TPTP Problem File: SLH0713^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Finite_Fields/0008_Card_Irreducible_Polynomials_Aux/prob_00383_013082__18380668_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1525 ( 447 unt; 248 typ; 0 def)
% Number of atoms : 3784 (1461 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 18195 ( 349 ~; 66 |; 148 &;15741 @)
% ( 0 <=>;1891 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 30 ( 29 usr)
% Number of type conns : 500 ( 500 >; 0 *; 0 +; 0 <<)
% Number of symbols : 222 ( 219 usr; 23 con; 0-4 aty)
% Number of variables : 2963 ( 69 ^;2839 !; 55 ?;2963 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:23:04.582
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
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% Explicit typings (219)
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thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
a_minu4820293213911669576t_unit: partia5333488208502193986t_unit > list_list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
a_minu2241224857956505934t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_001t__Product____Type__Ounit,type,
a_minu1178922365601208552t_unit: partia3473558348976337314t_unit > list_set_list_list_a > list_set_list_list_a > list_set_list_list_a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_minu3984020753470702548t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Ofield_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
field_1861437471013600865t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
field_6388047844668329575t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
field_1540243473349940225t_unit: partia4960592913263135132t_unit > $o ).
thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
field_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
add_li174743652000525320t_unit: partia2956882679547061052t_unit > list_list_a > list_list_a > list_list_a ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
zero_l317200538825487809t_unit: partia5333488208502193986t_unit > list_list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_l347298301471573063t_unit: partia2956882679547061052t_unit > list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_001t__Product____Type__Ounit,type,
zero_l1604441510127931233t_unit: partia3473558348976337314t_unit > list_set_list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
zero_l4142658623432671053t_unit: partia2670972154091845814t_unit > list_a ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
zero_s2920163772466840039t_unit: partia4960592913263135132t_unit > set_list_list_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring__Characteristic_Ochar_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_c500279861223467766t_unit: partia2670972154091845814t_unit > nat ).
thf(sy_c_Ring__Characteristic_Ochar_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
ring_c8395554250859618576t_unit: partia4960592913263135132t_unit > nat ).
thf(sy_c_Ring__Divisibility_Oeuclidean__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_e7478897652244013592t_unit: partia2670972154091845814t_unit > ( list_a > nat ) > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_f796907574329358751t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001tf__a_001tf__b,type,
ring_f5272581269873410839in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n4705423059119889713t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001tf__a_001tf__b,type,
ring_n4045954140777738665in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n5188127996776581661t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001tf__a_001tf__b,type,
ring_n3639167112692572309ng_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_001t__Product____Type__Ounit,type,
ring_r5224476855413033410t_unit: partia5333488208502193986t_unit > list_list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J_001t__Product____Type__Ounit,type,
ring_r97889109428395874t_unit: partia3473558348976337314t_unit > list_set_list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5437400583859147359t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
collect_list_list_a: ( list_list_a > $o ) > set_list_list_a ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
insert_list_list_a: list_list_a > set_list_list_a > set_list_list_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__List__Olist_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_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_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_a,type,
a2: nat ).
thf(sy_v_m,type,
m: nat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1271)
thf(fact_0_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_1_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_2_assms_I3_J,axiom,
ord_less_nat @ one_one_nat @ a2 ).
% assms(3)
thf(fact_3_p_Ofactorial__domain__axioms,axiom,
ring_f796907574329358751t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.factorial_domain_axioms
thf(fact_4_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_5_p_Onoetherian__domain__axioms,axiom,
ring_n4705423059119889713t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.noetherian_domain_axioms
thf(fact_6_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_7_calculation,axiom,
( ( polyno5814909790663948098es_a_b @ r @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ a2 @ n ) ) @ ( card_I2373409586816755191ly_a_b @ r @ ( power_power_nat @ a2 @ m ) ) )
= ( polyno5814909790663948098es_a_b @ r @ ( 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 @ ( power_power_nat @ a2 @ n ) @ one_one_nat ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( var_a_b @ r ) ) @ ( 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 @ ( power_power_nat @ a2 @ m ) @ one_one_nat ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( var_a_b @ r ) ) ) ) ).
% calculation
thf(fact_8_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_9_p_Onoetherian__ring__axioms,axiom,
ring_n5188127996776581661t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.noetherian_ring_axioms
thf(fact_10_p_Oprincipal__domain__axioms,axiom,
ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.principal_domain_axioms
thf(fact_11_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_12_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_13_var__pow__eq__one__iff,axiom,
! [K: nat] :
( ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_a_b @ r ) @ K )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( K = zero_zero_nat ) ) ).
% var_pow_eq_one_iff
thf(fact_14_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_15_a,axiom,
ord_less_nat @ one_one_nat @ ( power_power_nat @ a2 @ m ) ).
% a
thf(fact_16_b,axiom,
ord_less_nat @ one_one_nat @ ( power_power_nat @ a2 @ n ) ).
% b
thf(fact_17_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_18_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_19_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_20_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_21_assms_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ m ).
% assms(1)
thf(fact_22_assms_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ n ).
% assms(2)
thf(fact_23_degree__var,axiom,
( ( minus_minus_nat @ ( size_size_list_a @ ( var_a_b @ r ) ) @ one_one_nat )
= one_one_nat ) ).
% degree_var
thf(fact_24_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_25_degree__one,axiom,
! [K2: set_a] :
( ( minus_minus_nat @ ( size_size_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) @ one_one_nat )
= zero_zero_nat ) ).
% degree_one
thf(fact_26_b1,axiom,
ord_less_nat @ zero_zero_nat @ ( power_power_nat @ a2 @ n ) ).
% b1
thf(fact_27_a1,axiom,
ord_less_nat @ zero_zero_nat @ ( power_power_nat @ a2 @ m ) ).
% a1
thf(fact_28_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_29_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_30_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_31_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_32_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_33_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_34_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_35_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_36_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_37_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_38_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_39_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_40_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_41_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_42_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_43_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_44_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_45_power__strict__decreasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_46_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_47_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_48_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_49_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_50_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_51_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_52_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_53_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_54_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_55_power__strict__increasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_nat @ N @ N2 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_strict_increasing
thf(fact_56_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_57_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_58_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_59_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_60_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_61_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_62_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_63_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_64_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_65_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_66_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A: list_list_a,P: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_69_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_70_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_71_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_72_field_Ovar__pow__eq__one__iff,axiom,
! [R: partia4960592913263135132t_unit,K: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( ( pow_li5711373720449609902it_nat @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ ( var_se2996050386653789495t_unit @ R ) @ K )
= ( one_li3223383766543049405t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
= ( K = zero_zero_nat ) ) ) ).
% field.var_pow_eq_one_iff
thf(fact_73_field_Ovar__pow__eq__one__iff,axiom,
! [R: partia2175431115845679010xt_a_b,K: nat] :
( ( field_a_b @ R )
=> ( ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( var_a_b @ R ) @ K )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
= ( K = zero_zero_nat ) ) ) ).
% field.var_pow_eq_one_iff
thf(fact_74_field_Ovar__pow__eq__one__iff,axiom,
! [R: partia2670972154091845814t_unit,K: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( var_li8453953174693405341t_unit @ R ) @ K )
= ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
= ( K = zero_zero_nat ) ) ) ).
% field.var_pow_eq_one_iff
thf(fact_75_field_Ovar__pow__eq__one__iff,axiom,
! [R: partia2956882679547061052t_unit,K: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( var_li3532061862469730199t_unit @ R ) @ K )
= ( one_li8923720976704309949t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
= ( K = zero_zero_nat ) ) ) ).
% field.var_pow_eq_one_iff
thf(fact_76_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_77_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_78_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_79_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_80_gauss__poly__def,axiom,
( card_I2619780863984422015t_unit
= ( ^ [K3: partia2670972154091845814t_unit,N3: nat] : ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ K3 @ ( partia5361259788508890537t_unit @ K3 ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ K3 @ ( partia5361259788508890537t_unit @ K3 ) ) @ ( var_li8453953174693405341t_unit @ K3 ) @ N3 ) @ ( var_li8453953174693405341t_unit @ K3 ) ) ) ) ).
% gauss_poly_def
thf(fact_81_gauss__poly__def,axiom,
( card_I3787608780883923065t_unit
= ( ^ [K3: partia2956882679547061052t_unit,N3: nat] : ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ K3 @ ( partia2464479390973590831t_unit @ K3 ) ) @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ K3 @ ( partia2464479390973590831t_unit @ K3 ) ) @ ( var_li3532061862469730199t_unit @ K3 ) @ N3 ) @ ( var_li3532061862469730199t_unit @ K3 ) ) ) ) ).
% gauss_poly_def
thf(fact_82_gauss__poly__def,axiom,
( card_I2373409586816755191ly_a_b
= ( ^ [K3: partia2175431115845679010xt_a_b,N3: nat] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ K3 @ ( partia707051561876973205xt_a_b @ K3 ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ K3 @ ( partia707051561876973205xt_a_b @ K3 ) ) @ ( var_a_b @ K3 ) @ N3 ) @ ( var_a_b @ K3 ) ) ) ) ).
% gauss_poly_def
thf(fact_83_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_84_Group_Onat__pow__0,axiom,
! [G: partia2670972154091845814t_unit,X: list_a] :
( ( pow_li1142815632869257134it_nat @ G @ X @ zero_zero_nat )
= ( one_li8328186300101108157t_unit @ G ) ) ).
% Group.nat_pow_0
thf(fact_85_Group_Onat__pow__0,axiom,
! [G: partia2175431115845679010xt_a_b,X: a] :
( ( pow_a_1026414303147256608_b_nat @ G @ X @ zero_zero_nat )
= ( one_a_ring_ext_a_b @ G ) ) ).
% Group.nat_pow_0
thf(fact_86_Group_Onat__pow__0,axiom,
! [G: partia2956882679547061052t_unit,X: list_list_a] :
( ( pow_li488931774710091566it_nat @ G @ X @ zero_zero_nat )
= ( one_li8234411390022467901t_unit @ G ) ) ).
% Group.nat_pow_0
thf(fact_87_Group_Onat__pow__0,axiom,
! [G: partia4960592913263135132t_unit,X: set_list_list_a] :
( ( pow_se6773336042625134382it_nat @ G @ X @ zero_zero_nat )
= ( one_se2489417650821308733t_unit @ G ) ) ).
% Group.nat_pow_0
thf(fact_88_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_89_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_90_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_91_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_92_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_93_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_94_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_95_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_96_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_97_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_98_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_99_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_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_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_119_p_Odegree__one,axiom,
! [K2: set_list_a] :
( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) @ one_one_nat )
= zero_zero_nat ) ).
% p.degree_one
thf(fact_120_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_121_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 ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A3
!= ( 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 ) ) @ A3 @ 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_122_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_123_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_124_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_125_zero__diff,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ zero_z4454100511807792257list_a @ A )
= zero_z4454100511807792257list_a ) ).
% zero_diff
thf(fact_126_zero__diff,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ A )
= zero_zero_multiset_a ) ).
% zero_diff
thf(fact_127_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_128_diff__zero,axiom,
! [A: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A @ zero_z4454100511807792257list_a )
= A ) ).
% diff_zero
thf(fact_129_diff__zero,axiom,
! [A: multiset_a] :
( ( minus_3765977307040488491iset_a @ A @ zero_zero_multiset_a )
= A ) ).
% diff_zero
thf(fact_130_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_131_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_132_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_133_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_134_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_135_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_136_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_137_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_138_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_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_149_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_150_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_151_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_152_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_153_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ M )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_154_dvd__power__same,axiom,
! [X: nat,Y: nat,N: nat] :
( ( dvd_dvd_nat @ X @ Y )
=> ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_155_dvd__power__same,axiom,
! [X: int,Y: int,N: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) ) ) ).
% dvd_power_same
thf(fact_156_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_157_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_158_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_159_is__unit__power__iff,axiom,
! [A: int,N: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
| ( N = zero_zero_nat ) ) ) ).
% is_unit_power_iff
thf(fact_160_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_161_zero__reorient,axiom,
! [X: multiset_list_a] :
( ( zero_z4454100511807792257list_a = X )
= ( X = zero_z4454100511807792257list_a ) ) ).
% zero_reorient
thf(fact_162_zero__reorient,axiom,
! [X: multiset_a] :
( ( zero_zero_multiset_a = X )
= ( X = zero_zero_multiset_a ) ) ).
% zero_reorient
thf(fact_163_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_164_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_165_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_166_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_167_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_168_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_169_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_170_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_171_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_172_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_173_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_174_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N4 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_175_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ( P @ M2 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_176_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_177_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_178_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_179_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_180_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_181_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_182_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_183_dvd__power,axiom,
! [N: nat,X: int] :
( ( ( ord_less_nat @ zero_zero_nat @ N )
| ( X = one_one_int ) )
=> ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).
% dvd_power
thf(fact_184_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_185_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_186_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_187_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_188_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_189_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_190_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_191_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_192_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_193_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_194_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_195_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_196_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_197_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_198_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A4: int,B2: int] :
( ( minus_minus_int @ A4 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_199_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_200_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_201_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_202_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_203_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_204_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_205_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N4 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_206_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_207_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_208_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_209_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_210_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_211_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_212_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_213_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_214_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_215_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_216_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_217_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_218_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_219_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_220_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_221_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_222_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_223_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_224_p_Odegree__zero__imp__splitted,axiom,
! [P2: list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno6259083269128200473t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ).
% p.degree_zero_imp_splitted
thf(fact_225_degree__zero__imp__splitted,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P2 ) ) ) ).
% degree_zero_imp_splitted
thf(fact_226_degree__one__imp__splitted,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P2 ) ) ) ).
% degree_one_imp_splitted
thf(fact_227_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_228_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_229_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_230_pow__divides__pow__iff,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( dvd_dvd_int @ A @ B ) ) ) ).
% pow_divides_pow_iff
thf(fact_231_degree__zero__imp__not__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_232_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_233_p_Odegree__zero__imp__not__is__root,axiom,
! [P2: list_list_a,X: list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ X ) ) ) ).
% p.degree_zero_imp_not_is_root
thf(fact_234_p_Omonoid__cancelI,axiom,
( ! [A3: list_a,B3: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B3 ) )
=> ( ( member_list_a @ A3 @ ( 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 ) ) ) )
=> ( A3 = B3 ) ) ) ) )
=> ( ! [A3: list_a,B3: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B3 @ C2 ) )
=> ( ( member_list_a @ A3 @ ( 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 ) ) ) )
=> ( A3 = B3 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.monoid_cancelI
thf(fact_235_gauss__poly__div__gauss__poly__iff__2,axiom,
! [L: nat,A: int,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( dvd_dvd_int @ ( minus_minus_int @ ( power_power_int @ A @ L ) @ one_one_int ) @ ( minus_minus_int @ ( power_power_int @ A @ M ) @ one_one_int ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ).
% gauss_poly_div_gauss_poly_iff_2
thf(fact_236_is__root__poly__mult__imp__is__root,axiom,
! [P2: list_a,Q: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_237_p_Ois__root__poly__mult__imp__is__root,axiom,
! [P2: list_list_a,Q: list_list_a,X: list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q @ ( 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 ) ) ) ) @ P2 @ Q ) @ X )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ X )
| ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ X ) ) ) ) ) ).
% p.is_root_poly_mult_imp_is_root
thf(fact_238_gcd__nat_Oasym,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ~ ( ( dvd_dvd_nat @ B @ A )
& ( B != A ) ) ) ).
% gcd_nat.asym
thf(fact_239_gcd__nat_Orefl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% gcd_nat.refl
thf(fact_240_gcd__nat_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( dvd_dvd_nat @ A @ C ) ) ) ).
% gcd_nat.trans
thf(fact_241_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A4: nat,B2: nat] :
( ( dvd_dvd_nat @ A4 @ B2 )
& ( dvd_dvd_nat @ B2 @ A4 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_242_gcd__nat_Oirrefl,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ A @ A )
& ( A != A ) ) ).
% gcd_nat.irrefl
thf(fact_243_gcd__nat_Oantisym,axiom,
! [A: nat,B: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ B @ A )
=> ( A = B ) ) ) ).
% gcd_nat.antisym
thf(fact_244_gcd__nat_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_245_gcd__nat_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ B )
=> ( ( ( dvd_dvd_nat @ B @ C )
& ( B != C ) )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_246_gcd__nat_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( ( dvd_dvd_nat @ B @ C )
=> ( ( dvd_dvd_nat @ A @ C )
& ( A != C ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_247_gcd__nat_Ostrict__iff__not,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_248_gcd__nat_Oorder__iff__strict,axiom,
( dvd_dvd_nat
= ( ^ [A4: nat,B2: nat] :
( ( ( dvd_dvd_nat @ A4 @ B2 )
& ( A4 != B2 ) )
| ( A4 = B2 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_249_gcd__nat_Ostrict__iff__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
= ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_250_gcd__nat_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( dvd_dvd_nat @ A @ B ) ) ).
% gcd_nat.strict_implies_order
thf(fact_251_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) )
=> ( A != B ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_252_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( dvd_dvd_nat @ A @ B )
=> ( ( dvd_dvd_nat @ A @ B )
& ( A != B ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_253_field_Ogauss__poly__div__gauss__poly__iff__2,axiom,
! [R: partia2175431115845679010xt_a_b,L: nat,A: int,M: nat] :
( ( field_a_b @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( dvd_dvd_int @ ( minus_minus_int @ ( power_power_int @ A @ L ) @ one_one_int ) @ ( minus_minus_int @ ( power_power_int @ A @ M ) @ one_one_int ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ) ).
% field.gauss_poly_div_gauss_poly_iff_2
thf(fact_254_field_Ogauss__poly__div__gauss__poly__iff__2,axiom,
! [R: partia2670972154091845814t_unit,L: nat,A: int,M: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( dvd_dvd_int @ ( minus_minus_int @ ( power_power_int @ A @ L ) @ one_one_int ) @ ( minus_minus_int @ ( power_power_int @ A @ M ) @ one_one_int ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ) ).
