TPTP Problem File: SLH0306^1.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/0007_Monic_Polynomial_Factorization/prob_00487_016311__18313828_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1637 ( 248 unt; 358 typ; 0 def)
% Number of atoms : 4541 ( 998 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 23255 ( 362 ~; 68 |; 143 &;19988 @)
% ( 0 <=>;2694 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 41 ( 40 usr)
% Number of type conns : 633 ( 633 >; 0 *; 0 +; 0 <<)
% Number of symbols : 319 ( 318 usr; 12 con; 0-4 aty)
% Number of variables : 2723 ( 49 ^;2597 !; 77 ?;2723 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:22:33.872
%------------------------------------------------------------------------------
% Could-be-implicit typings (40)
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thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r1091214237498979717t_unit: partia7496981018696276118t_unit > set_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6795642478576035723t_unit: partia6043505979758434576t_unit > set_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Set_OCollect_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_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
insert8656068109252948922list_a: set_list_list_a > set_set_list_list_a > set_set_list_list_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
insert_set_list_a: set_list_a > set_set_list_a > set_set_list_a ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subfie1779122896746047282t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_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_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
member5524387281408368019list_a: list_set_list_a > set_list_set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
member_list_set_a: list_set_a > set_list_set_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member334759470184282131list_a: set_list_list_a > set_set_list_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_d____,type,
d: list_a ).
thf(sy_v_f,type,
f: list_a ).
% Relevant facts (1278)
thf(fact_0_assms,axiom,
monic_3145109188698636716ly_a_b @ r @ f ).
% assms
thf(fact_1_monic__poly__carr__2,axiom,
! [F: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( member_list_a @ F @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% monic_poly_carr_2
thf(fact_2_that,axiom,
monic_4919232885364369782ly_a_b @ r @ d ).
% that
thf(fact_3_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_4_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_5_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_6_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_7_p_Ofactorial__domain__axioms,axiom,
ring_f796907574329358751t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.factorial_domain_axioms
thf(fact_8_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_9_p_Onoetherian__domain__axioms,axiom,
ring_n4705423059119889713t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.noetherian_domain_axioms
thf(fact_10_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_11_p_Omult__of_Ocarrier__not__empty,axiom,
( ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
!= bot_bot_set_list_a ) ).
% p.mult_of.carrier_not_empty
thf(fact_12_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_13_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_14_p_Onoetherian__ring__axioms,axiom,
ring_n5188127996776581661t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.noetherian_ring_axioms
thf(fact_15_p_Oprincipal__domain__axioms,axiom,
ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.principal_domain_axioms
thf(fact_16_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_17_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_18_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_19_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_20_p_Ozeroprimeideal,axiom,
primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.zeroprimeideal
thf(fact_21_p_Oring__primeE_I1_J,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_primeE(1)
thf(fact_22_insert__Diff__single,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= ( insert_list_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_23_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_24_insert__Diff__single,axiom,
! [A: list_list_a,A2: set_list_list_a] :
( ( insert_list_list_a @ A @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) )
= ( insert_list_list_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_25_p_Ogenideal__zero,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% p.genideal_zero
thf(fact_26_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_27_empty__Collect__eq,axiom,
! [P2: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P2 ) )
= ( ! [X: list_a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_28_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X: a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_29_empty__Collect__eq,axiom,
! [P2: list_list_a > $o] :
( ( bot_bo1875519244922727510list_a
= ( collect_list_list_a @ P2 ) )
= ( ! [X: list_list_a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_30_Collect__empty__eq,axiom,
! [P2: list_a > $o] :
( ( ( collect_list_a @ P2 )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_31_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_32_Collect__empty__eq,axiom,
! [P2: list_list_a > $o] :
( ( ( collect_list_list_a @ P2 )
= bot_bo1875519244922727510list_a )
= ( ! [X: list_list_a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_33_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X: list_a] :
~ ( member_list_a @ X @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_34_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_35_all__not__in__conv,axiom,
! [A2: set_list_list_a] :
( ( ! [X: list_list_a] :
~ ( member_list_list_a @ X @ A2 ) )
= ( A2 = bot_bo1875519244922727510list_a ) ) ).
