TPTP Problem File: SLH0494^1.p
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%------------------------------------------------------------------------------
% 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/0008_Card_Irreducible_Polynomials_Aux/prob_00301_010445__18376754_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1618 ( 468 unt; 343 typ; 0 def)
% Number of atoms : 3682 (1463 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 18486 ( 331 ~; 42 |; 107 &;16151 @)
% ( 0 <=>;1855 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 42 ( 41 usr)
% Number of type conns : 668 ( 668 >; 0 *; 0 +; 0 <<)
% Number of symbols : 305 ( 302 usr; 23 con; 0-4 aty)
% Number of variables : 2945 ( 74 ^;2819 !; 52 ?;2945 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:23:00.719
%------------------------------------------------------------------------------
% Could-be-implicit typings (41)
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bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
member6842060177613954879list_a: list_l7815035709764258326list_a > set_li3407770045201608054list_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_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_J,type,
member6908278366116243871list_a: list_l8227414553961969782list_a > set_li3232697616086161366list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J_J,type,
member352051402189872281list_a: list_list_set_list_a > set_li664707282716828624list_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_It__List__Olist_Itf__a_J_J_J_J,type,
member6124916891863447321list_a: list_set_list_list_a > set_li7845362039408639568list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5524387281408368019list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member334759470184282131list_a: set_list_list_a > set_set_list_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1271)
thf(fact_0__092_060open_062_092_060ominus_062_092_060_094bsub_062P_092_060_094esub_062_AX_A_092_060in_062_Acarrier_AP_092_060close_062,axiom,
member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% \<open>\<ominus>\<^bsub>P\<^esub> X \<in> carrier P\<close>
thf(fact_1_assms,axiom,
ord_less_nat @ one_one_nat @ n ).
% assms
thf(fact_2__092_060open_062monic__poly_AR_A_IX_A_091_094_093_092_060_094bsub_062P_092_060_094esub_062_An_J_092_060close_062,axiom,
monic_3145109188698636716ly_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ n ) ).
% \<open>monic_poly R (X [^]\<^bsub>P\<^esub> n)\<close>
thf(fact_3_monic__poly__var,axiom,
monic_3145109188698636716ly_a_b @ r @ ( var_a_b @ r ) ).
% monic_poly_var
thf(fact_4_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_5_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_6_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_7_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_8_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_9_gauss__poly__carr,axiom,
! [N: nat] : ( member_list_a @ ( card_I2373409586816755191ly_a_b @ r @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% gauss_poly_carr
thf(fact_10_monic__poly__carr,axiom,
! [F: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% monic_poly_carr
thf(fact_11_monic__poly__pow,axiom,
! [F: list_a,N: nat] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( monic_3145109188698636716ly_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ N ) ) ) ).
% monic_poly_pow
thf(fact_12_gauss__poly__degree,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( card_I2373409586816755191ly_a_b @ r @ N ) ) @ one_one_nat )
= N ) ) ).
% gauss_poly_degree
thf(fact_13_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_14__092_060open_062degree_A_I_092_060ominus_062_092_060_094bsub_062P_092_060_094esub_062_AX_J_A_060_Adegree_A_IX_A_091_094_093_092_060_094bsub_062P_092_060_094esub_062_An_J_092_060close_062,axiom,
ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ n ) ) @ one_one_nat ) ).
% \<open>degree (\<ominus>\<^bsub>P\<^esub> X) < degree (X [^]\<^bsub>P\<^esub> n)\<close>
thf(fact_15_p_Ofactorial__domain__axioms,axiom,
ring_f796907574329358751t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.factorial_domain_axioms
thf(fact_16_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_17_abelian__monoid__axioms,axiom,
abelian_monoid_a_b @ r ).
% abelian_monoid_axioms
thf(fact_18_p_Onoetherian__domain__axioms,axiom,
ring_n4705423059119889713t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.noetherian_domain_axioms
thf(fact_19_var__closed_I1_J,axiom,
member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% var_closed(1)
thf(fact_20_degree__var,axiom,
( ( minus_minus_nat @ ( size_size_list_a @ ( var_a_b @ r ) ) @ one_one_nat )
= one_one_nat ) ).
% degree_var
thf(fact_21_var__pow__closed,axiom,
! [N: nat] : ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% var_pow_closed
thf(fact_22_univ__poly__a__inv__degree,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 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% univ_poly_a_inv_degree
thf(fact_23_var__pow__degree,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ N ) ) @ one_one_nat )
= N ) ).
% var_pow_degree
thf(fact_24_p_Ominus__minus,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= X ) ) ).
% p.minus_minus
thf(fact_25_p_Oadd_Oinv__closed,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.add.inv_closed
thf(fact_26_p_Onat__pow__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.nat_pow_closed
thf(fact_27_p_Onat__pow__eone,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ one_one_nat )
= X ) ) ).
% p.nat_pow_eone
thf(fact_28_field_Ogauss__poly__degree,axiom,
! [R: partia7496981018696276118t_unit,N: nat] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ ( card_I5471341911849640095t_unit @ R @ N ) ) @ one_one_nat )
= N ) ) ) ).
% field.gauss_poly_degree
thf(fact_29_field_Ogauss__poly__degree,axiom,
! [R: partia2956882679547061052t_unit,N: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ ( card_I3787608780883923065t_unit @ R @ N ) ) @ one_one_nat )
= N ) ) ) ).
% field.gauss_poly_degree
thf(fact_30_field_Ogauss__poly__degree,axiom,
! [R: partia4960592913263135132t_unit,N: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ ( card_I259811512781981209t_unit @ R @ N ) ) @ one_one_nat )
= N ) ) ) ).
% field.gauss_poly_degree
thf(fact_31_field_Ogauss__poly__degree,axiom,
! [R: partia2670972154091845814t_unit,N: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( card_I2619780863984422015t_unit @ R @ N ) ) @ one_one_nat )
= N ) ) ) ).
% field.gauss_poly_degree
thf(fact_32_field_Ogauss__poly__degree,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( field_a_b @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( card_I2373409586816755191ly_a_b @ R @ N ) ) @ one_one_nat )
= N ) ) ) ).
% field.gauss_poly_degree
thf(fact_33_field_Ogauss__poly__carr,axiom,
! [R: partia5333488208502193986t_unit,N: nat] :
( ( field_40794040841493211t_unit @ R )
=> ( member6842060177613954879list_a @ ( card_I8204580741270601971t_unit @ R @ N ) @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ R @ ( partia5038748322285217333t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_carr
thf(fact_34_field_Ogauss__poly__carr,axiom,
! [R: partia3473558348976337314t_unit,N: nat] :
( ( field_3650391457538083963t_unit @ R )
=> ( member6908278366116243871list_a @ ( card_I4320469302700135059t_unit @ R @ N ) @ ( partia6954261711924553499t_unit @ ( univ_p7468970815438322146t_unit @ R @ ( partia6708307881709191317t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_carr
thf(fact_35_field_Ogauss__poly__carr,axiom,
! [R: partia4556295656693239580t_unit,N: nat] :
( ( field_6241291760955494017t_unit @ R )
=> ( member352051402189872281list_a @ ( card_I4960859800387535001t_unit @ R @ N ) @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_carr
thf(fact_36_field_Ogauss__poly__carr,axiom,
! [R: partia7496981018696276118t_unit,N: nat] :
( ( field_26233345952514695t_unit @ R )
=> ( member5524387281408368019list_a @ ( card_I5471341911849640095t_unit @ R @ N ) @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_carr
thf(fact_37_field_Ogauss__poly__carr,axiom,
! [R: partia4960592913263135132t_unit,N: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( member6124916891863447321list_a @ ( card_I259811512781981209t_unit @ R @ N ) @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_carr
thf(fact_38_field_Ogauss__poly__carr,axiom,
! [R: partia2956882679547061052t_unit,N: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( member5342144027231129785list_a @ ( card_I3787608780883923065t_unit @ R @ N ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_carr
thf(fact_39_field_Ogauss__poly__carr,axiom,
! [R: partia2670972154091845814t_unit,N: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( member_list_list_a @ ( card_I2619780863984422015t_unit @ R @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_carr
thf(fact_40_field_Ogauss__poly__carr,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( field_a_b @ R )
=> ( member_list_a @ ( card_I2373409586816755191ly_a_b @ R @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% field.gauss_poly_carr
thf(fact_41_p_Oprincipal__domain__axioms,axiom,
ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.principal_domain_axioms
thf(fact_42_p_Onoetherian__ring__axioms,axiom,
ring_n5188127996776581661t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.noetherian_ring_axioms
thf(fact_43_p_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.onepideal
thf(fact_44_degree__one__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_one_imp_splitted
thf(fact_45_monic__poly__add__distinct,axiom,
! [F: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ G ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) )
=> ( monic_3145109188698636716ly_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ G ) ) ) ) ) ).
% monic_poly_add_distinct
thf(fact_46_polynomial__pow__degree,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% polynomial_pow_degree
thf(fact_47_gauss__poly__not__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( card_I2373409586816755191ly_a_b @ r @ N )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% gauss_poly_not_zero
thf(fact_48_degree__one__imp__pirreducible,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_49_field_Omonic__poly__pow,axiom,
! [R: partia5333488208502193986t_unit,F: list_l7815035709764258326list_a,N: nat] :
( ( field_40794040841493211t_unit @ R )
=> ( ( monic_7946909391392167976t_unit @ R @ F )
=> ( monic_7946909391392167976t_unit @ R @ ( pow_li2121314091559947310it_nat @ ( univ_p3766428211910075458t_unit @ R @ ( partia5038748322285217333t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_50_field_Omonic__poly__pow,axiom,
! [R: partia3473558348976337314t_unit,F: list_l8227414553961969782list_a,N: nat] :
( ( field_3650391457538083963t_unit @ R )
=> ( ( monic_248643784349308360t_unit @ R @ F )
=> ( monic_248643784349308360t_unit @ R @ ( pow_li8421058921225051182it_nat @ ( univ_p7468970815438322146t_unit @ R @ ( partia6708307881709191317t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_51_field_Omonic__poly__pow,axiom,
! [R: partia4556295656693239580t_unit,F: list_list_set_list_a,N: nat] :
( ( field_6241291760955494017t_unit @ R )
=> ( ( monic_3621385675998241358t_unit @ R @ F )
=> ( monic_3621385675998241358t_unit @ R @ ( pow_li6530163574085554350it_nat @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_52_field_Omonic__poly__pow,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a,N: nat] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_3395465470813675732t_unit @ R @ F )
=> ( monic_3395465470813675732t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_53_field_Omonic__poly__pow,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a,N: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_8143709425247463374t_unit @ R @ F )
=> ( monic_8143709425247463374t_unit @ R @ ( pow_li5711373720449609902it_nat @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_54_field_Omonic__poly__pow,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a,N: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_5008461317928820916t_unit @ R @ F )
=> ( monic_5008461317928820916t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_55_field_Omonic__poly__pow,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a,N: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_5986596350207772206t_unit @ R @ F )
=> ( monic_5986596350207772206t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_56_field_Omonic__poly__pow,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a,N: nat] :
( ( field_a_b @ R )
=> ( ( monic_3145109188698636716ly_a_b @ R @ F )
=> ( monic_3145109188698636716ly_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_57_field_Omonic__poly__carr,axiom,
! [R: partia5333488208502193986t_unit,F: list_l7815035709764258326list_a] :
( ( field_40794040841493211t_unit @ R )
=> ( ( monic_7946909391392167976t_unit @ R @ F )
=> ( member6842060177613954879list_a @ F @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ R @ ( partia5038748322285217333t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_58_field_Omonic__poly__carr,axiom,
! [R: partia3473558348976337314t_unit,F: list_l8227414553961969782list_a] :
( ( field_3650391457538083963t_unit @ R )
=> ( ( monic_248643784349308360t_unit @ R @ F )
=> ( member6908278366116243871list_a @ F @ ( partia6954261711924553499t_unit @ ( univ_p7468970815438322146t_unit @ R @ ( partia6708307881709191317t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_59_field_Omonic__poly__carr,axiom,
! [R: partia4556295656693239580t_unit,F: list_list_set_list_a] :
( ( field_6241291760955494017t_unit @ R )
=> ( ( monic_3621385675998241358t_unit @ R @ F )
=> ( member352051402189872281list_a @ F @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_60_field_Omonic__poly__carr,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_3395465470813675732t_unit @ R @ F )
=> ( member5524387281408368019list_a @ F @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_61_field_Omonic__poly__carr,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_8143709425247463374t_unit @ R @ F )
=> ( member6124916891863447321list_a @ F @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_62_field_Omonic__poly__carr,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_5986596350207772206t_unit @ R @ F )
=> ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_63_field_Omonic__poly__carr,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_5008461317928820916t_unit @ R @ F )
=> ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_64_field_Omonic__poly__carr,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a] :
( ( field_a_b @ R )
=> ( ( monic_3145109188698636716ly_a_b @ R @ F )
=> ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_65_pderiv__inv,axiom,
! [F: list_a] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( formal4452980811800949548iv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( formal4452980811800949548iv_a_b @ r @ F ) ) ) ) ).
% pderiv_inv
thf(fact_66_pderiv__zero,axiom,
! [K: set_a] :
( ( formal4452980811800949548iv_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% pderiv_zero
thf(fact_67_p_Oadd_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% p.add.l_cancel
thf(fact_68_p_Oadd_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% p.add.m_assoc
thf(fact_69_mem__Collect__eq,axiom,
! [A: list_list_list_a,P2: list_list_list_a > $o] :
( ( member5342144027231129785list_a @ A @ ( collec1292721268053437947list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A: list_set_list_list_a,P2: list_set_list_list_a > $o] :
( ( member6124916891863447321list_a @ A @ ( collec962832732428061787list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A: list_set_list_a,P2: list_set_list_a > $o] :
( ( member5524387281408368019list_a @ A @ ( collec5381118732811369429list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A: set_list_list_a,P2: set_list_list_a > $o] :
( ( member334759470184282131list_a @ A @ ( collec191490921587283541list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A: set_list_a,P2: set_list_a > $o] :
( ( member_set_list_a @ A @ ( collect_set_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A: list_a,P2: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
! [A: list_list_a,P2: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A2: set_list_list_list_a] :
( ( collec1292721268053437947list_a
@ ^ [X2: list_list_list_a] : ( member5342144027231129785list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A2: set_li7845362039408639568list_a] :
( ( collec962832732428061787list_a
@ ^ [X2: list_set_list_list_a] : ( member6124916891863447321list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
! [A2: set_list_set_list_a] :
( ( collec5381118732811369429list_a
@ ^ [X2: list_set_list_a] : ( member5524387281408368019list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A2: set_set_list_list_a] :
( ( collec191490921587283541list_a
@ ^ [X2: set_list_list_a] : ( member334759470184282131list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A2: set_set_list_a] :
( ( collect_set_list_a
@ ^ [X2: set_list_a] : ( member_set_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A2: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X2: list_list_a] : ( member_list_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_85_Collect__cong,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P2 )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_86_Collect__cong,axiom,
! [P2: list_list_a > $o,Q: list_list_a > $o] :
( ! [X3: list_list_a] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_list_a @ P2 )
= ( collect_list_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_87_Collect__cong,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_list_a @ P2 )
= ( collect_list_a @ Q ) ) ) ).
% Collect_cong
thf(fact_88_p_Oadd_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% p.add.m_comm
thf(fact_89_p_Oadd_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% p.add.m_lcomm
thf(fact_90_p_Oadd_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A = B ) ) ) ) ) ).
% p.add.r_cancel
thf(fact_91_var__neq__zero,axiom,
( ( var_a_b @ r )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% var_neq_zero
thf(fact_92_pderiv__carr,axiom,
! [F: list_a] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( formal4452980811800949548iv_a_b @ r @ F ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% pderiv_carr
thf(fact_93_p_Oadd_Oinv__comm,axiom,
! [X: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.add.inv_comm
thf(fact_94_p_Oadd_Ol__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.add.l_inv_ex
thf(fact_95_p_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.add.one_unique
thf(fact_96_p_Oadd_Or__inv__ex,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.add.r_inv_ex
thf(fact_97_p_Ominus__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% p.minus_unique
thf(fact_98_p_Oa__transpose__inv,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= Z )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% p.a_transpose_inv
thf(fact_99_p_Oadd_Oinv__mult__group,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) ) ) ) ) ).
% p.add.inv_mult_group
thf(fact_100_p_Oadd_Oinv__solve__left,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) @ C ) )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A ) ) ) ) ) ) ).
% p.add.inv_solve_left
thf(fact_101_p_Oadd_Oinv__solve__left_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) @ C )
= A )
= ( C
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A ) ) ) ) ) ) ).
% p.add.inv_solve_left'
thf(fact_102_p_Oadd_Oinv__solve__right,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C ) ) )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ) ) ).
% p.add.inv_solve_right
thf(fact_103_p_Oadd_Oinv__solve__right_H,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C ) )
= A )
= ( B
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ) ) ).
% p.add.inv_solve_right'
thf(fact_104_p_Ominus__add,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ) ) ).
% p.minus_add
thf(fact_105_p_Or__neg1,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= Y ) ) ) ).
% p.r_neg1
thf(fact_106_p_Or__neg2,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y ) )
= Y ) ) ) ).
% p.r_neg2
thf(fact_107_p_Opow__non__zero,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.pow_non_zero
thf(fact_108_p_Onat__pow__pow,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ M )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( times_times_nat @ N @ M ) ) ) ) ).
% p.nat_pow_pow
thf(fact_109_p_Oring__irreducibleE_I1_J,axiom,
! [R2: list_a] :
( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R2 )
=> ( R2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_irreducibleE(1)
thf(fact_110_pderiv__add,axiom,
! [F: list_a,G: list_a] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( formal4452980811800949548iv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ G ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( formal4452980811800949548iv_a_b @ r @ F ) @ ( formal4452980811800949548iv_a_b @ r @ G ) ) ) ) ) ).
% pderiv_add
thf(fact_111_p_Opolynomial__pow__degree,axiom,
! [P: list_list_a,N: nat] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ).
% p.polynomial_pow_degree
thf(fact_112_p_Ol__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.l_neg
thf(fact_113_p_Ominus__equality,axiom,
! [Y: list_a,X: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= Y ) ) ) ) ).
% p.minus_equality
thf(fact_114_p_Or__neg,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.r_neg
thf(fact_115_p_Osum__zero__eq__neg,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( X
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ) ) ).
% p.sum_zero_eq_neg
thf(fact_116_pirreducible__imp__not__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ).
% pirreducible_imp_not_splitted
thf(fact_117_p_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.zero_closed
thf(fact_118_p_Oadd_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.add.m_closed
thf(fact_119_p_Oadd_Oright__cancel,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% p.add.right_cancel
thf(fact_120_p_Ominus__zero,axiom,
( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.minus_zero
thf(fact_121_p_Oadd_Ol__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.add.l_cancel_one
thf(fact_122_p_Oadd_Ol__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.add.l_cancel_one'
thf(fact_123_p_Oadd_Or__cancel__one,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X )
= X )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.add.r_cancel_one
thf(fact_124_p_Oadd_Or__cancel__one_H,axiom,
! [X: list_a,A: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.add.r_cancel_one'
thf(fact_125_p_Ol__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% p.l_zero
thf(fact_126_p_Or__zero,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% p.r_zero
thf(fact_127_p_Oadd_Oinv__eq__1__iff,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( X
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.add.inv_eq_1_iff
thf(fact_128_field_Ovar__neq__zero,axiom,
! [R: partia5333488208502193986t_unit] :
( ( field_40794040841493211t_unit @ R )
=> ( ( var_li1858665414325356817t_unit @ R )
!= ( zero_l7603606658129810363t_unit @ ( univ_p3766428211910075458t_unit @ R @ ( partia5038748322285217333t_unit @ R ) ) ) ) ) ).
% field.var_neq_zero
thf(fact_129_field_Ovar__neq__zero,axiom,
! [R: partia3473558348976337314t_unit] :
( ( field_3650391457538083963t_unit @ R )
=> ( ( var_li7072070971431396017t_unit @ R )
!= ( zero_l5774950708658833755t_unit @ ( univ_p7468970815438322146t_unit @ R @ ( partia6708307881709191317t_unit @ R ) ) ) ) ) ).
% field.var_neq_zero
thf(fact_130_field_Ovar__neq__zero,axiom,
! [R: partia4556295656693239580t_unit] :
( ( field_6241291760955494017t_unit @ R )
=> ( ( var_li7697098674259343287t_unit @ R )
!= ( zero_l246158410283469281t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) ) ) ).
% field.var_neq_zero
thf(fact_131_field_Ovar__neq__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( var_se6008125447796440765t_unit @ R )
!= ( zero_l7621212060072393831t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ).
% field.var_neq_zero
thf(fact_132_field_Ovar__neq__zero,axiom,
! [R: partia4960592913263135132t_unit] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( var_se2996050386653789495t_unit @ R )
!= ( zero_l1604441510127931233t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) ) ).
% field.var_neq_zero
thf(fact_133_field_Ovar__neq__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( var_a_b @ R )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% field.var_neq_zero
thf(fact_134_field_Ovar__neq__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( var_li8453953174693405341t_unit @ R )
!= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% field.var_neq_zero
thf(fact_135_field_Ovar__neq__zero,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( var_li3532061862469730199t_unit @ R )
!= ( zero_l317200538825487809t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% field.var_neq_zero
thf(fact_136_field_Ogauss__poly__not__zero,axiom,
! [R: partia5333488208502193986t_unit,N: nat] :
( ( field_40794040841493211t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( card_I8204580741270601971t_unit @ R @ N )
!= ( zero_l7603606658129810363t_unit @ ( univ_p3766428211910075458t_unit @ R @ ( partia5038748322285217333t_unit @ R ) ) ) ) ) ) ).
% field.gauss_poly_not_zero
thf(fact_137_field_Ogauss__poly__not__zero,axiom,
! [R: partia3473558348976337314t_unit,N: nat] :
( ( field_3650391457538083963t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( card_I4320469302700135059t_unit @ R @ N )
!= ( zero_l5774950708658833755t_unit @ ( univ_p7468970815438322146t_unit @ R @ ( partia6708307881709191317t_unit @ R ) ) ) ) ) ) ).
% field.gauss_poly_not_zero
thf(fact_138_field_Ogauss__poly__not__zero,axiom,
! [R: partia4556295656693239580t_unit,N: nat] :
( ( field_6241291760955494017t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( card_I4960859800387535001t_unit @ R @ N )
!= ( zero_l246158410283469281t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) ) ) ) ).
% field.gauss_poly_not_zero
thf(fact_139_field_Ogauss__poly__not__zero,axiom,
! [R: partia7496981018696276118t_unit,N: nat] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( card_I5471341911849640095t_unit @ R @ N )
!= ( zero_l7621212060072393831t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ) ).
% field.gauss_poly_not_zero
thf(fact_140_field_Ogauss__poly__not__zero,axiom,
! [R: partia4960592913263135132t_unit,N: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( card_I259811512781981209t_unit @ R @ N )
!= ( zero_l1604441510127931233t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) ) ) ).
% field.gauss_poly_not_zero
thf(fact_141_field_Ogauss__poly__not__zero,axiom,
! [R: partia2670972154091845814t_unit,N: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( card_I2619780863984422015t_unit @ R @ N )
!= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% field.gauss_poly_not_zero
thf(fact_142_field_Ogauss__poly__not__zero,axiom,
! [R: partia2956882679547061052t_unit,N: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( card_I3787608780883923065t_unit @ R @ N )
!= ( zero_l317200538825487809t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ).
% field.gauss_poly_not_zero
thf(fact_143_field_Ogauss__poly__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( field_a_b @ R )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ( card_I2373409586816755191ly_a_b @ R @ N )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% field.gauss_poly_not_zero
thf(fact_144_field_Omonic__poly__add__distinct,axiom,
! [R: partia5333488208502193986t_unit,F: list_l7815035709764258326list_a,G: list_l7815035709764258326list_a] :
( ( field_40794040841493211t_unit @ R )
=> ( ( monic_7946909391392167976t_unit @ R @ F )
=> ( ( member6842060177613954879list_a @ G @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ R @ ( partia5038748322285217333t_unit @ R ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s3057721424629808770list_a @ G ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s3057721424629808770list_a @ F ) @ one_one_nat ) )
=> ( monic_7946909391392167976t_unit @ R @ ( add_li5031911511988408956t_unit @ ( univ_p3766428211910075458t_unit @ R @ ( partia5038748322285217333t_unit @ R ) ) @ F @ G ) ) ) ) ) ) ).