% field.gauss_poly_div_gauss_poly_iff_2
thf(fact_255_field_Ogauss__poly__div__gauss__poly__iff__2,axiom,
! [R: partia2956882679547061052t_unit,L: nat,A: int,M: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( dvd_dvd_int @ ( minus_minus_int @ ( power_power_int @ A @ L ) @ one_one_int ) @ ( minus_minus_int @ ( power_power_int @ A @ M ) @ one_one_int ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ) ).
% field.gauss_poly_div_gauss_poly_iff_2
thf(fact_256_field_Ogauss__poly__div__gauss__poly__iff__2,axiom,
! [R: partia4960592913263135132t_unit,L: nat,A: int,M: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( ord_less_nat @ zero_zero_nat @ L )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ( dvd_dvd_int @ ( minus_minus_int @ ( power_power_int @ A @ L ) @ one_one_int ) @ ( minus_minus_int @ ( power_power_int @ A @ M ) @ one_one_int ) )
= ( dvd_dvd_nat @ L @ M ) ) ) ) ) ).
% field.gauss_poly_div_gauss_poly_iff_2
thf(fact_257_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_258_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_259_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_260_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_261_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_262_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_263_p_Oalg__mult__gt__zero__iff__is__root,axiom,
! [P2: list_list_a,X: list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ X ) )
= ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ X ) ) ) ).
% p.alg_mult_gt_zero_iff_is_root
thf(fact_264_alg__mult__gt__zero__iff__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
= ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_265_p_Odegree__zero__imp__empty__roots,axiom,
! [P2: list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
= zero_z4454100511807792257list_a ) ) ) ).
% p.degree_zero_imp_empty_roots
thf(fact_266_p_Oring__primeI,axiom,
! [P2: list_a] :
( ( P2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ).
% p.ring_primeI
thf(fact_267_p_Oring__primeE_I3_J,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ).
% p.ring_primeE(3)
thf(fact_268_field_Odegree__one__imp__splitted,axiom,
! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( polyno3744827648284794291t_unit @ R @ P2 ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_269_field_Odegree__one__imp__splitted,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( polyno5970451904377802771t_unit @ R @ P2 ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_270_field_Odegree__one__imp__splitted,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( polyno6259083269128200473t_unit @ R @ P2 ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_271_field_Odegree__one__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( polyno8329700637149614481ed_a_b @ R @ P2 ) ) ) ) ).
% field.degree_one_imp_splitted
thf(fact_272_p_Opderiv__const,axiom,
! [X: list_list_a,K2: 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 ) ) @ K2 ) ) ) ) ).
% p.pderiv_const
thf(fact_273_p_Osplitted__on__def,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( polyno1986131841096413848t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P2 )
= ( ( size_s2335926164413107382list_a @ ( polyno5990348478334826338t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P2 ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ).
% p.splitted_on_def
thf(fact_274_p_Oring__primeE_I1_J,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( P2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_primeE(1)
thf(fact_275_p_Opderiv__zero,axiom,
! [K2: 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 ) ) @ K2 ) ) )
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ).
% p.pderiv_zero
thf(fact_276_p_Opderiv__var,axiom,
! [K2: 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 ) ) @ K2 ) ) ) ).
% p.pderiv_var
thf(fact_277_p_Osplitted__def,axiom,
! [P2: list_list_a] :
( ( polyno6259083269128200473t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
= ( ( size_s2335926164413107382list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ).
% p.splitted_def
thf(fact_278_ring_Osplitted__on_Ocong,axiom,
polyno1986131841096413848t_unit = polyno1986131841096413848t_unit ).
% ring.splitted_on.cong
thf(fact_279_ring_Osplitted__on_Ocong,axiom,
polyno2453258491555121552on_a_b = polyno2453258491555121552on_a_b ).
% ring.splitted_on.cong
thf(fact_280_ring_Oroots__on_Ocong,axiom,
polyno5990348478334826338t_unit = polyno5990348478334826338t_unit ).
% ring.roots_on.cong
thf(fact_281_ring_Oroots__on_Ocong,axiom,
polyno5714441830345289050on_a_b = polyno5714441830345289050on_a_b ).
% ring.roots_on.cong
thf(fact_282_ring_Oalg__mult_Ocong,axiom,
polyno4259638811958763678t_unit = polyno4259638811958763678t_unit ).
% ring.alg_mult.cong
thf(fact_283_ring_Oalg__mult_Ocong,axiom,
polyno4422430861927485590lt_a_b = polyno4422430861927485590lt_a_b ).
% ring.alg_mult.cong
thf(fact_284_ring_Oroots_Ocong,axiom,
polyno7858422826990252003t_unit = polyno7858422826990252003t_unit ).
% ring.roots.cong
thf(fact_285_ring_Oroots_Ocong,axiom,
polynomial_roots_a_b = polynomial_roots_a_b ).
% ring.roots.cong
thf(fact_286_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_287_ring_Ois__root_Ocong,axiom,
polyno6951661231331188332t_unit = polyno6951661231331188332t_unit ).
% ring.is_root.cong
thf(fact_288_ring_Osplitted_Ocong,axiom,
polyno6259083269128200473t_unit = polyno6259083269128200473t_unit ).
% ring.splitted.cong
thf(fact_289_ring_Osplitted_Ocong,axiom,
polyno8329700637149614481ed_a_b = polyno8329700637149614481ed_a_b ).
% ring.splitted.cong
thf(fact_290_p_Opirreducible__roots,axiom,
! [P2: list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( 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 ) ) ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
= zero_z4454100511807792257list_a ) ) ) ) ).
% p.pirreducible_roots
thf(fact_291_size__empty,axiom,
( ( size_s2335926164413107382list_a @ zero_z4454100511807792257list_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_292_size__empty,axiom,
( ( size_size_multiset_a @ zero_zero_multiset_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_293_size__eq__0__iff__empty,axiom,
! [M3: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ M3 )
= zero_zero_nat )
= ( M3 = zero_z4454100511807792257list_a ) ) ).
% size_eq_0_iff_empty
thf(fact_294_size__eq__0__iff__empty,axiom,
! [M3: multiset_a] :
( ( ( size_size_multiset_a @ M3 )
= zero_zero_nat )
= ( M3 = zero_zero_multiset_a ) ) ).
% size_eq_0_iff_empty
thf(fact_295_int_Onat__pow__0,axiom,
! [X: int] :
( ( power_power_int @ X @ zero_zero_nat )
= one_one_int ) ).
% int.nat_pow_0
thf(fact_296_p_Opolynomial__pow__degree,axiom,
! [P2: list_list_a,N: nat] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( 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 ) ) ) ) @ P2 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% p.polynomial_pow_degree
thf(fact_297_pderiv__const,axiom,
! [X: list_a,K2: 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 @ K2 ) ) ) ) ).
% pderiv_const
thf(fact_298_polynomial__pow__degree,axiom,
! [P2: list_a,N: nat] :
( ( member_list_a @ P2 @ ( 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 ) ) @ P2 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% polynomial_pow_degree
thf(fact_299_degree__zero__imp__empty__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= zero_zero_nat )
=> ( ( polynomial_roots_a_b @ r @ P2 )
= zero_zero_multiset_a ) ) ) ).
% degree_zero_imp_empty_roots
thf(fact_300_splitted__on__def,axiom,
! [K2: set_a,P2: list_a] :
( ( polyno2453258491555121552on_a_b @ r @ K2 @ P2 )
= ( ( size_size_multiset_a @ ( polyno5714441830345289050on_a_b @ r @ K2 @ P2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).
% splitted_on_def
thf(fact_301_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_302_pderiv__zero,axiom,
! [K2: set_a] :
( ( formal4452980811800949548iv_a_b @ r @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% pderiv_zero
thf(fact_303_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_304_pderiv__var,axiom,
! [K2: set_a] :
( ( formal4452980811800949548iv_a_b @ r @ ( var_a_b @ r ) )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ K2 ) ) ) ).
% pderiv_var
thf(fact_305_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_306_splitted__def,axiom,
! [P2: list_a] :
( ( polyno8329700637149614481ed_a_b @ r @ P2 )
= ( ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P2 ) )
= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).
% splitted_def
thf(fact_307_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_308_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_309_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_310_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_311_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_312_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_313_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_314_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_315_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_316_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_317_int_Onat__pow__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% int.nat_pow_one
thf(fact_318_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_319_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_320_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_321_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_322_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A4: nat,B2: nat] : ( times_times_nat @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_323_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A4: int,B2: int] : ( times_times_int @ B2 @ A4 ) ) ) ).
% mult.commute
thf(fact_324_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_325_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_326_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_327_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_328_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_329_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_330_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_331_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_332_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_333_power__commuting__commutes,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= ( times_times_int @ Y @ X ) )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
= ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).
% power_commuting_commutes
thf(fact_334_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_335_power__mult__distrib,axiom,
! [A: int,B: int,N: nat] :
( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
= ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).
% power_mult_distrib
thf(fact_336_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_337_power__commutes,axiom,
! [A: int,N: nat] :
( ( times_times_int @ ( power_power_int @ A @ N ) @ A )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_commutes
thf(fact_338_dvd__productE,axiom,
! [P2: nat,A: nat,B: nat] :
( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A @ B ) )
=> ~ ! [X3: nat,Y4: nat] :
( ( P2
= ( times_times_nat @ X3 @ Y4 ) )
=> ( ( dvd_dvd_nat @ X3 @ A )
=> ~ ( dvd_dvd_nat @ Y4 @ B ) ) ) ) ).
% dvd_productE
thf(fact_339_dvd__productE,axiom,
! [P2: int,A: int,B: int] :
( ( dvd_dvd_int @ P2 @ ( times_times_int @ A @ B ) )
=> ~ ! [X3: int,Y4: int] :
( ( P2
= ( times_times_int @ X3 @ Y4 ) )
=> ( ( dvd_dvd_int @ X3 @ A )
=> ~ ( dvd_dvd_int @ Y4 @ B ) ) ) ) ).
% dvd_productE
thf(fact_340_division__decomp,axiom,
! [A: nat,B: nat,C: nat] :
( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
=> ? [B4: nat,C3: nat] :
( ( A
= ( times_times_nat @ B4 @ C3 ) )
& ( dvd_dvd_nat @ B4 @ B )
& ( dvd_dvd_nat @ C3 @ C ) ) ) ).
% division_decomp
thf(fact_341_division__decomp,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
=> ? [B4: int,C3: int] :
( ( A
= ( times_times_int @ B4 @ C3 ) )
& ( dvd_dvd_int @ B4 @ B )
& ( dvd_dvd_int @ C3 @ C ) ) ) ).
% division_decomp
thf(fact_342_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_343_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_344_power__mult,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( times_times_nat @ M @ N ) )
= ( power_power_int @ ( power_power_int @ A @ M ) @ N ) ) ).
% power_mult
thf(fact_345_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_346_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_347_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_348_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_349_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_350_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_351_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_352_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_353_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_354_bezout1__nat,axiom,
! [A: nat,B: nat] :
? [D2: nat,X3: nat,Y4: nat] :
( ( dvd_dvd_nat @ D2 @ A )
& ( dvd_dvd_nat @ D2 @ B )
& ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y4 ) )
= D2 )
| ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y4 ) )
= D2 ) ) ) ).
% bezout1_nat
thf(fact_355_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_356_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_357_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_358_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_359_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_360_int_Ozero__not__one,axiom,
zero_zero_int != one_one_int ).
% int.zero_not_one
thf(fact_361_diff__empty,axiom,
! [M3: multiset_list_a] :
( ( ( minus_7431248565939055793list_a @ M3 @ zero_z4454100511807792257list_a )
= M3 )
& ( ( minus_7431248565939055793list_a @ zero_z4454100511807792257list_a @ M3 )
= zero_z4454100511807792257list_a ) ) ).
% diff_empty
thf(fact_362_diff__empty,axiom,
! [M3: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ M3 @ zero_zero_multiset_a )
= M3 )
& ( ( minus_3765977307040488491iset_a @ zero_zero_multiset_a @ M3 )
= zero_zero_multiset_a ) ) ).
% diff_empty
thf(fact_363_Multiset_Odiff__cancel,axiom,
! [A2: multiset_list_a] :
( ( minus_7431248565939055793list_a @ A2 @ A2 )
= zero_z4454100511807792257list_a ) ).
% Multiset.diff_cancel
thf(fact_364_Multiset_Odiff__cancel,axiom,
! [A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ A2 @ A2 )
= zero_zero_multiset_a ) ).
% Multiset.diff_cancel
thf(fact_365_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_366_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_367_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_368_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_369_power__eq__if,axiom,
( power_power_nat
= ( ^ [P3: nat,M4: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P3 @ ( power_power_nat @ P3 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_370_power__eq__if,axiom,
( power_power_int
= ( ^ [P3: int,M4: nat] : ( if_int @ ( M4 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P3 @ ( power_power_int @ P3 @ ( minus_minus_nat @ M4 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_371_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_372_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_373_int_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% int.nat_pow_zero
thf(fact_374_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( ring_r97889109428395874t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno3744827648284794291t_unit @ R @ P2 ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_375_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno5970451904377802771t_unit @ R @ P2 ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_376_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno6259083269128200473t_unit @ R @ P2 ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_377_field_Opirreducible__imp__not__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno8329700637149614481ed_a_b @ R @ P2 ) ) ) ) ) ).
% field.pirreducible_imp_not_splitted
thf(fact_378_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_379_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_380_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_381_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_382_pirreducible__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ( ( polynomial_roots_a_b @ r @ P2 )
= zero_zero_multiset_a ) ) ) ) ).
% pirreducible_roots
thf(fact_383_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_384_unit__prod,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% unit_prod
thf(fact_385_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_386_dvd__times__right__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_387_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_388_dvd__times__left__cancel__iff,axiom,
! [A: int,B: int,C: int] :
( ( A != zero_zero_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_389_dvd__mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_right
thf(fact_390_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_391_p_Oprimeness__condition,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
= ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ).
% p.primeness_condition
thf(fact_392_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_393_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_394_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_395_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_396_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_397_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_398_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_399_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_400_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_401_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_402_dvd__0__left__iff,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% dvd_0_left_iff
thf(fact_403_dvd__0__left__iff,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
= ( A = zero_zero_int ) ) ).
% dvd_0_left_iff
thf(fact_404_dvd__0__right,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% dvd_0_right
thf(fact_405_dvd__0__right,axiom,
! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% dvd_0_right
thf(fact_406_degree__one__imp__pirreducible,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ).
% degree_one_imp_pirreducible
thf(fact_407_pirreducible__imp__not__splitted,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno8329700637149614481ed_a_b @ r @ P2 ) ) ) ) ).
% pirreducible_imp_not_splitted
thf(fact_408_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_409_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_410_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_411_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_412_dvd__mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( dvd_dvd_int @ A @ B ) ) ) ).
% dvd_mult_cancel_left
thf(fact_413_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_414_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_415_dvd__trans,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ C )
=> ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_trans
thf(fact_416_dvd__refl,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% dvd_refl
thf(fact_417_dvd__refl,axiom,
! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% dvd_refl
thf(fact_418_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_419_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_420_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_421_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_422_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_423_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_424_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_425_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_426_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_427_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_428_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_429_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_430_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_431_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_432_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_433_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_434_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_435_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_436_dvd__0__left,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% dvd_0_left
thf(fact_437_dvd__0__left,axiom,
! [A: int] :
( ( dvd_dvd_int @ zero_zero_int @ A )
=> ( A = zero_zero_int ) ) ).
% dvd_0_left
thf(fact_438_one__dvd,axiom,
! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% one_dvd
thf(fact_439_one__dvd,axiom,
! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% one_dvd
thf(fact_440_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_441_unit__imp__dvd,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ B @ A ) ) ).
% unit_imp_dvd
thf(fact_442_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_443_dvd__unit__imp__unit,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ B @ one_one_int )
=> ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% dvd_unit_imp_unit
thf(fact_444_dvd__triv__right,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% dvd_triv_right
thf(fact_445_dvd__triv__right,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% dvd_triv_right
thf(fact_446_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_447_dvd__mult__right,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ B @ C ) ) ).
% dvd_mult_right
thf(fact_448_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_449_mult__dvd__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( dvd_dvd_int @ A @ B )
=> ( ( dvd_dvd_int @ C @ D )
=> ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% mult_dvd_mono
thf(fact_450_dvd__triv__left,axiom,
! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% dvd_triv_left
thf(fact_451_dvd__triv__left,axiom,
! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% dvd_triv_left
thf(fact_452_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_453_dvd__mult__left,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
=> ( dvd_dvd_int @ A @ C ) ) ).
% dvd_mult_left
thf(fact_454_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_455_dvd__mult2,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ B )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult2
thf(fact_456_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_457_dvd__mult,axiom,
! [A: int,C: int,B: int] :
( ( dvd_dvd_int @ A @ C )
=> ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% dvd_mult
thf(fact_458_dvd__def,axiom,
( dvd_dvd_nat
= ( ^ [B2: nat,A4: nat] :
? [K4: nat] :
( A4
= ( times_times_nat @ B2 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_459_dvd__def,axiom,
( dvd_dvd_int
= ( ^ [B2: int,A4: int] :
? [K4: int] :
( A4
= ( times_times_int @ B2 @ K4 ) ) ) ) ).
% dvd_def
thf(fact_460_dvdI,axiom,
! [A: nat,B: nat,K: nat] :
( ( A
= ( times_times_nat @ B @ K ) )
=> ( dvd_dvd_nat @ B @ A ) ) ).
% dvdI
thf(fact_461_dvdI,axiom,
! [A: int,B: int,K: int] :
( ( A
= ( times_times_int @ B @ K ) )
=> ( dvd_dvd_int @ B @ A ) ) ).
% dvdI
thf(fact_462_dvdE,axiom,
! [B: nat,A: nat] :
( ( dvd_dvd_nat @ B @ A )
=> ~ ! [K5: nat] :
( A
!= ( times_times_nat @ B @ K5 ) ) ) ).