% all_not_in_conv
thf(fact_36_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_37_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_38_empty__iff,axiom,
! [C: list_list_a] :
~ ( member_list_list_a @ C @ bot_bo1875519244922727510list_a ) ).
% empty_iff
thf(fact_39_insert__absorb2,axiom,
! [X2: list_a,A2: set_list_a] :
( ( insert_list_a @ X2 @ ( insert_list_a @ X2 @ A2 ) )
= ( insert_list_a @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_40_insert__absorb2,axiom,
! [X2: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A2 ) )
= ( insert_a @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_41_insert__absorb2,axiom,
! [X2: list_list_a,A2: set_list_list_a] :
( ( insert_list_list_a @ X2 @ ( insert_list_list_a @ X2 @ A2 ) )
= ( insert_list_list_a @ X2 @ A2 ) ) ).
% insert_absorb2
thf(fact_42_insert__iff,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_43_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_44_insert__iff,axiom,
! [A: list_list_a,B: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ ( insert_list_list_a @ B @ A2 ) )
= ( ( A = B )
| ( member_list_list_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_45_insertCI,axiom,
! [A: list_a,B2: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A @ B2 )
=> ( A = B ) )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_46_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_47_insertCI,axiom,
! [A: list_list_a,B2: set_list_list_a,B: list_list_a] :
( ( ~ ( member_list_list_a @ A @ B2 )
=> ( A = B ) )
=> ( member_list_list_a @ A @ ( insert_list_list_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_48_Diff__idemp,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ B2 )
= ( minus_646659088055828811list_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_49_Diff__idemp,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_50_Diff__idemp,axiom,
! [A2: set_list_list_a,B2: set_list_list_a] :
( ( minus_5335179877275218001list_a @ ( minus_5335179877275218001list_a @ A2 @ B2 ) @ B2 )
= ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_51_Diff__iff,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
= ( ( member_list_a @ C @ A2 )
& ~ ( member_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_52_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_53_Diff__iff,axiom,
! [C: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A2 @ B2 ) )
= ( ( member_list_list_a @ C @ A2 )
& ~ ( member_list_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_54_DiffI,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_55_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_56_DiffI,axiom,
! [C: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ A2 )
=> ( ~ ( member_list_list_a @ C @ B2 )
=> ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_57_p_Oring__primeE_I3_J,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.ring_primeE(3)
thf(fact_58_p_Oring__primeI,axiom,
! [P: list_a] :
( ( P
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.ring_primeI
thf(fact_59_p_Ogenideal__self_H,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% p.genideal_self'
thf(fact_60_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_61_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_62_singletonI,axiom,
! [A: list_list_a] : ( member_list_list_a @ A @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ).
% singletonI
thf(fact_63_Diff__cancel,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ A2 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_64_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_65_Diff__cancel,axiom,
! [A2: set_list_list_a] :
( ( minus_5335179877275218001list_a @ A2 @ A2 )
= bot_bo1875519244922727510list_a ) ).
% Diff_cancel
thf(fact_66_empty__Diff,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ bot_bot_set_list_a @ A2 )
= bot_bot_set_list_a ) ).
% empty_Diff
thf(fact_67_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_68_empty__Diff,axiom,
! [A2: set_list_list_a] :
( ( minus_5335179877275218001list_a @ bot_bo1875519244922727510list_a @ A2 )
= bot_bo1875519244922727510list_a ) ).
% empty_Diff
thf(fact_69_Diff__empty,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Diff_empty
thf(fact_70_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_71_Diff__empty,axiom,
! [A2: set_list_list_a] :
( ( minus_5335179877275218001list_a @ A2 @ bot_bo1875519244922727510list_a )
= A2 ) ).