% field.monic_poly_add_distinct
thf(fact_145_field_Omonic__poly__add__distinct,axiom,
! [R: partia3473558348976337314t_unit,F: list_l8227414553961969782list_a,G: list_l8227414553961969782list_a] :
( ( field_3650391457538083963t_unit @ R )
=> ( ( monic_248643784349308360t_unit @ R @ F )
=> ( ( member6908278366116243871list_a @ G @ ( partia6954261711924553499t_unit @ ( univ_p7468970815438322146t_unit @ R @ ( partia6708307881709191317t_unit @ R ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s2304674544134913250list_a @ G ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s2304674544134913250list_a @ F ) @ one_one_nat ) )
=> ( monic_248643784349308360t_unit @ R @ ( add_li7866670184587707420t_unit @ ( univ_p7468970815438322146t_unit @ R @ ( partia6708307881709191317t_unit @ R ) ) @ F @ G ) ) ) ) ) ) ).
% field.monic_poly_add_distinct
thf(fact_146_field_Omonic__poly__add__distinct,axiom,
! [R: partia4556295656693239580t_unit,F: list_list_set_list_a,G: list_list_set_list_a] :
( ( field_6241291760955494017t_unit @ R )
=> ( ( monic_3621385675998241358t_unit @ R @ F )
=> ( ( member352051402189872281list_a @ G @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ G ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s4069122225494125916list_a @ F ) @ one_one_nat ) )
=> ( monic_3621385675998241358t_unit @ R @ ( add_li1654802311551555874t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) @ F @ G ) ) ) ) ) ) ).
% field.monic_poly_add_distinct
thf(fact_147_field_Omonic__poly__add__distinct,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a,G: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_3395465470813675732t_unit @ R @ F )
=> ( ( member5524387281408368019list_a @ G @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ G ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ F ) @ one_one_nat ) )
=> ( monic_3395465470813675732t_unit @ R @ ( add_li1651944370010710056t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ F @ G ) ) ) ) ) ) ).
% field.monic_poly_add_distinct
thf(fact_148_field_Omonic__poly__add__distinct,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a,G: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_8143709425247463374t_unit @ R @ F )
=> ( ( member6124916891863447321list_a @ G @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s618615678312925148list_a @ G ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s618615678312925148list_a @ F ) @ one_one_nat ) )
=> ( monic_8143709425247463374t_unit @ R @ ( add_li3013085411396017826t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ F @ G ) ) ) ) ) ) ).
% field.monic_poly_add_distinct
thf(fact_149_field_Omonic__poly__add__distinct,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a,G: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_5986596350207772206t_unit @ R @ F )
=> ( ( member5342144027231129785list_a @ G @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ G ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ F ) @ one_one_nat ) )
=> ( monic_5986596350207772206t_unit @ R @ ( add_li5162926044081146114t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ F @ G ) ) ) ) ) ) ).
% field.monic_poly_add_distinct
thf(fact_150_field_Omonic__poly__add__distinct,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a,G: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_5008461317928820916t_unit @ R @ F )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ G ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat ) )
=> ( monic_5008461317928820916t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ G ) ) ) ) ) ) ).
% field.monic_poly_add_distinct
thf(fact_151_field_Omonic__poly__add__distinct,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a,G: list_a] :
( ( field_a_b @ R )
=> ( ( monic_3145109188698636716ly_a_b @ R @ F )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ G ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) )
=> ( monic_3145109188698636716ly_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ G ) ) ) ) ) ) ).
% field.monic_poly_add_distinct
thf(fact_152_field_Omonic__poly__var,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( monic_3395465470813675732t_unit @ R @ ( var_se6008125447796440765t_unit @ R ) ) ) ).
% field.monic_poly_var
thf(fact_153_field_Omonic__poly__var,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( monic_5986596350207772206t_unit @ R @ ( var_li3532061862469730199t_unit @ R ) ) ) ).
% field.monic_poly_var
thf(fact_154_field_Omonic__poly__var,axiom,
! [R: partia4960592913263135132t_unit] :
( ( field_1540243473349940225t_unit @ R )
=> ( monic_8143709425247463374t_unit @ R @ ( var_se2996050386653789495t_unit @ R ) ) ) ).
% field.monic_poly_var
thf(fact_155_field_Omonic__poly__var,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( monic_5008461317928820916t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) ) ) ).
% field.monic_poly_var
thf(fact_156_field_Omonic__poly__var,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( monic_3145109188698636716ly_a_b @ R @ ( var_a_b @ R ) ) ) ).
% field.monic_poly_var
thf(fact_157_p_OboundD__carrier,axiom,
! [N: nat,F: nat > list_a,M: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F @ M ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.boundD_carrier
thf(fact_158_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia5333488208502193986t_unit,P: list_l7815035709764258326list_a] :
( ( field_40794040841493211t_unit @ R )
=> ( ( member6842060177613954879list_a @ P @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ R @ ( partia5038748322285217333t_unit @ R ) ) ) )
=> ( ( ring_r1748661638422082364t_unit @ ( univ_p3766428211910075458t_unit @ R @ ( partia5038748322285217333t_unit @ R ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s3057721424629808770list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno1480074024635739021t_unit @ R @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_159_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia3473558348976337314t_unit,P: list_l8227414553961969782list_a] :
( ( field_3650391457538083963t_unit @ R )
=> ( ( member6908278366116243871list_a @ P @ ( partia6954261711924553499t_unit @ ( univ_p7468970815438322146t_unit @ R @ ( partia6708307881709191317t_unit @ R ) ) ) )
=> ( ( ring_r87237867408614620t_unit @ ( univ_p7468970815438322146t_unit @ R @ ( partia6708307881709191317t_unit @ R ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s2304674544134913250list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno484166042464326957t_unit @ R @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_160_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia4556295656693239580t_unit,P: list_list_set_list_a] :
( ( field_6241291760955494017t_unit @ R )
=> ( ( member352051402189872281list_a @ P @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) )
=> ( ( ring_r7962978046438709730t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno8445875935890348083t_unit @ R @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_161_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno7858167711734664505t_unit @ R @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_162_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia4960592913263135132t_unit,P: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( ring_r97889109428395874t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno3744827648284794291t_unit @ R @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_163_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno5970451904377802771t_unit @ R @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_164_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno8329700637149614481ed_a_b @ R @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_165_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno6259083269128200473t_unit @ R @ P ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_166_p_Ozero__is__prime_I1_J,axiom,
prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.zero_is_prime(1)
thf(fact_167_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_168_field_Odegree__one__imp__splitted,axiom,
! [R: partia5333488208502193986t_unit,P: list_l7815035709764258326list_a] :
( ( field_40794040841493211t_unit @ R )
=> ( ( member6842060177613954879list_a @ P @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ R @ ( partia5038748322285217333t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s3057721424629808770list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno1480074024635739021t_unit @ R @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_169_field_Odegree__one__imp__splitted,axiom,
! [R: partia3473558348976337314t_unit,P: list_l8227414553961969782list_a] :
( ( field_3650391457538083963t_unit @ R )
=> ( ( member6908278366116243871list_a @ P @ ( partia6954261711924553499t_unit @ ( univ_p7468970815438322146t_unit @ R @ ( partia6708307881709191317t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2304674544134913250list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno484166042464326957t_unit @ R @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_170_field_Odegree__one__imp__splitted,axiom,
! [R: partia4556295656693239580t_unit,P: list_list_set_list_a] :
( ( field_6241291760955494017t_unit @ R )
=> ( ( member352051402189872281list_a @ P @ ( partia2063761723659798037t_unit @ ( univ_p2555602637952293736t_unit @ R @ ( partia7265347635606999311t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s4069122225494125916list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno8445875935890348083t_unit @ R @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_171_field_Odegree__one__imp__splitted,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno7858167711734664505t_unit @ R @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_172_field_Odegree__one__imp__splitted,axiom,
! [R: partia4960592913263135132t_unit,P: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno3744827648284794291t_unit @ R @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_173_field_Odegree__one__imp__splitted,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno5970451904377802771t_unit @ R @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_174_field_Odegree__one__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno8329700637149614481ed_a_b @ R @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_175_field_Odegree__one__imp__splitted,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno6259083269128200473t_unit @ R @ P ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_176_p_Oprimeness__condition,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 )
= ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.primeness_condition
thf(fact_177_p_Oring__primeE_I1_J,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_primeE(1)
thf(fact_178_pderiv__const,axiom,
! [X: list_a,K: set_a] :
( ( ( minus_minus_nat @ ( size_size_list_a @ X ) @ one_one_nat )
= zero_zero_nat )
=> ( ( formal4452980811800949548iv_a_b @ r @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% pderiv_const
thf(fact_179_pirreducible__degree,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 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% pirreducible_degree
thf(fact_180_rupture__is__field__iff__pirreducible,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_181_p_Ominus__eq,axiom,
! [X: list_a,Y: list_a] :
( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) ) ) ).
% p.minus_eq
thf(fact_182_p_Odegree__var,axiom,
( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ one_one_nat )
= one_one_nat ) ).
% p.degree_var
thf(fact_183_p_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.nat_pow_zero
thf(fact_184_p_Oring__primeE_I3_J,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.ring_primeE(3)
thf(fact_185_p_Oring__primeI,axiom,
! [P: list_a] :
( ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.ring_primeI
thf(fact_186_p_Odegree__zero__imp__splitted,axiom,
! [P: list_list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno6259083269128200473t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.degree_zero_imp_splitted
thf(fact_187_p_Ominus__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.minus_closed
thf(fact_188_p_Or__right__minus__eq,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A = B ) ) ) ) ).
% p.r_right_minus_eq
thf(fact_189_ring_Osplitted_Ocong,axiom,
polyno5970451904377802771t_unit = polyno5970451904377802771t_unit ).
% ring.splitted.cong
thf(fact_190_ring_Osplitted_Ocong,axiom,
polyno3744827648284794291t_unit = polyno3744827648284794291t_unit ).
% ring.splitted.cong
thf(fact_191_ring_Osplitted_Ocong,axiom,
polyno7858167711734664505t_unit = polyno7858167711734664505t_unit ).
% ring.splitted.cong
thf(fact_192_ring_Osplitted_Ocong,axiom,
polyno8329700637149614481ed_a_b = polyno8329700637149614481ed_a_b ).
% ring.splitted.cong
thf(fact_193_ring_Osplitted_Ocong,axiom,
polyno6259083269128200473t_unit = polyno6259083269128200473t_unit ).
% ring.splitted.cong
thf(fact_194_gauss__poly__def,axiom,
( card_I8204580741270601971t_unit
= ( ^ [K2: partia5333488208502193986t_unit,N2: nat] : ( a_minu6308263530791455170t_unit @ ( univ_p3766428211910075458t_unit @ K2 @ ( partia5038748322285217333t_unit @ K2 ) ) @ ( pow_li2121314091559947310it_nat @ ( univ_p3766428211910075458t_unit @ K2 @ ( partia5038748322285217333t_unit @ K2 ) ) @ ( var_li1858665414325356817t_unit @ K2 ) @ N2 ) @ ( var_li1858665414325356817t_unit @ K2 ) ) ) ) ).
% gauss_poly_def
thf(fact_195_gauss__poly__def,axiom,
( card_I4320469302700135059t_unit
= ( ^ [K2: partia3473558348976337314t_unit,N2: nat] : ( a_minu8267672101703213922t_unit @ ( univ_p7468970815438322146t_unit @ K2 @ ( partia6708307881709191317t_unit @ K2 ) ) @ ( pow_li8421058921225051182it_nat @ ( univ_p7468970815438322146t_unit @ K2 @ ( partia6708307881709191317t_unit @ K2 ) ) @ ( var_li7072070971431396017t_unit @ K2 ) @ N2 ) @ ( var_li7072070971431396017t_unit @ K2 ) ) ) ) ).
% gauss_poly_def
thf(fact_196_gauss__poly__def,axiom,
( card_I4960859800387535001t_unit
= ( ^ [K2: partia4556295656693239580t_unit,N2: nat] : ( a_minu9044011302611522408t_unit @ ( univ_p2555602637952293736t_unit @ K2 @ ( partia7265347635606999311t_unit @ K2 ) ) @ ( pow_li6530163574085554350it_nat @ ( univ_p2555602637952293736t_unit @ K2 @ ( partia7265347635606999311t_unit @ K2 ) ) @ ( var_li7697098674259343287t_unit @ K2 ) @ N2 ) @ ( var_li7697098674259343287t_unit @ K2 ) ) ) ) ).
% gauss_poly_def
thf(fact_197_gauss__poly__def,axiom,
( card_I259811512781981209t_unit
= ( ^ [K2: partia4960592913263135132t_unit,N2: nat] : ( a_minu1178922365601208552t_unit @ ( univ_p7077926387201515752t_unit @ K2 @ ( partia3317168157747563407t_unit @ K2 ) ) @ ( pow_li5711373720449609902it_nat @ ( univ_p7077926387201515752t_unit @ K2 @ ( partia3317168157747563407t_unit @ K2 ) ) @ ( var_se2996050386653789495t_unit @ K2 ) @ N2 ) @ ( var_se2996050386653789495t_unit @ K2 ) ) ) ) ).
% gauss_poly_def
thf(fact_198_gauss__poly__def,axiom,
( card_I5471341911849640095t_unit
= ( ^ [K2: partia7496981018696276118t_unit,N2: nat] : ( a_minu6874796375791416686t_unit @ ( univ_p863672496597069550t_unit @ K2 @ ( partia141011252114345353t_unit @ K2 ) ) @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ K2 @ ( partia141011252114345353t_unit @ K2 ) ) @ ( var_se6008125447796440765t_unit @ K2 ) @ N2 ) @ ( var_se6008125447796440765t_unit @ K2 ) ) ) ) ).
% gauss_poly_def
thf(fact_199_gauss__poly__def,axiom,
( card_I2619780863984422015t_unit
= ( ^ [K2: partia2670972154091845814t_unit,N2: nat] : ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ K2 @ ( partia5361259788508890537t_unit @ K2 ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ K2 @ ( partia5361259788508890537t_unit @ K2 ) ) @ ( var_li8453953174693405341t_unit @ K2 ) @ N2 ) @ ( var_li8453953174693405341t_unit @ K2 ) ) ) ) ).
% gauss_poly_def
thf(fact_200_gauss__poly__def,axiom,
( card_I3787608780883923065t_unit
= ( ^ [K2: partia2956882679547061052t_unit,N2: nat] : ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ K2 @ ( partia2464479390973590831t_unit @ K2 ) ) @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ K2 @ ( partia2464479390973590831t_unit @ K2 ) ) @ ( var_li3532061862469730199t_unit @ K2 ) @ N2 ) @ ( var_li3532061862469730199t_unit @ K2 ) ) ) ) ).
% gauss_poly_def
thf(fact_201_gauss__poly__def,axiom,
( card_I2373409586816755191ly_a_b
= ( ^ [K2: partia2175431115845679010xt_a_b,N2: nat] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ K2 @ ( partia707051561876973205xt_a_b @ K2 ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ K2 @ ( partia707051561876973205xt_a_b @ K2 ) ) @ ( var_a_b @ K2 ) @ N2 ) @ ( var_a_b @ K2 ) ) ) ) ).
% gauss_poly_def
thf(fact_202_p_Odegree__zero__imp__not__is__root,axiom,
! [P: list_list_a,X: list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) ) ) ).
% p.degree_zero_imp_not_is_root
thf(fact_203_degree__zero__imp__not__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_204_monic__poly__min__degree,axiom,
! [F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ F )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) ) ) ).
% monic_poly_min_degree
thf(fact_205_nat__mult__le__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_206_mult__le__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_207_p_Opderiv__const,axiom,
! [X: list_list_a,K: set_list_a] :
( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ X ) @ one_one_nat )
= zero_zero_nat )
=> ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X )
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ).
% p.pderiv_const
thf(fact_208_nat__mult__less__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_209_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_210_mult__less__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_211_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_212_p_Opderiv__zero,axiom,
! [K: set_list_a] :
( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).
% p.pderiv_zero
thf(fact_213_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_214_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_215_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_216_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_217_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_218_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_219_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_220_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_221_mult__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ( times_times_nat @ M @ K3 )
= ( times_times_nat @ N @ K3 ) )
= ( ( M = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_222_mult__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( ( M = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_223_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_224_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_225_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_226_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_227_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_228_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_229_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_230_ring_Ois__root_Ocong,axiom,
polyno6951661231331188332t_unit = polyno6951661231331188332t_unit ).
% ring.is_root.cong
thf(fact_231_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_232_size__neq__size__imp__neq,axiom,
! [X: multis8176868331707208240list_a,Y: multis8176868331707208240list_a] :
( ( ( size_s82858050752783516list_a @ X )
!= ( size_s82858050752783516list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_233_size__neq__size__imp__neq,axiom,
! [X: multiset_set_list_a,Y: multiset_set_list_a] :
( ( ( size_s1226348209404258454list_a @ X )
!= ( size_s1226348209404258454list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_234_size__neq__size__imp__neq,axiom,
! [X: list_list_list_a,Y: list_list_list_a] :
( ( ( size_s2403821588304063868list_a @ X )
!= ( size_s2403821588304063868list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_235_size__neq__size__imp__neq,axiom,
! [X: list_set_list_list_a,Y: list_set_list_list_a] :
( ( ( size_s618615678312925148list_a @ X )
!= ( size_s618615678312925148list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_236_size__neq__size__imp__neq,axiom,
! [X: list_set_list_a,Y: list_set_list_a] :
( ( ( size_s1991367317912710102list_a @ X )
!= ( size_s1991367317912710102list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_237_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_238_size__neq__size__imp__neq,axiom,
! [X: list_list_a,Y: list_list_a] :
( ( ( size_s349497388124573686list_a @ X )
!= ( size_s349497388124573686list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_239_size__neq__size__imp__neq,axiom,
! [X: multiset_list_a,Y: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ X )
!= ( size_s2335926164413107382list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_240_size__neq__size__imp__neq,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( ( size_size_multiset_a @ X )
!= ( size_size_multiset_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_241_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_242_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_243_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P2 @ M2 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_244_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_245_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_246_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_247_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_248_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_249_diff__commute,axiom,
! [I: nat,J: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K3 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K3 ) @ J ) ) ).
% diff_commute
thf(fact_250_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K3: nat,B: nat] :
( ( P2 @ K3 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_251_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_252_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_253_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_254_le__trans,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K3 )
=> ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% le_trans
thf(fact_255_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_256_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_257_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_258_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_259_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_260_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_261_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_262_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_263_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_264_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_265_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_266_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_267_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_268_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_269_less__imp__diff__less,axiom,
! [J: nat,K3: nat,N: nat] :
( ( ord_less_nat @ J @ K3 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K3 ) ) ).
% less_imp_diff_less
thf(fact_270_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_271_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_272_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_273_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
thf(fact_274_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_275_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_276_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_277_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_278_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_279_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_280_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_281_Nat_Odiff__diff__eq,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_282_le__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_283_eq__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ( minus_minus_nat @ M @ K3 )
= ( minus_minus_nat @ N @ K3 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_284_monic__irreducible__poly__def,axiom,
( monic_4605209128504297202t_unit
= ( ^ [R3: partia5333488208502193986t_unit,F2: list_l7815035709764258326list_a] :
( ( monic_7946909391392167976t_unit @ R3 @ F2 )
& ( ring_r1748661638422082364t_unit @ ( univ_p3766428211910075458t_unit @ R3 @ ( partia5038748322285217333t_unit @ R3 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_285_monic__irreducible__poly__def,axiom,
( monic_4901478363665587346t_unit
= ( ^ [R3: partia3473558348976337314t_unit,F2: list_l8227414553961969782list_a] :
( ( monic_248643784349308360t_unit @ R3 @ F2 )
& ( ring_r87237867408614620t_unit @ ( univ_p7468970815438322146t_unit @ R3 @ ( partia6708307881709191317t_unit @ R3 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_286_monic__irreducible__poly__def,axiom,
( monic_5520764287479354904t_unit
= ( ^ [R3: partia4556295656693239580t_unit,F2: list_list_set_list_a] :
( ( monic_3621385675998241358t_unit @ R3 @ F2 )
& ( ring_r7962978046438709730t_unit @ ( univ_p2555602637952293736t_unit @ R3 @ ( partia7265347635606999311t_unit @ R3 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_287_monic__irreducible__poly__def,axiom,
( monic_819715999873801112t_unit
= ( ^ [R3: partia4960592913263135132t_unit,F2: list_set_list_list_a] :
( ( monic_8143709425247463374t_unit @ R3 @ F2 )
& ( ring_r97889109428395874t_unit @ ( univ_p7077926387201515752t_unit @ R3 @ ( partia3317168157747563407t_unit @ R3 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_288_monic__irreducible__poly__def,axiom,
( monic_2059080652700942750t_unit
= ( ^ [R3: partia7496981018696276118t_unit,F2: list_set_list_a] :
( ( monic_3395465470813675732t_unit @ R3 @ F2 )
& ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R3 @ ( partia141011252114345353t_unit @ R3 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_289_monic__irreducible__poly__def,axiom,
( monic_868474719114584568t_unit
= ( ^ [R3: partia2956882679547061052t_unit,F2: list_list_list_a] :
( ( monic_5986596350207772206t_unit @ R3 @ F2 )
& ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R3 @ ( partia2464479390973590831t_unit @ R3 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_290_monic__irreducible__poly__def,axiom,
( monic_104106837769529726t_unit
= ( ^ [R3: partia2670972154091845814t_unit,F2: list_list_a] :
( ( monic_5008461317928820916t_unit @ R3 @ F2 )
& ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ ( partia5361259788508890537t_unit @ R3 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_291_monic__irreducible__poly__def,axiom,
( monic_4919232885364369782ly_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,F2: list_a] :
( ( monic_3145109188698636716ly_a_b @ R3 @ F2 )
& ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R3 @ ( partia707051561876973205xt_a_b @ R3 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_292_nat__mult__eq__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( ( K3 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_293_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_294_diff__mult__distrib2,axiom,
! [K3: nat,M: nat,N: nat] :
( ( times_times_nat @ K3 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_295_diff__mult__distrib,axiom,
! [M: nat,N: nat,K3: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K3 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) ) ) ).
% diff_mult_distrib
thf(fact_296_mult__le__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ).
% mult_le_mono2
thf(fact_297_mult__le__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ).
% mult_le_mono1
thf(fact_298_mult__le__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_299_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_300_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_301_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_302_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_303_field_Omonic__poly__min__degree,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_2059080652700942750t_unit @ R @ F )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ F ) @ one_one_nat ) ) ) ) ).
% field.monic_poly_min_degree
thf(fact_304_field_Omonic__poly__min__degree,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_868474719114584568t_unit @ R @ F )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ F ) @ one_one_nat ) ) ) ) ).
% field.monic_poly_min_degree
thf(fact_305_field_Omonic__poly__min__degree,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_819715999873801112t_unit @ R @ F )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s618615678312925148list_a @ F ) @ one_one_nat ) ) ) ) ).
% field.monic_poly_min_degree
thf(fact_306_field_Omonic__poly__min__degree,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_104106837769529726t_unit @ R @ F )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat ) ) ) ) ).
% field.monic_poly_min_degree
thf(fact_307_field_Omonic__poly__min__degree,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a] :
( ( field_a_b @ R )
=> ( ( monic_4919232885364369782ly_a_b @ R @ F )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) ) ) ) ).
% field.monic_poly_min_degree
thf(fact_308_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_309_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K4 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_310_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_311_less__diff__iff,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_312_mult__less__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ) ).
% mult_less_mono1
thf(fact_313_mult__less__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_314_nat__mult__eq__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ( times_times_nat @ K3 @ M )
= ( times_times_nat @ K3 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_315_nat__mult__less__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_316_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_317_nat__mult__le__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_318_p_Oalg__mult__gt__zero__iff__is__root,axiom,
! [P: list_list_a,X: list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) )
= ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) ) ) ).