% dvdE
thf(fact_463_dvdE,axiom,
! [B: int,A: int] :
( ( dvd_dvd_int @ B @ A )
=> ~ ! [K5: int] :
( A
!= ( times_times_int @ B @ K5 ) ) ) ).
% dvdE
thf(fact_464_dvd__diff,axiom,
! [X: int,Y: int,Z: int] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( dvd_dvd_int @ X @ Z )
=> ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z ) ) ) ) ).
% dvd_diff
thf(fact_465_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_466_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_467_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_468_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_469_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_470_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_471_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_472_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_473_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_474_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_475_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_476_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_477_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_478_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_479_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_480_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_481_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_482_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_483_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_484_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_485_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_486_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_487_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_488_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_489_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_490_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_491_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_492_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_493_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_494_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_495_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_496_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_497_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_498_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_499_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_500_not__is__unit__0,axiom,
~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% not_is_unit_0
thf(fact_501_not__is__unit__0,axiom,
~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% not_is_unit_0
thf(fact_502_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_503_is__unit__mult__iff,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
= ( ( dvd_dvd_int @ A @ one_one_int )
& ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% is_unit_mult_iff
thf(fact_504_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_505_dvd__mult__unit__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff
thf(fact_506_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_507_mult__unit__dvd__iff,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% mult_unit_dvd_iff
thf(fact_508_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_509_dvd__mult__unit__iff_H,axiom,
! [B: int,A: int,C: int] :
( ( dvd_dvd_int @ B @ one_one_int )
=> ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
= ( dvd_dvd_int @ A @ C ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_510_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_511_mult__unit__dvd__iff_H,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
= ( dvd_dvd_int @ B @ C ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_512_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_513_unit__mult__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ A @ B )
= ( times_times_int @ A @ C ) )
= ( B = C ) ) ) ).
% unit_mult_left_cancel
thf(fact_514_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_515_unit__mult__right__cancel,axiom,
! [A: int,B: int,C: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ( ( ( times_times_int @ B @ A )
= ( times_times_int @ C @ A ) )
= ( B = C ) ) ) ).
% unit_mult_right_cancel
thf(fact_516_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_517_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_518_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_519_unit__dvdE,axiom,
! [A: int,B: int] :
( ( dvd_dvd_int @ A @ one_one_int )
=> ~ ( ( A != zero_zero_int )
=> ! [C2: int] :
( B
!= ( times_times_int @ A @ C2 ) ) ) ) ).
% unit_dvdE
thf(fact_520_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( dvd_dvd_nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_521_splitted__imp__trivial__factors,axiom,
! [P2: list_a,Q: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ P2 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q @ P2 )
=> ( ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ).
% splitted_imp_trivial_factors
thf(fact_522_trivial__factors__imp__splitted,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( 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 @ P2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno8329700637149614481ed_a_b @ r @ P2 ) ) ) ).
% trivial_factors_imp_splitted
thf(fact_523_size__roots__le__degree,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ ( polynomial_roots_a_b @ r @ P2 ) ) @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).
% size_roots_le_degree
thf(fact_524_p_Osubring__polynomial__pow__degree,axiom,
! [K2: set_list_a,P2: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% p.subring_polynomial_pow_degree
thf(fact_525_pirreducible__degree,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% pirreducible_degree
thf(fact_526_no__roots__imp__same__roots,axiom,
! [P2: list_a,Q: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( polynomial_roots_a_b @ r @ P2 )
= zero_zero_multiset_a )
=> ( ( polynomial_roots_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) )
= ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% no_roots_imp_same_roots
thf(fact_527_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_528_zero__pdivides,axiom,
! [P2: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P2 )
= ( P2 = nil_a ) ) ).
% zero_pdivides
thf(fact_529_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_530_p_Ouniv__poly__not__field,axiom,
! [K2: set_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ~ ( field_1861437471013600865t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ).
% p.univ_poly_not_field
thf(fact_531_polynomial__pow__not__zero,axiom,
! [P2: list_a,N: nat] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_532_pdivides__zero,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P2 @ nil_a ) ) ).
% pdivides_zero
thf(fact_533_p_Ouniv__poly__a__minus__consistent,axiom,
! [K2: set_list_a,Q: list_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 @ Q )
= ( 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 ) ) ) ) @ P2 @ Q ) ) ) ) ).
% p.univ_poly_a_minus_consistent
thf(fact_534_polynomial__pow__division,axiom,
! [P2: list_a,N: nat,M: nat] :
( ( member_list_a @ P2 @ ( 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 ) ) @ P2 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ M ) ) ) ) ).
% polynomial_pow_division
thf(fact_535_p_Ovar__closed_I1_J,axiom,
! [K2: set_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) ) ) ) ).
% p.var_closed(1)
thf(fact_536_p_Opderiv__carr,axiom,
! [K2: set_list_a,F: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( 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 ) ) @ K2 ) ) ) ) ) ).
% p.pderiv_carr
thf(fact_537_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_538_pdivides__imp__splitted,axiom,
! [P2: list_a,Q: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
=> ( polyno8329700637149614481ed_a_b @ r @ P2 ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_539_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_540_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_541_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_542_p_Opolynomial__pow__division,axiom,
! [P2: list_list_a,N: nat,M: nat] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( 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 ) ) ) ) @ P2 @ 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 ) ) ) ) @ P2 @ M ) ) ) ) ).
% p.polynomial_pow_division
thf(fact_543_p_Ovar__pow__closed,axiom,
! [K2: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) @ ( 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 ) ) @ K2 ) ) ) ) ).
% p.var_pow_closed
thf(fact_544_pdivides__imp__degree__le,axiom,
! [P2: list_a,Q: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ).
% pdivides_imp_degree_le
thf(fact_545_p_Ovar__pow__degree,axiom,
! [K2: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) ) @ one_one_nat )
= N ) ) ).
% p.var_pow_degree
thf(fact_546_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_547_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_548_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_549_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_550_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_551_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_552_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_553_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_554_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_555_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_556_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_557_p_Ocarrier__polynomial__shell,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.carrier_polynomial_shell
thf(fact_558_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_559_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_560_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_561_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_562_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_563_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_564_power__increasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_565_power__increasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% power_increasing
thf(fact_566_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_567_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_568_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_569_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_570_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_571_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_572_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_573_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_574_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_575_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_576_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_577_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_578_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_579_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_580_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_581_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_582_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_583_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_584_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_585_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_586_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_587_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_588_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_589_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_590_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_591_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_592_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_593_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_594_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_595_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_596_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_597_power__decreasing,axiom,
! [N: nat,N2: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_598_power__decreasing,axiom,
! [N: nat,N2: nat,A: int] :
( ( ord_less_eq_nat @ N @ N2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_599_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_600_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_601_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_602_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_603_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_604_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_605_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_606_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_607_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_608_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_609_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_610_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_611_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_612_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_613_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_614_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_615_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_616_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_617_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_618_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_619_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_620_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_621_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_622_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_623_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_624_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_625_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_626_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_627_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_628_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_629_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_630_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_631_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_632_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_633_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_634_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_635_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_636_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_637_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_638_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_639_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_640_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_641_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K5: nat] :
( ( ord_less_eq_nat @ K5 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K5 )
=> ~ ( P @ I3 ) )
& ( P @ K5 ) ) ) ) ).
% ex_least_nat_le
thf(fact_642_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_643_dvd__power__le,axiom,
! [X: int,Y: int,N: nat,M: nat] :
( ( dvd_dvd_int @ X @ Y )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M ) ) ) ) ).
% dvd_power_le
thf(fact_644_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_645_power__le__dvd,axiom,
! [A: int,N: nat,B: int,M: nat] :
( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% power_le_dvd
thf(fact_646_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_647_le__imp__power__dvd,axiom,
! [M: nat,N: nat,A: int] :
( ( ord_less_eq_nat @ M @ N )
=> ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% le_imp_power_dvd
thf(fact_648_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_649_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_650_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_651_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_652_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_653_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_654_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_655_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_656_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_657_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_658_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_659_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_660_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_661_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_662_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_663_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_664_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_665_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_666_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_667_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_668_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_669_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_670_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_671_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_672_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_673_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_674_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_675_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_676_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_677_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_678_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_679_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_680_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_681_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_682_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_683_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_684_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_685_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_686_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_687_diff__size__le__size__Diff,axiom,
! [M3: multiset_list_a,M5: multiset_list_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2335926164413107382list_a @ M3 ) @ ( size_s2335926164413107382list_a @ M5 ) ) @ ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M3 @ M5 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_688_diff__size__le__size__Diff,axiom,
! [M3: multiset_a,M5: multiset_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_multiset_a @ M3 ) @ ( size_size_multiset_a @ M5 ) ) @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M3 @ M5 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_689_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_690_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_691_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_692_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_693_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_694_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_695_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_696_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_697_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_698_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_699_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_700_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_701_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_702_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_703_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_704_dvd__power__iff,axiom,
! [X: int,M: nat,N: nat] :
( ( X != zero_zero_int )
=> ( ( dvd_dvd_int @ ( power_power_int @ X @ M ) @ ( power_power_int @ X @ N ) )
= ( ( dvd_dvd_int @ X @ one_one_int )
| ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% dvd_power_iff
thf(fact_705_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_706_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_707_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_708_field_Opdivides__imp__splitted,axiom,
! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a,Q: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( member6124916891863447321list_a @ Q @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( Q != nil_set_list_list_a )
=> ( ( polyno3744827648284794291t_unit @ R @ Q )
=> ( ( polyno3637028486239637860t_unit @ R @ P2 @ Q )
=> ( polyno3744827648284794291t_unit @ R @ P2 ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_709_field_Opdivides__imp__splitted,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a,Q: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( Q != nil_list_list_a )
=> ( ( polyno5970451904377802771t_unit @ R @ Q )
=> ( ( polyno4453881341673752516t_unit @ R @ P2 @ Q )
=> ( polyno5970451904377802771t_unit @ R @ P2 ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_710_field_Opdivides__imp__splitted,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6259083269128200473t_unit @ R @ Q )
=> ( ( polyno8016796738000020810t_unit @ R @ P2 @ Q )
=> ( polyno6259083269128200473t_unit @ R @ P2 ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_711_field_Opdivides__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ R @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R @ P2 @ Q )
=> ( polyno8329700637149614481ed_a_b @ R @ P2 ) ) ) ) ) ) ) ).
% field.pdivides_imp_splitted
thf(fact_712_field_Ono__roots__imp__same__roots,axiom,
! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a,Q: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( P2 != nil_set_list_list_a )
=> ( ( member6124916891863447321list_a @ Q @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( ( polyno2127442156181624701t_unit @ R @ P2 )
= zero_z6145066983645916903list_a )
=> ( ( polyno2127442156181624701t_unit @ R @ ( mult_l7436655221470123345t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ P2 @ Q ) )
= ( polyno2127442156181624701t_unit @ R @ Q ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_713_field_Ono__roots__imp__same__roots,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a,Q: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P2 != nil_list_list_a )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ( polyno3707469075594375645t_unit @ R @ P2 )
= zero_z1542645121299710087list_a )
=> ( ( polyno3707469075594375645t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P2 @ Q ) )
= ( polyno3707469075594375645t_unit @ R @ Q ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_714_field_Ono__roots__imp__same__roots,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P2 != nil_list_a )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ( polyno7858422826990252003t_unit @ R @ P2 )
= zero_z4454100511807792257list_a )
=> ( ( polyno7858422826990252003t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P2 @ Q ) )
= ( polyno7858422826990252003t_unit @ R @ Q ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_715_field_Ono__roots__imp__same__roots,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ( polynomial_roots_a_b @ R @ P2 )
= zero_zero_multiset_a )
=> ( ( polynomial_roots_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P2 @ Q ) )
= ( polynomial_roots_a_b @ R @ Q ) ) ) ) ) ) ) ).
% field.no_roots_imp_same_roots
thf(fact_716_field_Osize__roots__le__degree,axiom,
! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ord_less_eq_nat @ ( size_s82858050752783516list_a @ ( polyno2127442156181624701t_unit @ R @ P2 ) ) @ ( minus_minus_nat @ ( size_s618615678312925148list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_717_field_Osize__roots__le__degree,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ord_less_eq_nat @ ( size_s8523483970790017596list_a @ ( polyno3707469075594375645t_unit @ R @ P2 ) ) @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_718_field_Osize__roots__le__degree,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ord_less_eq_nat @ ( size_s2335926164413107382list_a @ ( polyno7858422826990252003t_unit @ R @ P2 ) ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_719_field_Osize__roots__le__degree,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ ( polynomial_roots_a_b @ R @ P2 ) ) @ ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% field.size_roots_le_degree
thf(fact_720_field_Otrivial__factors__imp__splitted,axiom,
! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P2 @ ( 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 @ P2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s618615678312925148list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno3744827648284794291t_unit @ R @ P2 ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_721_field_Otrivial__factors__imp__splitted,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( 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 @ P2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno5970451904377802771t_unit @ R @ P2 ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_722_field_Otrivial__factors__imp__splitted,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( 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 @ P2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno6259083269128200473t_unit @ R @ P2 ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_723_field_Otrivial__factors__imp__splitted,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P2 @ ( 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 @ P2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Q2 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno8329700637149614481ed_a_b @ R @ P2 ) ) ) ) ).
% field.trivial_factors_imp_splitted
thf(fact_724_field_Osplitted__imp__trivial__factors,axiom,
! [R: partia4960592913263135132t_unit,P2: list_set_list_list_a,Q: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( member6124916891863447321list_a @ P2 @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( P2 != nil_set_list_list_a )
=> ( ( polyno3744827648284794291t_unit @ R @ P2 )
=> ( ( member6124916891863447321list_a @ Q @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) )
=> ( ( ring_r97889109428395874t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ Q )
=> ( ( polyno3637028486239637860t_unit @ R @ Q @ P2 )
=> ( ( minus_minus_nat @ ( size_s618615678312925148list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_725_field_Osplitted__imp__trivial__factors,axiom,
! [R: partia2956882679547061052t_unit,P2: list_list_list_a,Q: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( member5342144027231129785list_a @ P2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( P2 != nil_list_list_a )
=> ( ( polyno5970451904377802771t_unit @ R @ P2 )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ Q )
=> ( ( polyno4453881341673752516t_unit @ R @ Q @ P2 )
=> ( ( minus_minus_nat @ ( size_s2403821588304063868list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_726_field_Osplitted__imp__trivial__factors,axiom,
! [R: partia2670972154091845814t_unit,P2: list_list_a,Q: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( P2 != nil_list_a )
=> ( ( polyno6259083269128200473t_unit @ R @ P2 )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ Q )
=> ( ( polyno8016796738000020810t_unit @ R @ Q @ P2 )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_727_field_Osplitted__imp__trivial__factors,axiom,
! [R: partia2175431115845679010xt_a_b,P2: list_a,Q: list_a] :
( ( field_a_b @ R )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ R @ P2 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ Q )
=> ( ( polyno5814909790663948098es_a_b @ R @ Q @ P2 )
=> ( ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ) ).
% field.splitted_imp_trivial_factors
thf(fact_728_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_729_p_Ounitary__monom__eq__var__pow,axiom,
! [K2: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) ) ) ).
% p.unitary_monom_eq_var_pow
thf(fact_730_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_731_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_732_p_Opdivides__imp__degree__le,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% p.pdivides_imp_degree_le
thf(fact_733_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_734_length__0__conv,axiom,
! [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_a ) ) ).
% length_0_conv
thf(fact_735_p_Ozero__pdivides,axiom,
! [P2: list_list_a] :
( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P2 )
= ( P2 = nil_list_a ) ) ).
% p.zero_pdivides
thf(fact_736_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_737_p_Opolynomial__pow__not__zero,axiom,
! [P2: list_list_a,N: nat] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P2 != 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 ) ) ) ) @ P2 @ N )
!= nil_list_a ) ) ) ).
% p.polynomial_pow_not_zero
thf(fact_738_p_Opdivides__zero,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ nil_list_a ) ) ) ).
% p.pdivides_zero
thf(fact_739_p_Osubring__polynomial__pow__not__zero,axiom,
! [K2: set_list_a,P2: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( P2 != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 @ N )
!= nil_list_a ) ) ) ) ).
% p.subring_polynomial_pow_not_zero
thf(fact_740_p_OpirreducibleE_I1_J,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 )
=> ( P2 != nil_list_a ) ) ) ) ).
% p.pirreducibleE(1)
thf(fact_741_int_Olless__eq,axiom,
( ord_less_int
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ( X2 != Y6 ) ) ) ) ).
% int.lless_eq
thf(fact_742_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_743_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_744_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_745_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_a] :
( ( size_s349497388124573686list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_746_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_747_length__induct,axiom,
! [P: list_list_a > $o,Xs: list_list_a] :
( ! [Xs2: list_list_a] :
( ! [Ys2: list_list_a] :
( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ys2 ) @ ( size_s349497388124573686list_a @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_748_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_749_list_Osize_I3_J,axiom,
( ( size_s349497388124573686list_a @ nil_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_750_p_Odegree__oneE,axiom,
! [P2: list_list_a,K2: set_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A3: list_a] :
( ( member_list_a @ A3 @ K2 )
=> ( ( A3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ! [B3: list_a] :
( ( member_list_a @ B3 @ K2 )
=> ( P2
!= ( cons_list_a @ A3 @ ( cons_list_a @ B3 @ nil_list_a ) ) ) ) ) ) ) ) ).
% p.degree_oneE
thf(fact_751_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_752_p_Oconst__term__simprules__shell_I2_J,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ P2 @ Q ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ) ).
% p.const_term_simprules_shell(2)
thf(fact_753_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K2 ) ) ) ).
% univ_poly_zero_closed
thf(fact_754_univ__poly__zero__closed,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) ) ) ).
% univ_poly_zero_closed
thf(fact_755_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_756_p_Oconst__term__not__zero,axiom,
! [P2: list_list_a] :
( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P2 != nil_list_a ) ) ).