% Diff_empty
thf(fact_72_insert__Diff1,axiom,
! [X2: list_a,B2: set_list_a,A2: set_list_a] :
( ( member_list_a @ X2 @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X2 @ A2 ) @ B2 )
= ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_73_insert__Diff1,axiom,
! [X2: a,B2: set_a,A2: set_a] :
( ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_74_insert__Diff1,axiom,
! [X2: list_list_a,B2: set_list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ X2 @ B2 )
=> ( ( minus_5335179877275218001list_a @ ( insert_list_list_a @ X2 @ A2 ) @ B2 )
= ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_75_Diff__insert0,axiom,
! [X2: list_a,A2: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X2 @ A2 )
=> ( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ B2 ) )
= ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_76_Diff__insert0,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_77_Diff__insert0,axiom,
! [X2: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ~ ( member_list_list_a @ X2 @ A2 )
=> ( ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ X2 @ B2 ) )
= ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_78_p_Omult__of_Olcmof__exists,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ? [C2: list_a] :
( ( member_list_a @ C2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( islcm_4429780156966476082t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ C2 @ A @ B ) ) ) ) ).
% p.mult_of.lcmof_exists
thf(fact_79_mem__Collect__eq,axiom,
! [A: list_a,P2: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
! [A: list_list_a,P2: list_list_a > $o] :
( ( member_list_list_a @ A @ ( collect_list_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A2: set_list_list_a] :
( ( collect_list_list_a
@ ^ [X: list_list_a] : ( member_list_list_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_85_p_Omult__of_Odivisor__chain__condition__monoid__axioms,axiom,
diviso7418750317856897340t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.mult_of.divisor_chain_condition_monoid_axioms
thf(fact_86_p_Omult__of_Oprimeness__condition__monoid__axioms,axiom,
primen7481049025794127375t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.mult_of.primeness_condition_monoid_axioms
thf(fact_87_field_Omonic__poly__carr__2,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_8143709425247463374t_unit @ R @ F )
=> ( member6124916891863447321list_a @ F @ ( partia4786195206251613344t_unit @ ( ring_m6059372198833198775t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) ) ) ) ).
% field.monic_poly_carr_2
thf(fact_88_field_Omonic__poly__carr__2,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_3395465470813675732t_unit @ R @ F )
=> ( member5524387281408368019list_a @ F @ ( partia4809544917648604204t_unit @ ( ring_m9145517834477070013t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ) ) ).
% field.monic_poly_carr_2
thf(fact_89_field_Omonic__poly__carr__2,axiom,
! [R: partia6043505979758434576t_unit,F: list_set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( monic_6852076386464302682t_unit @ R @ F )
=> ( member_list_set_a @ F @ ( partia270248232491039416t_unit @ ( ring_m3856952734742274883t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ) ).
% field.monic_poly_carr_2
thf(fact_90_field_Omonic__poly__carr__2,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_5008461317928820916t_unit @ R @ F )
=> ( member_list_list_a @ F @ ( partia7172811403827572716t_unit @ ( ring_m2698952956457993885t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).
% field.monic_poly_carr_2
thf(fact_91_field_Omonic__poly__carr__2,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_5986596350207772206t_unit @ R @ F )
=> ( member5342144027231129785list_a @ F @ ( partia3655612087661131104t_unit @ ( ring_m4345264959146251543t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ).
% field.monic_poly_carr_2
thf(fact_92_field_Omonic__poly__carr__2,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a] :
( ( field_a_b @ R )
=> ( ( monic_3145109188698636716ly_a_b @ R @ F )
=> ( member_list_a @ F @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% field.monic_poly_carr_2
thf(fact_93_field_Omonic__poly__carr,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_8143709425247463374t_unit @ R @ F )
=> ( member6124916891863447321list_a @ F @ ( partia6708307881709191317t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_94_field_Omonic__poly__carr,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_3395465470813675732t_unit @ R @ F )
=> ( member5524387281408368019list_a @ F @ ( partia7265347635606999311t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_95_field_Omonic__poly__carr,axiom,
! [R: partia6043505979758434576t_unit,F: list_set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( monic_6852076386464302682t_unit @ R @ F )
=> ( member_list_set_a @ F @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_96_field_Omonic__poly__carr,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_5986596350207772206t_unit @ R @ F )
=> ( member5342144027231129785list_a @ F @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_97_field_Omonic__poly__carr,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_5008461317928820916t_unit @ R @ F )
=> ( member_list_list_a @ F @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_98_field_Omonic__poly__carr,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a] :
( ( field_a_b @ R )
=> ( ( monic_3145109188698636716ly_a_b @ R @ F )
=> ( member_list_a @ F @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% field.monic_poly_carr
thf(fact_99_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X: list_a] : ( member_list_a @ X @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_100_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X: a] : ( member_a @ X @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_101_ex__in__conv,axiom,
! [A2: set_list_list_a] :
( ( ? [X: list_list_a] : ( member_list_list_a @ X @ A2 ) )
= ( A2 != bot_bo1875519244922727510list_a ) ) ).