% p.alg_mult_gt_zero_iff_is_root
thf(fact_319_alg__mult__gt__zero__iff__is__root,axiom,
! [P: list_a,X: 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 @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_320_p_Ofreshmans__dream,axiom,
! [X: list_a,Y: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% p.freshmans_dream
thf(fact_321_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_322_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_323_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_324_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_325_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_326_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_327_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_328_zero__diff,axiom,
! [A: multiset_nat] :
( ( minus_8522176038001411705et_nat @ zero_z7348594199698428585et_nat @ A )
= zero_z7348594199698428585et_nat ) ).
% zero_diff
thf(fact_329_zero__diff,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ zero_z4454100511807792257list_a @ A )
= zero_z4454100511807792257list_a ) ).
% zero_diff
thf(fact_330_zero__diff,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ A )
= zero_zero_multiset_a ) ).
% zero_diff
thf(fact_331_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_332_diff__zero,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A @ zero_z4454100511807792257list_a )
= A ) ).
% diff_zero
thf(fact_333_diff__zero,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ zero_zero_multiset_a )
= A ) ).
% diff_zero
thf(fact_334_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_335_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A @ A )
= zero_z4454100511807792257list_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_336_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ A )
= zero_zero_multiset_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_337_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_338_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_339_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_340_ring_Oalg__mult_Ocong,axiom,
polyno4259638811958763678t_unit = polyno4259638811958763678t_unit ).
% ring.alg_mult.cong
thf(fact_341_ring_Oalg__mult_Ocong,axiom,
polyno4422430861927485590lt_a_b = polyno4422430861927485590lt_a_b ).
% ring.alg_mult.cong
thf(fact_342_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_343_zero__reorient,axiom,
! [X: multiset_list_a] :
( ( zero_z4454100511807792257list_a = X )
= ( X = zero_z4454100511807792257list_a ) ) ).
% zero_reorient
thf(fact_344_zero__reorient,axiom,
! [X: multiset_a] :
( ( zero_zero_multiset_a = X )
= ( X = zero_zero_multiset_a ) ) ).
% zero_reorient
thf(fact_345_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_346_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_347_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_348_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).
% mult.commute
thf(fact_349_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_350_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_351_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_352_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_353_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_354_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_355_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_356_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_357_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_358_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_359_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_360_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_361_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_362_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_363_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_364_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_365_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_366_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_367_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_368_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_369_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_370_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_371_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_372_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_373_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_374_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_375_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_376_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_377_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_378_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_379_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_380_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_381_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_382_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_383_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_384_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_385_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_386_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_387_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_388_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_389_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_390_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_391_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_392_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_393_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_394_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_395_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_396_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_397_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_398_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_399_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_400_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_401_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_402_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_403_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_404_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_405_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_406_p_Odegree__zero__imp__empty__roots,axiom,
! [P: list_list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= zero_z4454100511807792257list_a ) ) ) ).
% p.degree_zero_imp_empty_roots
thf(fact_407_p_Opirreducible__roots,axiom,
! [P: list_list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= zero_z4454100511807792257list_a ) ) ) ) ).
% p.pirreducible_roots
thf(fact_408_p_Osplitted__on__def,axiom,
! [K: set_list_a,P: list_list_a] :
( ( polyno1986131841096413848t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
= ( ( size_s2335926164413107382list_a @ ( polyno5990348478334826338t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ).
% p.splitted_on_def
thf(fact_409_p_Osplitted__def,axiom,
! [P: list_list_a] :
( ( polyno6259083269128200473t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( ( size_s2335926164413107382list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ).
% p.splitted_def
thf(fact_410_p_Ofreshmans__dream__ext,axiom,
! [X: list_a,Y: list_a,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( power_power_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( power_power_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( power_power_nat @ ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M ) ) ) ) ) ) ) ).
% p.freshmans_dream_ext
thf(fact_411_p_Osubring__polynomial__pow__degree,axiom,
! [K: set_list_a,P: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% p.subring_polynomial_pow_degree
thf(fact_412_trivial__factors__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [Q2: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q2 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% trivial_factors_imp_splitted
thf(fact_413_p_Ocarrier__is__subring,axiom,
subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.carrier_is_subring
thf(fact_414_p_Ouniv__poly__not__field,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ~ ( field_1861437471013600865t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).
% p.univ_poly_not_field
thf(fact_415_p_Ouniv__poly__a__minus__consistent,axiom,
! [K: set_list_a,Q3: list_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q3 )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ Q3 ) ) ) ) ).
% p.univ_poly_a_minus_consistent
thf(fact_416_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_417_p_Ovar__closed_I1_J,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ).
% p.var_closed(1)
thf(fact_418_p_Opderiv__carr,axiom,
! [K: set_list_a,F: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_list_a @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ) ).
% p.pderiv_carr
thf(fact_419_p_Opderiv__add,axiom,
! [K: set_list_a,F: list_list_a,G: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ F @ G ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G ) ) ) ) ) ) ).
% p.pderiv_add
thf(fact_420_p_Ovar__pow__closed,axiom,
! [K: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ).
% p.var_pow_closed
thf(fact_421_p_Ovar__pow__degree,axiom,
! [K: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) ) @ one_one_nat )
= N ) ) ).
% p.var_pow_degree
thf(fact_422_p_Ocarrier__polynomial__shell,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_list_a @ 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 ) ) ) ) ) ) ) ) ).
% p.carrier_polynomial_shell
thf(fact_423_ring_Oroots_Ocong,axiom,
polyno7858422826990252003t_unit = polyno7858422826990252003t_unit ).
% ring.roots.cong
thf(fact_424_ring_Oroots_Ocong,axiom,
polynomial_roots_a_b = polynomial_roots_a_b ).
% ring.roots.cong
thf(fact_425_ring_Oroots__on_Ocong,axiom,
polyno5990348478334826338t_unit = polyno5990348478334826338t_unit ).
% ring.roots_on.cong
thf(fact_426_ring_Oroots__on_Ocong,axiom,
polyno5714441830345289050on_a_b = polyno5714441830345289050on_a_b ).
% ring.roots_on.cong
thf(fact_427_ring_Osplitted__on_Ocong,axiom,
polyno1986131841096413848t_unit = polyno1986131841096413848t_unit ).
% ring.splitted_on.cong
thf(fact_428_ring_Osplitted__on_Ocong,axiom,
polyno2453258491555121552on_a_b = polyno2453258491555121552on_a_b ).
% ring.splitted_on.cong
thf(fact_429_field_Osize__roots__le__degree,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ord_less_eq_nat @ ( size_s1226348209404258454list_a @ ( polyno4169377219242390531t_unit @ R @ P ) ) @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ P ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_430_field_Osize__roots__le__degree,axiom,
! [R: partia4960592913263135132t_unit,P: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ord_less_eq_nat @ ( size_s82858050752783516list_a @ ( polyno2127442156181624701t_unit @ R @ P ) ) @ ( minus_minus_nat @ ( size_s618615678312925148list_a @ P ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_431_field_Osize__roots__le__degree,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ord_less_eq_nat @ ( size_s8523483970790017596list_a @ ( polyno3707469075594375645t_unit @ R @ P ) ) @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_432_field_Osize__roots__le__degree,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ord_less_eq_nat @ ( size_s2335926164413107382list_a @ ( polyno7858422826990252003t_unit @ R @ P ) ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_433_field_Osize__roots__le__degree,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ ( polynomial_roots_a_b @ R @ P ) ) @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_434_field_Otrivial__factors__imp__splitted,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ! [Q2: list_set_list_a] :
( ( member5524387281408368019list_a @ Q2 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ Q2 )
=> ( ( polyno9075941895896075626t_unit @ R @ Q2 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s1991367317912710102list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno7858167711734664505t_unit @ R @ P ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_435_field_Otrivial__factors__imp__splitted,axiom,
! [R: partia4960592913263135132t_unit,P: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ! [Q2: list_set_list_list_a] :
( ( member6124916891863447321list_a @ Q2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( ring_r97889109428395874t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ Q2 )
=> ( ( polyno3637028486239637860t_unit @ R @ Q2 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s618615678312925148list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno3744827648284794291t_unit @ R @ P ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_436_field_Otrivial__factors__imp__splitted,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ! [Q2: list_list_list_a] :
( ( member5342144027231129785list_a @ Q2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ Q2 )
=> ( ( polyno4453881341673752516t_unit @ R @ Q2 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno5970451904377802771t_unit @ R @ P ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_437_field_Otrivial__factors__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ! [Q2: list_a] :
( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ R @ Q2 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno8329700637149614481ed_a_b @ R @ P ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_438_field_Otrivial__factors__imp__splitted,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ! [Q2: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ Q2 )
=> ( ( polyno8016796738000020810t_unit @ R @ Q2 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno6259083269128200473t_unit @ R @ P ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_439_divides__monic__poly,axiom,
! [F: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G )
=> ( ! [D2: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ D2 )
=> ( ord_less_eq_nat @ ( monic_5301438133677370042lt_a_b @ r @ D2 @ F ) @ ( monic_5301438133677370042lt_a_b @ r @ D2 @ G ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ F @ G ) ) ) ) ).
% divides_monic_poly
thf(fact_440_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_441_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_442_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_443_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_444_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_445_splitted__on__def,axiom,
! [K: set_a,P: list_a] :
( ( polyno2453258491555121552on_a_b @ r @ K @ P )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ r @ K @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% splitted_on_def
thf(fact_446_splitted__def,axiom,
! [P: list_a] :
( ( polyno8329700637149614481ed_a_b @ r @ P )
= ( ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P ) )
= ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% splitted_def
thf(fact_447_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_448_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_449_size__roots__le__degree,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P ) ) @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ).
% size_roots_le_degree
thf(fact_450_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_451_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_452_p_Opolynomial__pow__division,axiom,
! [P: list_list_a,N: nat,M: nat] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ M ) ) ) ) ).
% p.polynomial_pow_division
thf(fact_453_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_454_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_455_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_456_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_457_power__commutes,axiom,
! [A: nat,N: nat] :
( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_commutes
thf(fact_458_power__mult__distrib,axiom,
! [A: nat,B: nat,N: nat] :
( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_459_power__commuting__commutes,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= ( times_times_nat @ Y @ X ) )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
= ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_460_power__mult,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_nat @ ( power_power_nat @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_461_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_462_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_463_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_464_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_465_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_466_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_467_field_Odivides__monic__poly,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a,G: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_3395465470813675732t_unit @ R @ F )
=> ( ( monic_3395465470813675732t_unit @ R @ G )
=> ( ! [D2: list_set_list_a] :
( ( monic_2059080652700942750t_unit @ R @ D2 )
=> ( ord_less_eq_nat @ ( monic_1916329280030354658t_unit @ R @ D2 @ F ) @ ( monic_1916329280030354658t_unit @ R @ D2 @ G ) ) )
=> ( polyno9075941895896075626t_unit @ R @ F @ G ) ) ) ) ) ).
% field.divides_monic_poly
thf(fact_468_field_Odivides__monic__poly,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a,G: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_5986596350207772206t_unit @ R @ F )
=> ( ( monic_5986596350207772206t_unit @ R @ G )
=> ( ! [D2: list_list_list_a] :
( ( monic_868474719114584568t_unit @ R @ D2 )
=> ( ord_less_eq_nat @ ( monic_6498620745769757500t_unit @ R @ D2 @ F ) @ ( monic_6498620745769757500t_unit @ R @ D2 @ G ) ) )
=> ( polyno4453881341673752516t_unit @ R @ F @ G ) ) ) ) ) ).
% field.divides_monic_poly
thf(fact_469_field_Odivides__monic__poly,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a,G: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_8143709425247463374t_unit @ R @ F )
=> ( ( monic_8143709425247463374t_unit @ R @ G )
=> ( ! [D2: list_set_list_list_a] :
( ( monic_819715999873801112t_unit @ R @ D2 )
=> ( ord_less_eq_nat @ ( monic_8185780642485851356t_unit @ R @ D2 @ F ) @ ( monic_8185780642485851356t_unit @ R @ D2 @ G ) ) )
=> ( polyno3637028486239637860t_unit @ R @ F @ G ) ) ) ) ) ).
% field.divides_monic_poly
thf(fact_470_field_Odivides__monic__poly,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a,G: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_5008461317928820916t_unit @ R @ F )
=> ( ( monic_5008461317928820916t_unit @ R @ G )
=> ( ! [D2: list_list_a] :
( ( monic_104106837769529726t_unit @ R @ D2 )
=> ( ord_less_eq_nat @ ( monic_2223747685970961602t_unit @ R @ D2 @ F ) @ ( monic_2223747685970961602t_unit @ R @ D2 @ G ) ) )
=> ( polyno8016796738000020810t_unit @ R @ F @ G ) ) ) ) ) ).
% field.divides_monic_poly
thf(fact_471_field_Odivides__monic__poly,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a,G: list_a] :
( ( field_a_b @ R )
=> ( ( monic_3145109188698636716ly_a_b @ R @ F )
=> ( ( monic_3145109188698636716ly_a_b @ R @ G )
=> ( ! [D2: list_a] :
( ( monic_4919232885364369782ly_a_b @ R @ D2 )
=> ( ord_less_eq_nat @ ( monic_5301438133677370042lt_a_b @ R @ D2 @ F ) @ ( monic_5301438133677370042lt_a_b @ R @ D2 @ G ) ) )
=> ( polyno5814909790663948098es_a_b @ R @ F @ G ) ) ) ) ) ).
% field.divides_monic_poly
thf(fact_472_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_473_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_474_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_475_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_476_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_477_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_478_power__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_479_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_480_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_481_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_482_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_483_power__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_484_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_485_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_486_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_487_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_488_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_489_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_490_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_491_power__eq__if,axiom,
( power_power_nat
= ( ^ [P3: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_492_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_493_splitted__imp__trivial__factors,axiom,
! [P: list_a,Q3: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ P )
=> ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q3 @ P )
=> ( ( minus_minus_nat @ ( size_size_list_a @ Q3 ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ).
% splitted_imp_trivial_factors
thf(fact_494_p_Opdivides__imp__degree__le,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( Q3 != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q3 ) @ one_one_nat ) ) ) ) ) ) ) ).
% p.pdivides_imp_degree_le
thf(fact_495_pdivides__imp__degree__le,axiom,
! [P: list_a,Q3: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q3 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q3 ) @ one_one_nat ) ) ) ) ) ) ).
% pdivides_imp_degree_le
thf(fact_496_size__eq__0__iff__empty,axiom,
! [M4: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ M4 )
= zero_zero_nat )
= ( M4 = zero_z4454100511807792257list_a ) ) ).
% size_eq_0_iff_empty
thf(fact_497_size__eq__0__iff__empty,axiom,
! [M4: multiset_a] :
( ( ( size_size_multiset_a @ M4 )
= zero_zero_nat )
= ( M4 = zero_zero_multiset_a ) ) ).
% size_eq_0_iff_empty
thf(fact_498_size__empty,axiom,
( ( size_s2335926164413107382list_a @ zero_z4454100511807792257list_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_499_size__empty,axiom,
( ( size_size_multiset_a @ zero_zero_multiset_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_500_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_501_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_502_p_Ozero__pdivides,axiom,
! [P: list_list_a] :
( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= ( P = nil_list_a ) ) ).
% p.zero_pdivides
thf(fact_503_p_Ozero__pdivides__zero,axiom,
polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ nil_list_a ).
% p.zero_pdivides_zero
thf(fact_504_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_505_pdivides__zero,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ).
% pdivides_zero
thf(fact_506_pdivides__imp__splitted,axiom,
! [P: list_a,Q3: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q3 != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q3 )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q3 )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_507_p_Opolynomial__pow__not__zero,axiom,
! [P: list_list_a,N: nat] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ N )
!= nil_list_a ) ) ) ).
% p.polynomial_pow_not_zero
thf(fact_508_p_Opdivides__zero,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a ) ) ) ).
% p.pdivides_zero
thf(fact_509_p_Osubring__polynomial__pow__not__zero,axiom,
! [K: set_list_a,P: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ N )
!= nil_list_a ) ) ) ) ).
% p.subring_polynomial_pow_not_zero
thf(fact_510_p_OpirreducibleE_I1_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ).
% p.pirreducibleE(1)
thf(fact_511_diff__empty,axiom,
! [M4: multiset_list_a] :
( ( ( minus_7431248565939055793list_a @ M4 @ zero_z4454100511807792257list_a )
= M4 )
& ( ( minus_7431248565939055793list_a @ zero_z4454100511807792257list_a @ M4 )
= zero_z4454100511807792257list_a ) ) ).
% diff_empty
thf(fact_512_diff__empty,axiom,
! [M4: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ M4 @ zero_zero_multiset_a )
= M4 )
& ( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ M4 )
= zero_zero_multiset_a ) ) ).
% diff_empty
thf(fact_513_Multiset_Odiff__cancel,axiom,
! [A2: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A2 @ A2 )
= zero_z4454100511807792257list_a ) ).
% Multiset.diff_cancel
thf(fact_514_Multiset_Odiff__cancel,axiom,
! [A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ A2 @ A2 )
= zero_zero_multiset_a ) ).
% Multiset.diff_cancel
thf(fact_515_field_Opdivides__imp__splitted,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,Q3: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( member5524387281408368019list_a @ Q3 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( Q3 != nil_set_list_a )
=> ( ( polyno7858167711734664505t_unit @ R @ Q3 )
=> ( ( polyno9075941895896075626t_unit @ R @ P @ Q3 )
=> ( polyno7858167711734664505t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_516_field_Opdivides__imp__splitted,axiom,
! [R: partia4960592913263135132t_unit,P: list_set_list_list_a,Q3: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( member6124916891863447321list_a @ Q3 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( Q3 != nil_set_list_list_a )
=> ( ( polyno3744827648284794291t_unit @ R @ Q3 )
=> ( ( polyno3637028486239637860t_unit @ R @ P @ Q3 )
=> ( polyno3744827648284794291t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_517_field_Opdivides__imp__splitted,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q3: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( Q3 != nil_list_list_a )
=> ( ( polyno5970451904377802771t_unit @ R @ Q3 )
=> ( ( polyno4453881341673752516t_unit @ R @ P @ Q3 )
=> ( polyno5970451904377802771t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_518_field_Opdivides__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q3: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( Q3 != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ R @ Q3 )
=> ( ( polyno5814909790663948098es_a_b @ R @ P @ Q3 )
=> ( polyno8329700637149614481ed_a_b @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_519_field_Opdivides__imp__splitted,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q3: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( Q3 != nil_list_a )
=> ( ( polyno6259083269128200473t_unit @ R @ Q3 )
=> ( ( polyno8016796738000020810t_unit @ R @ P @ Q3 )
=> ( polyno6259083269128200473t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_520_diff__size__le__size__Diff,axiom,
! [M4: multiset_list_a,M5: multiset_list_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2335926164413107382list_a @ M4 ) @ ( size_s2335926164413107382list_a @ M5 ) ) @ ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M4 @ M5 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_521_diff__size__le__size__Diff,axiom,
! [M4: multiset_a,M5: multiset_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_multiset_a @ M4 ) @ ( size_size_multiset_a @ M5 ) ) @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M4 @ M5 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_522_field_Osplitted__imp__trivial__factors,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,Q3: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( ( polyno7858167711734664505t_unit @ R @ P )
=> ( ( member5524387281408368019list_a @ Q3 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ring_r7392830359377363176t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ Q3 )
=> ( ( polyno9075941895896075626t_unit @ R @ Q3 @ P )
=> ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ Q3 ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_523_field_Osplitted__imp__trivial__factors,axiom,
! [R: partia4960592913263135132t_unit,P: list_set_list_list_a,Q3: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( P != nil_set_list_list_a )
=> ( ( polyno3744827648284794291t_unit @ R @ P )
=> ( ( member6124916891863447321list_a @ Q3 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( ring_r97889109428395874t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ Q3 )
=> ( ( polyno3637028486239637860t_unit @ R @ Q3 @ P )
=> ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ Q3 ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_524_field_Osplitted__imp__trivial__factors,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q3: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( polyno5970451904377802771t_unit @ R @ P )
=> ( ( member5342144027231129785list_a @ Q3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ Q3 )
=> ( ( polyno4453881341673752516t_unit @ R @ Q3 @ P )
=> ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Q3 ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_525_field_Osplitted__imp__trivial__factors,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q3: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ R @ P )
=> ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ Q3 )
=> ( ( polyno5814909790663948098es_a_b @ R @ Q3 @ P )
=> ( ( minus_minus_nat @ ( size_size_list_a @ Q3 ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_526_field_Osplitted__imp__trivial__factors,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q3: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( polyno6259083269128200473t_unit @ R @ P )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ Q3 )
=> ( ( polyno8016796738000020810t_unit @ R @ Q3 @ P )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q3 ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_527_nonempty__has__size,axiom,
! [S2: multiset_list_a] :
( ( S2 != zero_z4454100511807792257list_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s2335926164413107382list_a @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_528_nonempty__has__size,axiom,
! [S2: multiset_a] :
( ( S2 != zero_zero_multiset_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_size_multiset_a @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_529_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_530_length__greater__0__conv,axiom,
! [Xs: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) )
= ( Xs != nil_list_a ) ) ).
% length_greater_0_conv
thf(fact_531_p_Odegree__oneE,axiom,
! [P: list_list_a,K: set_list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A4: list_a] :
( ( member_list_a @ A4 @ K )
=> ( ( A4
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ! [B3: list_a] :
( ( member_list_a @ B3 @ K )
=> ( P
!= ( cons_list_a @ A4 @ ( cons_list_a @ B3 @ nil_list_a ) ) ) ) ) ) ) ) ).
% p.degree_oneE
thf(fact_532_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_533_length__0__conv,axiom,
! [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_a ) ) ).
% length_0_conv
thf(fact_534_p_Oconst__term__simprules__shell_I3_J,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q3 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 ) ) ) ) ) ) ).
% p.const_term_simprules_shell(3)
thf(fact_535_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_536_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_537_p_Onormalize_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ~ ! [V: list_a,Va: list_list_a] :
( X
!= ( cons_list_a @ V @ Va ) ) ) ).
% p.normalize.cases
thf(fact_538_p_Oconst__term__not__zero,axiom,
! [P: list_list_a] :
( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P != nil_list_a ) ) ).
% p.const_term_not_zero
thf(fact_539_p_Oconst__term__simprules__shell_I1_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ K ) ) ) ).
% p.const_term_simprules_shell(1)
thf(fact_540_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P2: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_541_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P2: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y3: list_a,Ys2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_542_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P2: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_543_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_list_a,P2: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P2 @ nil_list_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: list_a,Ys2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P2 @ Xs2 @ Ys2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_544_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P2: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z2: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_545_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,P2: list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z2: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_546_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,P2: list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: list_a,Ys2: list_list_a,Z2: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_547_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,P2: list_a > list_list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y3: list_a,Ys2: list_list_a,Z2: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_548_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,P2: list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a,Z2: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_549_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,P2: list_list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a,Z2: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_550_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,P2: list_list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: list_a,Ys2: list_list_a,Z2: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_551_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,P2: list_list_a > list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( P2 @ nil_list_a @ nil_list_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: list_a,Ys2: list_list_a,Z2: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_552_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_553_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P2: list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z2: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_554_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P2: list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z2: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_555_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_a,P2: list_a > list_list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: list_a,Ys2: list_list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_556_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_a,P2: list_list_a > list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_557_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_list_a,P2: list_a > list_a > list_list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y3: a,Ys2: list_a,Z2: list_a,Zs2: list_list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_558_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_list_a,P2: list_a > list_list_a > list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y3: list_a,Ys2: list_list_a,Z2: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_559_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,Ws: list_a,P2: list_a > list_list_a > list_list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( size_s349497388124573686list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_a @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y3: list_a,Ys2: list_list_a,Z2: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y3 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_560_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P2: list_list_a > list_a > list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_s349497388124573686list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a,Z2: a,Zs2: list_a,W: list_a,Ws2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_list_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_561_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P2: list_list_a > list_a > list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_s349497388124573686list_a @ Zs ) )
=> ( ( ( size_s349497388124573686list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P2 @ nil_list_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y3: a,Ys2: list_a,Z2: list_a,Zs2: list_list_a,W: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P2 @ Xs2 @ Ys2 @ Zs2 @ Ws2 )
=> ( P2 @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y3 @ Ys2 ) @ ( cons_list_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_562_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_563_impossible__Cons,axiom,
! [Xs: list_list_a,Ys: list_list_a,X: list_a] :
( ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs
!= ( cons_list_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_564_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_565_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_a] :
( ( size_s349497388124573686list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_566_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_567_neq__if__length__neq,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
!= ( size_s349497388124573686list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_568_length__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys3: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_569_length__induct,axiom,
! [P2: list_list_a > $o,Xs: list_list_a] :
( ! [Xs2: list_list_a] :
( ! [Ys3: list_list_a] :
( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ys3 ) @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_570_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_571_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_572_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_573_list_Osize_I3_J,axiom,
( ( size_s349497388124573686list_a @ nil_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_574_p_Oalg__multE_I2_J,axiom,
! [X: list_a,P: list_list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ nil_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) ) ) ) ) ) ).