% p.const_term_not_zero
thf(fact_757_p_Oconst__term__simprules__shell_I1_J,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) @ K2 ) ) ) ).
% p.const_term_simprules_shell(1)
thf(fact_758_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_759_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_760_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_761_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_list_a,P: list_list_a > list_list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y4: list_a,Ys3: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_762_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_763_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys3: list_a,Z3: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_764_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_765_list__induct3,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a,Z3: list_a,Zs2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_766_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,P: 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 ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_767_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,P: 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 ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a,Z3: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_768_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_a,P: 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 ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y4: list_a,Ys3: list_list_a,Z3: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_769_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_list_a,Zs: list_list_a,P: 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 ) )
=> ( ( P @ nil_list_a @ nil_list_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y4: list_a,Ys3: list_list_a,Z3: list_a,Zs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_770_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a,W2: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_771_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a,W2: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_list_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_772_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys3: list_a,Z3: list_a,Zs2: list_list_a,W2: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_773_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a,Z3: a,Zs2: list_a,W2: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_774_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: 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 ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a,W2: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_775_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_list_a,Ws: list_list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_a @ nil_list_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y4: a,Ys3: list_a,Z3: list_a,Zs2: list_list_a,W2: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z3 @ Zs2 ) @ ( cons_list_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_776_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_a,Ws: list_list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a,Z3: a,Zs2: list_a,W2: list_a,Ws2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_list_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_777_list__induct4,axiom,
! [Xs: list_a,Ys: list_list_a,Zs: list_list_a,Ws: list_a,P: 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 ) )
=> ( ( P @ nil_a @ nil_list_a @ nil_list_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y4: list_a,Ys3: list_list_a,Z3: list_a,Zs2: list_list_a,W2: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys3 ) )
=> ( ( ( size_s349497388124573686list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_778_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,Ws: list_list_a,P: 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 ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_a @ nil_list_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a,W2: list_a,Ws2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_s349497388124573686list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_list_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_779_list__induct4,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_list_a,Ws: list_a,P: 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 ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y4: a,Ys3: list_a,Z3: list_a,Zs2: list_list_a,W2: a,Ws2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_s349497388124573686list_a @ Zs2 ) )
=> ( ( ( size_s349497388124573686list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_list_a @ Z3 @ Zs2 ) @ ( cons_a @ W2 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_780_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_781_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_782_univ__poly__one,axiom,
! [R: partia2956882679547061052t_unit,K2: set_list_list_a] :
( ( one_li8923720976704309949t_unit @ ( univ_p2250591967980070728t_unit @ R @ K2 ) )
= ( cons_list_list_a @ ( one_li8234411390022467901t_unit @ R ) @ nil_list_list_a ) ) ).
% univ_poly_one
thf(fact_783_univ__poly__one,axiom,
! [R: partia4960592913263135132t_unit,K2: set_set_list_list_a] :
( ( one_li3223383766543049405t_unit @ ( univ_p7077926387201515752t_unit @ R @ K2 ) )
= ( cons_set_list_list_a @ ( one_se2489417650821308733t_unit @ R ) @ nil_set_list_list_a ) ) ).
% univ_poly_one
thf(fact_784_univ__poly__one,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ K2 ) )
= ( cons_a @ ( one_a_ring_ext_a_b @ R ) @ nil_a ) ) ).
% univ_poly_one
thf(fact_785_univ__poly__one,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) )
= ( cons_list_a @ ( one_li8328186300101108157t_unit @ R ) @ nil_list_a ) ) ).
% univ_poly_one
thf(fact_786_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_787_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_788_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_789_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_790_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_791_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_792_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_793_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_794_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_795_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_796_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_797_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_798_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_799_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_800_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_801_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_802_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K2: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K2 ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_803_univ__poly__zero,axiom,
! [R: partia2670972154091845814t_unit,K2: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K2 ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_804_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_805_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_806_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_807_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd_int @ Z @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_808_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_809_p_Oalg__multE_I2_J,axiom,
! [X: list_a,P2: list_list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P2 != 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 ) @ P2 )
=> ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ X ) ) ) ) ) ) ).
% p.alg_multE(2)
thf(fact_810_p_Ole__alg__mult__imp__pdivides,axiom,
! [X: list_a,P2: list_list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( 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 ) ) @ P2 @ 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 ) @ P2 ) ) ) ) ).
% p.le_alg_mult_imp_pdivides
thf(fact_811_p_Oalg__multE_I1_J,axiom,
! [X: list_a,P2: list_list_a] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P2 != 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 ) ) @ P2 @ X ) ) @ P2 ) ) ) ) ).
% p.alg_multE(1)
thf(fact_812_p_Opoly__mult__var,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( P2 = nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= nil_list_a ) )
& ( ( P2 != nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( append_list_a @ P2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ) ).
% p.poly_mult_var
thf(fact_813_normalize_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [V: a,Va: list_a] :
( X
!= ( cons_a @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_814_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_815_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_816_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_817_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_818_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_819_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_820_degree__oneE,axiom,
! [P2: list_a,K2: set_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ K2 )
=> ( ( A3
!= ( zero_a_b @ r ) )
=> ! [B3: a] :
( ( member_a @ B3 @ K2 )
=> ( P2
!= ( cons_a @ A3 @ ( cons_a @ B3 @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_821_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_822_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_823_append__eq__append__conv,axiom,
! [Xs: list_list_a,Ys: list_list_a,Us: list_list_a,Vs: list_list_a] :
( ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
| ( ( size_s349497388124573686list_a @ Us )
= ( size_s349497388124573686list_a @ Vs ) ) )
=> ( ( ( append_list_a @ Xs @ Us )
= ( append_list_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_824_univ__poly__a__inv__degree,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( 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 ) ) @ P2 ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_size_list_a @ P2 ) @ one_one_nat ) ) ) ).
% univ_poly_a_inv_degree
thf(fact_825_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_826_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_827_p_Opdivides__imp__is__root,axiom,
! [P2: list_list_a,X: list_a] :
( ( P2 != 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 ) ) @ P2 )
=> ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ X ) ) ) ) ).
% p.pdivides_imp_is_root
thf(fact_828_p_Ois__root__imp__pdivides,axiom,
! [P2: list_list_a,X: list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ 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 ) ) @ P2 ) ) ) ).
% p.is_root_imp_pdivides
thf(fact_829_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_830_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_831_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_832_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_833_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,Y4: a,Ys4: list_a] :
( ( X3 != Y4 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs3 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_834_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,Y4: list_a,Ys4: list_list_a] :
( ( X3 != Y4 )
& ( 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 @ Y4 @ nil_list_a ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_835_p_Oroots__inclI,axiom,
! [P2: list_list_a,Q: list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q @ ( 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 ) ) ) ) ) )
=> ( ( Q != nil_list_a )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != 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 ) ) @ A3 ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ A3 ) ) @ Q ) ) )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ) ).
% p.roots_inclI
thf(fact_836_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_837_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_838_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_839_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_840_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_841_p_Onot__empty__rootsE,axiom,
! [P2: list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
!= zero_z4454100511807792257list_a )
=> ~ ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A3 @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) )
=> ( ( 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 ) ) @ A3 ) @ 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 ) ) @ A3 ) @ nil_list_a ) ) @ P2 ) ) ) ) ) ) ).
% p.not_empty_rootsE
thf(fact_842_subset__mset_Odual__order_Orefl,axiom,
! [A: multiset_list_a] : ( subseteq_mset_list_a @ A @ A ) ).
% subset_mset.dual_order.refl
thf(fact_843_subset__mset_Odual__order_Orefl,axiom,
! [A: multiset_a] : ( subseteq_mset_a @ A @ A ) ).
% subset_mset.dual_order.refl
thf(fact_844_subset__mset_Oorder__refl,axiom,
! [X: multiset_list_a] : ( subseteq_mset_list_a @ X @ X ) ).
% subset_mset.order_refl
thf(fact_845_subset__mset_Oorder__refl,axiom,
! [X: multiset_a] : ( subseteq_mset_a @ X @ X ) ).
% subset_mset.order_refl
thf(fact_846_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_847_p_Ouniv__poly__a__inv__consistent,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 )
= ( 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 ) ) ) ) @ P2 ) ) ) ) ).
% p.univ_poly_a_inv_consistent
thf(fact_848_p_Ouniv__poly__a__inv__length,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 ) )
= ( size_s349497388124573686list_a @ P2 ) ) ) ) ).
% p.univ_poly_a_inv_length
thf(fact_849_p_Opderiv__inv,axiom,
! [K2: set_list_a,F: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ F ) )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) ) ) ) ) ).
% p.pderiv_inv
thf(fact_850_pdivides__imp__is__root,axiom,
! [P2: list_a,X: a] :
( ( P2 != nil_a )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 )
=> ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_851_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_852_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_853_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_854_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_855_p_Oconst__term__simprules__shell_I4_J,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ P2 ) )
= ( 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 ) ) @ P2 ) ) ) ) ) ).
% p.const_term_simprules_shell(4)
thf(fact_856_p_Ouniv__poly__a__inv__degree,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 ) ) @ one_one_nat )
= ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ).
% p.univ_poly_a_inv_degree
thf(fact_857_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_858_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_859_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_860_is__root__imp__pdivides,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ P2 ) ) ) ).
% is_root_imp_pdivides
thf(fact_861_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_862_p_Oroots__mem__iff__is__root,axiom,
! [P2: list_list_a,X: list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_a @ X @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) )
= ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ X ) ) ) ).
% p.roots_mem_iff_is_root
thf(fact_863_alg__multE_I1_J,axiom,
! [X: a,P2: list_a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) ) @ P2 ) ) ) ) ).
% alg_multE(1)
thf(fact_864_p_Opdivides__imp__roots__incl,axiom,
! [P2: list_list_a,Q: list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q @ ( 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 ) ) ) ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ) ).
% p.pdivides_imp_roots_incl
thf(fact_865_le__alg__mult__imp__pdivides,axiom,
! [X: a,P2: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N ) @ P2 ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_866_alg__multE_I2_J,axiom,
! [X: a,P2: list_a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X ) @ nil_a ) ) @ N ) @ P2 )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ X ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_867_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_868_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_869_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_870_multiset__nonemptyE,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
=> ~ ! [X3: a] :
~ ( member_a @ X3 @ ( set_mset_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_871_subset__mset_Ozero__le,axiom,
! [X: multiset_list_a] : ( subseteq_mset_list_a @ zero_z4454100511807792257list_a @ X ) ).
% subset_mset.zero_le
thf(fact_872_subset__mset_Ozero__le,axiom,
! [X: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ X ) ).
% subset_mset.zero_le
thf(fact_873_subset__mset_Obot__least,axiom,
! [A: multiset_list_a] : ( subseteq_mset_list_a @ zero_z4454100511807792257list_a @ A ) ).
% subset_mset.bot_least
thf(fact_874_subset__mset_Obot__least,axiom,
! [A: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A ) ).
% subset_mset.bot_least
thf(fact_875_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_876_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_877_empty__le,axiom,
! [A2: multiset_list_a] : ( subseteq_mset_list_a @ zero_z4454100511807792257list_a @ A2 ) ).
% empty_le
thf(fact_878_empty__le,axiom,
! [A2: multiset_a] : ( subseteq_mset_a @ zero_zero_multiset_a @ A2 ) ).
% empty_le
thf(fact_879_in__diffD,axiom,
! [A: list_list_a,M3: multiset_list_list_a,N2: multiset_list_list_a] :
( ( member_list_list_a @ A @ ( set_mset_list_list_a @ ( minus_5831295526526677175list_a @ M3 @ N2 ) ) )
=> ( member_list_list_a @ A @ ( set_mset_list_list_a @ M3 ) ) ) ).
% in_diffD
thf(fact_880_in__diffD,axiom,
! [A: list_a,M3: multiset_list_a,N2: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M3 @ N2 ) ) )
=> ( member_list_a @ A @ ( set_mset_list_a @ M3 ) ) ) ).
% in_diffD
thf(fact_881_in__diffD,axiom,
! [A: a,M3: multiset_a,N2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M3 @ N2 ) ) )
=> ( member_a @ A @ ( set_mset_a @ M3 ) ) ) ).
% in_diffD
thf(fact_882_diff__subset__eq__self,axiom,
! [M3: multiset_list_a,N2: multiset_list_a] : ( subseteq_mset_list_a @ ( minus_7431248565939055793list_a @ M3 @ N2 ) @ M3 ) ).
% diff_subset_eq_self
thf(fact_883_diff__subset__eq__self,axiom,
! [M3: multiset_a,N2: multiset_a] : ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ M3 @ N2 ) @ M3 ) ).
% diff_subset_eq_self
thf(fact_884_mset__subset__eqD,axiom,
! [A2: multiset_list_list_a,B5: multiset_list_list_a,X: list_list_a] :
( ( subset8447756916971205105list_a @ A2 @ B5 )
=> ( ( member_list_list_a @ X @ ( set_mset_list_list_a @ A2 ) )
=> ( member_list_list_a @ X @ ( set_mset_list_list_a @ B5 ) ) ) ) ).
% mset_subset_eqD
thf(fact_885_mset__subset__eqD,axiom,
! [A2: multiset_list_a,B5: multiset_list_a,X: list_a] :
( ( subseteq_mset_list_a @ A2 @ B5 )
=> ( ( member_list_a @ X @ ( set_mset_list_a @ A2 ) )
=> ( member_list_a @ X @ ( set_mset_list_a @ B5 ) ) ) ) ).
% mset_subset_eqD
thf(fact_886_mset__subset__eqD,axiom,
! [A2: multiset_a,B5: multiset_a,X: a] :
( ( subseteq_mset_a @ A2 @ B5 )
=> ( ( member_a @ X @ ( set_mset_a @ A2 ) )
=> ( member_a @ X @ ( set_mset_a @ B5 ) ) ) ) ).
% mset_subset_eqD
thf(fact_887_subset__mset_Odual__order_Oantisym,axiom,
! [B: multiset_list_a,A: multiset_list_a] :
( ( subseteq_mset_list_a @ B @ A )
=> ( ( subseteq_mset_list_a @ A @ B )
=> ( A = B ) ) ) ).
% subset_mset.dual_order.antisym
thf(fact_888_subset__mset_Odual__order_Oantisym,axiom,
! [B: multiset_a,A: multiset_a] :
( ( subseteq_mset_a @ B @ A )
=> ( ( subseteq_mset_a @ A @ B )
=> ( A = B ) ) ) ).
% subset_mset.dual_order.antisym
thf(fact_889_subset__mset_Odual__order_Oeq__iff,axiom,
( ( ^ [Y3: multiset_list_a,Z2: multiset_list_a] : ( Y3 = Z2 ) )
= ( ^ [A4: multiset_list_a,B2: multiset_list_a] :
( ( subseteq_mset_list_a @ B2 @ A4 )
& ( subseteq_mset_list_a @ A4 @ B2 ) ) ) ) ).
% subset_mset.dual_order.eq_iff
thf(fact_890_subset__mset_Odual__order_Oeq__iff,axiom,
( ( ^ [Y3: multiset_a,Z2: multiset_a] : ( Y3 = Z2 ) )
= ( ^ [A4: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ B2 @ A4 )
& ( subseteq_mset_a @ A4 @ B2 ) ) ) ) ).
% subset_mset.dual_order.eq_iff
thf(fact_891_subset__mset_Odual__order_Otrans,axiom,
! [B: multiset_list_a,A: multiset_list_a,C: multiset_list_a] :
( ( subseteq_mset_list_a @ B @ A )
=> ( ( subseteq_mset_list_a @ C @ B )
=> ( subseteq_mset_list_a @ C @ A ) ) ) ).
% subset_mset.dual_order.trans
thf(fact_892_subset__mset_Odual__order_Otrans,axiom,
! [B: multiset_a,A: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ B @ A )
=> ( ( subseteq_mset_a @ C @ B )
=> ( subseteq_mset_a @ C @ A ) ) ) ).
% subset_mset.dual_order.trans
thf(fact_893_subset__mset_Oord__le__eq__trans,axiom,
! [A: multiset_list_a,B: multiset_list_a,C: multiset_list_a] :
( ( subseteq_mset_list_a @ A @ B )
=> ( ( B = C )
=> ( subseteq_mset_list_a @ A @ C ) ) ) ).
% subset_mset.ord_le_eq_trans
thf(fact_894_subset__mset_Oord__le__eq__trans,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( B = C )
=> ( subseteq_mset_a @ A @ C ) ) ) ).
% subset_mset.ord_le_eq_trans
thf(fact_895_subset__mset_Oord__eq__le__trans,axiom,
! [A: multiset_list_a,B: multiset_list_a,C: multiset_list_a] :
( ( A = B )
=> ( ( subseteq_mset_list_a @ B @ C )
=> ( subseteq_mset_list_a @ A @ C ) ) ) ).
% subset_mset.ord_eq_le_trans
thf(fact_896_subset__mset_Oord__eq__le__trans,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( A = B )
=> ( ( subseteq_mset_a @ B @ C )
=> ( subseteq_mset_a @ A @ C ) ) ) ).
% subset_mset.ord_eq_le_trans
thf(fact_897_subset__mset_Oorder__antisym,axiom,
! [X: multiset_list_a,Y: multiset_list_a] :
( ( subseteq_mset_list_a @ X @ Y )
=> ( ( subseteq_mset_list_a @ Y @ X )
=> ( X = Y ) ) ) ).
% subset_mset.order_antisym
thf(fact_898_subset__mset_Oorder__antisym,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( subseteq_mset_a @ X @ Y )
=> ( ( subseteq_mset_a @ Y @ X )
=> ( X = Y ) ) ) ).
% subset_mset.order_antisym
thf(fact_899_subset__mset_Oorder__eq__iff,axiom,
( ( ^ [Y3: multiset_list_a,Z2: multiset_list_a] : ( Y3 = Z2 ) )
= ( ^ [X2: multiset_list_a,Y6: multiset_list_a] :
( ( subseteq_mset_list_a @ X2 @ Y6 )
& ( subseteq_mset_list_a @ Y6 @ X2 ) ) ) ) ).