% ex_in_conv
thf(fact_102_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y: list_a] :
~ ( member_list_a @ Y @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_103_equals0I,axiom,
! [A2: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_104_equals0I,axiom,
! [A2: set_list_list_a] :
( ! [Y: list_list_a] :
~ ( member_list_list_a @ Y @ A2 )
=> ( A2 = bot_bo1875519244922727510list_a ) ) ).
% equals0I
thf(fact_105_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_106_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_107_equals0D,axiom,
! [A2: set_list_list_a,A: list_list_a] :
( ( A2 = bot_bo1875519244922727510list_a )
=> ~ ( member_list_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_108_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_109_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_110_emptyE,axiom,
! [A: list_list_a] :
~ ( member_list_list_a @ A @ bot_bo1875519244922727510list_a ) ).
% emptyE
thf(fact_111_mk__disjoint__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ? [B3: set_list_a] :
( ( A2
= ( insert_list_a @ A @ B3 ) )
& ~ ( member_list_a @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_112_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B3: set_a] :
( ( A2
= ( insert_a @ A @ B3 ) )
& ~ ( member_a @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_113_mk__disjoint__insert,axiom,
! [A: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ A2 )
=> ? [B3: set_list_list_a] :
( ( A2
= ( insert_list_list_a @ A @ B3 ) )
& ~ ( member_list_list_a @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_114_insert__commute,axiom,
! [X2: list_a,Y2: list_a,A2: set_list_a] :
( ( insert_list_a @ X2 @ ( insert_list_a @ Y2 @ A2 ) )
= ( insert_list_a @ Y2 @ ( insert_list_a @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_115_insert__commute,axiom,
! [X2: a,Y2: a,A2: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y2 @ A2 ) )
= ( insert_a @ Y2 @ ( insert_a @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_116_insert__commute,axiom,
! [X2: list_list_a,Y2: list_list_a,A2: set_list_list_a] :
( ( insert_list_list_a @ X2 @ ( insert_list_list_a @ Y2 @ A2 ) )
= ( insert_list_list_a @ Y2 @ ( insert_list_list_a @ X2 @ A2 ) ) ) ).
% insert_commute
thf(fact_117_insert__eq__iff,axiom,
! [A: list_a,A2: set_list_a,B: list_a,B2: set_list_a] :
( ~ ( member_list_a @ A @ A2 )
=> ( ~ ( member_list_a @ B @ B2 )
=> ( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_list_a] :
( ( A2
= ( insert_list_a @ B @ C3 ) )
& ~ ( member_list_a @ B @ C3 )
& ( B2
= ( insert_list_a @ A @ C3 ) )
& ~ ( member_list_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_118_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A2
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B2
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_119_insert__eq__iff,axiom,
! [A: list_list_a,A2: set_list_list_a,B: list_list_a,B2: set_list_list_a] :
( ~ ( member_list_list_a @ A @ A2 )
=> ( ~ ( member_list_list_a @ B @ B2 )
=> ( ( ( insert_list_list_a @ A @ A2 )
= ( insert_list_list_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C3: set_list_list_a] :
( ( A2
= ( insert_list_list_a @ B @ C3 ) )
& ~ ( member_list_list_a @ B @ C3 )
& ( B2
= ( insert_list_list_a @ A @ C3 ) )
& ~ ( member_list_list_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_120_insert__absorb,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_121_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_122_insert__absorb,axiom,
! [A: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ A2 )
=> ( ( insert_list_list_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_123_insert__ident,axiom,
! [X2: list_a,A2: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X2 @ A2 )
=> ( ~ ( member_list_a @ X2 @ B2 )
=> ( ( ( insert_list_a @ X2 @ A2 )
= ( insert_list_a @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_124_insert__ident,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ~ ( member_a @ X2 @ B2 )
=> ( ( ( insert_a @ X2 @ A2 )
= ( insert_a @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_125_insert__ident,axiom,
! [X2: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ~ ( member_list_list_a @ X2 @ A2 )
=> ( ~ ( member_list_list_a @ X2 @ B2 )
=> ( ( ( insert_list_list_a @ X2 @ A2 )