% p.alg_multE(2)
thf(fact_575_p_Ole__alg__mult__imp__pdivides,axiom,
! [X: list_a,P: list_list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ nil_list_a ) ) @ N ) @ P ) ) ) ) ).
% p.le_alg_mult_imp_pdivides
thf(fact_576_p_Oalg__multE_I1_J,axiom,
! [X: list_a,P: list_list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P != nil_list_a )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) ) @ P ) ) ) ) ).
% p.alg_multE(1)
thf(fact_577_p_Ois__root__imp__pdivides,axiom,
! [P: list_list_a,X: list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ nil_list_a ) ) @ P ) ) ) ).
% p.is_root_imp_pdivides
thf(fact_578_p_Opirreducible__degree,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% p.pirreducible_degree
thf(fact_579_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_580_p_Osubring__props_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.subring_props(1)
thf(fact_581_p_Ozero__not__one,axiom,
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.zero_not_one
thf(fact_582_p_Osubring__props_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).
% p.subring_props(2)
thf(fact_583_p_Osubring__props_I7_J,axiom,
! [K: set_list_a,H1: list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H2 @ K )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 ) @ K ) ) ) ) ).
% p.subring_props(7)
thf(fact_584_p_Osubring__props_I3_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).
% p.subring_props(3)
thf(fact_585_p_Osubring__props_I5_J,axiom,
! [K: set_list_a,H: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H @ K )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ K ) ) ) ).
% p.subring_props(5)
thf(fact_586_monic__poly__one,axiom,
monic_3145109188698636716ly_a_b @ r @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monic_poly_one
thf(fact_587_pderiv__var,axiom,
! [K: set_a] :
( ( formal4452980811800949548iv_a_b @ r @ ( var_a_b @ r ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% pderiv_var
thf(fact_588_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_589_var__pow__eq__one__iff,axiom,
! [K3: nat] :
( ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ K3 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( K3 = zero_zero_nat ) ) ).
% var_pow_eq_one_iff
thf(fact_590_p_Omonic__degree__one__root__condition,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) @ nil_list_a ) ) @ B )
= ( A = B ) ) ) ).
% p.monic_degree_one_root_condition
thf(fact_591_p_Odegree__one__imp__pirreducible,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) ) ) ) ).
% p.degree_one_imp_pirreducible
thf(fact_592_p_Opdivides__imp__is__root,axiom,
! [P: list_list_a,X: list_a] :
( ( P != nil_list_a )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ nil_list_a ) ) @ P )
=> ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) ) ) ) ).
% p.pdivides_imp_is_root
thf(fact_593_p_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.one_closed
thf(fact_594_p_Onat__pow__one,axiom,
! [N: nat] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.nat_pow_one
thf(fact_595_p_Onat__pow__0,axiom,
! [X: list_a] :
( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ zero_zero_nat )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.nat_pow_0
thf(fact_596_univ__poly__one,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a] :
( ( one_li8923720976704309949t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) )
= ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ nil_list_list_a ) ) ).
% univ_poly_one
thf(fact_597_univ__poly__one,axiom,
! [R: partia4960592913263135132t_unit,K: set_set_list_list_a] :
( ( one_li3223383766543049405t_unit @ ( univ_p7077926387201515752t_unit @ R @ K ) )
= ( cons_set_list_list_a @ ( one_se2489417650821308733t_unit @ R ) @ nil_set_list_list_a ) ) ).
% univ_poly_one
thf(fact_598_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_599_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_600_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_601_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_602_var__def,axiom,
( var_li3532061862469730199t_unit
= ( ^ [R3: partia2956882679547061052t_unit] : ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R3 ) @ ( cons_list_list_a @ ( zero_l347298301471573063t_unit @ R3 ) @ nil_list_list_a ) ) ) ) ).
% var_def
thf(fact_603_var__def,axiom,
( var_se2996050386653789495t_unit
= ( ^ [R3: partia4960592913263135132t_unit] : ( cons_set_list_list_a @ ( one_se2489417650821308733t_unit @ R3 ) @ ( cons_set_list_list_a @ ( zero_s2920163772466840039t_unit @ R3 ) @ nil_set_list_list_a ) ) ) ) ).
% var_def
thf(fact_604_field_Omonic__poly__one,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( monic_3395465470813675732t_unit @ R @ ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_605_field_Omonic__poly__one,axiom,
! [R: partia4960592913263135132t_unit] :
( ( field_1540243473349940225t_unit @ R )
=> ( monic_8143709425247463374t_unit @ R @ ( one_li3223383766543049405t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_606_field_Omonic__poly__one,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( monic_5008461317928820916t_unit @ R @ ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_607_field_Omonic__poly__one,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( monic_5986596350207772206t_unit @ R @ ( one_li8923720976704309949t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_608_field_Omonic__poly__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( monic_3145109188698636716ly_a_b @ R @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_609_field_Ovar__pow__eq__one__iff,axiom,
! [R: partia7496981018696276118t_unit,K3: nat] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( var_se6008125447796440765t_unit @ R ) @ K3 )
= ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
= ( K3 = zero_zero_nat ) ) ) ).
% field.var_pow_eq_one_iff
thf(fact_610_field_Ovar__pow__eq__one__iff,axiom,
! [R: partia4960592913263135132t_unit,K3: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( ( pow_li5711373720449609902it_nat @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ ( var_se2996050386653789495t_unit @ R ) @ K3 )
= ( one_li3223383766543049405t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
= ( K3 = zero_zero_nat ) ) ) ).
% field.var_pow_eq_one_iff
thf(fact_611_field_Ovar__pow__eq__one__iff,axiom,
! [R: partia2175431115845679010xt_a_b,K3: nat] :
( ( field_a_b @ R )
=> ( ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( var_a_b @ R ) @ K3 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
= ( K3 = zero_zero_nat ) ) ) ).
% field.var_pow_eq_one_iff
thf(fact_612_field_Ovar__pow__eq__one__iff,axiom,
! [R: partia2670972154091845814t_unit,K3: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( var_li8453953174693405341t_unit @ R ) @ K3 )
= ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
= ( K3 = zero_zero_nat ) ) ) ).
% field.var_pow_eq_one_iff
thf(fact_613_field_Ovar__pow__eq__one__iff,axiom,
! [R: partia2956882679547061052t_unit,K3: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( var_li3532061862469730199t_unit @ R ) @ K3 )
= ( one_li8923720976704309949t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
= ( K3 = zero_zero_nat ) ) ) ).
% field.var_pow_eq_one_iff
thf(fact_614_field_Odegree__one__monic__poly,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ( monic_2059080652700942750t_unit @ R @ F )
& ( ( minus_minus_nat @ ( size_s1991367317912710102list_a @ F ) @ one_one_nat )
= one_one_nat ) )
= ( ? [X2: set_list_a] :
( ( member_set_list_a @ X2 @ ( partia141011252114345353t_unit @ R ) )
& ( F
= ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ X2 ) @ nil_set_list_a ) ) ) ) ) ) ) ).
% field.degree_one_monic_poly
thf(fact_615_field_Odegree__one__monic__poly,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( ( monic_819715999873801112t_unit @ R @ F )
& ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ F ) @ one_one_nat )
= one_one_nat ) )
= ( ? [X2: set_list_list_a] :
( ( member334759470184282131list_a @ X2 @ ( partia3317168157747563407t_unit @ R ) )
& ( F
= ( cons_set_list_list_a @ ( one_se2489417650821308733t_unit @ R ) @ ( cons_set_list_list_a @ ( a_inv_6360815108636782831t_unit @ R @ X2 ) @ nil_set_list_list_a ) ) ) ) ) ) ) ).
% field.degree_one_monic_poly
thf(fact_616_field_Odegree__one__monic__poly,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ( monic_104106837769529726t_unit @ R @ F )
& ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat )
= one_one_nat ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
& ( F
= ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ X2 ) @ nil_list_a ) ) ) ) ) ) ) ).
% field.degree_one_monic_poly
thf(fact_617_field_Odegree__one__monic__poly,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ( monic_868474719114584568t_unit @ R @ F )
& ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ F ) @ one_one_nat )
= one_one_nat ) )
= ( ? [X2: list_list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
& ( F
= ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ X2 ) @ nil_list_list_a ) ) ) ) ) ) ) ).
% field.degree_one_monic_poly
thf(fact_618_field_Odegree__one__monic__poly,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a] :
( ( field_a_b @ R )
=> ( ( ( monic_4919232885364369782ly_a_b @ R @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= one_one_nat ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( F
= ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ ( cons_a @ ( a_inv_a_b @ R @ X2 ) @ nil_a ) ) ) ) ) ) ) ).
% field.degree_one_monic_poly
thf(fact_619_p_Ouniv__poly__is__principal,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).
% p.univ_poly_is_principal
thf(fact_620_p_Oexists__unique__long__division,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( Q3 != nil_list_a )
=> ? [X3: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 @ X3 )
& ! [Y4: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 @ Y4 )
=> ( Y4 = X3 ) ) ) ) ) ) ) ).
% p.exists_unique_long_division
thf(fact_621_p_Opprime__iff__pirreducible,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) ) ) ) ).
% p.pprime_iff_pirreducible
thf(fact_622_p_Opmod__const_I1_J,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q3 ) @ one_one_nat ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 )
= nil_list_a ) ) ) ) ) ).
% p.pmod_const(1)
thf(fact_623_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_624_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_625_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_626_p_Opderiv__var,axiom,
! [K: set_list_a] :
( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).
% p.pderiv_var
thf(fact_627_pdivides__imp__is__root,axiom,
! [P: list_a,X: a] :
( ( P != nil_a )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_628_p_Ouniv__poly__a__inv__consistent,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P ) ) ) ) ).
% p.univ_poly_a_inv_consistent
thf(fact_629_p_Ouniv__poly__a__inv__length,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) )
= ( size_s349497388124573686list_a @ P ) ) ) ) ).
% p.univ_poly_a_inv_length
thf(fact_630_p_Opderiv__inv,axiom,
! [K: set_list_a,F: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ F ) )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) ) ) ) ) ).
% p.pderiv_inv
thf(fact_631_p_Olong__division__a__inv_I1_J,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) @ Q3 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) ) ) ) ) ) ).
% p.long_division_a_inv(1)
thf(fact_632_p_Olong__division__closed_I1_J,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ) ) ).
% p.long_division_closed(1)
thf(fact_633_p_Odegree__one,axiom,
! [K: set_list_a] :
( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) @ one_one_nat )
= zero_zero_nat ) ).
% p.degree_one
thf(fact_634_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 )
=> ~ ! [A4: a] :
( ( member_a @ A4 @ K )
=> ( ( A4
!= ( zero_a_b @ r ) )
=> ! [B3: a] :
( ( member_a @ B3 @ K )
=> ( P
!= ( cons_a @ A4 @ ( cons_a @ B3 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_635_p_Oconst__term__simprules__shell_I4_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ) ).
% p.const_term_simprules_shell(4)
thf(fact_636_p_Olong__division__zero_I1_J,axiom,
! [K: set_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Q3 )
= nil_list_a ) ) ) ).
% p.long_division_zero(1)
thf(fact_637_is__root__imp__pdivides,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P ) ) ) ).
% is_root_imp_pdivides
thf(fact_638_degree__one__monic__poly,axiom,
! [F: list_a] :
( ( ( monic_4919232885364369782ly_a_b @ r @ F )
& ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
= one_one_nat ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( F
= ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X2 ) @ nil_a ) ) ) ) ) ) ).
% degree_one_monic_poly
thf(fact_639_p_Olong__division__add_I1_J,axiom,
! [K: set_list_a,A: list_list_a,B: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ A @ B ) @ Q3 )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q3 ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q3 ) ) ) ) ) ) ) ).
% p.long_division_add(1)
thf(fact_640_p_OpprimeE_I1_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ).
% p.pprimeE(1)
thf(fact_641_p_Ouniv__poly__a__inv__degree,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) ) ) ) ).
% p.univ_poly_a_inv_degree
thf(fact_642_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_643_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_644_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_645_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_646_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_647_alg__multE_I1_J,axiom,
! [X: a,P: list_a] :
( ( member_a @ X @ ( 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 @ X ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) ) @ P ) ) ) ) ).
% alg_multE(1)
thf(fact_648_alg__multE_I2_J,axiom,
! [X: a,P: list_a,N: nat] :
( ( member_a @ X @ ( 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 @ X ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_649_le__alg__mult__imp__pdivides,axiom,
! [X: a,P: list_a,N: nat] :
( ( member_a @ X @ ( 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 @ X ) )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N ) @ P ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_650_ring_Olong__divides_Ocong,axiom,
polyno6947042923167803568t_unit = polyno6947042923167803568t_unit ).
% ring.long_divides.cong
thf(fact_651_ring_Olong__divides_Ocong,axiom,
polyno2806191415236617128es_a_b = polyno2806191415236617128es_a_b ).
% ring.long_divides.cong
thf(fact_652_ring_Opdiv_Ocong,axiom,
polyno5893782122288709345t_unit = polyno5893782122288709345t_unit ).
% ring.pdiv.cong
thf(fact_653_p_Orupture__one__not__zero,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) )
=> ( ( one_se2489417650821308733t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P ) )
!= ( zero_s2920163772466840039t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P ) ) ) ) ) ) ).
% p.rupture_one_not_zero
thf(fact_654_p_Oroots__inclI,axiom,
! [P: list_list_a,Q3: list_list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( member_list_list_a @ Q3 @ ( 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 ) ) ) ) ) )
=> ( ( Q3 != nil_list_a )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_list_a )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A4 ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A4 ) ) @ Q3 ) ) )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 ) ) ) ) ) ) ).
% p.roots_inclI
thf(fact_655_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_656_gauss__poly__factor,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( card_I2373409586816755191ly_a_b @ r @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( var_a_b @ r ) ) ) ) ).
% gauss_poly_factor
thf(fact_657_p_Om__assoc,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% p.m_assoc
thf(fact_658_p_Om__comm,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) ) ) ) ).
% p.m_comm
thf(fact_659_p_Om__lcomm,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) ) ) ) ) ) ).
% p.m_lcomm
thf(fact_660_p_Osubring__props_I6_J,axiom,
! [K: set_list_a,H1: list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H2 @ K )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 ) @ K ) ) ) ) ).
% p.subring_props(6)
thf(fact_661_monic__poly__mult,axiom,
! [F: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G )
=> ( monic_3145109188698636716ly_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ G ) ) ) ) ).
% monic_poly_mult
thf(fact_662_p_Ointegral,axiom,
! [A: list_a,B: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( B
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.integral
thf(fact_663_p_Ointegral__iff,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( B
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.integral_iff
thf(fact_664_p_Om__lcancel,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( A
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% p.m_lcancel
thf(fact_665_p_Om__rcancel,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( A
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% p.m_rcancel
thf(fact_666_p_Ol__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% p.l_distr
thf(fact_667_p_Or__distr,axiom,
! [X: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ Y ) ) ) ) ) ) ).
% p.r_distr
thf(fact_668_p_Oinv__unique,axiom,
! [Y: list_a,X: list_a,Y2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% p.inv_unique
thf(fact_669_p_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.one_unique
thf(fact_670_p_Ol__minus,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X ) @ Y )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) ) ) ) ).
% p.l_minus
thf(fact_671_p_Or__minus,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) ) ) ) ) ).
% p.r_minus
thf(fact_672_p_Ogroup__commutes__pow,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) ) ) ) ) ).
% p.group_commutes_pow
thf(fact_673_p_Onat__pow__comm,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) ) ) ).
% p.nat_pow_comm
thf(fact_674_p_Onat__pow__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N ) ) ) ) ) ).
% p.nat_pow_distrib
thf(fact_675_p_Opow__mult__distrib,axiom,
! [X: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X ) )
=> ( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N ) ) ) ) ) ) ).
% p.pow_mult_distrib
thf(fact_676_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q3: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q3 @ ( 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 @ Q3 ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q3 @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_677_p_Ocring__fieldI2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A4
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A4 @ X4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.cring_fieldI2
thf(fact_678_p_Osquare__eq__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ X )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( X
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.square_eq_one
thf(fact_679_pderiv__mult,axiom,
! [F: list_a,G: list_a] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( formal4452980811800949548iv_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ G ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( formal4452980811800949548iv_a_b @ r @ F ) @ G ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( formal4452980811800949548iv_a_b @ r @ G ) ) ) ) ) ) ).
% pderiv_mult
thf(fact_680_subset__mset_Ole__zero__eq,axiom,
! [N: multiset_list_a] :
( ( subseteq_mset_list_a @ N @ zero_z4454100511807792257list_a )
= ( N = zero_z4454100511807792257list_a ) ) ).
% subset_mset.le_zero_eq
thf(fact_681_subset__mset_Ole__zero__eq,axiom,
! [N: multiset_a] :
( ( subseteq_mset_a @ N @ zero_zero_multiset_a )
= ( N = zero_zero_multiset_a ) ) ).
% subset_mset.le_zero_eq
thf(fact_682_subset__mset_Oextremum__unique,axiom,
! [A: multiset_list_a] :
( ( subseteq_mset_list_a @ A @ zero_z4454100511807792257list_a )
= ( A = zero_z4454100511807792257list_a ) ) ).
% subset_mset.extremum_unique
thf(fact_683_subset__mset_Oextremum__unique,axiom,
! [A: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
= ( A = zero_zero_multiset_a ) ) ).
% subset_mset.extremum_unique
thf(fact_684_no__roots__imp__same__roots,axiom,
! [P: list_a,Q3: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( polynomial_roots_a_b @ r @ P )
= zero_zero_multiset_a )
=> ( ( polynomial_roots_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) )
= ( polynomial_roots_a_b @ r @ Q3 ) ) ) ) ) ) ).
% no_roots_imp_same_roots
thf(fact_685_p_Orupture__is__field__iff__pirreducible,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( field_1540243473349940225t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P ) )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) ) ) ) ).
% p.rupture_is_field_iff_pirreducible
thf(fact_686_p_OsubringI,axiom,
! [H3: set_list_a] :
( ( ord_le8861187494160871172list_a @ H3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ H3 )
=> ( ! [H4: list_a] :
( ( member_list_a @ H4 @ H3 )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H4 ) @ H3 ) )
=> ( ! [H12: list_a,H22: list_a] :
( ( member_list_a @ H12 @ H3 )
=> ( ( member_list_a @ H22 @ H3 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H22 ) @ H3 ) ) )
=> ( ! [H12: list_a,H22: list_a] :
( ( member_list_a @ H12 @ H3 )
=> ( ( member_list_a @ H22 @ H3 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H12 @ H22 ) @ H3 ) ) )
=> ( subrin6918843898125473962t_unit @ H3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.subringI
thf(fact_687_p_Opdivides__imp__roots__incl,axiom,
! [P: list_list_a,Q3: list_list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( member_list_list_a @ Q3 @ ( 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 ) ) ) ) ) )
=> ( ( Q3 != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 ) ) ) ) ) ) ).
% p.pdivides_imp_roots_incl
thf(fact_688_p_Om__closed,axiom,
! [X: list_a,Y: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.m_closed
thf(fact_689_p_Ol__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.l_null
thf(fact_690_p_Or__null,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.r_null
thf(fact_691_p_Ol__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X )
= X ) ) ).
% p.l_one
thf(fact_692_p_Or__one,axiom,
! [X: list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X ) ) ).
% p.r_one
thf(fact_693_diff__subset__eq__self,axiom,
! [M4: multiset_list_a,N4: multiset_list_a] : ( subseteq_mset_list_a @ ( minus_7431248565939055793list_a @ M4 @ N4 ) @ M4 ) ).
% diff_subset_eq_self
thf(fact_694_diff__subset__eq__self,axiom,
! [M4: multiset_a,N4: multiset_a] : ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ M4 @ N4 ) @ M4 ) ).
% diff_subset_eq_self
thf(fact_695_empty__le,axiom,
! [A2: multiset_list_a] : ( subseteq_mset_list_a @ zero_z4454100511807792257list_a @ A2 ) ).
% empty_le
thf(fact_696_empty__le,axiom,
! [A2: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A2 ) ).
% empty_le
thf(fact_697_subset__mset_Oextremum__uniqueI,axiom,
! [A: multiset_list_a] :
( ( subseteq_mset_list_a @ A @ zero_z4454100511807792257list_a )
=> ( A = zero_z4454100511807792257list_a ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_698_subset__mset_Oextremum__uniqueI,axiom,
! [A: multiset_a] :
( ( subseteq_mset_a @ A @ zero_zero_multiset_a )
=> ( A = zero_zero_multiset_a ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_699_subset__mset_Obot__least,axiom,
! [A: multiset_list_a] : ( subseteq_mset_list_a @ zero_z4454100511807792257list_a @ A ) ).
% subset_mset.bot_least
thf(fact_700_subset__mset_Obot__least,axiom,
! [A: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A ) ).
% subset_mset.bot_least
thf(fact_701_subset__mset_Ozero__le,axiom,
! [X: multiset_list_a] : ( subseteq_mset_list_a @ zero_z4454100511807792257list_a @ X ) ).
% subset_mset.zero_le
thf(fact_702_subset__mset_Ozero__le,axiom,
! [X: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ X ) ).
% subset_mset.zero_le
thf(fact_703_Diff__eq__empty__iff__mset,axiom,
! [A2: multiset_list_a,B4: multiset_list_a] :
( ( ( minus_7431248565939055793list_a @ A2 @ B4 )
= zero_z4454100511807792257list_a )
= ( subseteq_mset_list_a @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_704_Diff__eq__empty__iff__mset,axiom,
! [A2: multiset_a,B4: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ A2 @ B4 )
= zero_zero_multiset_a )
= ( subseteq_mset_a @ A2 @ B4 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_705_size__mset__mono,axiom,
! [A2: multiset_list_a,B4: multiset_list_a] :
( ( subseteq_mset_list_a @ A2 @ B4 )
=> ( ord_less_eq_nat @ ( size_s2335926164413107382list_a @ A2 ) @ ( size_s2335926164413107382list_a @ B4 ) ) ) ).
% size_mset_mono
thf(fact_706_size__mset__mono,axiom,
! [A2: multiset_a,B4: multiset_a] :
( ( subseteq_mset_a @ A2 @ B4 )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ A2 ) @ ( size_size_multiset_a @ B4 ) ) ) ).
% size_mset_mono
thf(fact_707_size__Diff__submset,axiom,
! [M4: multiset_list_a,M5: multiset_list_a] :
( ( subseteq_mset_list_a @ M4 @ M5 )
=> ( ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M5 @ M4 ) )
= ( minus_minus_nat @ ( size_s2335926164413107382list_a @ M5 ) @ ( size_s2335926164413107382list_a @ M4 ) ) ) ) ).
% size_Diff_submset
thf(fact_708_size__Diff__submset,axiom,
! [M4: multiset_a,M5: multiset_a] :
( ( subseteq_mset_a @ M4 @ M5 )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ M4 ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M5 ) @ ( size_size_multiset_a @ M4 ) ) ) ) ).