% subset_mset.order_eq_iff
thf(fact_900_subset__mset_Oorder__eq__iff,axiom,
( ( ^ [Y3: multiset_a,Z2: multiset_a] : ( Y3 = Z2 ) )
= ( ^ [X2: multiset_a,Y6: multiset_a] :
( ( subseteq_mset_a @ X2 @ Y6 )
& ( subseteq_mset_a @ Y6 @ X2 ) ) ) ) ).
% subset_mset.order_eq_iff
thf(fact_901_subset__mset_Oantisym__conv,axiom,
! [Y: multiset_list_a,X: multiset_list_a] :
( ( subseteq_mset_list_a @ Y @ X )
=> ( ( subseteq_mset_list_a @ X @ Y )
= ( X = Y ) ) ) ).
% subset_mset.antisym_conv
thf(fact_902_subset__mset_Oantisym__conv,axiom,
! [Y: multiset_a,X: multiset_a] :
( ( subseteq_mset_a @ Y @ X )
=> ( ( subseteq_mset_a @ X @ Y )
= ( X = Y ) ) ) ).
% subset_mset.antisym_conv
thf(fact_903_subset__mset_Oorder__trans,axiom,
! [X: multiset_list_a,Y: multiset_list_a,Z: multiset_list_a] :
( ( subseteq_mset_list_a @ X @ Y )
=> ( ( subseteq_mset_list_a @ Y @ Z )
=> ( subseteq_mset_list_a @ X @ Z ) ) ) ).
% subset_mset.order_trans
thf(fact_904_subset__mset_Oorder__trans,axiom,
! [X: multiset_a,Y: multiset_a,Z: multiset_a] :
( ( subseteq_mset_a @ X @ Y )
=> ( ( subseteq_mset_a @ Y @ Z )
=> ( subseteq_mset_a @ X @ Z ) ) ) ).
% subset_mset.order_trans
thf(fact_905_subset__mset_Oeq__refl,axiom,
! [X: multiset_list_a,Y: multiset_list_a] :
( ( X = Y )
=> ( subseteq_mset_list_a @ X @ Y ) ) ).
% subset_mset.eq_refl
thf(fact_906_subset__mset_Oeq__refl,axiom,
! [X: multiset_a,Y: multiset_a] :
( ( X = Y )
=> ( subseteq_mset_a @ X @ Y ) ) ).
% subset_mset.eq_refl
thf(fact_907_subset__mset_Oantisym,axiom,
! [A: multiset_list_a,B: multiset_list_a] :
( ( subseteq_mset_list_a @ A @ B )
=> ( ( subseteq_mset_list_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_mset.antisym
thf(fact_908_subset__mset_Oantisym,axiom,
! [A: multiset_a,B: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ B @ A )
=> ( A = B ) ) ) ).
% subset_mset.antisym
thf(fact_909_subset__mset_Oeq__iff,axiom,
( ( ^ [Y3: multiset_list_a,Z2: multiset_list_a] : ( Y3 = Z2 ) )
= ( ^ [A4: multiset_list_a,B2: multiset_list_a] :
( ( subseteq_mset_list_a @ A4 @ B2 )
& ( subseteq_mset_list_a @ B2 @ A4 ) ) ) ) ).
% subset_mset.eq_iff
thf(fact_910_subset__mset_Oeq__iff,axiom,
( ( ^ [Y3: multiset_a,Z2: multiset_a] : ( Y3 = Z2 ) )
= ( ^ [A4: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A4 @ B2 )
& ( subseteq_mset_a @ B2 @ A4 ) ) ) ) ).
% subset_mset.eq_iff
thf(fact_911_subset__mset_Otrans,axiom,
! [A: multiset_list_a,B: multiset_list_a,C: multiset_list_a] :
( ( subseteq_mset_list_a @ A @ B )
=> ( ( subseteq_mset_list_a @ B @ C )
=> ( subseteq_mset_list_a @ A @ C ) ) ) ).
% subset_mset.trans
thf(fact_912_subset__mset_Otrans,axiom,
! [A: multiset_a,B: multiset_a,C: multiset_a] :
( ( subseteq_mset_a @ A @ B )
=> ( ( subseteq_mset_a @ B @ C )
=> ( subseteq_mset_a @ A @ C ) ) ) ).
% subset_mset.trans
thf(fact_913_set__mset__mono,axiom,
! [A2: multiset_list_a,B5: multiset_list_a] :
( ( subseteq_mset_list_a @ A2 @ B5 )
=> ( ord_le8861187494160871172list_a @ ( set_mset_list_a @ A2 ) @ ( set_mset_list_a @ B5 ) ) ) ).
% set_mset_mono
thf(fact_914_set__mset__mono,axiom,
! [A2: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ A2 @ B5 )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A2 ) @ ( set_mset_a @ B5 ) ) ) ).
% set_mset_mono
thf(fact_915_Diff__eq__empty__iff__mset,axiom,
! [A2: multiset_list_a,B5: multiset_list_a] :
( ( ( minus_7431248565939055793list_a @ A2 @ B5 )
= zero_z4454100511807792257list_a )
= ( subseteq_mset_list_a @ A2 @ B5 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_916_Diff__eq__empty__iff__mset,axiom,
! [A2: multiset_a,B5: multiset_a] :
( ( ( minus_3765977307040488491iset_a @ A2 @ B5 )
= zero_zero_multiset_a )
= ( subseteq_mset_a @ A2 @ B5 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_917_size__mset__mono,axiom,
! [A2: multiset_list_a,B5: multiset_list_a] :
( ( subseteq_mset_list_a @ A2 @ B5 )
=> ( ord_less_eq_nat @ ( size_s2335926164413107382list_a @ A2 ) @ ( size_s2335926164413107382list_a @ B5 ) ) ) ).
% size_mset_mono
thf(fact_918_size__mset__mono,axiom,
! [A2: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ A2 @ B5 )
=> ( ord_less_eq_nat @ ( size_size_multiset_a @ A2 ) @ ( size_size_multiset_a @ B5 ) ) ) ).
% size_mset_mono
thf(fact_919_size__Diff__submset,axiom,
! [M3: multiset_list_a,M5: multiset_list_a] :
( ( subseteq_mset_list_a @ M3 @ M5 )
=> ( ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M5 @ M3 ) )
= ( minus_minus_nat @ ( size_s2335926164413107382list_a @ M5 ) @ ( size_s2335926164413107382list_a @ M3 ) ) ) ) ).
% size_Diff_submset
thf(fact_920_size__Diff__submset,axiom,
! [M3: multiset_a,M5: multiset_a] :
( ( subseteq_mset_a @ M3 @ M5 )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M5 @ M3 ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M5 ) @ ( size_size_multiset_a @ M3 ) ) ) ) ).
% size_Diff_submset
thf(fact_921_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_922_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_923_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_924_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_925_ring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( P2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_926_roots__inclI,axiom,
! [P2: list_a,Q: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P2 != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A3 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ A3 ) ) @ Q ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% roots_inclI
thf(fact_927_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_928_add__mset__add__mset__same__iff,axiom,
! [A: list_a,A2: multiset_list_a,B5: multiset_list_a] :
( ( ( add_mset_list_a @ A @ A2 )
= ( add_mset_list_a @ A @ B5 ) )
= ( A2 = B5 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_929_add__mset__add__mset__same__iff,axiom,
! [A: a,A2: multiset_a,B5: multiset_a] :
( ( ( add_mset_a @ A @ A2 )
= ( add_mset_a @ A @ B5 ) )
= ( A2 = B5 ) ) ).
% add_mset_add_mset_same_iff
thf(fact_930_multi__self__add__other__not__self,axiom,
! [M3: multiset_list_a,X: list_a] :
( M3
!= ( add_mset_list_a @ X @ M3 ) ) ).
% multi_self_add_other_not_self
thf(fact_931_multi__self__add__other__not__self,axiom,
! [M3: multiset_a,X: a] :
( M3
!= ( add_mset_a @ X @ M3 ) ) ).
% multi_self_add_other_not_self
thf(fact_932_roots__mem__iff__is__root,axiom,
! [P2: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ X @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) ) )
= ( polyno4133073214067823460ot_a_b @ r @ P2 @ X ) ) ) ).
% roots_mem_iff_is_root
thf(fact_933_add__mset__eq__singleton__iff,axiom,
! [X: list_a,M3: multiset_list_a,Y: list_a] :
( ( ( add_mset_list_a @ X @ M3 )
= ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) )
= ( ( M3 = zero_z4454100511807792257list_a )
& ( X = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_934_add__mset__eq__singleton__iff,axiom,
! [X: a,M3: multiset_a,Y: a] :
( ( ( add_mset_a @ X @ M3 )
= ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( ( M3 = zero_zero_multiset_a )
& ( X = Y ) ) ) ).
% add_mset_eq_singleton_iff
thf(fact_935_single__eq__add__mset,axiom,
! [A: list_a,B: list_a,M3: multiset_list_a] :
( ( ( add_mset_list_a @ A @ zero_z4454100511807792257list_a )
= ( add_mset_list_a @ B @ M3 ) )
= ( ( B = A )
& ( M3 = zero_z4454100511807792257list_a ) ) ) ).
% single_eq_add_mset
thf(fact_936_single__eq__add__mset,axiom,
! [A: a,B: a,M3: multiset_a] :
( ( ( add_mset_a @ A @ zero_zero_multiset_a )
= ( add_mset_a @ B @ M3 ) )
= ( ( B = A )
& ( M3 = zero_zero_multiset_a ) ) ) ).
% single_eq_add_mset
thf(fact_937_add__mset__eq__single,axiom,
! [B: list_a,M3: multiset_list_a,A: list_a] :
( ( ( add_mset_list_a @ B @ M3 )
= ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) )
= ( ( B = A )
& ( M3 = zero_z4454100511807792257list_a ) ) ) ).
% add_mset_eq_single
thf(fact_938_add__mset__eq__single,axiom,
! [B: a,M3: multiset_a,A: a] :
( ( ( add_mset_a @ B @ M3 )
= ( add_mset_a @ A @ zero_zero_multiset_a ) )
= ( ( B = A )
& ( M3 = zero_zero_multiset_a ) ) ) ).
% add_mset_eq_single
thf(fact_939_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_940_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_941_pdivides__imp__roots__incl,axiom,
! [P2: list_a,Q: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ ( polynomial_roots_a_b @ r @ Q ) ) ) ) ) ) ).
% pdivides_imp_roots_incl
thf(fact_942_add__mset__subseteq__single__iff,axiom,
! [A: list_a,M3: multiset_list_a,B: list_a] :
( ( subseteq_mset_list_a @ ( add_mset_list_a @ A @ M3 ) @ ( add_mset_list_a @ B @ zero_z4454100511807792257list_a ) )
= ( ( M3 = zero_z4454100511807792257list_a )
& ( A = B ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_943_add__mset__subseteq__single__iff,axiom,
! [A: a,M3: multiset_a,B: a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ M3 ) @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= ( ( M3 = zero_zero_multiset_a )
& ( A = B ) ) ) ).
% add_mset_subseteq_single_iff
thf(fact_944_not__empty__rootsE,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( polynomial_roots_a_b @ r @ P2 )
!= zero_zero_multiset_a )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A3 @ ( set_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) ) )
=> ( ( member_list_a @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ A3 ) @ 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 @ A3 ) @ nil_a ) ) @ P2 ) ) ) ) ) ) ).
% not_empty_rootsE
thf(fact_945_add__mset__remove__trivial,axiom,
! [X: list_a,M3: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ X @ M3 ) @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) )
= M3 ) ).
% add_mset_remove_trivial
thf(fact_946_add__mset__remove__trivial,axiom,
! [X: a,M3: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X @ M3 ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M3 ) ).
% add_mset_remove_trivial
thf(fact_947_diff__add__mset__swap,axiom,
! [B: list_list_a,A2: multiset_list_list_a,M3: multiset_list_list_a] :
( ~ ( member_list_list_a @ B @ ( set_mset_list_list_a @ A2 ) )
=> ( ( minus_5831295526526677175list_a @ ( add_mset_list_list_a @ B @ M3 ) @ A2 )
= ( add_mset_list_list_a @ B @ ( minus_5831295526526677175list_a @ M3 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_948_diff__add__mset__swap,axiom,
! [B: list_a,A2: multiset_list_a,M3: multiset_list_a] :
( ~ ( member_list_a @ B @ ( set_mset_list_a @ A2 ) )
=> ( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ B @ M3 ) @ A2 )
= ( add_mset_list_a @ B @ ( minus_7431248565939055793list_a @ M3 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_949_diff__add__mset__swap,axiom,
! [B: a,A2: multiset_a,M3: multiset_a] :
( ~ ( member_a @ B @ ( set_mset_a @ A2 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M3 ) @ A2 )
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M3 @ A2 ) ) ) ) ).
% diff_add_mset_swap
thf(fact_950_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_951_single__subset__iff,axiom,
! [A: list_list_a,M3: multiset_list_list_a] :
( ( subset8447756916971205105list_a @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) @ M3 )
= ( member_list_list_a @ A @ ( set_mset_list_list_a @ M3 ) ) ) ).
% single_subset_iff
thf(fact_952_single__subset__iff,axiom,
! [A: list_a,M3: multiset_list_a] :
( ( subseteq_mset_list_a @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) @ M3 )
= ( member_list_a @ A @ ( set_mset_list_a @ M3 ) ) ) ).
% single_subset_iff
thf(fact_953_single__subset__iff,axiom,
! [A: a,M3: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ M3 )
= ( member_a @ A @ ( set_mset_a @ M3 ) ) ) ).
% single_subset_iff
thf(fact_954_insert__DiffM,axiom,
! [X: list_list_a,M3: multiset_list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M3 ) )
=> ( ( add_mset_list_list_a @ X @ ( minus_5831295526526677175list_a @ M3 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) )
= M3 ) ) ).
% insert_DiffM
thf(fact_955_insert__DiffM,axiom,
! [X: list_a,M3: multiset_list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M3 ) )
=> ( ( add_mset_list_a @ X @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) )
= M3 ) ) ).
% insert_DiffM
thf(fact_956_insert__DiffM,axiom,
! [X: a,M3: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M3 ) )
=> ( ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= M3 ) ) ).
% insert_DiffM
thf(fact_957_diff__union__swap2,axiom,
! [Y: list_list_a,M3: multiset_list_list_a,X: list_list_a] :
( ( member_list_list_a @ Y @ ( set_mset_list_list_a @ M3 ) )
=> ( ( minus_5831295526526677175list_a @ ( add_mset_list_list_a @ X @ M3 ) @ ( add_mset_list_list_a @ Y @ zero_z1542645121299710087list_a ) )
= ( add_mset_list_list_a @ X @ ( minus_5831295526526677175list_a @ M3 @ ( add_mset_list_list_a @ Y @ zero_z1542645121299710087list_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_958_diff__union__swap2,axiom,
! [Y: list_a,M3: multiset_list_a,X: list_a] :
( ( member_list_a @ Y @ ( set_mset_list_a @ M3 ) )
=> ( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ X @ M3 ) @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) )
= ( add_mset_list_a @ X @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_959_diff__union__swap2,axiom,
! [Y: a,M3: multiset_a,X: a] :
( ( member_a @ Y @ ( set_mset_a @ M3 ) )
=> ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X @ M3 ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
= ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).
% diff_union_swap2
thf(fact_960_mset__subset__eq__add__mset__cancel,axiom,
! [A: list_a,A2: multiset_list_a,B5: multiset_list_a] :
( ( subseteq_mset_list_a @ ( add_mset_list_a @ A @ A2 ) @ ( add_mset_list_a @ A @ B5 ) )
= ( subseteq_mset_list_a @ A2 @ B5 ) ) ).
% mset_subset_eq_add_mset_cancel
thf(fact_961_mset__subset__eq__add__mset__cancel,axiom,
! [A: a,A2: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ A2 ) @ ( add_mset_a @ A @ B5 ) )
= ( subseteq_mset_a @ A2 @ B5 ) ) ).
% mset_subset_eq_add_mset_cancel
thf(fact_962_union__single__eq__member,axiom,
! [X: list_list_a,M3: multiset_list_list_a,N2: multiset_list_list_a] :
( ( ( add_mset_list_list_a @ X @ M3 )
= N2 )
=> ( member_list_list_a @ X @ ( set_mset_list_list_a @ N2 ) ) ) ).
% union_single_eq_member
thf(fact_963_union__single__eq__member,axiom,
! [X: list_a,M3: multiset_list_a,N2: multiset_list_a] :
( ( ( add_mset_list_a @ X @ M3 )
= N2 )
=> ( member_list_a @ X @ ( set_mset_list_a @ N2 ) ) ) ).
% union_single_eq_member
thf(fact_964_union__single__eq__member,axiom,
! [X: a,M3: multiset_a,N2: multiset_a] :
( ( ( add_mset_a @ X @ M3 )
= N2 )
=> ( member_a @ X @ ( set_mset_a @ N2 ) ) ) ).
% union_single_eq_member
thf(fact_965_insert__noteq__member,axiom,
! [B: list_list_a,B5: multiset_list_list_a,C: list_list_a,C4: multiset_list_list_a] :
( ( ( add_mset_list_list_a @ B @ B5 )
= ( add_mset_list_list_a @ C @ C4 ) )
=> ( ( B != C )
=> ( member_list_list_a @ C @ ( set_mset_list_list_a @ B5 ) ) ) ) ).
% insert_noteq_member
thf(fact_966_insert__noteq__member,axiom,
! [B: list_a,B5: multiset_list_a,C: list_a,C4: multiset_list_a] :
( ( ( add_mset_list_a @ B @ B5 )
= ( add_mset_list_a @ C @ C4 ) )
=> ( ( B != C )
=> ( member_list_a @ C @ ( set_mset_list_a @ B5 ) ) ) ) ).
