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K2 @ ( embedded_combine_a_b @ r @ Ks2 @ Us4 ) )
= ( embedded_combine_a_b @ r @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ K2 ) @ Ks2 ) @ Us4 ) ) ) ) ) ).
% combine_r_distr
thf(fact_1190_combine__append,axiom,
! [Ks2: list_a,Us4: list_a,Ks3: list_a,Vs2: list_a] :
( ( ( size_size_list_a @ Ks2 )
= ( size_size_list_a @ Us4 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( embedded_combine_a_b @ r @ Ks2 @ Us4 ) @ ( embedded_combine_a_b @ r @ Ks3 @ Vs2 ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks2 @ Ks3 ) @ ( append_a @ Us4 @ Vs2 ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_1191_combine__replicate,axiom,
! [Us4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( replicate_a @ ( size_size_list_a @ Us4 ) @ ( zero_a_b @ r ) ) @ Us4 )
= ( zero_a_b @ r ) ) ) ).
% combine_replicate
thf(fact_1192_combine__append__replicate,axiom,
! [Us4: list_a,Ks2: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks2 @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) @ Us4 )
= ( embedded_combine_a_b @ r @ Ks2 @ Us4 ) ) ) ).
% combine_append_replicate
thf(fact_1193_combine__in__carrier,axiom,
! [Ks2: list_a,Us4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( embedded_combine_a_b @ r @ Ks2 @ Us4 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% combine_in_carrier
thf(fact_1194_long__dividesI,axiom,
! [B: list_a,R2: list_a,P: list_a,Q: list_a] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ B ) @ R2 ) )
=> ( ( ( R2 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) ) ) ) ) ) ).
% long_dividesI
thf(fact_1195_field__long__division__theorem,axiom,
! [K: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P )
=> ( ( polynomial_a_b @ r @ K @ B )
=> ( ( B != nil_a )
=> ? [Q2: list_a,R4: list_a] :
( ( polynomial_a_b @ r @ K @ Q2 )
& ( polynomial_a_b @ r @ K @ R4 )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ Q2 ) @ R4 ) )
& ( ( R4 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% field_long_division_theorem
thf(fact_1196_degree__var,axiom,
( ( minus_minus_nat @ ( size_size_list_a @ ( var_a_b @ r ) ) @ one_one_nat )
= one_one_nat ) ).
% degree_var
thf(fact_1197_degree__one,axiom,
! [K: set_a] :
( ( minus_minus_nat @ ( size_size_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ K ) ) ) @ one_one_nat )
= zero_zero_nat ) ).
% degree_one
thf(fact_1198_coeff__degree,axiom,
! [P: list_a,I: nat] :
( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ I )
=> ( ( coeff_a_b @ r @ P @ I )
= ( zero_a_b @ r ) ) ) ).
% coeff_degree
thf(fact_1199_coeff__nth,axiom,
! [I: nat,P: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ P ) )
=> ( ( coeff_a_b @ r @ P @ I )
= ( nth_a @ P @ ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ I ) ) ) ) ).
% coeff_nth
thf(fact_1200_degree__one__imp__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_1201_univ__poly__a__inv__degree,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% univ_poly_a_inv_degree
thf(fact_1202_var__pow__degree,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) @ one_one_nat )
= N ) ) ).
% var_pow_degree
thf(fact_1203_degree__oneE,axiom,
! [P: list_a,K: set_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A2: a] :
( ( member_a @ A2 @ K )
=> ( ( A2
!= ( zero_a_b @ r ) )
=> ! [B2: a] :
( ( member_a @ B2 @ K )
=> ( P
!= ( cons_a @ A2 @ ( cons_a @ B2 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_1204_d,axiom,
( ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ d ) @ one_one_nat ) @ k2 )
= j ) ).
% d
thf(fact_1205_degree__zero__imp__not__is__root,axiom,
! [P: list_a,X3: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P @ X3 ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_1206_pmod__const_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pmod_a_b @ r @ P @ Q )
= P ) ) ) ) ) ).
% pmod_const(2)
thf(fact_1207_pirreducible__degree,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% pirreducible_degree
thf(fact_1208_pdivides__imp__degree__le,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pdivides_imp_degree_le
thf(fact_1209_pmod__degree,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q )
= nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pmod_degree
thf(fact_1210_univ__poly__units_H,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
= ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( P != nil_a )
& ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat ) ) ) ) ).
% univ_poly_units'
thf(fact_1211_pmod__const_I1_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) )
=> ( ( polynomial_pdiv_a_b @ r @ P @ Q )
= nil_a ) ) ) ) ) ).