% size_Diff_submset
thf(fact_709_field_Omonic__poly__mult,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a,G: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_3395465470813675732t_unit @ R @ F )
=> ( ( monic_3395465470813675732t_unit @ R @ G )
=> ( monic_3395465470813675732t_unit @ R @ ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_710_field_Omonic__poly__mult,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a,G: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_8143709425247463374t_unit @ R @ F )
=> ( ( monic_8143709425247463374t_unit @ R @ G )
=> ( monic_8143709425247463374t_unit @ R @ ( mult_l7436655221470123345t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_711_field_Omonic__poly__mult,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a,G: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_5008461317928820916t_unit @ R @ F )
=> ( ( monic_5008461317928820916t_unit @ R @ G )
=> ( monic_5008461317928820916t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_712_field_Omonic__poly__mult,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a,G: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_5986596350207772206t_unit @ R @ F )
=> ( ( monic_5986596350207772206t_unit @ R @ G )
=> ( monic_5986596350207772206t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_713_field_Omonic__poly__mult,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a,G: list_a] :
( ( field_a_b @ R )
=> ( ( monic_3145109188698636716ly_a_b @ R @ F )
=> ( ( monic_3145109188698636716ly_a_b @ R @ G )
=> ( monic_3145109188698636716ly_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_714_field_Ono__roots__imp__same__roots,axiom,
! [R: partia7496981018696276118t_unit,P: list_set_list_a,Q3: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( ( member5524387281408368019list_a @ Q3 @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( ( polyno4169377219242390531t_unit @ R @ P )
= zero_z7061913751530476641list_a )
=> ( ( polyno4169377219242390531t_unit @ R @ ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ P @ Q3 ) )
= ( polyno4169377219242390531t_unit @ R @ Q3 ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_715_field_Ono__roots__imp__same__roots,axiom,
! [R: partia4960592913263135132t_unit,P: list_set_list_list_a,Q3: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( P != nil_set_list_list_a )
=> ( ( member6124916891863447321list_a @ Q3 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( ( polyno2127442156181624701t_unit @ R @ P )
= zero_z6145066983645916903list_a )
=> ( ( polyno2127442156181624701t_unit @ R @ ( mult_l7436655221470123345t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ P @ Q3 ) )
= ( polyno2127442156181624701t_unit @ R @ Q3 ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_716_field_Ono__roots__imp__same__roots,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q3: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( member5342144027231129785list_a @ Q3 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ( polyno3707469075594375645t_unit @ R @ P )
= zero_z1542645121299710087list_a )
=> ( ( polyno3707469075594375645t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q3 ) )
= ( polyno3707469075594375645t_unit @ R @ Q3 ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_717_field_Ono__roots__imp__same__roots,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q3: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( polyno7858422826990252003t_unit @ R @ P )
= zero_z4454100511807792257list_a )
=> ( ( polyno7858422826990252003t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q3 ) )
= ( polyno7858422826990252003t_unit @ R @ Q3 ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_718_field_Ono__roots__imp__same__roots,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q3: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P != nil_a )
=> ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( polynomial_roots_a_b @ R @ P )
= zero_zero_multiset_a )
=> ( ( polynomial_roots_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q3 ) )
= ( polynomial_roots_a_b @ R @ Q3 ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_719_field_Ogauss__poly__factor,axiom,
! [R: partia7496981018696276118t_unit,N: nat] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( card_I5471341911849640095t_unit @ R @ N )
= ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( a_minu6874796375791416686t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( var_se6008125447796440765t_unit @ R ) @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) @ ( var_se6008125447796440765t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_factor
thf(fact_720_field_Ogauss__poly__factor,axiom,
! [R: partia4960592913263135132t_unit,N: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( card_I259811512781981209t_unit @ R @ N )
= ( mult_l7436655221470123345t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ ( a_minu1178922365601208552t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ ( pow_li5711373720449609902it_nat @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ ( var_se2996050386653789495t_unit @ R ) @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( one_li3223383766543049405t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) @ ( var_se2996050386653789495t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_factor
thf(fact_721_field_Ogauss__poly__factor,axiom,
! [R: partia2670972154091845814t_unit,N: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( card_I2619780863984422015t_unit @ R @ N )
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( var_li8453953174693405341t_unit @ R ) @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) @ ( var_li8453953174693405341t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_factor
thf(fact_722_field_Ogauss__poly__factor,axiom,
! [R: partia2956882679547061052t_unit,N: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( card_I3787608780883923065t_unit @ R @ N )
= ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( var_li3532061862469730199t_unit @ R ) @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( one_li8923720976704309949t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) @ ( var_li3532061862469730199t_unit @ R ) ) ) ) ) ).
% field.gauss_poly_factor
thf(fact_723_field_Ogauss__poly__factor,axiom,
! [R: partia2175431115845679010xt_a_b,N: nat] :
( ( field_a_b @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( card_I2373409586816755191ly_a_b @ R @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( var_a_b @ R ) @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) @ ( var_a_b @ R ) ) ) ) ) ).
% field.gauss_poly_factor
thf(fact_724_p_Orupture__char,axiom,
! [K: set_list_a,F: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat ) )
=> ( ( ring_c8395554250859618576t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ F ) )
= ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.rupture_char
thf(fact_725_p_Ounitary__monom__eq__var__pow,axiom,
! [K: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N )
= ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) ) ) ).
% p.unitary_monom_eq_var_pow
thf(fact_726_roots__inclI,axiom,
! [P: list_a,Q3: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q3 != nil_a )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( 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 @ A4 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ A4 ) ) @ Q3 ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q3 ) ) ) ) ) ) ).
% roots_inclI
thf(fact_727_p_Onot__empty__rootsE,axiom,
! [P: list_list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
!= zero_z4454100511807792257list_a )
=> ~ ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A4 @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) )
=> ( ( member_list_list_a @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A4 ) @ nil_list_a ) ) @ ( 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 ) ) ) ) ) )
=> ~ ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A4 ) @ nil_list_a ) ) @ P ) ) ) ) ) ) ).
% p.not_empty_rootsE
thf(fact_728_p_Ois__root__poly__mult__imp__is__root,axiom,
! [P: list_list_a,Q3: list_list_a,X: list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( member_list_list_a @ Q3 @ ( 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 ) ) ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ Q3 ) @ X )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X )
| ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 @ X ) ) ) ) ) ).
% p.is_root_poly_mult_imp_is_root
thf(fact_729_p_Oconst__term__simprules__shell_I2_J,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q3 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 ) ) ) ) ) ) ).
% p.const_term_simprules_shell(2)
thf(fact_730_p_Opderiv__mult,axiom,
! [K: set_list_a,F: list_list_a,G: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ G @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ F @ G ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) @ G ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ F @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G ) ) ) ) ) ) ) ).
% p.pderiv_mult
thf(fact_731_p_OpprimeE_I3_J,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a,R2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q3 @ R2 ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 )
| ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ R2 ) ) ) ) ) ) ) ) ).
% p.pprimeE(3)
thf(fact_732_pdivides__imp__roots__incl,axiom,
! [P: list_a,Q3: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q3 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q3 )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ ( polynomial_roots_a_b @ r @ Q3 ) ) ) ) ) ) ).
% pdivides_imp_roots_incl
thf(fact_733_p_Opirreducible__pow__pdivides__iff,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a,R2: list_list_a,N: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( ~ ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q3 @ R2 ) )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% p.pirreducible_pow_pdivides_iff
thf(fact_734_p_Oroots__mem__iff__is__root,axiom,
! [P: list_list_a,X: list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( member_list_a @ X @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) )
= ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X ) ) ) ).
% p.roots_mem_iff_is_root
thf(fact_735_p_Osubfield__long__division__theorem__shell,axiom,
! [K: set_list_a,P: list_list_a,B: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( B
!= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ? [Q2: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
& ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
& ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ B @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% p.subfield_long_division_theorem_shell
thf(fact_736_multiset__nonemptyE,axiom,
! [A2: multiset_list_list_a] :
( ( A2 != zero_z1542645121299710087list_a )
=> ~ ! [X3: list_list_a] :
~ ( member_list_list_a @ X3 @ ( set_mset_list_list_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_737_multiset__nonemptyE,axiom,
! [A2: multiset_list_a] :
( ( A2 != zero_z4454100511807792257list_a )
=> ~ ! [X3: list_a] :
~ ( member_list_a @ X3 @ ( set_mset_list_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_738_multiset__nonemptyE,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
=> ~ ! [X3: a] :
~ ( member_a @ X3 @ ( set_mset_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_739_in__diffD,axiom,
! [A: list_list_a,M4: multiset_list_list_a,N4: multiset_list_list_a] :
( ( member_list_list_a @ A @ ( set_mset_list_list_a @ ( minus_5831295526526677175list_a @ M4 @ N4 ) ) )
=> ( member_list_list_a @ A @ ( set_mset_list_list_a @ M4 ) ) ) ).
% in_diffD
thf(fact_740_in__diffD,axiom,
! [A: list_a,M4: multiset_list_a,N4: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M4 @ N4 ) ) )
=> ( member_list_a @ A @ ( set_mset_list_a @ M4 ) ) ) ).
% in_diffD
thf(fact_741_in__diffD,axiom,
! [A: a,M4: multiset_a,N4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M4 @ N4 ) ) )
=> ( member_a @ A @ ( set_mset_a @ M4 ) ) ) ).
% in_diffD
thf(fact_742_p_Olong__dividesI,axiom,
! [B: list_list_a,R2: list_list_a,P: list_list_a,Q3: list_list_a] :
( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ R2 @ ( 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 ) ) ) ) ) )
=> ( ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ Q3 @ B ) @ R2 ) )
=> ( ( ( R2 = nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q3 ) @ one_one_nat ) ) )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 @ ( produc8696003437204565271list_a @ B @ R2 ) ) ) ) ) ) ).
% p.long_dividesI
thf(fact_743_p_Opmod__degree,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( Q3 != nil_list_a )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 )
= nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q3 ) @ one_one_nat ) ) ) ) ) ) ) ).
% p.pmod_degree
thf(fact_744_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_745_p_Opoly__add_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
~ ! [P1: list_list_a,P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ P1 @ P22 ) ) ).
% p.poly_add.cases
thf(fact_746_p_Ocombine_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [K4: list_a,Ks: list_list_a,U2: list_a,Us: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ K4 @ Ks ) @ ( cons_list_a @ U2 @ Us ) ) )
=> ( ! [Us: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Us ) )
=> ~ ! [Ks: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).
% p.combine.cases
thf(fact_747_p_Opoly__mult_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
=> ~ ! [V: list_a,Va: list_list_a,P22: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ V @ Va ) @ P22 ) ) ) ).
% p.poly_mult.cases
thf(fact_748_roots__mem__iff__is__root,axiom,
! [P: list_a,X: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ X @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P ) ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X ) ) ) ).
% roots_mem_iff_is_root
thf(fact_749_p_Olong__division__closed_I2_J,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ) ) ).
% p.long_division_closed(2)
thf(fact_750_p_Olong__division__zero_I2_J,axiom,
! [K: set_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Q3 )
= nil_list_a ) ) ) ).
% p.long_division_zero(2)
thf(fact_751_p_Olong__division__add__iff,axiom,
! [K: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q3 )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q3 ) )
= ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ A @ C ) @ Q3 )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ B @ C ) @ Q3 ) ) ) ) ) ) ) ) ).
% p.long_division_add_iff
thf(fact_752_p_Olong__division__add_I2_J,axiom,
! [K: set_list_a,A: list_list_a,B: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ A @ B ) @ Q3 )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q3 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q3 ) ) ) ) ) ) ) ).
% p.long_division_add(2)
thf(fact_753_p_Olong__division__a__inv_I2_J,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) @ Q3 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) ) ) ) ) ) ).
% p.long_division_a_inv(2)
thf(fact_754_p_Opmod__zero__iff__pdivides,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 @ P ) ) ) ) ) ).
% p.pmod_zero_iff_pdivides
thf(fact_755_p_Oexists__long__division,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( Q3 != nil_list_a )
=> ~ ! [B3: list_list_a] :
( ( member_list_list_a @ B3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ! [R4: list_list_a] :
( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ~ ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 @ ( produc8696003437204565271list_a @ B3 @ R4 ) ) ) ) ) ) ) ) ).
% p.exists_long_division
thf(fact_756_p_Osame__pmod__iff__pdivides,axiom,
! [K: set_list_a,A: list_list_a,B: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q3 )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q3 ) )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ A @ B ) ) ) ) ) ) ) ).
% p.same_pmod_iff_pdivides
thf(fact_757_p_Opdiv__pmod,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q3 @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) ) ) ) ) ) ).
% p.pdiv_pmod
thf(fact_758_p_Opmod__const_I2_J,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q3 ) @ one_one_nat ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 )
= P ) ) ) ) ) ).
% p.pmod_const(2)
thf(fact_759_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 )
=> ~ ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A4 @ ( 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 @ A4 ) @ 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 @ A4 ) @ nil_a ) ) @ P ) ) ) ) ) ) ).
% not_empty_rootsE
thf(fact_760_p_Olong__divisionI,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a,B: list_list_a,R2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( Q3 != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 @ ( produc8696003437204565271list_a @ B @ R2 ) )
=> ( ( produc8696003437204565271list_a @ B @ R2 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) ) ) ) ) ) ) ) ).
% p.long_divisionI
thf(fact_761_p_Olong__divisionE,axiom,
! [K: set_list_a,P: list_list_a,Q3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( Q3 != nil_list_a )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 ) ) ) ) ) ) ) ).
% p.long_divisionE
thf(fact_762_ring_Opmod_Ocong,axiom,
polyno1727750685288865234t_unit = polyno1727750685288865234t_unit ).
% ring.pmod.cong
thf(fact_763_primeness__condition,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeness_condition
thf(fact_764_long__dividesI,axiom,
! [B: list_a,R2: list_a,P: list_a,Q3: 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 ) ) @ Q3 @ 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 @ Q3 ) @ one_one_nat ) ) )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q3 @ ( produc6837034575241423639list_a @ B @ R2 ) ) ) ) ) ) ).
% long_dividesI
thf(fact_765_p_Omonoid__cancelI,axiom,
( ! [A4: list_a,B3: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A4 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B3 ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A4 = B3 ) ) ) ) )
=> ( ! [A4: list_a,B3: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A4 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ C2 ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( A4 = B3 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.monoid_cancelI
thf(fact_766_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_767_combine_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [K4: a,Ks: list_a,U2: a,Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ K4 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_768_poly__mult_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V: a,Va: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_769_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_770_gauss__poly__div__gauss__poly__iff__1,axiom,
! [L: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ L ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ M ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ).
% gauss_poly_div_gauss_poly_iff_1
thf(fact_771_p_Opoly__mult__var,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( P = nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= nil_list_a ) )
& ( ( P != nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ) ).
% p.poly_mult_var
thf(fact_772_p_Odegree__one__roots,axiom,
! [A: list_a,A5: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A5 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) )
= ( add_mset_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ B ) ) @ zero_z4454100511807792257list_a ) ) ) ) ) ) ).
% p.degree_one_roots
thf(fact_773_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_774_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_775_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us2: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us2 )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us2 )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_776_append__eq__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
| ( ( size_s349497388124573686list_a @ Us2 )
= ( size_s349497388124573686list_a @ Vs ) ) )
=> ( ( ( append_list_a @ Xs @ Us2 )
= ( append_list_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us2 = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_777_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_778_add__mset__eq__singleton__iff,axiom,
! [X: list_a,M4: multiset_list_a,Y: list_a] :
( ( ( add_mset_list_a @ X @ M4 )
= ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) )
= ( ( M4 = zero_z4454100511807792257list_a )
& ( X = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_779_add__mset__eq__singleton__iff,axiom,
! [X: a,M4: multiset_a,Y: a] :
( ( ( add_mset_a @ X @ M4 )
= ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( ( M4 = zero_zero_multiset_a )
& ( X = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_780_single__eq__add__mset,axiom,
! [A: list_a,B: list_a,M4: multiset_list_a] :
( ( ( add_mset_list_a @ A @ zero_z4454100511807792257list_a )
= ( add_mset_list_a @ B @ M4 ) )
= ( ( B = A )
& ( M4 = zero_z4454100511807792257list_a ) ) ) ).
% single_eq_add_mset
thf(fact_781_single__eq__add__mset,axiom,
! [A: a,B: a,M4: multiset_a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( add_mset_a @ B @ M4 ) )
= ( ( B = A )
& ( M4 = zero_zero_multiset_a ) ) ) ).
% single_eq_add_mset
thf(fact_782_add__mset__eq__single,axiom,
! [B: list_a,M4: multiset_list_a,A: list_a] :
( ( ( add_mset_list_a @ B @ M4 )
= ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) )
= ( ( B = A )
& ( M4 = zero_z4454100511807792257list_a ) ) ) ).
% add_mset_eq_single
thf(fact_783_add__mset__eq__single,axiom,
! [B: a,M4: multiset_a,A: a] :
( ( ( add_mset_a @ B @ M4 )
= ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( ( B = A )
& ( M4 = zero_zero_multiset_a ) ) ) ).
% add_mset_eq_single
thf(fact_784_single__eq__single,axiom,
! [A: list_a,B: list_a] :
( ( ( add_mset_list_a @ A @ zero_z4454100511807792257list_a )
= ( add_mset_list_a @ B @ zero_z4454100511807792257list_a ) )
= ( A = B ) ) ).
% single_eq_single
thf(fact_785_single__eq__single,axiom,
! [A: a,B: a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( A = B ) ) ).
% single_eq_single
thf(fact_786_dvd__times__left__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_787_dvd__times__right__cancel__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( A != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_788_unit__prod,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% unit_prod
thf(fact_789_nat__mult__dvd__cancel__disj,axiom,
! [K3: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( ( K3 = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_790_add__mset__subseteq__single__iff,axiom,
! [A: list_a,M4: multiset_list_a,B: list_a] :
( ( subseteq_mset_list_a @ ( add_mset_list_a @ A @ M4 ) @ ( add_mset_list_a @ B @ zero_z4454100511807792257list_a ) )
= ( ( M4 = zero_z4454100511807792257list_a )
& ( A = B ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_791_add__mset__subseteq__single__iff,axiom,
! [A: a,M4: multiset_a,B: a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ M4 ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( ( M4 = zero_zero_multiset_a )
& ( A = B ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_792_add__mset__remove__trivial,axiom,
! [X: list_a,M4: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ X @ M4 ) @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) )
= M4 ) ).
% add_mset_remove_trivial
thf(fact_793_add__mset__remove__trivial,axiom,
! [X: a,M4: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X @ M4 ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M4 ) ).
% add_mset_remove_trivial
thf(fact_794_diff__add__mset__swap,axiom,
! [B: list_list_a,A2: multiset_list_list_a,M4: multiset_list_list_a] :
( ~ ( member_list_list_a @ B @ ( set_mset_list_list_a @ A2 ) )
=> ( ( minus_5831295526526677175list_a @ ( add_mset_list_list_a @ B @ M4 ) @ A2 )
= ( add_mset_list_list_a @ B @ ( minus_5831295526526677175list_a @ M4 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_795_diff__add__mset__swap,axiom,
! [B: list_a,A2: multiset_list_a,M4: multiset_list_a] :
( ~ ( member_list_a @ B @ ( set_mset_list_a @ A2 ) )
=> ( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ B @ M4 ) @ A2 )
= ( add_mset_list_a @ B @ ( minus_7431248565939055793list_a @ M4 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_796_diff__add__mset__swap,axiom,
! [B: a,A2: multiset_a,M4: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A2 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M4 ) @ A2 )
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M4 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_797_p_Omonic__degree__one__roots,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) @ nil_list_a ) ) )
= ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) ).
% p.monic_degree_one_roots
thf(fact_798_single__subset__iff,axiom,
! [A: list_list_a,M4: multiset_list_list_a] :
( ( subset8447756916971205105list_a @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) @ M4 )
= ( member_list_list_a @ A @ ( set_mset_list_list_a @ M4 ) ) ) ).
% single_subset_iff
thf(fact_799_single__subset__iff,axiom,
! [A: list_a,M4: multiset_list_a] :
( ( subseteq_mset_list_a @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) @ M4 )
= ( member_list_a @ A @ ( set_mset_list_a @ M4 ) ) ) ).
% single_subset_iff
thf(fact_800_single__subset__iff,axiom,
! [A: a,M4: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M4 )
= ( member_a @ A @ ( set_mset_a @ M4 ) ) ) ).
% single_subset_iff
thf(fact_801_diff__union__swap2,axiom,
! [Y: list_list_a,M4: multiset_list_list_a,X: list_list_a] :
( ( member_list_list_a @ Y @ ( set_mset_list_list_a @ M4 ) )
=> ( ( minus_5831295526526677175list_a @ ( add_mset_list_list_a @ X @ M4 ) @ ( add_mset_list_list_a @ Y @ zero_z1542645121299710087list_a ) )
= ( add_mset_list_list_a @ X @ ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ Y @ zero_z1542645121299710087list_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_802_diff__union__swap2,axiom,
! [Y: list_a,M4: multiset_list_a,X: list_a] :
( ( member_list_a @ Y @ ( set_mset_list_a @ M4 ) )
=> ( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ X @ M4 ) @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) )
= ( add_mset_list_a @ X @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_803_diff__union__swap2,axiom,
! [Y: a,M4: multiset_a,X: a] :
( ( member_a @ Y @ ( set_mset_a @ M4 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X @ M4 ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_804_insert__DiffM,axiom,
! [X: list_list_a,M4: multiset_list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M4 ) )
=> ( ( add_mset_list_list_a @ X @ ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) )
= M4 ) ) ).
% insert_DiffM
thf(fact_805_insert__DiffM,axiom,
! [X: list_a,M4: multiset_list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M4 ) )
=> ( ( add_mset_list_a @ X @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) )
= M4 ) ) ).
% insert_DiffM
thf(fact_806_insert__DiffM,axiom,
! [X: a,M4: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M4 ) )
=> ( ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= M4 ) ) ).
% insert_DiffM
thf(fact_807_add__mset__diff__bothsides,axiom,
! [A: list_a,M4: multiset_list_a,A2: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ A @ M4 ) @ ( add_mset_list_a @ A @ A2 ) )
= ( minus_7431248565939055793list_a @ M4 @ A2 ) ) ).
% add_mset_diff_bothsides
thf(fact_808_add__mset__diff__bothsides,axiom,
! [A: a,M4: multiset_a,A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ A @ M4 ) @ ( add_mset_a @ A @ A2 ) )
= ( minus_3765977307040488491iset_a @ M4 @ A2 ) ) ).
% add_mset_diff_bothsides
thf(fact_809_dvd__diff__nat,axiom,
! [K3: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K3 @ M )
=> ( ( dvd_dvd_nat @ K3 @ N )
=> ( dvd_dvd_nat @ K3 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_810_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_811_dvd__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_trans
thf(fact_812_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_813_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_814_dvd__mult__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ B @ C ) ) ).
% dvd_mult_right
thf(fact_815_mult__dvd__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ C @ D )
=> ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_816_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_817_dvd__mult__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ).
% dvd_mult_left
thf(fact_818_dvd__mult2,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_819_dvd__mult,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ C )
=> ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% dvd_mult
thf(fact_820_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B2: nat,A3: nat] :
? [K5: nat] :
( A3
= ( times_times_nat @ B2 @ K5 ) ) ) ) ).
% dvd_def
thf(fact_821_dvdI,axiom,
! [A: nat,B: nat,K3: nat] :
( ( A
= ( times_times_nat @ B @ K3 ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_822_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K4: nat] :
( A
!= ( times_times_nat @ B @ K4 ) ) ) ).
% dvdE
thf(fact_823_dvd__unit__imp__unit,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% dvd_unit_imp_unit
thf(fact_824_unit__imp__dvd,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_825_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_826_multi__nonempty__split,axiom,
! [M4: multiset_list_a] :
( ( M4 != zero_z4454100511807792257list_a )
=> ? [A6: multiset_list_a,A4: list_a] :
( M4
= ( add_mset_list_a @ A4 @ A6 ) ) ) ).
% multi_nonempty_split
thf(fact_827_multi__nonempty__split,axiom,
! [M4: multiset_a] :
( ( M4 != zero_zero_multiset_a )
=> ? [A6: multiset_a,A4: a] :
( M4
= ( add_mset_a @ A4 @ A6 ) ) ) ).
% multi_nonempty_split
thf(fact_828_empty__not__add__mset,axiom,
! [A: list_a,A2: multiset_list_a] :
( zero_z4454100511807792257list_a
!= ( add_mset_list_a @ A @ A2 ) ) ).
% empty_not_add_mset
thf(fact_829_empty__not__add__mset,axiom,
! [A: a,A2: multiset_a] :
( zero_zero_multiset_a
!= ( add_mset_a @ A @ A2 ) ) ).
% empty_not_add_mset
thf(fact_830_multiset__induct2,axiom,
! [P2: multiset_list_a > multiset_list_a > $o,M4: multiset_list_a,N4: multiset_list_a] :
( ( P2 @ zero_z4454100511807792257list_a @ zero_z4454100511807792257list_a )
=> ( ! [A4: list_a,M6: multiset_list_a,N5: multiset_list_a] :
( ( P2 @ M6 @ N5 )
=> ( P2 @ ( add_mset_list_a @ A4 @ M6 ) @ N5 ) )
=> ( ! [A4: list_a,M6: multiset_list_a,N5: multiset_list_a] :
( ( P2 @ M6 @ N5 )
=> ( P2 @ M6 @ ( add_mset_list_a @ A4 @ N5 ) ) )
=> ( P2 @ M4 @ N4 ) ) ) ) ).