% insert_noteq_member
thf(fact_967_insert__noteq__member,axiom,
! [B: a,B5: multiset_a,C: a,C4: multiset_a] :
( ( ( add_mset_a @ B @ B5 )
= ( add_mset_a @ C @ C4 ) )
=> ( ( B != C )
=> ( member_a @ C @ ( set_mset_a @ B5 ) ) ) ) ).
% insert_noteq_member
thf(fact_968_multi__member__split,axiom,
! [X: list_list_a,M3: multiset_list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M3 ) )
=> ? [A6: multiset_list_list_a] :
( M3
= ( add_mset_list_list_a @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_969_multi__member__split,axiom,
! [X: list_a,M3: multiset_list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M3 ) )
=> ? [A6: multiset_list_a] :
( M3
= ( add_mset_list_a @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_970_multi__member__split,axiom,
! [X: a,M3: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M3 ) )
=> ? [A6: multiset_a] :
( M3
= ( add_mset_a @ X @ A6 ) ) ) ).
% multi_member_split
thf(fact_971_mset__add,axiom,
! [A: list_list_a,A2: multiset_list_list_a] :
( ( member_list_list_a @ A @ ( set_mset_list_list_a @ A2 ) )
=> ~ ! [B6: multiset_list_list_a] :
( A2
!= ( add_mset_list_list_a @ A @ B6 ) ) ) ).
% mset_add
thf(fact_972_mset__add,axiom,
! [A: list_a,A2: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ A2 ) )
=> ~ ! [B6: multiset_list_a] :
( A2
!= ( add_mset_list_a @ A @ B6 ) ) ) ).
% mset_add
thf(fact_973_mset__add,axiom,
! [A: a,A2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ A2 ) )
=> ~ ! [B6: multiset_a] :
( A2
!= ( add_mset_a @ A @ B6 ) ) ) ).
% mset_add
thf(fact_974_add__eq__conv__ex,axiom,
! [A: list_a,M3: multiset_list_a,B: list_a,N2: multiset_list_a] :
( ( ( add_mset_list_a @ A @ M3 )
= ( add_mset_list_a @ B @ N2 ) )
= ( ( ( M3 = N2 )
& ( A = B ) )
| ? [K3: multiset_list_a] :
( ( M3
= ( add_mset_list_a @ B @ K3 ) )
& ( N2
= ( add_mset_list_a @ A @ K3 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_975_add__eq__conv__ex,axiom,
! [A: a,M3: multiset_a,B: a,N2: multiset_a] :
( ( ( add_mset_a @ A @ M3 )
= ( add_mset_a @ B @ N2 ) )
= ( ( ( M3 = N2 )
& ( A = B ) )
| ? [K3: multiset_a] :
( ( M3
= ( add_mset_a @ B @ K3 ) )
& ( N2
= ( add_mset_a @ A @ K3 ) ) ) ) ) ).
% add_eq_conv_ex
thf(fact_976_add__mset__commute,axiom,
! [X: list_a,Y: list_a,M3: multiset_list_a] :
( ( add_mset_list_a @ X @ ( add_mset_list_a @ Y @ M3 ) )
= ( add_mset_list_a @ Y @ ( add_mset_list_a @ X @ M3 ) ) ) ).
% add_mset_commute
thf(fact_977_add__mset__commute,axiom,
! [X: a,Y: a,M3: multiset_a] :
( ( add_mset_a @ X @ ( add_mset_a @ Y @ M3 ) )
= ( add_mset_a @ Y @ ( add_mset_a @ X @ M3 ) ) ) ).
% add_mset_commute
thf(fact_978_add__mset__diff__bothsides,axiom,
! [A: list_a,M3: multiset_list_a,A2: multiset_list_a] :
( ( minus_7431248565939055793list_a @ ( add_mset_list_a @ A @ M3 ) @ ( add_mset_list_a @ A @ A2 ) )
= ( minus_7431248565939055793list_a @ M3 @ A2 ) ) ).
% add_mset_diff_bothsides
thf(fact_979_add__mset__diff__bothsides,axiom,
! [A: a,M3: multiset_a,A2: multiset_a] :
( ( minus_3765977307040488491iset_a @ ( add_mset_a @ A @ M3 ) @ ( add_mset_a @ A @ A2 ) )
= ( minus_3765977307040488491iset_a @ M3 @ A2 ) ) ).
% add_mset_diff_bothsides
thf(fact_980_multi__nonempty__split,axiom,
! [M3: multiset_list_a] :
( ( M3 != zero_z4454100511807792257list_a )
=> ? [A6: multiset_list_a,A3: list_a] :
( M3
= ( add_mset_list_a @ A3 @ A6 ) ) ) ).
% multi_nonempty_split
thf(fact_981_multi__nonempty__split,axiom,
! [M3: multiset_a] :
( ( M3 != zero_zero_multiset_a )
=> ? [A6: multiset_a,A3: a] :
( M3
= ( add_mset_a @ A3 @ A6 ) ) ) ).
% multi_nonempty_split
thf(fact_982_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_983_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_984_multiset__induct2,axiom,
! [P: multiset_list_a > multiset_list_a > $o,M3: multiset_list_a,N2: multiset_list_a] :
( ( P @ zero_z4454100511807792257list_a @ zero_z4454100511807792257list_a )
=> ( ! [A3: list_a,M6: multiset_list_a,N5: multiset_list_a] :
( ( P @ M6 @ N5 )
=> ( P @ ( add_mset_list_a @ A3 @ M6 ) @ N5 ) )
=> ( ! [A3: list_a,M6: multiset_list_a,N5: multiset_list_a] :
( ( P @ M6 @ N5 )
=> ( P @ M6 @ ( add_mset_list_a @ A3 @ N5 ) ) )
=> ( P @ M3 @ N2 ) ) ) ) ).
% multiset_induct2
thf(fact_985_multiset__induct2,axiom,
! [P: multiset_list_a > multiset_a > $o,M3: multiset_list_a,N2: multiset_a] :
( ( P @ zero_z4454100511807792257list_a @ zero_zero_multiset_a )
=> ( ! [A3: list_a,M6: multiset_list_a,N5: multiset_a] :
( ( P @ M6 @ N5 )
=> ( P @ ( add_mset_list_a @ A3 @ M6 ) @ N5 ) )
=> ( ! [A3: a,M6: multiset_list_a,N5: multiset_a] :
( ( P @ M6 @ N5 )
=> ( P @ M6 @ ( add_mset_a @ A3 @ N5 ) ) )
=> ( P @ M3 @ N2 ) ) ) ) ).
% multiset_induct2
thf(fact_986_multiset__induct2,axiom,
! [P: multiset_a > multiset_list_a > $o,M3: multiset_a,N2: multiset_list_a] :
( ( P @ zero_zero_multiset_a @ zero_z4454100511807792257list_a )
=> ( ! [A3: a,M6: multiset_a,N5: multiset_list_a] :
( ( P @ M6 @ N5 )
=> ( P @ ( add_mset_a @ A3 @ M6 ) @ N5 ) )
=> ( ! [A3: list_a,M6: multiset_a,N5: multiset_list_a] :
( ( P @ M6 @ N5 )
=> ( P @ M6 @ ( add_mset_list_a @ A3 @ N5 ) ) )
=> ( P @ M3 @ N2 ) ) ) ) ).
% multiset_induct2
thf(fact_987_multiset__induct2,axiom,
! [P: multiset_a > multiset_a > $o,M3: multiset_a,N2: multiset_a] :
( ( P @ zero_zero_multiset_a @ zero_zero_multiset_a )
=> ( ! [A3: a,M6: multiset_a,N5: multiset_a] :
( ( P @ M6 @ N5 )
=> ( P @ ( add_mset_a @ A3 @ M6 ) @ N5 ) )
=> ( ! [A3: a,M6: multiset_a,N5: multiset_a] :
( ( P @ M6 @ N5 )
=> ( P @ M6 @ ( add_mset_a @ A3 @ N5 ) ) )
=> ( P @ M3 @ N2 ) ) ) ) ).
% multiset_induct2
thf(fact_988_multiset__induct,axiom,
! [P: multiset_list_a > $o,M3: multiset_list_a] :
( ( P @ zero_z4454100511807792257list_a )
=> ( ! [X3: list_a,M6: multiset_list_a] :
( ( P @ M6 )
=> ( P @ ( add_mset_list_a @ X3 @ M6 ) ) )
=> ( P @ M3 ) ) ) ).
% multiset_induct
thf(fact_989_multiset__induct,axiom,
! [P: multiset_a > $o,M3: multiset_a] :
( ( P @ zero_zero_multiset_a )
=> ( ! [X3: a,M6: multiset_a] :
( ( P @ M6 )
=> ( P @ ( add_mset_a @ X3 @ M6 ) ) )
=> ( P @ M3 ) ) ) ).
% multiset_induct
thf(fact_990_multiset__cases,axiom,
! [M3: multiset_list_a] :
( ( M3 != zero_z4454100511807792257list_a )
=> ~ ! [X3: list_a,N5: multiset_list_a] :
( M3
!= ( add_mset_list_a @ X3 @ N5 ) ) ) ).
% multiset_cases
thf(fact_991_multiset__cases,axiom,
! [M3: multiset_a] :
( ( M3 != zero_zero_multiset_a )
=> ~ ! [X3: a,N5: multiset_a] :
( M3
!= ( add_mset_a @ X3 @ N5 ) ) ) ).
% multiset_cases
thf(fact_992_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_993_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_994_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_995_diff__union__swap,axiom,
! [A: list_a,B: list_a,M3: multiset_list_a] :
( ( A != B )
=> ( ( add_mset_list_a @ B @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) )
= ( minus_7431248565939055793list_a @ ( add_mset_list_a @ B @ M3 ) @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) ) ).
% diff_union_swap
thf(fact_996_diff__union__swap,axiom,
! [A: a,B: a,M3: multiset_a] :
( ( A != B )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M3 ) @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ).
% diff_union_swap
thf(fact_997_add__eq__conv__diff,axiom,
! [A: list_a,M3: multiset_list_a,B: list_a,N2: multiset_list_a] :
( ( ( add_mset_list_a @ A @ M3 )
= ( add_mset_list_a @ B @ N2 ) )
= ( ( ( M3 = N2 )
& ( A = B ) )
| ( ( M3
= ( add_mset_list_a @ B @ ( minus_7431248565939055793list_a @ N2 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) )
& ( N2
= ( add_mset_list_a @ A @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ B @ zero_z4454100511807792257list_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_998_add__eq__conv__diff,axiom,
! [A: a,M3: multiset_a,B: a,N2: multiset_a] :
( ( ( add_mset_a @ A @ M3 )
= ( add_mset_a @ B @ N2 ) )
= ( ( ( M3 = N2 )
& ( A = B ) )
| ( ( M3
= ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
& ( N2
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) ) ) ) ) ).
% add_eq_conv_diff
thf(fact_999_union__single__eq__diff,axiom,
! [X: list_a,M3: multiset_list_a,N2: multiset_list_a] :
( ( ( add_mset_list_a @ X @ M3 )
= N2 )
=> ( M3
= ( minus_7431248565939055793list_a @ N2 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_1000_union__single__eq__diff,axiom,
! [X: a,M3: multiset_a,N2: multiset_a] :
( ( ( add_mset_a @ X @ M3 )
= N2 )
=> ( M3
= ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ) ).
% union_single_eq_diff
thf(fact_1001_multiset__induct__min,axiom,
! [P: multiset_nat > $o,M3: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X3: nat,M6: multiset_nat] :
( ( P @ M6 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M6 ) )
=> ( ord_less_eq_nat @ X3 @ Xa ) )
=> ( P @ ( add_mset_nat @ X3 @ M6 ) ) ) )
=> ( P @ M3 ) ) ) ).
% multiset_induct_min
thf(fact_1002_multiset__induct__min,axiom,
! [P: multiset_int > $o,M3: multiset_int] :
( ( P @ zero_z3170743180189231877et_int )
=> ( ! [X3: int,M6: multiset_int] :
( ( P @ M6 )
=> ( ! [Xa: int] :
( ( member_int @ Xa @ ( set_mset_int @ M6 ) )
=> ( ord_less_eq_int @ X3 @ Xa ) )
=> ( P @ ( add_mset_int @ X3 @ M6 ) ) ) )
=> ( P @ M3 ) ) ) ).
% multiset_induct_min
thf(fact_1003_multiset__induct__max,axiom,
! [P: multiset_nat > $o,M3: multiset_nat] :
( ( P @ zero_z7348594199698428585et_nat )
=> ( ! [X3: nat,M6: multiset_nat] :
( ( P @ M6 )
=> ( ! [Xa: nat] :
( ( member_nat @ Xa @ ( set_mset_nat @ M6 ) )
=> ( ord_less_eq_nat @ Xa @ X3 ) )
=> ( P @ ( add_mset_nat @ X3 @ M6 ) ) ) )
=> ( P @ M3 ) ) ) ).
% multiset_induct_max
thf(fact_1004_multiset__induct__max,axiom,
! [P: multiset_int > $o,M3: multiset_int] :
( ( P @ zero_z3170743180189231877et_int )
=> ( ! [X3: int,M6: multiset_int] :
( ( P @ M6 )
=> ( ! [Xa: int] :
( ( member_int @ Xa @ ( set_mset_int @ M6 ) )
=> ( ord_less_eq_int @ Xa @ X3 ) )
=> ( P @ ( add_mset_int @ X3 @ M6 ) ) ) )
=> ( P @ M3 ) ) ) ).
% multiset_induct_max
thf(fact_1005_multi__subset__induct,axiom,
! [F2: multiset_list_list_a,A2: multiset_list_list_a,P: multiset_list_list_a > $o] :
( ( subset8447756916971205105list_a @ F2 @ A2 )
=> ( ( P @ zero_z1542645121299710087list_a )
=> ( ! [A3: list_list_a,F3: multiset_list_list_a] :
( ( member_list_list_a @ A3 @ ( set_mset_list_list_a @ A2 ) )
=> ( ( P @ F3 )
=> ( P @ ( add_mset_list_list_a @ A3 @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_1006_multi__subset__induct,axiom,
! [F2: multiset_list_a,A2: multiset_list_a,P: multiset_list_a > $o] :
( ( subseteq_mset_list_a @ F2 @ A2 )
=> ( ( P @ zero_z4454100511807792257list_a )
=> ( ! [A3: list_a,F3: multiset_list_a] :
( ( member_list_a @ A3 @ ( set_mset_list_a @ A2 ) )
=> ( ( P @ F3 )
=> ( P @ ( add_mset_list_a @ A3 @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_1007_multi__subset__induct,axiom,
! [F2: multiset_a,A2: multiset_a,P: multiset_a > $o] :
( ( subseteq_mset_a @ F2 @ A2 )
=> ( ( P @ zero_zero_multiset_a )
=> ( ! [A3: a,F3: multiset_a] :
( ( member_a @ A3 @ ( set_mset_a @ A2 ) )
=> ( ( P @ F3 )
=> ( P @ ( add_mset_a @ A3 @ F3 ) ) ) )
=> ( P @ F2 ) ) ) ) ).
% multi_subset_induct
thf(fact_1008_mset__subset__eq__single,axiom,
! [A: list_list_a,B5: multiset_list_list_a] :
( ( member_list_list_a @ A @ ( set_mset_list_list_a @ B5 ) )
=> ( subset8447756916971205105list_a @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) @ B5 ) ) ).
% mset_subset_eq_single
thf(fact_1009_mset__subset__eq__single,axiom,
! [A: list_a,B5: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ B5 ) )
=> ( subseteq_mset_list_a @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) @ B5 ) ) ).
% mset_subset_eq_single
thf(fact_1010_mset__subset__eq__single,axiom,
! [A: a,B5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ B5 ) )
=> ( subseteq_mset_a @ ( add_mset_a @ A @ zero_zero_multiset_a ) @ B5 ) ) ).
% mset_subset_eq_single
thf(fact_1011_diff__single__trivial,axiom,
! [X: list_list_a,M3: multiset_list_list_a] :
( ~ ( member_list_list_a @ X @ ( set_mset_list_list_a @ M3 ) )
=> ( ( minus_5831295526526677175list_a @ M3 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) )
= M3 ) ) ).
% diff_single_trivial
thf(fact_1012_diff__single__trivial,axiom,
! [X: list_a,M3: multiset_list_a] :
( ~ ( member_list_a @ X @ ( set_mset_list_a @ M3 ) )
=> ( ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) )
= M3 ) ) ).
% diff_single_trivial
thf(fact_1013_diff__single__trivial,axiom,
! [X: a,M3: multiset_a] :
( ~ ( member_a @ X @ ( set_mset_a @ M3 ) )
=> ( ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= M3 ) ) ).
% diff_single_trivial
thf(fact_1014_diff__single__eq__union,axiom,
! [X: list_list_a,M3: multiset_list_list_a,N2: multiset_list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M3 ) )
=> ( ( ( minus_5831295526526677175list_a @ M3 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) )
= N2 )
= ( M3
= ( add_mset_list_list_a @ X @ N2 ) ) ) ) ).
% diff_single_eq_union
thf(fact_1015_diff__single__eq__union,axiom,
! [X: list_a,M3: multiset_list_a,N2: multiset_list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M3 ) )
=> ( ( ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) )
= N2 )
= ( M3
= ( add_mset_list_a @ X @ N2 ) ) ) ) ).
% diff_single_eq_union
thf(fact_1016_diff__single__eq__union,axiom,
! [X: a,M3: multiset_a,N2: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M3 ) )
=> ( ( ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
= N2 )
= ( M3
= ( add_mset_a @ X @ N2 ) ) ) ) ).
% diff_single_eq_union
thf(fact_1017_multi__drop__mem__not__eq,axiom,
! [C: list_list_a,B5: multiset_list_list_a] :
( ( member_list_list_a @ C @ ( set_mset_list_list_a @ B5 ) )
=> ( ( minus_5831295526526677175list_a @ B5 @ ( add_mset_list_list_a @ C @ zero_z1542645121299710087list_a ) )
!= B5 ) ) ).