% pmod_const(1)
thf(fact_1212_subfield__long__division__theorem__shell,axiom,
! [K: set_a,P: list_a,B: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( B
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ? [Q2: list_a,R4: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
& ( P
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% subfield_long_division_theorem_shell
thf(fact_1213_degree__zero__imp__empty__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a ) ) ) ).
% degree_zero_imp_empty_roots
thf(fact_1214_pirreducible__roots,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a ) ) ) ) ).
% pirreducible_roots
thf(fact_1215_nat__pow__eone,axiom,
! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X3 @ one_one_nat )
= X3 ) ) ).
% nat_pow_eone
thf(fact_1216_True,axiom,
ord_less_nat @ ( plus_plus_nat @ k2 @ one_one_nat ) @ ( size_size_list_a @ f ) ).
% True
thf(fact_1217_degree__zero__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_zero_imp_splitted
thf(fact_1218_rupture__one__not__zero,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) )
=> ( ( one_se1127990129394575805t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) )
!= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P ) ) ) ) ) ) ).
% rupture_one_not_zero
thf(fact_1219_nat__pow__mult,axiom,
! [X3: a,N: nat,M: nat] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X3 @ M ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X3 @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% nat_pow_mult
thf(fact_1220_a,axiom,
ord_less_nat @ ( plus_plus_nat @ j @ one_one_nat ) @ ( size_size_list_a @ f ) ).
% a
thf(fact_1221_f,axiom,
( ( minus_minus_nat @ ( minus_minus_nat @ ( size_size_list_a @ f ) @ j ) @ one_one_nat )
= ( plus_plus_nat @ k2 @ one_one_nat ) ) ).
% f
thf(fact_1222_poly__mult__degree__eq,axiom,
! [K: set_a,P1: list_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( polynomial_a_b @ r @ K @ P1 )
=> ( ( polynomial_a_b @ r @ K @ P2 )
=> ( ( ( ( P1 = nil_a )
| ( P2 = nil_a ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_mult_a_b @ r @ P1 @ P2 ) ) @ one_one_nat )
= zero_zero_nat ) )
& ( ~ ( ( P1 = nil_a )
| ( P2 = nil_a ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( poly_mult_a_b @ r @ P1 @ P2 ) ) @ one_one_nat )
= ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ P1 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% poly_mult_degree_eq
thf(fact_1223_map__norm__in__poly__ring__carrier,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ).
% map_norm_in_poly_ring_carrier
thf(fact_1224_eval__rewrite,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ K ) @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( var_a_b @ r ) ) ) ) ) ).
% eval_rewrite
thf(fact_1225_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K5: a] : ( normalize_a_b @ r @ ( cons_a @ K5 @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_1226_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1227_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1228_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1229_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1230_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1231_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1232_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1233_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1234_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1235_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1236_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1237_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1238_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1239_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1240_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1241_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1242_linorder__neqE__nat,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_1243_infinite__descent,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P3 @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P3 @ M4 ) ) )
=> ( P3 @ N ) ) ).
% infinite_descent
thf(fact_1244_nat__less__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( P3 @ M4 ) )
=> ( P3 @ N3 ) )
=> ( P3 @ N ) ) ).
% nat_less_induct
thf(fact_1245_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1246_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_1247_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1248_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1249_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1250_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K2: nat,B: nat] :
( ( P3 @ K2 )
=> ( ! [Y5: nat] :
( ( P3 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B ) )
=> ? [X4: nat] :
( ( P3 @ X4 )
& ! [Y4: nat] :
( ( P3 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1251_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1252_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1253_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1254_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_1255_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1256_infinite__descent0,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P3 @ N3 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
& ~ ( P3 @ M4 ) ) ) )
=> ( P3 @ N ) ) ) ).
% infinite_descent0
thf(fact_1257_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1258_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1259_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1260_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1261_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1262_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1263_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1264_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1265_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1266_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1267_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1268_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1269_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1270_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
% Helper facts (7)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $true @ X3 @ Y2 )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X3: list_a,Y2: list_a] :
( ( if_list_a @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X3: list_a,Y2: list_a] :
( ( if_list_a @ $true @ X3 @ Y2 )
= X3 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
! [X3: set_list_a,Y2: set_list_a] :
( ( if_set_list_a @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_Itf__a_J_J_T,axiom,
! [X3: set_list_a,Y2: set_list_a] :
( ( if_set_list_a @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( coeff_a_b @ r @ d @ k2 )
= ( nth_a @ d @ j ) ) ).
%------------------------------------------------------------------------------