% multiset_induct2
thf(fact_831_multiset__induct2,axiom,
! [P2: multiset_list_a > multiset_a > $o,M4: multiset_list_a,N4: multiset_a] :
( ( P2 @ zero_z4454100511807792257list_a @ zero_zero_multiset_a )
=> ( ! [A4: list_a,M6: multiset_list_a,N5: multiset_a] :
( ( P2 @ M6 @ N5 )
=> ( P2 @ ( add_mset_list_a @ A4 @ M6 ) @ N5 ) )
=> ( ! [A4: a,M6: multiset_list_a,N5: multiset_a] :
( ( P2 @ M6 @ N5 )
=> ( P2 @ M6 @ ( add_mset_a @ A4 @ N5 ) ) )
=> ( P2 @ M4 @ N4 ) ) ) ) ).
% multiset_induct2
thf(fact_832_multiset__induct2,axiom,
! [P2: multiset_a > multiset_list_a > $o,M4: multiset_a,N4: multiset_list_a] :
( ( P2 @ zero_zero_multiset_a @ zero_z4454100511807792257list_a )
=> ( ! [A4: a,M6: multiset_a,N5: multiset_list_a] :
( ( P2 @ M6 @ N5 )
=> ( P2 @ ( add_mset_a @ A4 @ M6 ) @ N5 ) )
=> ( ! [A4: list_a,M6: multiset_a,N5: multiset_list_a] :
( ( P2 @ M6 @ N5 )
=> ( P2 @ M6 @ ( add_mset_list_a @ A4 @ N5 ) ) )
=> ( P2 @ M4 @ N4 ) ) ) ) ).
% multiset_induct2
thf(fact_833_multiset__induct2,axiom,
! [P2: multiset_a > multiset_a > $o,M4: multiset_a,N4: multiset_a] :
( ( P2 @ zero_zero_multiset_a @ zero_zero_multiset_a )
=> ( ! [A4: a,M6: multiset_a,N5: multiset_a] :
( ( P2 @ M6 @ N5 )
=> ( P2 @ ( add_mset_a @ A4 @ M6 ) @ N5 ) )
=> ( ! [A4: a,M6: multiset_a,N5: multiset_a] :
( ( P2 @ M6 @ N5 )
=> ( P2 @ M6 @ ( add_mset_a @ A4 @ N5 ) ) )
=> ( P2 @ M4 @ N4 ) ) ) ) ).
% multiset_induct2
thf(fact_834_multiset__induct,axiom,
! [P2: multiset_list_a > $o,M4: multiset_list_a] :
( ( P2 @ zero_z4454100511807792257list_a )
=> ( ! [X3: list_a,M6: multiset_list_a] :
( ( P2 @ M6 )
=> ( P2 @ ( add_mset_list_a @ X3 @ M6 ) ) )
=> ( P2 @ M4 ) ) ) ).
% multiset_induct
thf(fact_835_multiset__induct,axiom,
! [P2: multiset_a > $o,M4: multiset_a] :
( ( P2 @ zero_zero_multiset_a )
=> ( ! [X3: a,M6: multiset_a] :
( ( P2 @ M6 )
=> ( P2 @ ( add_mset_a @ X3 @ M6 ) ) )
=> ( P2 @ M4 ) ) ) ).
% multiset_induct
thf(fact_836_multiset__cases,axiom,
! [M4: multiset_list_a] :
( ( M4 != zero_z4454100511807792257list_a )
=> ~ ! [X3: list_a,N5: multiset_list_a] :
( M4
!= ( add_mset_list_a @ X3 @ N5 ) ) ) ).
% multiset_cases
thf(fact_837_multiset__cases,axiom,
! [M4: multiset_a] :
( ( M4 != zero_zero_multiset_a )
=> ~ ! [X3: a,N5: multiset_a] :
( M4
!= ( add_mset_a @ X3 @ N5 ) ) ) ).
% multiset_cases
thf(fact_838_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_839_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_840_is__unit__mult__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
& ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% is_unit_mult_iff
thf(fact_841_dvd__mult__unit__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_842_mult__unit__dvd__iff,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_843_dvd__mult__unit__iff_H,axiom,
! [B: nat,A: nat,C: nat] :
( ( dvd_dvd_nat @ B @ one_one_nat )
=> ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
= ( dvd_dvd_nat @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_844_mult__unit__dvd__iff_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
= ( dvd_dvd_nat @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_845_unit__mult__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ A @ B )
= ( times_times_nat @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_846_unit__mult__right__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ( ( ( times_times_nat @ B @ A )
= ( times_times_nat @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_847_multi__member__last,axiom,
! [X: list_list_a] : ( member_list_list_a @ X @ ( set_mset_list_list_a @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) ) ).
% multi_member_last
thf(fact_848_multi__member__last,axiom,
! [X: list_a] : ( member_list_a @ X @ ( set_mset_list_a @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) ) ).
% multi_member_last
thf(fact_849_multi__member__last,axiom,
! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).
% multi_member_last
thf(fact_850_dvd__power__le,axiom,
! [X: nat,Y: nat,N: nat,M: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_851_power__le__dvd,axiom,
! [A: nat,N: nat,B: nat,M: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_852_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_853_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_854_diff__union__swap,axiom,
! [A: list_a,B: list_a,M4: multiset_list_a] :
( ( A != B )
=> ( ( add_mset_list_a @ B @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) )
= ( minus_7431248565939055793list_a @ ( add_mset_list_a @ B @ M4 ) @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) ) ).
% diff_union_swap
thf(fact_855_diff__union__swap,axiom,
! [A: a,B: a,M4: multiset_a] :
( ( A != B )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M4 ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ).
% diff_union_swap
thf(fact_856_add__eq__conv__diff,axiom,
! [A: list_a,M4: multiset_list_a,B: list_a,N4: multiset_list_a] :
( ( ( add_mset_list_a @ A @ M4 )
= ( add_mset_list_a @ B @ N4 ) )
= ( ( ( M4 = N4 )
& ( A = B ) )
| ( ( M4
= ( add_mset_list_a @ B @ ( minus_7431248565939055793list_a @ N4 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) )
& ( N4
= ( add_mset_list_a @ A @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ B @ zero_z4454100511807792257list_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_857_add__eq__conv__diff,axiom,
! [A: a,M4: multiset_a,B: a,N4: multiset_a] :
( ( ( add_mset_a @ A @ M4 )
= ( add_mset_a @ B @ N4 ) )
= ( ( ( M4 = N4 )
& ( A = B ) )
| ( ( M4
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
& ( N4
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_858_union__single__eq__diff,axiom,
! [X: list_a,M4: multiset_list_a,N4: multiset_list_a] :
( ( ( add_mset_list_a @ X @ M4 )
= N4 )
=> ( M4
= ( minus_7431248565939055793list_a @ N4 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_859_union__single__eq__diff,axiom,
! [X: a,M4: multiset_a,N4: multiset_a] :
( ( ( add_mset_a @ X @ M4 )
= N4 )
=> ( M4
= ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_860_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_861_dvd__diffD,axiom,
! [K3: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K3 @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K3 @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K3 @ M ) ) ) ) ).
% dvd_diffD
thf(fact_862_dvd__diffD1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K3 @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K3 @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K3 @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_863_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_864_multiset__induct__max,axiom,
! [P2: multiset_nat > $o,M4: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X3: nat,M6: multiset_nat] :
( ( P2 @ M6 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M6 ) )
=> ( ord_less_eq_nat @ Xa @ X3 ) )
=> ( P2 @ ( add_mset_nat @ X3 @ M6 ) ) ) )
=> ( P2 @ M4 ) ) ) ).
% multiset_induct_max
thf(fact_865_multiset__induct__min,axiom,
! [P2: multiset_nat > $o,M4: multiset_nat] :
( ( P2 @ zero_z7348594199698428585et_nat )
=> ( ! [X3: nat,M6: multiset_nat] :
( ( P2 @ M6 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M6 ) )
=> ( ord_less_eq_nat @ X3 @ Xa ) )
=> ( P2 @ ( add_mset_nat @ X3 @ M6 ) ) ) )
=> ( P2 @ M4 ) ) ) ).
% multiset_induct_min
thf(fact_866_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X3: a,Xs3: list_a,Y3: a,Ys4: list_a] :
( ( X3 != Y3 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_867_same__length__different,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != Ys )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ? [Pre: list_list_a,X3: list_a,Xs3: list_list_a,Y3: list_a,Ys4: list_list_a] :
( ( X3 != Y3 )
& ( Xs
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ X3 @ nil_list_a ) @ Xs3 ) ) )
& ( Ys
= ( append_list_a @ Pre @ ( append_list_a @ ( cons_list_a @ Y3 @ nil_list_a ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_868_unit__dvdE,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ one_one_nat )
=> ~ ( ( A != zero_zero_nat )
=> ! [C2: nat] :
( B
!= ( times_times_nat @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_869_multi__subset__induct,axiom,
! [F3: multiset_list_list_a,A2: multiset_list_list_a,P2: multiset_list_list_a > $o] :
( ( subset8447756916971205105list_a @ F3 @ A2 )
=> ( ( P2 @ zero_z1542645121299710087list_a )
=> ( ! [A4: list_list_a,F4: multiset_list_list_a] :
( ( member_list_list_a @ A4 @ ( set_mset_list_list_a @ A2 ) )
=> ( ( P2 @ F4 )
=> ( P2 @ ( add_mset_list_list_a @ A4 @ F4 ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% multi_subset_induct
thf(fact_870_multi__subset__induct,axiom,
! [F3: multiset_list_a,A2: multiset_list_a,P2: multiset_list_a > $o] :
( ( subseteq_mset_list_a @ F3 @ A2 )
=> ( ( P2 @ zero_z4454100511807792257list_a )
=> ( ! [A4: list_a,F4: multiset_list_a] :
( ( member_list_a @ A4 @ ( set_mset_list_a @ A2 ) )
=> ( ( P2 @ F4 )
=> ( P2 @ ( add_mset_list_a @ A4 @ F4 ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% multi_subset_induct
thf(fact_871_multi__subset__induct,axiom,
! [F3: multiset_a,A2: multiset_a,P2: multiset_a > $o] :
( ( subseteq_mset_a @ F3 @ A2 )
=> ( ( P2 @ zero_zero_multiset_a )
=> ( ! [A4: a,F4: multiset_a] :
( ( member_a @ A4 @ ( set_mset_a @ A2 ) )
=> ( ( P2 @ F4 )
=> ( P2 @ ( add_mset_a @ A4 @ F4 ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% multi_subset_induct
thf(fact_872_mset__subset__eq__single,axiom,
! [A: list_list_a,B4: multiset_list_list_a] :
( ( member_list_list_a @ A @ ( set_mset_list_list_a @ B4 ) )
=> ( subset8447756916971205105list_a @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) @ B4 ) ) ).
% mset_subset_eq_single
thf(fact_873_mset__subset__eq__single,axiom,
! [A: list_a,B4: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ B4 ) )
=> ( subseteq_mset_list_a @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) @ B4 ) ) ).
% mset_subset_eq_single
thf(fact_874_mset__subset__eq__single,axiom,
! [A: a,B4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ B4 ) )
=> ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ B4 ) ) ).
% mset_subset_eq_single
thf(fact_875_is__unit__power__iff,axiom,
! [A: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ one_one_nat )
= ( ( dvd_dvd_nat @ A @ one_one_nat )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_876_more__than__one__mset__mset__diff,axiom,
! [A: list_list_a,M4: multiset_list_list_a] :
( ( member_list_list_a @ A @ ( set_mset_list_list_a @ ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) ) )
=> ( ( set_mset_list_list_a @ ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) )
= ( set_mset_list_list_a @ M4 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_877_more__than__one__mset__mset__diff,axiom,
! [A: list_a,M4: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) )
=> ( ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) )
= ( set_mset_list_a @ M4 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_878_more__than__one__mset__mset__diff,axiom,
! [A: a,M4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( set_mset_a @ M4 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_879_multiset__add__sub__el__shuffle,axiom,
! [C: list_list_a,B4: multiset_list_list_a,B: list_list_a] :
( ( member_list_list_a @ C @ ( set_mset_list_list_a @ B4 ) )
=> ( ( B != C )
=> ( ( add_mset_list_list_a @ B @ ( minus_5831295526526677175list_a @ B4 @ ( add_mset_list_list_a @ C @ zero_z1542645121299710087list_a ) ) )
= ( minus_5831295526526677175list_a @ ( add_mset_list_list_a @ B @ B4 ) @ ( add_mset_list_list_a @ C @ zero_z1542645121299710087list_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_880_multiset__add__sub__el__shuffle,axiom,
! [C: list_a,B4: multiset_list_a,B: list_a] :
( ( member_list_a @ C @ ( set_mset_list_a @ B4 ) )
=> ( ( B != C )
=> ( ( add_mset_list_a @ B @ ( minus_7431248565939055793list_a @ B4 @ ( add_mset_list_a @ C @ zero_z4454100511807792257list_a ) ) )
= ( minus_7431248565939055793list_a @ ( add_mset_list_a @ B @ B4 ) @ ( add_mset_list_a @ C @ zero_z4454100511807792257list_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_881_multiset__add__sub__el__shuffle,axiom,
! [C: a,B4: multiset_a,B: a] :
( ( member_a @ C @ ( set_mset_a @ B4 ) )
=> ( ( B != C )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B4 @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ B4 ) @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_882_add__mset__remove__trivial__eq,axiom,
! [N4: multiset_list_list_a,A: list_list_a] :
( ( N4
= ( add_mset_list_list_a @ A @ ( minus_5831295526526677175list_a @ N4 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) ) )
= ( member_list_list_a @ A @ ( set_mset_list_list_a @ N4 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_883_add__mset__remove__trivial__eq,axiom,
! [N4: multiset_list_a,A: list_a] :
( ( N4
= ( add_mset_list_a @ A @ ( minus_7431248565939055793list_a @ N4 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) )
= ( member_list_a @ A @ ( set_mset_list_a @ N4 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_884_add__mset__remove__trivial__eq,axiom,
! [N4: multiset_a,A: a] :
( ( N4
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A @ ( set_mset_a @ N4 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_885_add__mset__remove__trivial__If,axiom,
! [A: list_list_a,N4: multiset_list_list_a] :
( ( ( member_list_list_a @ A @ ( set_mset_list_list_a @ N4 ) )
=> ( ( add_mset_list_list_a @ A @ ( minus_5831295526526677175list_a @ N4 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) )
= N4 ) )
& ( ~ ( member_list_list_a @ A @ ( set_mset_list_list_a @ N4 ) )
=> ( ( add_mset_list_list_a @ A @ ( minus_5831295526526677175list_a @ N4 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) )
= ( add_mset_list_list_a @ A @ N4 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_886_add__mset__remove__trivial__If,axiom,
! [A: list_a,N4: multiset_list_a] :
( ( ( member_list_a @ A @ ( set_mset_list_a @ N4 ) )
=> ( ( add_mset_list_a @ A @ ( minus_7431248565939055793list_a @ N4 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) )
= N4 ) )
& ( ~ ( member_list_a @ A @ ( set_mset_list_a @ N4 ) )
=> ( ( add_mset_list_a @ A @ ( minus_7431248565939055793list_a @ N4 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) )
= ( add_mset_list_a @ A @ N4 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_887_add__mset__remove__trivial__If,axiom,
! [A: a,N4: multiset_a] :
( ( ( member_a @ A @ ( set_mset_a @ N4 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= N4 ) )
& ( ~ ( member_a @ A @ ( set_mset_a @ N4 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( add_mset_a @ A @ N4 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_888_multi__drop__mem__not__eq,axiom,
! [C: list_list_a,B4: multiset_list_list_a] :
( ( member_list_list_a @ C @ ( set_mset_list_list_a @ B4 ) )
=> ( ( minus_5831295526526677175list_a @ B4 @ ( add_mset_list_list_a @ C @ zero_z1542645121299710087list_a ) )
!= B4 ) ) ).
% multi_drop_mem_not_eq
thf(fact_889_multi__drop__mem__not__eq,axiom,
! [C: list_a,B4: multiset_list_a] :
( ( member_list_a @ C @ ( set_mset_list_a @ B4 ) )
=> ( ( minus_7431248565939055793list_a @ B4 @ ( add_mset_list_a @ C @ zero_z4454100511807792257list_a ) )
!= B4 ) ) ).
% multi_drop_mem_not_eq
thf(fact_890_multi__drop__mem__not__eq,axiom,
! [C: a,B4: multiset_a] :
( ( member_a @ C @ ( set_mset_a @ B4 ) )
=> ( ( minus_3765977307040488491iset_a @ B4 @ ( add_mset_a @ C @ zero_zero_multiset_a ) )
!= B4 ) ) ).
% multi_drop_mem_not_eq
thf(fact_891_diff__single__eq__union,axiom,
! [X: list_list_a,M4: multiset_list_list_a,N4: multiset_list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M4 ) )
=> ( ( ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) )
= N4 )
= ( M4
= ( add_mset_list_list_a @ X @ N4 ) ) ) ) ).
% diff_single_eq_union
thf(fact_892_diff__single__eq__union,axiom,
! [X: list_a,M4: multiset_list_a,N4: multiset_list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M4 ) )
=> ( ( ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) )
= N4 )
= ( M4
= ( add_mset_list_a @ X @ N4 ) ) ) ) ).
% diff_single_eq_union
thf(fact_893_diff__single__eq__union,axiom,
! [X: a,M4: multiset_a,N4: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M4 ) )
=> ( ( ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= N4 )
= ( M4
= ( add_mset_a @ X @ N4 ) ) ) ) ).
% diff_single_eq_union
thf(fact_894_diff__single__trivial,axiom,
! [X: list_list_a,M4: multiset_list_list_a] :
( ~ ( member_list_list_a @ X @ ( set_mset_list_list_a @ M4 ) )
=> ( ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) )
= M4 ) ) ).
% diff_single_trivial
thf(fact_895_diff__single__trivial,axiom,
! [X: list_a,M4: multiset_list_a] :
( ~ ( member_list_a @ X @ ( set_mset_list_a @ M4 ) )
=> ( ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) )
= M4 ) ) ).
% diff_single_trivial
thf(fact_896_diff__single__trivial,axiom,
! [X: a,M4: multiset_a] :
( ~ ( member_a @ X @ ( set_mset_a @ M4 ) )
=> ( ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M4 ) ) ).
% diff_single_trivial
thf(fact_897_size__1__singleton__mset,axiom,
! [M4: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ M4 )
= one_one_nat )
=> ? [A4: list_a] :
( M4
= ( add_mset_list_a @ A4 @ zero_z4454100511807792257list_a ) ) ) ).
% size_1_singleton_mset
thf(fact_898_size__1__singleton__mset,axiom,
! [M4: multiset_a] :
( ( ( size_size_multiset_a @ M4 )
= one_one_nat )
=> ? [A4: a] :
( M4
= ( add_mset_a @ A4 @ zero_zero_multiset_a ) ) ) ).
% size_1_singleton_mset
thf(fact_899_size__single,axiom,
! [B: list_a] :
( ( size_s2335926164413107382list_a @ ( add_mset_list_a @ B @ zero_z4454100511807792257list_a ) )
= one_one_nat ) ).
% size_single
thf(fact_900_size__single,axiom,
! [B: a] :
( ( size_size_multiset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= one_one_nat ) ).
% size_single
thf(fact_901_dvd__imp__le,axiom,
! [K3: nat,N: nat] :
( ( dvd_dvd_nat @ K3 @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K3 @ N ) ) ) ).
% dvd_imp_le
thf(fact_902_dvd__mult__cancel,axiom,
! [K3: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_903_nat__mult__dvd__cancel1,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_904_dvd__power__iff,axiom,
! [X: nat,M: nat,N: nat] :
( ( X != zero_zero_nat )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M ) @ ( power_power_nat @ X @ N ) )
= ( ( dvd_dvd_nat @ X @ one_one_nat )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_905_dvd__power,axiom,
! [N: nat,X: nat] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_nat ) )
=> ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).
% dvd_power
thf(fact_906_insert__subset__eq__iff,axiom,
! [A: list_list_a,A2: multiset_list_list_a,B4: multiset_list_list_a] :
( ( subset8447756916971205105list_a @ ( add_mset_list_list_a @ A @ A2 ) @ B4 )
= ( ( member_list_list_a @ A @ ( set_mset_list_list_a @ B4 ) )
& ( subset8447756916971205105list_a @ A2 @ ( minus_5831295526526677175list_a @ B4 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_907_insert__subset__eq__iff,axiom,
! [A: list_a,A2: multiset_list_a,B4: multiset_list_a] :
( ( subseteq_mset_list_a @ ( add_mset_list_a @ A @ A2 ) @ B4 )
= ( ( member_list_a @ A @ ( set_mset_list_a @ B4 ) )
& ( subseteq_mset_list_a @ A2 @ ( minus_7431248565939055793list_a @ B4 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_908_insert__subset__eq__iff,axiom,
! [A: a,A2: multiset_a,B4: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ A2 ) @ B4 )
= ( ( member_a @ A @ ( set_mset_a @ B4 ) )
& ( subseteq_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ B4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_909_size__Diff1__le,axiom,
! [M4: multiset_list_a,X: list_a] : ( ord_less_eq_nat @ ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) ) @ ( size_s2335926164413107382list_a @ M4 ) ) ).
% size_Diff1_le
thf(fact_910_size__Diff1__le,axiom,
! [M4: multiset_a,X: a] : ( ord_less_eq_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M4 ) ) ).
% size_Diff1_le
thf(fact_911_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_912_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
= ( N = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_913_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_914_size__Diff1__less,axiom,
! [X: list_list_a,M4: multiset_list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M4 ) )
=> ( ord_less_nat @ ( size_s8523483970790017596list_a @ ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) ) @ ( size_s8523483970790017596list_a @ M4 ) ) ) ).
% size_Diff1_less
thf(fact_915_size__Diff1__less,axiom,
! [X: list_a,M4: multiset_list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M4 ) )
=> ( ord_less_nat @ ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) ) @ ( size_s2335926164413107382list_a @ M4 ) ) ) ).
% size_Diff1_less
thf(fact_916_size__Diff1__less,axiom,
! [X: a,M4: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M4 ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M4 ) ) ) ).
% size_Diff1_less
thf(fact_917_size__Diff2__less,axiom,
! [X: list_list_a,M4: multiset_list_list_a,Y: list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M4 ) )
=> ( ( member_list_list_a @ Y @ ( set_mset_list_list_a @ M4 ) )
=> ( ord_less_nat @ ( size_s8523483970790017596list_a @ ( minus_5831295526526677175list_a @ ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) @ ( add_mset_list_list_a @ Y @ zero_z1542645121299710087list_a ) ) ) @ ( size_s8523483970790017596list_a @ M4 ) ) ) ) ).
% size_Diff2_less
thf(fact_918_size__Diff2__less,axiom,
! [X: list_a,M4: multiset_list_a,Y: list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M4 ) )
=> ( ( member_list_a @ Y @ ( set_mset_list_a @ M4 ) )
=> ( ord_less_nat @ ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) ) ) @ ( size_s2335926164413107382list_a @ M4 ) ) ) ) ).
% size_Diff2_less
thf(fact_919_size__Diff2__less,axiom,
! [X: a,M4: multiset_a,Y: a] :
( ( member_a @ X @ ( set_mset_a @ M4 ) )
=> ( ( member_a @ Y @ ( set_mset_a @ M4 ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M4 ) ) ) ) ).