% multi_drop_mem_not_eq
thf(fact_1018_multi__drop__mem__not__eq,axiom,
! [C: list_a,B5: multiset_list_a] :
( ( member_list_a @ C @ ( set_mset_list_a @ B5 ) )
=> ( ( minus_7431248565939055793list_a @ B5 @ ( add_mset_list_a @ C @ zero_z4454100511807792257list_a ) )
!= B5 ) ) ).
% multi_drop_mem_not_eq
thf(fact_1019_multi__drop__mem__not__eq,axiom,
! [C: a,B5: multiset_a] :
( ( member_a @ C @ ( set_mset_a @ B5 ) )
=> ( ( minus_3765977307040488491iset_a @ B5 @ ( add_mset_a @ C @ zero_zero_multiset_a ) )
!= B5 ) ) ).
% multi_drop_mem_not_eq
thf(fact_1020_add__mset__remove__trivial__If,axiom,
! [A: list_list_a,N2: multiset_list_list_a] :
( ( ( member_list_list_a @ A @ ( set_mset_list_list_a @ N2 ) )
=> ( ( add_mset_list_list_a @ A @ ( minus_5831295526526677175list_a @ N2 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) )
= N2 ) )
& ( ~ ( member_list_list_a @ A @ ( set_mset_list_list_a @ N2 ) )
=> ( ( add_mset_list_list_a @ A @ ( minus_5831295526526677175list_a @ N2 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) )
= ( add_mset_list_list_a @ A @ N2 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_1021_add__mset__remove__trivial__If,axiom,
! [A: list_a,N2: multiset_list_a] :
( ( ( member_list_a @ A @ ( set_mset_list_a @ N2 ) )
=> ( ( add_mset_list_a @ A @ ( minus_7431248565939055793list_a @ N2 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) )
= N2 ) )
& ( ~ ( member_list_a @ A @ ( set_mset_list_a @ N2 ) )
=> ( ( add_mset_list_a @ A @ ( minus_7431248565939055793list_a @ N2 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) )
= ( add_mset_list_a @ A @ N2 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_1022_add__mset__remove__trivial__If,axiom,
! [A: a,N2: multiset_a] :
( ( ( member_a @ A @ ( set_mset_a @ N2 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= N2 ) )
& ( ~ ( member_a @ A @ ( set_mset_a @ N2 ) )
=> ( ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( add_mset_a @ A @ N2 ) ) ) ) ).
% add_mset_remove_trivial_If
thf(fact_1023_add__mset__remove__trivial__eq,axiom,
! [N2: multiset_list_list_a,A: list_list_a] :
( ( N2
= ( add_mset_list_list_a @ A @ ( minus_5831295526526677175list_a @ N2 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) ) )
= ( member_list_list_a @ A @ ( set_mset_list_list_a @ N2 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_1024_add__mset__remove__trivial__eq,axiom,
! [N2: multiset_list_a,A: list_a] :
( ( N2
= ( add_mset_list_a @ A @ ( minus_7431248565939055793list_a @ N2 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) )
= ( member_list_a @ A @ ( set_mset_list_a @ N2 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_1025_add__mset__remove__trivial__eq,axiom,
! [N2: multiset_a,A: a] :
( ( N2
= ( add_mset_a @ A @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
= ( member_a @ A @ ( set_mset_a @ N2 ) ) ) ).
% add_mset_remove_trivial_eq
thf(fact_1026_multiset__add__sub__el__shuffle,axiom,
! [C: list_list_a,B5: multiset_list_list_a,B: list_list_a] :
( ( member_list_list_a @ C @ ( set_mset_list_list_a @ B5 ) )
=> ( ( B != C )
=> ( ( add_mset_list_list_a @ B @ ( minus_5831295526526677175list_a @ B5 @ ( add_mset_list_list_a @ C @ zero_z1542645121299710087list_a ) ) )
= ( minus_5831295526526677175list_a @ ( add_mset_list_list_a @ B @ B5 ) @ ( add_mset_list_list_a @ C @ zero_z1542645121299710087list_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_1027_multiset__add__sub__el__shuffle,axiom,
! [C: list_a,B5: multiset_list_a,B: list_a] :
( ( member_list_a @ C @ ( set_mset_list_a @ B5 ) )
=> ( ( B != C )
=> ( ( add_mset_list_a @ B @ ( minus_7431248565939055793list_a @ B5 @ ( add_mset_list_a @ C @ zero_z4454100511807792257list_a ) ) )
= ( minus_7431248565939055793list_a @ ( add_mset_list_a @ B @ B5 ) @ ( add_mset_list_a @ C @ zero_z4454100511807792257list_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_1028_multiset__add__sub__el__shuffle,axiom,
! [C: a,B5: multiset_a,B: a] :
( ( member_a @ C @ ( set_mset_a @ B5 ) )
=> ( ( B != C )
=> ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B5 @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) )
= ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ B5 ) @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ) ).
% multiset_add_sub_el_shuffle
thf(fact_1029_more__than__one__mset__mset__diff,axiom,
! [A: list_list_a,M3: multiset_list_list_a] :
( ( member_list_list_a @ A @ ( set_mset_list_list_a @ ( minus_5831295526526677175list_a @ M3 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) ) )
=> ( ( set_mset_list_list_a @ ( minus_5831295526526677175list_a @ M3 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) )
= ( set_mset_list_list_a @ M3 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_1030_more__than__one__mset__mset__diff,axiom,
! [A: list_a,M3: multiset_list_a] :
( ( member_list_a @ A @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) )
=> ( ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) )
= ( set_mset_list_a @ M3 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_1031_more__than__one__mset__mset__diff,axiom,
! [A: a,M3: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) )
=> ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) )
= ( set_mset_a @ M3 ) ) ) ).
% more_than_one_mset_mset_diff
thf(fact_1032_size__single,axiom,
! [B: list_a] :
( ( size_s2335926164413107382list_a @ ( add_mset_list_a @ B @ zero_z4454100511807792257list_a ) )
= one_one_nat ) ).
% size_single
thf(fact_1033_size__single,axiom,
! [B: a] :
( ( size_size_multiset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
= one_one_nat ) ).
% size_single
thf(fact_1034_size__1__singleton__mset,axiom,
! [M3: multiset_list_a] :
( ( ( size_s2335926164413107382list_a @ M3 )
= one_one_nat )
=> ? [A3: list_a] :
( M3
= ( add_mset_list_a @ A3 @ zero_z4454100511807792257list_a ) ) ) ).
% size_1_singleton_mset
thf(fact_1035_size__1__singleton__mset,axiom,
! [M3: multiset_a] :
( ( ( size_size_multiset_a @ M3 )
= one_one_nat )
=> ? [A3: a] :
( M3
= ( add_mset_a @ A3 @ zero_zero_multiset_a ) ) ) ).
% size_1_singleton_mset
thf(fact_1036_insert__subset__eq__iff,axiom,
! [A: list_list_a,A2: multiset_list_list_a,B5: multiset_list_list_a] :
( ( subset8447756916971205105list_a @ ( add_mset_list_list_a @ A @ A2 ) @ B5 )
= ( ( member_list_list_a @ A @ ( set_mset_list_list_a @ B5 ) )
& ( subset8447756916971205105list_a @ A2 @ ( minus_5831295526526677175list_a @ B5 @ ( add_mset_list_list_a @ A @ zero_z1542645121299710087list_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_1037_insert__subset__eq__iff,axiom,
! [A: list_a,A2: multiset_list_a,B5: multiset_list_a] :
( ( subseteq_mset_list_a @ ( add_mset_list_a @ A @ A2 ) @ B5 )
= ( ( member_list_a @ A @ ( set_mset_list_a @ B5 ) )
& ( subseteq_mset_list_a @ A2 @ ( minus_7431248565939055793list_a @ B5 @ ( add_mset_list_a @ A @ zero_z4454100511807792257list_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_1038_insert__subset__eq__iff,axiom,
! [A: a,A2: multiset_a,B5: multiset_a] :
( ( subseteq_mset_a @ ( add_mset_a @ A @ A2 ) @ B5 )
= ( ( member_a @ A @ ( set_mset_a @ B5 ) )
& ( subseteq_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ B5 @ ( add_mset_a @ A @ zero_zero_multiset_a ) ) ) ) ) ).
% insert_subset_eq_iff
thf(fact_1039_size__Diff1__le,axiom,
! [M3: multiset_list_a,X: list_a] : ( ord_less_eq_nat @ ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) ) @ ( size_s2335926164413107382list_a @ M3 ) ) ).
% size_Diff1_le
thf(fact_1040_size__Diff1__le,axiom,
! [M3: multiset_a,X: a] : ( ord_less_eq_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M3 ) ) ).
% size_Diff1_le
thf(fact_1041_size__Diff1__less,axiom,
! [X: list_list_a,M3: multiset_list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M3 ) )
=> ( ord_less_nat @ ( size_s8523483970790017596list_a @ ( minus_5831295526526677175list_a @ M3 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) ) @ ( size_s8523483970790017596list_a @ M3 ) ) ) ).
% size_Diff1_less
thf(fact_1042_size__Diff1__less,axiom,
! [X: list_a,M3: multiset_list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M3 ) )
=> ( ord_less_nat @ ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) ) @ ( size_s2335926164413107382list_a @ M3 ) ) ) ).
% size_Diff1_less
thf(fact_1043_size__Diff1__less,axiom,
! [X: a,M3: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M3 ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M3 ) ) ) ).
% size_Diff1_less
thf(fact_1044_size__Diff2__less,axiom,
! [X: list_list_a,M3: multiset_list_list_a,Y: list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M3 ) )
=> ( ( member_list_list_a @ Y @ ( set_mset_list_list_a @ M3 ) )
=> ( ord_less_nat @ ( size_s8523483970790017596list_a @ ( minus_5831295526526677175list_a @ ( minus_5831295526526677175list_a @ M3 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) @ ( add_mset_list_list_a @ Y @ zero_z1542645121299710087list_a ) ) ) @ ( size_s8523483970790017596list_a @ M3 ) ) ) ) ).
% size_Diff2_less
thf(fact_1045_size__Diff2__less,axiom,
! [X: list_a,M3: multiset_list_a,Y: list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M3 ) )
=> ( ( member_list_a @ Y @ ( set_mset_list_a @ M3 ) )
=> ( ord_less_nat @ ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) @ ( add_mset_list_a @ Y @ zero_z4454100511807792257list_a ) ) ) @ ( size_s2335926164413107382list_a @ M3 ) ) ) ) ).
% size_Diff2_less
thf(fact_1046_size__Diff2__less,axiom,
! [X: a,M3: multiset_a,Y: a] :
( ( member_a @ X @ ( set_mset_a @ M3 ) )
=> ( ( member_a @ Y @ ( set_mset_a @ M3 ) )
=> ( ord_less_nat @ ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) @ ( size_size_multiset_a @ M3 ) ) ) ) ).
% size_Diff2_less
thf(fact_1047_size__Diff__singleton,axiom,
! [X: list_list_a,M3: multiset_list_list_a] :
( ( member_list_list_a @ X @ ( set_mset_list_list_a @ M3 ) )
=> ( ( size_s8523483970790017596list_a @ ( minus_5831295526526677175list_a @ M3 @ ( add_mset_list_list_a @ X @ zero_z1542645121299710087list_a ) ) )
= ( minus_minus_nat @ ( size_s8523483970790017596list_a @ M3 ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_1048_size__Diff__singleton,axiom,
! [X: list_a,M3: multiset_list_a] :
( ( member_list_a @ X @ ( set_mset_list_a @ M3 ) )
=> ( ( size_s2335926164413107382list_a @ ( minus_7431248565939055793list_a @ M3 @ ( add_mset_list_a @ X @ zero_z4454100511807792257list_a ) ) )
= ( minus_minus_nat @ ( size_s2335926164413107382list_a @ M3 ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_1049_size__Diff__singleton,axiom,
! [X: a,M3: multiset_a] :
( ( member_a @ X @ ( set_mset_a @ M3 ) )
=> ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M3 @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
= ( minus_minus_nat @ ( size_size_multiset_a @ M3 ) @ one_one_nat ) ) ) ).
% size_Diff_singleton
thf(fact_1050_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_1051_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_1052_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_1053_primeness__condition,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeness_condition
thf(fact_1054_ring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI
thf(fact_1055_ring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).
% ring_primeE(3)
thf(fact_1056_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_1057_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_1058_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_1059_poly__mult__degree__one__monic__imp__same__roots,axiom,
! [A: a,P2: list_a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != 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 ) ) @ P2 ) )
= ( add_mset_a @ A @ ( polynomial_roots_a_b @ r @ P2 ) ) ) ) ) ) ).
% poly_mult_degree_one_monic_imp_same_roots
thf(fact_1060_p_Oroots__inclI_H,axiom,
! [P2: list_list_a,M: multiset_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_list_a )
=> ( ord_less_eq_nat @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ A3 ) @ ( count_list_a @ M @ A3 ) ) ) )
=> ( subseteq_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) @ M ) ) ) ).
% p.roots_inclI'
thf(fact_1061_p_Opirreducible__degree,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 )
=> ( ord_less_eq_nat @ one_one_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) ) ) ) ) ).
% p.pirreducible_degree
thf(fact_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_p_Osubring__props_I1_J,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.subring_props(1)
thf(fact_1080_p_Osubring__props_I2_J,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K2 ) ) ).
% p.subring_props(2)
thf(fact_1081_p_Osubring__props_I6_J,axiom,
! [K2: set_list_a,H1: list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K2 )
=> ( ( member_list_a @ H2 @ K2 )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 ) @ K2 ) ) ) ) ).
% p.subring_props(6)
thf(fact_1082_p_Osubring__props_I3_J,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K2 ) ) ).
% p.subring_props(3)
thf(fact_1083_p_Osubring__props_I5_J,axiom,
! [K2: set_list_a,H: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H @ K2 )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ K2 ) ) ) ).
% p.subring_props(5)
thf(fact_1084_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_1085_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A3
!= ( zero_a_b @ r ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A3 @ X4 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_1086_p_Odegree__one__imp__pirreducible,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat )
= one_one_nat )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 ) ) ) ) ).
% p.degree_one_imp_pirreducible
thf(fact_1087_p_Oalg__mult__eq__count__roots,axiom,
! [P2: list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
= ( count_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ) ).
% p.alg_mult_eq_count_roots
thf(fact_1088_p_Opirreducible__pow__pdivides__iff,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a,R2: list_list_a,N: nat] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 )
=> ( ~ ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q )
=> ( ( 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 ) ) @ K2 ) @ P2 @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Q @ 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 ) ) @ K2 ) @ P2 @ N ) @ R2 ) ) ) ) ) ) ) ) ).
% p.pirreducible_pow_pdivides_iff
thf(fact_1089_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_1090_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_1091_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_1092_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_1093_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_1094_p_Ouniv__poly__is__principal,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ).
% p.univ_poly_is_principal
thf(fact_1095_p_Oexists__unique__long__division,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ? [X3: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q @ X3 )
& ! [Y5: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q @ Y5 )
=> ( Y5 = X3 ) ) ) ) ) ) ) ).
% p.exists_unique_long_division
thf(fact_1096_p_OpprimeE_I3_J,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a,R2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ R2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Q @ R2 ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q )
| ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ R2 ) ) ) ) ) ) ) ) ).
% p.pprimeE(3)
thf(fact_1097_alg__mult__eq__count__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4422430861927485590lt_a_b @ r @ P2 )
= ( count_a @ ( polynomial_roots_a_b @ r @ P2 ) ) ) ) ).
% alg_mult_eq_count_roots
thf(fact_1098_p_OpprimeE_I1_J,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 )
=> ( P2 != nil_list_a ) ) ) ) ).
% p.pprimeE(1)
thf(fact_1099_p_Opprime__iff__pirreducible,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 ) ) ) ) ).
% p.pprime_iff_pirreducible
thf(fact_1100_roots__inclI_H,axiom,
! [P2: list_a,M: multiset_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P2 != nil_a )
=> ( ord_less_eq_nat @ ( polyno4422430861927485590lt_a_b @ r @ P2 @ A3 ) @ ( count_a @ M @ A3 ) ) ) )
=> ( subseteq_mset_a @ ( polynomial_roots_a_b @ r @ P2 ) @ M ) ) ) ).
% roots_inclI'
thf(fact_1101_p_Opmod__const_I1_J,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q )
= nil_list_a ) ) ) ) ) ).
% p.pmod_const(1)
thf(fact_1102_p_Olong__division__a__inv_I1_J,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ P2 ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) ) ) ) ) ) ).
% p.long_division_a_inv(1)
thf(fact_1103_p_Olong__division__closed_I1_J,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ) ) ) ).
% p.long_division_closed(1)
thf(fact_1104_p_Olong__division__zero_I1_J,axiom,
! [K2: set_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Q )
= nil_list_a ) ) ) ).
% p.long_division_zero(1)
thf(fact_1105_p_Orupture__one__not__zero,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) )
=> ( ( one_se2489417650821308733t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P2 ) )
!= ( zero_s2920163772466840039t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P2 ) ) ) ) ) ) ).
% p.rupture_one_not_zero
thf(fact_1106_p_Opmod__degree,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q )
= nil_list_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% p.pmod_degree
thf(fact_1107_p_Olong__division__closed_I2_J,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) ) ) ) ) ).
% p.long_division_closed(2)
thf(fact_1108_p_Olong__division__zero_I2_J,axiom,
! [K2: set_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Q )
= nil_list_a ) ) ) ).
% p.long_division_zero(2)
thf(fact_1109_p_Olong__division__a__inv_I2_J,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ P2 ) @ Q )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) ) ) ) ) ) ).
% p.long_division_a_inv(2)
thf(fact_1110_p_Orupture__is__field__iff__pirreducible,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( field_1540243473349940225t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P2 ) )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ P2 ) ) ) ) ).
% p.rupture_is_field_iff_pirreducible
thf(fact_1111_p_Opmod__zero__iff__pdivides,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ P2 ) ) ) ) ) ).
% p.pmod_zero_iff_pdivides
thf(fact_1112_p_Osame__pmod__iff__pdivides,axiom,
! [K2: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q ) )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ A @ B ) ) ) ) ) ) ) ).
% p.same_pmod_iff_pdivides
thf(fact_1113_p_Opmod__const_I2_J,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ P2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ Q ) @ one_one_nat ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q )
= P2 ) ) ) ) ) ).
% p.pmod_const(2)
thf(fact_1114_p_Osubfield__long__division__theorem__shell,axiom,
! [K2: set_list_a,P2: list_list_a,B: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( B
!= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ? [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 ) ) @ K2 ) ) )
& ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
& ( P2
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ B @ Q2 ) @ R4 ) )
& ( ( R4
= ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
| ( 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_1115_p_Olong__divisionE,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) ) ) ) ) ) ) ).
% p.long_divisionE
thf(fact_1116_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_1117_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_1118_p_Ocombine_Ocases,axiom,
! [X: produc7709606177366032167list_a] :
( ! [K5: list_a,Ks: list_list_a,U2: list_a,Us2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ ( cons_list_a @ K5 @ Ks ) @ ( cons_list_a @ U2 @ Us2 ) ) )
=> ( ! [Us2: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ nil_list_a @ Us2 ) )
=> ~ ! [Ks: list_list_a] :
( X
!= ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).
% p.combine.cases
thf(fact_1119_p_Olong__division__add_I2_J,axiom,
! [K2: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q ) ) ) ) ) ) ) ).
% p.long_division_add(2)
thf(fact_1120_p_Olong__division__add__iff,axiom,
! [K2: set_list_a,A: list_list_a,B: list_list_a,C: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q ) )
= ( ( 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 ) ) @ K2 ) @ A @ C ) @ Q )
= ( 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 ) ) @ K2 ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% p.long_division_add_iff
thf(fact_1121_p_Opderiv__add,axiom,
! [K2: set_list_a,F: list_list_a,G2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ G2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ F @ G2 ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G2 ) ) ) ) ) ) ).
% p.pderiv_add
thf(fact_1122_p_Olong__division__add_I1_J,axiom,
! [K2: set_list_a,A: list_list_a,B: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ A @ B ) @ Q )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q ) ) ) ) ) ) ) ).
% p.long_division_add(1)
thf(fact_1123_p_Oexists__long__division,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( Q != 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 ) ) @ K2 ) ) )
=> ! [R4: list_list_a] :
( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ~ ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q @ ( produc8696003437204565271list_a @ B3 @ R4 ) ) ) ) ) ) ) ) ).
% p.exists_long_division
thf(fact_1124_p_Opderiv__mult,axiom,
! [K2: set_list_a,F: list_list_a,G2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ G2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ F @ G2 ) )
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F ) @ G2 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ F @ ( formal6075833236969493044t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G2 ) ) ) ) ) ) ) ).
% p.pderiv_mult
thf(fact_1125_p_Opdiv__pmod,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( P2
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Q @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) ) ) ) ) ) ).
% p.pdiv_pmod
thf(fact_1126_p_Olong__divisionI,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a,B: list_list_a,R2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( Q != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q @ ( produc8696003437204565271list_a @ B @ R2 ) )
=> ( ( produc8696003437204565271list_a @ B @ R2 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) ) ) ) ) ) ) ) ).
% p.long_divisionI
thf(fact_1127_p_Olong__dividesI,axiom,
! [B: list_list_a,R2: list_list_a,P2: list_list_a,Q: 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 ) ) ) ) ) )
=> ( ( P2
= ( 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 ) ) ) ) @ Q @ 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 @ Q ) @ one_one_nat ) ) )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q @ ( produc8696003437204565271list_a @ B @ R2 ) ) ) ) ) ) ).
% p.long_dividesI
thf(fact_1128_p_Ofield__long__division__theorem,axiom,
! [K2: set_list_a,P2: list_list_a,B: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P2 )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ B )
=> ( ( B != nil_list_a )
=> ? [Q2: list_list_a,R4: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ Q2 )
& ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ R4 )
& ( P2
= ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ 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_1129_p_Oline__extension__smult__closed,axiom,
! [K2: set_list_a,E: set_list_a,A: list_a,K: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K5: list_a,V: list_a] :
( ( member_list_a @ K5 @ K2 )
=> ( ( member_list_a @ V @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K5 @ 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 @ K @ K2 )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A @ E ) ) ) ) ) ) ) ) ).
% p.line_extension_smult_closed
thf(fact_1130_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_1131_combine_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [K5: a,Ks: list_a,U2: a,Us2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ K5 @ Ks ) @ ( cons_a @ U2 @ Us2 ) ) )
=> ( ! [Us2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Us2 ) )
=> ~ ! [Ks: list_a] :
( X
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_1132_p_Oline__extension__in__carrier,axiom,
! [K2: set_list_a,A: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( 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 ) ) @ K2 @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.line_extension_in_carrier
thf(fact_1133_p_Ovar__closed_I2_J,axiom,
! [K2: set_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.var_closed(2)
thf(fact_1134_p_Oconst__term__zero,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P2 )
=> ( ( P2 != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
= ( 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 ) ) @ K2 @ P4 )
=> ( ( P4 != nil_list_a )
=> ( P2
!= ( 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_1135_p_Ozero__is__polynomial,axiom,
! [K2: set_list_a] : ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ nil_list_a ) ).
% p.zero_is_polynomial
thf(fact_1136_p_Ocarrier__polynomial,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P2 )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P2 ) ) ) ).
% p.carrier_polynomial
thf(fact_1137_p_Oone__is__polynomial,axiom,
! [K2: set_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ).
% p.one_is_polynomial
thf(fact_1138_long__dividesI,axiom,
! [B: list_a,R2: list_a,P2: list_a,Q: list_a] :
( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ B ) @ R2 ) )
=> ( ( ( R2 = nil_a )
| ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ R2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) )
=> ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ ( produc6837034575241423639list_a @ B @ R2 ) ) ) ) ) ) ).
% long_dividesI
thf(fact_1139_p_Oconst__term__explicit,axiom,
! [P2: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P2 != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 )
= A )
=> ~ ! [P4: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P2
!= ( append_list_a @ P4 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ).
% p.const_term_explicit
thf(fact_1140_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_1141_var__closed_I2_J,axiom,
polynomial_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ ( var_a_b @ r ) ).
% var_closed(2)
thf(fact_1142_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_1143_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_1144_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_1145_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_1146_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_1147_p_Osubring__props_I7_J,axiom,
! [K2: set_list_a,H1: list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K2 )
=> ( ( member_list_a @ H2 @ K2 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 ) @ K2 ) ) ) ) ).
% p.subring_props(7)
thf(fact_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_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_1163_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_1164_p_Opolynomial__incl,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P2 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ K2 ) ) ).
% p.polynomial_incl
thf(fact_1165_pderiv__add,axiom,
! [F: list_a,G2: list_a] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G2 @ ( 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 @ G2 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( formal4452980811800949548iv_a_b @ r @ F ) @ ( formal4452980811800949548iv_a_b @ r @ G2 ) ) ) ) ) ).
% pderiv_add
thf(fact_1166_p_Oline__extension__mem__iff,axiom,
! [U: list_a,K2: 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 ) ) @ K2 @ A @ E ) )
= ( ? [X2: list_a] :
( ( member_list_a @ X2 @ K2 )
& ? [Y6: list_a] :
( ( member_list_a @ Y6 @ 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 ) @ Y6 ) ) ) ) ) ) ).
% p.line_extension_mem_iff
thf(fact_1167_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_1168_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_1169_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_1170_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_1171_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_1172_p_Oconst__term__simprules_I1_J,axiom,
! [P2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.const_term_simprules(1)
thf(fact_1173_pderiv__mult,axiom,
! [F: list_a,G2: list_a] :
( ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G2 @ ( 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 @ G2 ) )
= ( 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 ) @ G2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ ( formal4452980811800949548iv_a_b @ r @ G2 ) ) ) ) ) ) ).
% pderiv_mult
thf(fact_1174_p_Oconst__term__simprules__shell_I3_J,axiom,
! [K2: set_list_a,P2: list_list_a,Q: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 ) @ P2 @ Q ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ) ).
% p.const_term_simprules_shell(3)
thf(fact_1175_zero__is__polynomial,axiom,
! [K2: set_a] : ( polynomial_a_b @ r @ K2 @ nil_a ) ).
% zero_is_polynomial
thf(fact_1176_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_1177_p_Oconst__term__eq__last,axiom,
! [P2: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( 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 ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P2 @ ( cons_list_a @ A @ nil_list_a ) ) )
= A ) ) ) ).
% p.const_term_eq_last
thf(fact_1178_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_1179_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_1180_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_1181_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_1182_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_1183_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_1184_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_1185_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_1186_p_Omonom__in__carrier,axiom,
! [A: list_a,N: nat] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.monom_in_carrier
thf(fact_1187_p_Opolynomial__in__carrier,axiom,
! [K2: set_list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ P2 )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.polynomial_in_carrier
thf(fact_1188_p_Oexp__base__closed,axiom,
! [X: list_a,N: nat] :
( ( member_list_a @ X @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.exp_base_closed
thf(fact_1189_monic__poly__add__distinct,axiom,
! [F: list_a,G2: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( member_list_a @ G2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ ( minus_minus_nat @ ( size_size_list_a @ G2 ) @ 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 @ G2 ) ) ) ) ) ).
% monic_poly_add_distinct
thf(fact_1190_polynomial__incl,axiom,
! [K2: set_a,P2: list_a] :
( ( polynomial_a_b @ r @ K2 @ P2 )
=> ( ord_less_eq_set_a @ ( set_a2 @ P2 ) @ K2 ) ) ).
% polynomial_incl
thf(fact_1191_monic__poly__var,axiom,
monic_3145109188698636716ly_a_b @ r @ ( var_a_b @ r ) ).
% monic_poly_var
thf(fact_1192_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_1193_monic__poly__mult,axiom,
! [F: list_a,G2: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G2 )
=> ( monic_3145109188698636716ly_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ G2 ) ) ) ) ).
% monic_poly_mult
thf(fact_1194_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_1195_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_1196_gauss__poly__monic,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( monic_3145109188698636716ly_a_b @ r @ ( card_I2373409586816755191ly_a_b @ r @ N ) ) ) ).
% gauss_poly_monic
thf(fact_1197_divides__monic__poly,axiom,
! [F: list_a,G2: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G2 )
=> ( ! [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 @ G2 ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ F @ G2 ) ) ) ) ).
% divides_monic_poly
thf(fact_1198_exp__base__closed,axiom,
! [X: a,N: nat] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_1199_p_Ofactors__mult,axiom,
! [Fa: list_list_a,A: list_a,Fb: list_list_a,B: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fa @ A )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fb @ B )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fa ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fb ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Fa @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ).
% p.factors_mult
thf(fact_1200_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_1201_p_Ofactors__closed,axiom,
! [Fs: list_list_a,A: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( 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 ) ) ) ) ) ) ).
% p.factors_closed
thf(fact_1202_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_1203_p_Orupture__char,axiom,
! [K2: set_list_a,F: list_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( 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 ) ) @ K2 ) ) )
=> ( ( 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 ) ) @ K2 @ F ) )
= ( ring_c500279861223467766t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.rupture_char
thf(fact_1204_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 )
=> ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ H3 )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ Xa2 ) @ H3 ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ H3 ) ) ) ) ) ).
% p.add.one_in_subset
thf(fact_1205_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_1206_p_Osubring__props_I4_J,axiom,
! [K2: set_list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( K2 != bot_bot_set_list_a ) ) ).
% p.subring_props(4)
thf(fact_1207_p_Omonom__eq__var__pow,axiom,
! [K2: set_list_a,A: list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) @ ( cons_list_a @ A @ nil_list_a ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) ) ) ) ) ).
% p.monom_eq_var_pow
thf(fact_1208_multiplicity__ge__iff,axiom,
! [D: list_a,F: list_a,K: 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 @ K @ ( monic_5301438133677370042lt_a_b @ r @ D @ F ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ D @ K ) @ F ) ) ) ) ).
% multiplicity_ge_iff
thf(fact_1209_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_1210_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_1211_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_1212_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_1213_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_1214_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_1215_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_1216_p_Olead__coeff__not__zero,axiom,
! [K2: set_list_a,A: list_a,P2: list_list_a] :
( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( cons_list_a @ A @ P2 ) )
=> ( member_list_a @ A @ ( minus_646659088055828811list_a @ K2 @ ( 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_1217_p_Osubfield__m__inv__simprule,axiom,
! [K2: set_list_a,K: list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K @ ( minus_646659088055828811list_a @ K2 @ ( 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 ) ) @ K @ A ) @ K2 )
=> ( member_list_a @ A @ K2 ) ) ) ) ) ).
% p.subfield_m_inv_simprule
thf(fact_1218_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_1219_p_Olead__coeff__in__carrier,axiom,
! [K2: set_list_a,A: list_a,P2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( polyno1315193887021588240t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( cons_list_a @ A @ P2 ) )
=> ( 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_1220_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_1221_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_1222_p_Oconst__is__polynomial,axiom,
! [A: list_a,K2: set_list_a] :
( ( member_list_a @ A @ ( minus_646659088055828811list_a @ K2 @ ( 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 ) ) @ K2 @ ( cons_list_a @ A @ nil_list_a ) ) ) ).
% p.const_is_polynomial
thf(fact_1223_p_Omonom__is__polynomial,axiom,
! [K2: set_list_a,A: list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ K2 @ ( 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 ) ) @ K2 @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N ) ) ) ) ).
% p.monom_is_polynomial
thf(fact_1224_p_Oeuclidean__domainI,axiom,
! [Phi: list_a > nat] :
( ! [A3: list_a,B3: list_a] :
( ( member_list_a @ A3 @ ( 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 ) ) )
=> ? [Q3: list_a,R5: list_a] :
( ( member_list_a @ Q3 @ ( 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 ) ) ) )
& ( A3
= ( 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 @ Q3 ) @ 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_1225_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_1226_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_1227_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_1228_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_1229_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_1230_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_1231_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_1232_lead__coeff__not__zero,axiom,
! [K2: set_a,A: a,P2: list_a] :
( ( polynomial_a_b @ r @ K2 @ ( cons_a @ A @ P2 ) )
=> ( member_a @ A @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% lead_coeff_not_zero
thf(fact_1233_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_1234_p_Ovar__carr,axiom,
! [K2: set_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% p.var_carr
thf(fact_1235_p_Ovar__pow__carr,axiom,
! [K2: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) @ ( 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 ) ) @ K2 ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% p.var_pow_carr
thf(fact_1236_p_Odegree__pow,axiom,
! [K2: set_list_a,F: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ F @ N ) ) @ one_one_nat )
= ( times_times_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat ) @ N ) ) ) ) ).
% p.degree_pow
thf(fact_1237_const__is__polynomial,axiom,
! [A: a,K2: set_a] :
( ( member_a @ A @ ( minus_minus_set_a @ K2 @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( polynomial_a_b @ r @ K2 @ ( cons_a @ A @ nil_a ) ) ) ).
% const_is_polynomial
thf(fact_1238_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_1239_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_1240_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_1241_p_Odegree__mult,axiom,
! [K2: set_list_a,F: list_list_a,G2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( member_list_list_a @ G2 @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ F @ G2 ) ) @ one_one_nat )
= ( plus_plus_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ G2 ) @ one_one_nat ) ) ) ) ) ) ).
% p.degree_mult
thf(fact_1242_p_Odegree__add__distinct,axiom,
! [K2: set_list_a,F: list_list_a,G2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K2 @ ( 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 ) ) @ K2 ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( member_list_list_a @ G2 @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat )
!= ( minus_minus_nat @ ( size_s349497388124573686list_a @ G2 ) @ one_one_nat ) )
=> ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ F @ G2 ) ) @ one_one_nat )
= ( ord_max_nat @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ F ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s349497388124573686list_a @ G2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% p.degree_add_distinct
thf(fact_1243_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_1244_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_1245_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_1246_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1247_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1248_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1249_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1250_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_1251_max__nat_Oleft__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ zero_zero_nat @ A )
= A ) ).
% max_nat.left_neutral
thf(fact_1252_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_1253_max__nat_Oright__neutral,axiom,
! [A: nat] :
( ( ord_max_nat @ A @ zero_zero_nat )
= A ) ).
% max_nat.right_neutral
thf(fact_1254_max__0L,axiom,
! [N: nat] :
( ( ord_max_nat @ zero_zero_nat @ N )
= N ) ).
% max_0L
thf(fact_1255_max__0R,axiom,
! [N: nat] :
( ( ord_max_nat @ N @ zero_zero_nat )
= N ) ).
% max_0R
thf(fact_1256_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_1257_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1258_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1259_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1260_degree__add__distinct,axiom,
! [F: list_a,G2: 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 @ G2 @ ( 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 @ G2 ) @ one_one_nat ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ G2 ) ) @ one_one_nat )
= ( ord_max_nat @ ( minus_minus_nat @ ( size_size_list_a @ F ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ G2 ) @ one_one_nat ) ) ) ) ) ) ).
% degree_add_distinct
thf(fact_1261_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1262_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1263_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1264_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1265_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1266_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1267_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1268_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1269_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1270_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
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $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,
( ( polyno5814909790663948098es_a_b @ r @ ( 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 @ ( power_power_nat @ a2 @ n ) @ one_one_nat ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( var_a_b @ r ) ) @ ( 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 @ ( power_power_nat @ a2 @ m ) @ one_one_nat ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( var_a_b @ r ) ) )
= ( 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 ) @ ( minus_minus_nat @ ( power_power_nat @ a2 @ n ) @ one_one_nat ) ) @ ( 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 ) @ ( minus_minus_nat @ ( power_power_nat @ a2 @ m ) @ one_one_nat ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
%------------------------------------------------------------------------------