% size_Diff2_less
thf(fact_920_size__Diff__singleton__if,axiom,
! [X: list_list_a,A2: multiset_list_list_a] :
( ( ( member_list_list_a @ X @ ( set_mset_list_list_a @ A2 ) )
=> ( ( size_s8523483970790017596list_a @ ( minus_5831295526526677175list_a @ A2 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) )
= ( minus_minus_nat @ ( size_s8523483970790017596list_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_list_list_a @ X @ ( set_mset_list_list_a @ A2 ) )
=> ( ( size_s8523483970790017596list_a @ ( minus_5831295526526677175list_a @ A2 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) )
= ( size_s8523483970790017596list_a @ A2 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_921_size__Diff__singleton__if,axiom,
! [X: list_a,A2: multiset_list_a] :
( ( ( member_list_a @ X @ ( set_mset_list_a @ A2 ) )
=> ( ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ A2 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) )
= ( minus_minus_nat @ ( size_s2335926164413107382list_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_list_a @ X @ ( set_mset_list_a @ A2 ) )
=> ( ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ A2 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) )
= ( size_s2335926164413107382list_a @ A2 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_922_size__Diff__singleton__if,axiom,
! [X: a,A2: multiset_a] :
( ( ( member_a @ X @ ( set_mset_a @ A2 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A2 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ ( set_mset_a @ A2 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A2 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( size_size_multiset_a @ A2 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_923_size__Diff__singleton,axiom,
! [X: list_list_a,M4: multiset_list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M4 ) )
=> ( ( size_s8523483970790017596list_a @ ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) )
= ( minus_minus_nat @ ( size_s8523483970790017596list_a @ M4 ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_924_size__Diff__singleton,axiom,
! [X: list_a,M4: multiset_list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M4 ) )
=> ( ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) )
= ( minus_minus_nat @ ( size_s2335926164413107382list_a @ M4 ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_925_size__Diff__singleton,axiom,
! [X: a,M4: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M4 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M4 ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_926_field_Opoly__mult__degree__one__monic__imp__same__roots,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,P: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member5524387281408368019list_a @ P @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) )
=> ( ( P != nil_set_list_a )
=> ( ( polyno4169377219242390531t_unit @ R @ ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( cons_set_list_a @ ( one_se1127990129394575805t_unit @ R ) @ ( cons_set_list_a @ ( a_inv_5715216516650856053t_unit @ R @ A ) @ nil_set_list_a ) ) @ P ) )
= ( add_mset_set_list_a @ A @ ( polyno4169377219242390531t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.poly_mult_degree_one_monic_imp_same_roots
thf(fact_927_field_Opoly__mult__degree__one__monic__imp__same__roots,axiom,
! [R: partia4960592913263135132t_unit,A: set_list_list_a,P: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member334759470184282131list_a @ A @ ( partia3317168157747563407t_unit @ R ) )
=> ( ( member6124916891863447321list_a @ P @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( P != nil_set_list_list_a )
=> ( ( polyno2127442156181624701t_unit @ R @ ( mult_l7436655221470123345t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ ( cons_set_list_list_a @ ( one_se2489417650821308733t_unit @ R ) @ ( cons_set_list_list_a @ ( a_inv_6360815108636782831t_unit @ R @ A ) @ nil_set_list_list_a ) ) @ P ) )
= ( add_ms1779951704026522312list_a @ A @ ( polyno2127442156181624701t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.poly_mult_degree_one_monic_imp_same_roots
thf(fact_928_field_Opoly__mult__degree__one__monic__imp__same__roots,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,P: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P != nil_list_list_a )
=> ( ( polyno3707469075594375645t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ ( cons_list_list_a @ ( a_inv_7033018035630854991t_unit @ R @ A ) @ nil_list_list_a ) ) @ P ) )
= ( add_mset_list_list_a @ A @ ( polyno3707469075594375645t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.poly_mult_degree_one_monic_imp_same_roots
thf(fact_929_field_Opoly__mult__degree__one__monic__imp__same__roots,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,P: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( polyno7858422826990252003t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ R @ A ) @ nil_list_a ) ) @ P ) )
= ( add_mset_list_a @ A @ ( polyno7858422826990252003t_unit @ R @ P ) ) ) ) ) ) ) ).
% field.poly_mult_degree_one_monic_imp_same_roots
thf(fact_930_field_Opoly__mult__degree__one__monic__imp__same__roots,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,P: list_a] :
( ( field_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P != nil_a )
=> ( ( polynomial_roots_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( 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 @ A ) @ nil_a ) ) @ P ) )
= ( add_mset_a @ A @ ( polynomial_roots_a_b @ R @ P ) ) ) ) ) ) ) ).
% field.poly_mult_degree_one_monic_imp_same_roots
thf(fact_931_field_Ogauss__poly__div__gauss__poly__iff__1,axiom,
! [R: partia7496981018696276118t_unit,L: nat,M: nat] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( polyno9075941895896075626t_unit @ R @ ( a_minu6874796375791416686t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( var_se6008125447796440765t_unit @ R ) @ L ) @ ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) @ ( a_minu6874796375791416686t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ ( var_se6008125447796440765t_unit @ R ) @ M ) @ ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ).
% field.gauss_poly_div_gauss_poly_iff_1
thf(fact_932_field_Ogauss__poly__div__gauss__poly__iff__1,axiom,
! [R: partia4960592913263135132t_unit,L: nat,M: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( polyno3637028486239637860t_unit @ R @ ( a_minu1178922365601208552t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ ( pow_li5711373720449609902it_nat @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ ( var_se2996050386653789495t_unit @ R ) @ L ) @ ( one_li3223383766543049405t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) @ ( a_minu1178922365601208552t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ ( pow_li5711373720449609902it_nat @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ ( var_se2996050386653789495t_unit @ R ) @ M ) @ ( one_li3223383766543049405t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ).
% field.gauss_poly_div_gauss_poly_iff_1
thf(fact_933_field_Ogauss__poly__div__gauss__poly__iff__1,axiom,
! [R: partia2175431115845679010xt_a_b,L: nat,M: nat] :
( ( field_a_b @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( polyno5814909790663948098es_a_b @ R @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( var_a_b @ R ) @ L ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( var_a_b @ R ) @ M ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ).
% field.gauss_poly_div_gauss_poly_iff_1
thf(fact_934_field_Ogauss__poly__div__gauss__poly__iff__1,axiom,
! [R: partia2670972154091845814t_unit,L: nat,M: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( polyno8016796738000020810t_unit @ R @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( var_li8453953174693405341t_unit @ R ) @ L ) @ ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( var_li8453953174693405341t_unit @ R ) @ M ) @ ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ).
% field.gauss_poly_div_gauss_poly_iff_1
thf(fact_935_field_Ogauss__poly__div__gauss__poly__iff__1,axiom,
! [R: partia2956882679547061052t_unit,L: nat,M: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( polyno4453881341673752516t_unit @ R @ ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( var_li3532061862469730199t_unit @ R ) @ L ) @ ( one_li8923720976704309949t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) @ ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( var_li3532061862469730199t_unit @ R ) @ M ) @ ( one_li8923720976704309949t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ).
% field.gauss_poly_div_gauss_poly_iff_1
thf(fact_936_pow__divides__pow__iff,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( dvd_dvd_nat @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_937_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_938_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_939_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_940_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_941_poly__mult__degree__one__monic__imp__same__roots,axiom,
! [A: a,P: list_a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polynomial_roots_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( 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 @ A ) @ nil_a ) ) @ P ) )
= ( add_mset_a @ A @ ( polynomial_roots_a_b @ r @ P ) ) ) ) ) ) ).
% poly_mult_degree_one_monic_imp_same_roots
thf(fact_942_dvd__productE,axiom,
! [P: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P @ ( times_times_nat @ A @ B ) )
=> ~ ! [X3: nat,Y3: nat] :
( ( P
= ( times_times_nat @ X3 @ Y3 ) )
=> ( ( dvd_dvd_nat @ X3 @ A )
=> ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).
% dvd_productE
thf(fact_943_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
=> ? [B5: nat,C3: nat] :
( ( A
= ( times_times_nat @ B5 @ C3 ) )
& ( dvd_dvd_nat @ B5 @ B )
& ( dvd_dvd_nat @ C3 @ C ) ) ) ).
% division_decomp
thf(fact_944_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_945_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_946_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_947_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_948_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_949_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_950_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D2: nat,X3: nat,Y3: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
= D2 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
= D2 ) ) ) ).
% bezout1_nat
thf(fact_951_p_Ofield__long__division__theorem,axiom,
! [K: set_list_a,P: list_list_a,B: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ B )
=> ( ( B != nil_list_a )
=> ? [Q2: list_list_a,R4: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Q2 )
& ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ R4 )
& ( P
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ B @ Q2 ) @ R4 ) )
& ( ( R4 = nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ R4 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ B ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% p.field_long_division_theorem
thf(fact_952_p_Oroots__inclI_H,axiom,
! [P: list_list_a,M: multiset_list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_list_a )
=> ( ord_less_eq_nat @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A4 ) @ ( count_list_a @ M @ A4 ) ) ) )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ M ) ) ) ).
% p.roots_inclI'
thf(fact_953_p_Oline__extension__smult__closed,axiom,
! [K: set_list_a,E: set_list_a,A: list_a,K3: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K4: list_a,V: list_a] :
( ( member_list_a @ K4 @ K )
=> ( ( member_list_a @ V @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ V ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) ) ) ) ) ) ) ) ).
% p.line_extension_smult_closed
thf(fact_954_p_Oline__extension__in__carrier,axiom,
! [K: set_list_a,A: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.line_extension_in_carrier
thf(fact_955_p_Oline__extension__mem__iff,axiom,
! [U: list_a,K: set_list_a,A: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ K )
& ? [Y5: list_a] :
( ( member_list_a @ Y5 @ E )
& ( U
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ A ) @ Y5 ) ) ) ) ) ) ).
% p.line_extension_mem_iff
thf(fact_956_p_Ovar__closed_I2_J,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.var_closed(2)
thf(fact_957_p_Oalg__mult__eq__count__roots,axiom,
! [P: list_list_a] :
( ( member_list_list_a @ 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 ) ) ) ) ) )
=> ( ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( count_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ).
% p.alg_mult_eq_count_roots
thf(fact_958_p_Oconst__term__zero,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ! [P4: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P4 )
=> ( ( P4 != nil_list_a )
=> ( P
!= ( append_list_a @ P4 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ) ) ) ).
% p.const_term_zero
thf(fact_959_count__empty,axiom,
! [A: list_a] :
( ( count_list_a @ zero_z4454100511807792257list_a @ A )
= zero_zero_nat ) ).
% count_empty
thf(fact_960_count__empty,axiom,
! [A: a] :
( ( count_a @ zero_zero_multiset_a @ A )
= zero_zero_nat ) ).
% count_empty
thf(fact_961_count__diff,axiom,
! [M4: multiset_list_a,N4: multiset_list_a,A: list_a] :
( ( count_list_a @ ( minus_7431248565939055793list_a @ M4 @ N4 ) @ A )
= ( minus_minus_nat @ ( count_list_a @ M4 @ A ) @ ( count_list_a @ N4 @ A ) ) ) ).
% count_diff
thf(fact_962_count__diff,axiom,
! [M4: multiset_a,N4: multiset_a,A: a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ M4 @ N4 ) @ A )
= ( minus_minus_nat @ ( count_a @ M4 @ A ) @ ( count_a @ N4 @ A ) ) ) ).
% count_diff
thf(fact_963_count__greater__zero__iff,axiom,
! [M4: multiset_list_list_a,X: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_list_list_a @ M4 @ X ) )
= ( member_list_list_a @ X @ ( set_mset_list_list_a @ M4 ) ) ) ).
% count_greater_zero_iff
thf(fact_964_count__greater__zero__iff,axiom,
! [M4: multiset_list_a,X: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_list_a @ M4 @ X ) )
= ( member_list_a @ X @ ( set_mset_list_a @ M4 ) ) ) ).
% count_greater_zero_iff
thf(fact_965_count__greater__zero__iff,axiom,
! [M4: multiset_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_a @ M4 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M4 ) ) ) ).
% count_greater_zero_iff
thf(fact_966_count__greater__eq__one__iff,axiom,
! [M4: multiset_list_list_a,X: list_list_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_list_list_a @ M4 @ X ) )
= ( member_list_list_a @ X @ ( set_mset_list_list_a @ M4 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_967_count__greater__eq__one__iff,axiom,
! [M4: multiset_list_a,X: list_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_list_a @ M4 @ X ) )
= ( member_list_a @ X @ ( set_mset_list_a @ M4 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_968_count__greater__eq__one__iff,axiom,
! [M4: multiset_a,X: a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_a @ M4 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M4 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_969_p_Ozero__is__polynomial,axiom,
! [K: set_list_a] : ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ nil_list_a ) ).
% p.zero_is_polynomial
thf(fact_970_p_Ocarrier__polynomial,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P ) ) ) ).
% p.carrier_polynomial
thf(fact_971_p_Oone__is__polynomial,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ).
% p.one_is_polynomial
thf(fact_972_univ__poly__carrier,axiom,
( polynomial_a_b
= ( ^ [R3: partia2175431115845679010xt_a_b,K2: set_a,P3: list_a] : ( member_list_a @ P3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R3 @ K2 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_973_univ__poly__carrier,axiom,
( polyno1315193887021588240t_unit
= ( ^ [R3: partia2670972154091845814t_unit,K2: set_list_a,P3: list_list_a] : ( member_list_list_a @ P3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R3 @ K2 ) ) ) ) ) ).
% univ_poly_carrier
thf(fact_974_mset__subset__eqI,axiom,
! [A2: multiset_list_a,B4: multiset_list_a] :
( ! [A4: list_a] : ( ord_less_eq_nat @ ( count_list_a @ A2 @ A4 ) @ ( count_list_a @ B4 @ A4 ) )
=> ( subseteq_mset_list_a @ A2 @ B4 ) ) ).
% mset_subset_eqI
thf(fact_975_mset__subset__eqI,axiom,
! [A2: multiset_a,B4: multiset_a] :
( ! [A4: a] : ( ord_less_eq_nat @ ( count_a @ A2 @ A4 ) @ ( count_a @ B4 @ A4 ) )
=> ( subseteq_mset_a @ A2 @ B4 ) ) ).
% mset_subset_eqI
thf(fact_976_subseteq__mset__def,axiom,
( subseteq_mset_list_a
= ( ^ [A7: multiset_list_a,B6: multiset_list_a] :
! [A3: list_a] : ( ord_less_eq_nat @ ( count_list_a @ A7 @ A3 ) @ ( count_list_a @ B6 @ A3 ) ) ) ) ).
% subseteq_mset_def
thf(fact_977_subseteq__mset__def,axiom,
( subseteq_mset_a
= ( ^ [A7: multiset_a,B6: multiset_a] :
! [A3: a] : ( ord_less_eq_nat @ ( count_a @ A7 @ A3 ) @ ( count_a @ B6 @ A3 ) ) ) ) ).
% subseteq_mset_def
thf(fact_978_mset__subset__eq__count,axiom,
! [A2: multiset_list_a,B4: multiset_list_a,A: list_a] :
( ( subseteq_mset_list_a @ A2 @ B4 )
=> ( ord_less_eq_nat @ ( count_list_a @ A2 @ A ) @ ( count_list_a @ B4 @ A ) ) ) ).
% mset_subset_eq_count
thf(fact_979_mset__subset__eq__count,axiom,
! [A2: multiset_a,B4: multiset_a,A: a] :
( ( subseteq_mset_a @ A2 @ B4 )
=> ( ord_less_eq_nat @ ( count_a @ A2 @ A ) @ ( count_a @ B4 @ A ) ) ) ).
% mset_subset_eq_count
thf(fact_980_zero__multiset_Orep__eq,axiom,
( ( count_list_a @ zero_z4454100511807792257list_a )
= ( ^ [A3: list_a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_981_zero__multiset_Orep__eq,axiom,
( ( count_a @ zero_zero_multiset_a )
= ( ^ [A3: a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_982_count__inI,axiom,
! [M4: multiset_list_list_a,X: list_list_a] :
( ( ( count_list_list_a @ M4 @ X )
!= zero_zero_nat )
=> ( member_list_list_a @ X @ ( set_mset_list_list_a @ M4 ) ) ) ).
% count_inI
thf(fact_983_count__inI,axiom,
! [M4: multiset_list_a,X: list_a] :
( ( ( count_list_a @ M4 @ X )
!= zero_zero_nat )
=> ( member_list_a @ X @ ( set_mset_list_a @ M4 ) ) ) ).
% count_inI
thf(fact_984_count__inI,axiom,
! [M4: multiset_a,X: a] :
( ( ( count_a @ M4 @ X )
!= zero_zero_nat )
=> ( member_a @ X @ ( set_mset_a @ M4 ) ) ) ).
% count_inI
thf(fact_985_count__eq__zero__iff,axiom,
! [M4: multiset_list_list_a,X: list_list_a] :
( ( ( count_list_list_a @ M4 @ X )
= zero_zero_nat )
= ( ~ ( member_list_list_a @ X @ ( set_mset_list_list_a @ M4 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_986_count__eq__zero__iff,axiom,
! [M4: multiset_list_a,X: list_a] :
( ( ( count_list_a @ M4 @ X )
= zero_zero_nat )
= ( ~ ( member_list_a @ X @ ( set_mset_list_a @ M4 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_987_count__eq__zero__iff,axiom,
! [M4: multiset_a,X: a] :
( ( ( count_a @ M4 @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_mset_a @ M4 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_988_minus__multiset_Orep__eq,axiom,
! [X: multiset_list_a,Xa2: multiset_list_a] :
( ( count_list_a @ ( minus_7431248565939055793list_a @ X @ Xa2 ) )
= ( ^ [A3: list_a] : ( minus_minus_nat @ ( count_list_a @ X @ A3 ) @ ( count_list_a @ Xa2 @ A3 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_989_minus__multiset_Orep__eq,axiom,
! [X: multiset_a,Xa2: multiset_a] :
( ( count_a @ ( minus_3765977307040488491iset_a @ X @ Xa2 ) )
= ( ^ [A3: a] : ( minus_minus_nat @ ( count_a @ X @ A3 ) @ ( count_a @ Xa2 @ A3 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_990_in__diff__count,axiom,
! [A: list_list_a,M4: multiset_list_list_a,N4: multiset_list_list_a] :
( ( member_list_list_a @ A @ ( set_mset_list_list_a @ ( minus_5831295526526677175list_a @ M4 @ N4 ) ) )
= ( ord_less_nat @ ( count_list_list_a @ N4 @ A ) @ ( count_list_list_a @ M4 @ A ) ) ) ).
% in_diff_count
thf(fact_991_in__diff__count,axiom,
! [A: list_a,M4: multiset_list_a,N4: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M4 @ N4 ) ) )
= ( ord_less_nat @ ( count_list_a @ N4 @ A ) @ ( count_list_a @ M4 @ A ) ) ) ).
% in_diff_count
thf(fact_992_in__diff__count,axiom,
! [A: a,M4: multiset_a,N4: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M4 @ N4 ) ) )
= ( ord_less_nat @ ( count_a @ N4 @ A ) @ ( count_a @ M4 @ A ) ) ) ).
% in_diff_count
thf(fact_993_count__single,axiom,
! [B: list_a,A: list_a] :
( ( ( B = A )
=> ( ( count_list_a @ ( add_mset_list_a @ B @ zero_z4454100511807792257list_a ) @ A )
= one_one_nat ) )
& ( ( B != A )
=> ( ( count_list_a @ ( add_mset_list_a @ B @ zero_z4454100511807792257list_a ) @ A )
= zero_zero_nat ) ) ) ).
% count_single
thf(fact_994_count__single,axiom,
! [B: a,A: a] :
( ( ( B = A )
=> ( ( count_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) @ A )
= one_one_nat ) )
& ( ( B != A )
=> ( ( count_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) @ A )
= zero_zero_nat ) ) ) ).
% count_single
thf(fact_995_degree__one__roots,axiom,
! [A: a,A5: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A5 )
= ( 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 @ A5 @ B ) ) @ zero_zero_multiset_a ) ) ) ) ) ) ).
% degree_one_roots
thf(fact_996_p_Oadd_Oone__in__subset,axiom,
! [H3: set_list_a] :
( ( ord_le8861187494160871172list_a @ H3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( H3 != bot_bot_set_list_a )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ H3 )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) @ H3 ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ H3 )
=> ! [Xa3: list_a] :
( ( member_list_a @ Xa3 @ H3 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Xa3 ) @ H3 ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ H3 ) ) ) ) ) ).
% p.add.one_in_subset
thf(fact_997_Group_Onat__pow__0,axiom,
! [G2: partia2670972154091845814t_unit,X: list_a] :
( ( pow_li1142815632869257134it_nat @ G2 @ X @ zero_zero_nat )
= ( one_li8328186300101108157t_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_998_Group_Onat__pow__0,axiom,
! [G2: partia2175431115845679010xt_a_b,X: a] :
( ( pow_a_1026414303147256608_b_nat @ G2 @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_999_Group_Onat__pow__0,axiom,
! [G2: partia2956882679547061052t_unit,X: list_list_a] :
( ( pow_li488931774710091566it_nat @ G2 @ X @ zero_zero_nat )
= ( one_li8234411390022467901t_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_1000_Group_Onat__pow__0,axiom,
! [G2: partia4960592913263135132t_unit,X: set_list_list_a] :
( ( pow_se6773336042625134382it_nat @ G2 @ X @ zero_zero_nat )
= ( one_se2489417650821308733t_unit @ G2 ) ) ).
% Group.nat_pow_0
thf(fact_1001_pow__non__zero,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ N )
!= ( zero_a_b @ r ) ) ) ) ).
% pow_non_zero
thf(fact_1002_nat__pow__pow,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ M )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( times_times_nat @ N @ M ) ) ) ) ).
% nat_pow_pow
thf(fact_1003_pow__mult__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_1004_nat__pow__distrib,axiom,
! [X: a,Y: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_1005_nat__pow__comm,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_1006_group__commutes__pow,axiom,
! [X: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_1007_m__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_1008_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_1009_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_1010_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_1011_var__closed_I2_J,axiom,
polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ ( var_a_b @ r ) ).
% var_closed(2)
thf(fact_1012_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_1013_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_1014_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_1015_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_1016_f__comm__group__1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ( ( Y
!= ( zero_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% f_comm_group_1
thf(fact_1017_one__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% one_unique
thf(fact_1018_inv__unique,axiom,
! [Y: a,X: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y2 ) ) ) ) ) ) ).
% inv_unique
thf(fact_1019_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_1020_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_1021_p_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= bot_bot_set_list_a ) ).
% p.carrier_not_empty
thf(fact_1022_p_Osubring__props_I4_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( K != bot_bot_set_list_a ) ) ).
% p.subring_props(4)
thf(fact_1023_square__eq__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ X )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( X
= ( one_a_ring_ext_a_b @ r ) )
| ( X
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% square_eq_one
thf(fact_1024_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A4
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A4 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_1025_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_1026_nat__pow__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_1027_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_1028_zero__is__polynomial,axiom,
! [K: set_a] : ( polynomial_a_b @ r @ K @ nil_a ) ).
% zero_is_polynomial
thf(fact_1029_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 ) ) ) )
=> ( ! [A4: a] :
( ( member_a @ A4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ A4 ) @ ( count_a @ M @ A4 ) ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P ) @ M ) ) ) ).
% roots_inclI'
thf(fact_1030_set__mset__empty,axiom,
( ( set_mset_list_a @ zero_z4454100511807792257list_a )
= bot_bot_set_list_a ) ).
% set_mset_empty
thf(fact_1031_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_1032_set__mset__empty,axiom,
( ( set_mset_list_list_a @ zero_z1542645121299710087list_a )
= bot_bo1875519244922727510list_a ) ).
% set_mset_empty
thf(fact_1033_set__mset__eq__empty__iff,axiom,
! [M4: multiset_list_a] :
( ( ( set_mset_list_a @ M4 )
= bot_bot_set_list_a )
= ( M4 = zero_z4454100511807792257list_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_1034_set__mset__eq__empty__iff,axiom,
! [M4: multiset_a] :
( ( ( set_mset_a @ M4 )
= bot_bot_set_a )
= ( M4 = zero_zero_multiset_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_1035_set__mset__eq__empty__iff,axiom,
! [M4: multiset_list_list_a] :
( ( ( set_mset_list_list_a @ M4 )
= bot_bo1875519244922727510list_a )
= ( M4 = zero_z1542645121299710087list_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_1036_nat__pow__eone,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X @ one_one_nat )
= X ) ) ).
% nat_pow_eone
thf(fact_1037_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_1038_local_Onat__pow__0,axiom,
! [X: a] :
( ( pow_a_1026414303147256608_b_nat @ r @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ r ) ) ).
% local.nat_pow_0
thf(fact_1039_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_1040_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_1041_r__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( one_a_ring_ext_a_b @ r ) )
= X ) ) ).
% r_one
thf(fact_1042_l__one,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X )
= X ) ) ).
% l_one
thf(fact_1043_p_Omonom__eq__var__pow,axiom,
! [K: set_list_a,A: list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N )
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( cons_list_a @ A @ nil_list_a ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) ) ) ) ) ).
% p.monom_eq_var_pow
thf(fact_1044_multiplicity__ge__iff,axiom,
! [D: list_a,F: list_a,K3: nat] :
( ( monic_4919232885364369782ly_a_b @ r @ D )
=> ( ( member_list_a @ F @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( ord_less_eq_nat @ K3 @ ( monic_5301438133677370042lt_a_b @ r @ D @ F ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ D @ K3 ) @ F ) ) ) ) ).
% multiplicity_ge_iff
thf(fact_1045_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_1046_p_Ozeropideal,axiom,
princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.zeropideal
thf(fact_1047_p_Oone__zeroI,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.one_zeroI
thf(fact_1048_p_Oone__zeroD,axiom,
( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% p.one_zeroD
thf(fact_1049_p_Ocarrier__one__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.carrier_one_zero
thf(fact_1050_p_Ocarrier__one__not__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.carrier_one_not_zero
thf(fact_1051_var__carr,axiom,
member_list_a @ ( var_a_b @ r ) @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% var_carr
thf(fact_1052_p_Olead__coeff__not__zero,axiom,
! [K: set_list_a,A: list_a,P: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ A @ P ) )
=> ( member_list_a @ A @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ).
% p.lead_coeff_not_zero
thf(fact_1053_p_Osubfield__m__inv__simprule,axiom,
! [K: set_list_a,K3: list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K3 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ A ) @ K )
=> ( member_list_a @ A @ K ) ) ) ) ) ).
% p.subfield_m_inv_simprule
thf(fact_1054_var__pow__carr,axiom,
! [N: nat] : ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ N ) @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% var_pow_carr
thf(fact_1055_p_Olead__coeff__in__carrier,axiom,
! [K: set_list_a,A: list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ A @ P ) )
=> ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ).
% p.lead_coeff_in_carrier
thf(fact_1056_degree__pow,axiom,
! [F: list_a,N: nat] :
( ( member_list_a @ F @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ N ) ) @ one_one_nat )
= ( times_times_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) @ N ) ) ) ).
% degree_pow
thf(fact_1057_multiplicity__ge__1__iff__pdivides,axiom,
! [D: list_a,F: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ D )
=> ( ( member_list_a @ F @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ ( monic_5301438133677370042lt_a_b @ r @ D @ F ) )
= ( polyno5814909790663948098es_a_b @ r @ D @ F ) ) ) ) ).
% multiplicity_ge_1_iff_pdivides
thf(fact_1058_p_Oconst__is__polynomial,axiom,
! [A: list_a,K: set_list_a] :
( ( member_list_a @ A @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ A @ nil_list_a ) ) ) ).
% p.const_is_polynomial
thf(fact_1059_p_Omonom__is__polynomial,axiom,
! [K: set_list_a,A: list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N ) ) ) ) ).
% p.monom_is_polynomial
thf(fact_1060_p_Oeuclidean__domainI,axiom,
! [Phi: list_a > nat] :
( ! [A4: list_a,B3: list_a] :
( ( member_list_a @ A4 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ B3 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ? [Q4: list_a,R5: list_a] :
( ( member_list_a @ Q4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( member_list_a @ R5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( A4
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ Q4 ) @ R5 ) )
& ( ( R5
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B3 ) ) ) ) ) )
=> ( ring_e7478897652244013592t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Phi ) ) ).
% p.euclidean_domainI
thf(fact_1061_set__mset__single,axiom,
! [B: list_a] :
( ( set_mset_list_a @ ( add_mset_list_a @ B @ zero_z4454100511807792257list_a ) )
= ( insert_list_a @ B @ bot_bot_set_list_a ) ) ).
% set_mset_single
thf(fact_1062_set__mset__single,axiom,
! [B: a] :
( ( set_mset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( insert_a @ B @ bot_bot_set_a ) ) ).
% set_mset_single
thf(fact_1063_set__mset__single,axiom,
! [B: list_list_a] :
( ( set_mset_list_list_a @ ( add_mset_list_list_a @ B @ zero_z1542645121299710087list_a ) )
= ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a ) ) ).
% set_mset_single
thf(fact_1064_at__most__one__mset__mset__diff,axiom,
! [A: list_a,M4: multiset_list_a] :
( ~ ( member_list_a @ A @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) )
=> ( ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M4 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) )
= ( minus_646659088055828811list_a @ ( set_mset_list_a @ M4 ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_1065_at__most__one__mset__mset__diff,axiom,
! [A: a,M4: multiset_a] :
( ~ ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M4 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( minus_minus_set_a @ ( set_mset_a @ M4 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_1066_at__most__one__mset__mset__diff,axiom,
! [A: list_list_a,M4: multiset_list_list_a] :
( ~ ( member_list_list_a @ A @ ( set_mset_list_list_a @ ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) ) )
=> ( ( set_mset_list_list_a @ ( minus_5831295526526677175list_a @ M4 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) )
= ( minus_5335179877275218001list_a @ ( set_mset_list_list_a @ M4 ) @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ) ) ).
% at_most_one_mset_mset_diff
thf(fact_1067_field_Omultiplicity__ge__1__iff__pdivides,axiom,
! [R: partia7496981018696276118t_unit,D: list_set_list_a,F: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_2059080652700942750t_unit @ R @ D )
=> ( ( member5524387281408368019list_a @ F @ ( minus_4926175736359902257list_a @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) @ ( insert4622323883622259002list_a @ ( zero_l7621212060072393831t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) @ bot_bo743868357060641846list_a ) ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ ( monic_1916329280030354658t_unit @ R @ D @ F ) )
= ( polyno9075941895896075626t_unit @ R @ D @ F ) ) ) ) ) ).
% field.multiplicity_ge_1_iff_pdivides
thf(fact_1068_field_Omultiplicity__ge__1__iff__pdivides,axiom,
! [R: partia4960592913263135132t_unit,D: list_set_list_list_a,F: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_819715999873801112t_unit @ R @ D )
=> ( ( member6124916891863447321list_a @ F @ ( minus_3081475696310877495list_a @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) @ ( insert2341036838393066304list_a @ ( zero_l1604441510127931233t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) @ bot_bo6714768636134629052list_a ) ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ ( monic_8185780642485851356t_unit @ R @ D @ F ) )
= ( polyno3637028486239637860t_unit @ R @ D @ F ) ) ) ) ) ).
% field.multiplicity_ge_1_iff_pdivides
thf(fact_1069_field_Omultiplicity__ge__1__iff__pdivides,axiom,
! [R: partia2956882679547061052t_unit,D: list_list_list_a,F: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_868474719114584568t_unit @ R @ D )
=> ( ( member5342144027231129785list_a @ F @ ( minus_4283034900646138583list_a @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) @ ( insert5239656212080388320list_a @ ( zero_l317200538825487809t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) @ bot_bo3073669775042201692list_a ) ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ ( monic_6498620745769757500t_unit @ R @ D @ F ) )
= ( polyno4453881341673752516t_unit @ R @ D @ F ) ) ) ) ) ).
% field.multiplicity_ge_1_iff_pdivides
thf(fact_1070_field_Omultiplicity__ge__1__iff__pdivides,axiom,
! [R: partia2670972154091845814t_unit,D: list_list_a,F: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_104106837769529726t_unit @ R @ D )
=> ( ( member_list_list_a @ F @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ ( monic_2223747685970961602t_unit @ R @ D @ F ) )
= ( polyno8016796738000020810t_unit @ R @ D @ F ) ) ) ) ) ).
% field.multiplicity_ge_1_iff_pdivides
thf(fact_1071_field_Omultiplicity__ge__1__iff__pdivides,axiom,
! [R: partia2175431115845679010xt_a_b,D: list_a,F: list_a] :
( ( field_a_b @ R )
=> ( ( monic_4919232885364369782ly_a_b @ R @ D )
=> ( ( member_list_a @ F @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( ord_less_eq_nat @ one_one_nat @ ( monic_5301438133677370042lt_a_b @ R @ D @ F ) )
= ( polyno5814909790663948098es_a_b @ R @ D @ F ) ) ) ) ) ).
% field.multiplicity_ge_1_iff_pdivides
thf(fact_1072_field_Omultiplicity__ge__iff,axiom,
! [R: partia7496981018696276118t_unit,D: list_set_list_a,F: list_set_list_a,K3: nat] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_2059080652700942750t_unit @ R @ D )
=> ( ( member5524387281408368019list_a @ F @ ( minus_4926175736359902257list_a @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) @ ( insert4622323883622259002list_a @ ( zero_l7621212060072393831t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) @ bot_bo743868357060641846list_a ) ) )
=> ( ( ord_less_eq_nat @ K3 @ ( monic_1916329280030354658t_unit @ R @ D @ F ) )
= ( polyno9075941895896075626t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ D @ K3 ) @ F ) ) ) ) ) ).
% field.multiplicity_ge_iff
thf(fact_1073_field_Omultiplicity__ge__iff,axiom,
! [R: partia4960592913263135132t_unit,D: list_set_list_list_a,F: list_set_list_list_a,K3: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_819715999873801112t_unit @ R @ D )
=> ( ( member6124916891863447321list_a @ F @ ( minus_3081475696310877495list_a @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) @ ( insert2341036838393066304list_a @ ( zero_l1604441510127931233t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) @ bot_bo6714768636134629052list_a ) ) )
=> ( ( ord_less_eq_nat @ K3 @ ( monic_8185780642485851356t_unit @ R @ D @ F ) )
= ( polyno3637028486239637860t_unit @ R @ ( pow_li5711373720449609902it_nat @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ D @ K3 ) @ F ) ) ) ) ) ).
% field.multiplicity_ge_iff
thf(fact_1074_field_Omultiplicity__ge__iff,axiom,
! [R: partia2956882679547061052t_unit,D: list_list_list_a,F: list_list_list_a,K3: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_868474719114584568t_unit @ R @ D )
=> ( ( member5342144027231129785list_a @ F @ ( minus_4283034900646138583list_a @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) @ ( insert5239656212080388320list_a @ ( zero_l317200538825487809t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) @ bot_bo3073669775042201692list_a ) ) )
=> ( ( ord_less_eq_nat @ K3 @ ( monic_6498620745769757500t_unit @ R @ D @ F ) )
= ( polyno4453881341673752516t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ D @ K3 ) @ F ) ) ) ) ) ).
% field.multiplicity_ge_iff
thf(fact_1075_field_Omultiplicity__ge__iff,axiom,
! [R: partia2670972154091845814t_unit,D: list_list_a,F: list_list_a,K3: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_104106837769529726t_unit @ R @ D )
=> ( ( member_list_list_a @ F @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( ord_less_eq_nat @ K3 @ ( monic_2223747685970961602t_unit @ R @ D @ F ) )
= ( polyno8016796738000020810t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ D @ K3 ) @ F ) ) ) ) ) ).
% field.multiplicity_ge_iff
thf(fact_1076_field_Omultiplicity__ge__iff,axiom,
! [R: partia2175431115845679010xt_a_b,D: list_a,F: list_a,K3: nat] :
( ( field_a_b @ R )
=> ( ( monic_4919232885364369782ly_a_b @ R @ D )
=> ( ( member_list_a @ F @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( ord_less_eq_nat @ K3 @ ( monic_5301438133677370042lt_a_b @ R @ D @ F ) )
= ( polyno5814909790663948098es_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ D @ K3 ) @ F ) ) ) ) ) ).
% field.multiplicity_ge_iff
thf(fact_1077_p_Ozeromaximalideal__fieldI,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.zeromaximalideal_fieldI
thf(fact_1078_p_Ozeromaximalideal__eq__field,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.zeromaximalideal_eq_field
thf(fact_1079_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_1080_carrier__one__not__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
!= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
!= ( zero_a_b @ r ) ) ) ).
% carrier_one_not_zero
thf(fact_1081_carrier__one__zero,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% carrier_one_zero
thf(fact_1082_one__zeroD,axiom,
( ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) )
=> ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% one_zeroD
thf(fact_1083_one__zeroI,axiom,
( ( ( partia707051561876973205xt_a_b @ r )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
=> ( ( one_a_ring_ext_a_b @ r )
= ( zero_a_b @ r ) ) ) ).
% one_zeroI
thf(fact_1084_lead__coeff__not__zero,axiom,
! [K: set_a,A: a,P: list_a] :
( ( polynomial_a_b @ r @ K @ ( cons_a @ A @ P ) )
=> ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% lead_coeff_not_zero
thf(fact_1085_f__comm__group__2,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% f_comm_group_2
thf(fact_1086_p_Ovar__carr,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% p.var_carr
thf(fact_1087_p_Ovar__pow__carr,axiom,
! [K: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% p.var_pow_carr
thf(fact_1088_p_Odegree__pow,axiom,
! [K: set_list_a,F: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ F @ N ) ) @ one_one_nat )
= ( times_times_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat ) @ N ) ) ) ) ).
% p.degree_pow
thf(fact_1089_const__is__polynomial,axiom,
! [A: a,K: set_a] :
( ( member_a @ A @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K @ ( cons_a @ A @ nil_a ) ) ) ).
% const_is_polynomial
thf(fact_1090_zeromaximalideal__fieldI,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( field_a_b @ r ) ) ).
% zeromaximalideal_fieldI
thf(fact_1091_zeromaximalideal__eq__field,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
= ( field_a_b @ r ) ) ).
% zeromaximalideal_eq_field
thf(fact_1092_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_1093_p_Odegree__mult,axiom,
! [K: set_list_a,F: list_list_a,G: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( member_list_list_a @ G @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ F @ G ) ) @ one_one_nat )
= ( plus_plus_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ G ) @ one_one_nat ) ) ) ) ) ) ).
% p.degree_mult
thf(fact_1094_p_Odegree__add__distinct,axiom,
! [K: set_list_a,F: list_list_a,G: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ F @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( member_list_list_a @ G @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat )
!= ( minus_minus_nat @ ( size_s349497388124573686list_a @ G ) @ one_one_nat ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ F @ G ) ) @ one_one_nat )
= ( ord_max_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ G ) @ one_one_nat ) ) ) ) ) ) ) ).
% p.degree_add_distinct
thf(fact_1095_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1096_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1097_nat__pow__mult,axiom,
! [X: a,N: nat,M: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X @ M ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% nat_pow_mult
thf(fact_1098_p_Onat__pow__mult,axiom,
! [X: list_a,N: nat,M: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ M ) )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% p.nat_pow_mult
thf(fact_1099_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1100_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1101_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1102_add__0,axiom,
! [A: multiset_list_a] :
( ( plus_p690419498615200257list_a @ zero_z4454100511807792257list_a @ A )
= A ) ).
% add_0
thf(fact_1103_add__0,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
= A ) ).
% add_0
thf(fact_1104_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1105_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1106_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1107_add__cancel__right__right,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( A
= ( plus_p690419498615200257list_a @ A @ B ) )
= ( B = zero_z4454100511807792257list_a ) ) ).
% add_cancel_right_right
thf(fact_1108_add__cancel__right__right,axiom,
! [A: multiset_a,B: multiset_a] :
( ( A
= ( plus_plus_multiset_a @ A @ B ) )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_right_right
thf(fact_1109_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1110_add__cancel__right__left,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( A
= ( plus_p690419498615200257list_a @ B @ A ) )
= ( B = zero_z4454100511807792257list_a ) ) ).
% add_cancel_right_left
thf(fact_1111_add__cancel__right__left,axiom,
! [A: multiset_a,B: multiset_a] :
( ( A
= ( plus_plus_multiset_a @ B @ A ) )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_right_left
thf(fact_1112_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1113_add__cancel__left__right,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ A @ B )
= A )
= ( B = zero_z4454100511807792257list_a ) ) ).
% add_cancel_left_right
thf(fact_1114_add__cancel__left__right,axiom,
! [A: multiset_a,B: multiset_a] :
( ( ( plus_plus_multiset_a @ A @ B )
= A )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_left_right
thf(fact_1115_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1116_add__cancel__left__left,axiom,
! [B: multiset_list_a,A: multiset_list_a] :
( ( ( plus_p690419498615200257list_a @ B @ A )
= A )
= ( B = zero_z4454100511807792257list_a ) ) ).
% add_cancel_left_left
thf(fact_1117_add__cancel__left__left,axiom,
! [B: multiset_a,A: multiset_a] :
( ( ( plus_plus_multiset_a @ B @ A )
= A )
= ( B = zero_zero_multiset_a ) ) ).
% add_cancel_left_left
thf(fact_1118_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1119_add_Oright__neutral,axiom,
! [A: multiset_list_a] :
( ( plus_p690419498615200257list_a @ A @ zero_z4454100511807792257list_a )
= A ) ).
% add.right_neutral
thf(fact_1120_add_Oright__neutral,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% add.right_neutral
thf(fact_1121_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1122_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1123_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1124_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1125_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1126_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1127_dvd__add__triv__right__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_right_iff
thf(fact_1128_dvd__add__triv__left__iff,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_triv_left_iff
thf(fact_1129_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1130_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1131_nat__add__left__cancel__less,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1132_diff__diff__left,axiom,
! [I: nat,J: nat,K3: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K3 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K3 ) ) ) ).
% diff_diff_left
thf(fact_1133_nat__add__left__cancel__le,axiom,
! [K3: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1134_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_1135_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_1136_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_1137_max__nat_Oneutr__eq__iff,axiom,
! [A: nat,B: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A @ B ) )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_1138_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_1139_max__nat_Oeq__neutr__iff,axiom,
! [A: nat,B: nat] :
( ( ( ord_max_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_1140_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_1141_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_1142_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1143_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1144_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1145_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1146_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_1147_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_1148_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1149_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1150_diff__add__zero,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A @ ( plus_p690419498615200257list_a @ A @ B ) )
= zero_z4454100511807792257list_a ) ).
% diff_add_zero
thf(fact_1151_diff__add__zero,axiom,
! [A: multiset_a,B: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ ( plus_plus_multiset_a @ A @ B ) )
= zero_zero_multiset_a ) ).
% diff_add_zero
thf(fact_1152_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1153_dvd__add__times__triv__right__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1154_dvd__add__times__triv__left__iff,axiom,
! [A: nat,C: nat,B: nat] :
( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
= ( dvd_dvd_nat @ A @ B ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1155_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_1156_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_1157_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_1158_Nat_Odiff__diff__right,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1159_Nat_Oadd__diff__assoc2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K3 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K3 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1160_Nat_Oadd__diff__assoc,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K3 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1161_length__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).
% length_append
thf(fact_1162_length__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( size_s349497388124573686list_a @ ( append_list_a @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s349497388124573686list_a @ Xs ) @ ( size_s349497388124573686list_a @ Ys ) ) ) ).
% length_append
thf(fact_1163_degree__add__distinct,axiom,
! [F: list_a,G: list_a] :
( ( member_list_a @ F @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ G @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat )
!= ( minus_minus_nat @ ( size_size_list_a @ G ) @ one_one_nat ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ G ) ) @ one_one_nat )
= ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ G ) @ one_one_nat ) ) ) ) ) ) ).
% degree_add_distinct
thf(fact_1164_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1165_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1166_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1167_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1168_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1169_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1170_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1171_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1172_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1173_add__le__add__imp__diff__le,axiom,
! [I: nat,K3: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K3 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K3 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K3 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1174_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1175_add__le__imp__le__diff,axiom,
! [I: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K3 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1176_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1177_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1178_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1179_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1180_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1181_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1182_diff__cancel2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K3 ) @ ( plus_plus_nat @ N @ K3 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1183_Nat_Odiff__cancel,axiom,
! [K3: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K3 @ M ) @ ( plus_plus_nat @ K3 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1184_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
= ( ord_max_nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_1185_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K3 )
= ( J
= ( plus_plus_nat @ K3 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1186_Nat_Odiff__add__assoc2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K3 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K3 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1187_Nat_Odiff__add__assoc,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K3 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K3 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1188_Nat_Ole__diff__conv2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1189_le__diff__conv,axiom,
! [J: nat,K3: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K3 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K3 ) ) ) ).
% le_diff_conv
thf(fact_1190_less__diff__conv,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ).
% less_diff_conv
thf(fact_1191_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1192_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K3: nat] :
( ! [M7: nat,N3: nat] :
( ( ord_less_nat @ M7 @ N3 )
=> ( ord_less_nat @ ( F @ M7 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K3 ) @ ( F @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1193_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1194_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_list_a] :
( ( plus_p690419498615200257list_a @ zero_z4454100511807792257list_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1195_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_1196_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1197_add_Ocomm__neutral,axiom,
! [A: multiset_list_a] :
( ( plus_p690419498615200257list_a @ A @ zero_z4454100511807792257list_a )
= A ) ).
% add.comm_neutral
thf(fact_1198_add_Ocomm__neutral,axiom,
! [A: multiset_a] :
( ( plus_plus_multiset_a @ A @ zero_zero_multiset_a )
= A ) ).
% add.comm_neutral
thf(fact_1199_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1200_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1201_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K3 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1202_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1203_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1204_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1205_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1206_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1207_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1208_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_1209_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1210_combine__common__factor,axiom,
! [A: nat,E2: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% combine_common_factor
thf(fact_1211_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1212_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1213_less__add__eq__less,axiom,
! [K3: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K3 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K3 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1214_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1215_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1216_add__less__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ K3 ) ) ) ).
% add_less_mono1
thf(fact_1217_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1218_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1219_add__less__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K3 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1220_add__lessD1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K3 )
=> ( ord_less_nat @ I @ K3 ) ) ).
% add_lessD1
thf(fact_1221_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K3: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K3 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K3 ) ) ).
% left_add_mult_distrib
thf(fact_1222_add__mult__distrib2,axiom,
! [K3: nat,M: nat,N: nat] :
( ( times_times_nat @ K3 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K3 @ M ) @ ( times_times_nat @ K3 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1223_add__mult__distrib,axiom,
! [M: nat,N: nat,K3: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K3 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K3 ) @ ( times_times_nat @ N @ K3 ) ) ) ).
% add_mult_distrib
thf(fact_1224_nat__add__max__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_1225_nat__add__max__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
= ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_1226_max__add__distrib__right,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).
% max_add_distrib_right
thf(fact_1227_max__add__distrib__left,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
= ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).
% max_add_distrib_left
thf(fact_1228_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1229_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1230_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1231_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_1232_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_1233_group__cancel_Oadd2,axiom,
! [B4: nat,K3: nat,B: nat,A: nat] :
( ( B4
= ( plus_plus_nat @ K3 @ B ) )
=> ( ( plus_plus_nat @ A @ B4 )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1234_group__cancel_Oadd1,axiom,
! [A2: nat,K3: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K3 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K3 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1235_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( I = J )
& ( K3 = L ) )
=> ( ( plus_plus_nat @ I @ K3 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1236_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1237_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
= ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% nat_mult_max_right
thf(fact_1238_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q3: nat] :
( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
= ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% nat_mult_max_left
thf(fact_1239_add__leE,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K3 @ N ) ) ) ).
% add_leE
thf(fact_1240_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1241_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1242_add__leD1,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1243_add__leD2,axiom,
! [M: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K3 ) @ N )
=> ( ord_less_eq_nat @ K3 @ N ) ) ).
% add_leD2
thf(fact_1244_le__Suc__ex,axiom,
! [K3: nat,L: nat] :
( ( ord_less_eq_nat @ K3 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K3 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1245_add__le__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1246_add__le__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ K3 ) ) ) ).
% add_le_mono1
thf(fact_1247_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1248_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1249_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K5: nat] :
( N2
= ( plus_plus_nat @ M3 @ K5 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1250_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1251_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1252_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C4: nat] :
( B2
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_1253_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1254_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_1255_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1256_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1257_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1258_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1259_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K3 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1260_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less_nat @ zero_zero_nat @ K4 )
& ( ( plus_plus_nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1261_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_1262_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1263_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1264_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1265_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1266_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1267_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1268_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1269_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K3 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1270_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
monic_3145109188698636716ly_a_b @ r @ ( card_I2373409586816755191ly_a_b @ r @ n ) ).
%------------------------------------------------------------------------------