= ( insert_list_list_a @ X2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_126_Set_Oset__insert,axiom,
! [X2: list_a,A2: set_list_a] :
( ( member_list_a @ X2 @ A2 )
=> ~ ! [B3: set_list_a] :
( ( A2
= ( insert_list_a @ X2 @ B3 ) )
=> ( member_list_a @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_127_Set_Oset__insert,axiom,
! [X2: a,A2: set_a] :
( ( member_a @ X2 @ A2 )
=> ~ ! [B3: set_a] :
( ( A2
= ( insert_a @ X2 @ B3 ) )
=> ( member_a @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_128_Set_Oset__insert,axiom,
! [X2: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ X2 @ A2 )
=> ~ ! [B3: set_list_list_a] :
( ( A2
= ( insert_list_list_a @ X2 @ B3 ) )
=> ( member_list_list_a @ X2 @ B3 ) ) ) ).
% Set.set_insert
thf(fact_129_insertI2,axiom,
! [A: list_a,B2: set_list_a,B: list_a] :
( ( member_list_a @ A @ B2 )
=> ( member_list_a @ A @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_130_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_131_insertI2,axiom,
! [A: list_list_a,B2: set_list_list_a,B: list_list_a] :
( ( member_list_list_a @ A @ B2 )
=> ( member_list_list_a @ A @ ( insert_list_list_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_132_insertI1,axiom,
! [A: list_a,B2: set_list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ B2 ) ) ).
% insertI1
thf(fact_133_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_134_insertI1,axiom,
! [A: list_list_a,B2: set_list_list_a] : ( member_list_list_a @ A @ ( insert_list_list_a @ A @ B2 ) ) ).
% insertI1
thf(fact_135_insertE,axiom,
! [A: list_a,B: list_a,A2: set_list_a] :
( ( member_list_a @ A @ ( insert_list_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_list_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_136_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_137_insertE,axiom,
! [A: list_list_a,B: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ ( insert_list_list_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_list_list_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_138_DiffD2,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ~ ( member_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_139_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_140_DiffD2,axiom,
! [C: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A2 @ B2 ) )
=> ~ ( member_list_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_141_DiffD1,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ( member_list_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_142_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_143_DiffD1,axiom,
! [C: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A2 @ B2 ) )
=> ( member_list_list_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_144_DiffE,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
=> ~ ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_145_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_146_DiffE,axiom,
! [C: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( member_list_list_a @ C @ ( minus_5335179877275218001list_a @ A2 @ B2 ) )
=> ~ ( ( member_list_list_a @ C @ A2 )
=> ( member_list_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_147_singleton__inject,axiom,
! [A: list_a,B: list_a] :
( ( ( insert_list_a @ A @ bot_bot_set_list_a )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_148_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_149_singleton__inject,axiom,
! [A: list_list_a,B: list_list_a] :
( ( ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a )
= ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_150_insert__not__empty,axiom,
! [A: list_a,A2: set_list_a] :
( ( insert_list_a @ A @ A2 )
!= bot_bot_set_list_a ) ).
% insert_not_empty
thf(fact_151_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_152_insert__not__empty,axiom,
! [A: list_list_a,A2: set_list_list_a] :
( ( insert_list_list_a @ A @ A2 )
!= bot_bo1875519244922727510list_a ) ).
% insert_not_empty
thf(fact_153_doubleton__eq__iff,axiom,
! [A: list_a,B: list_a,C: list_a,D: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( insert_list_a @ C @ ( insert_list_a @ D @ bot_bot_set_list_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_154_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_155_doubleton__eq__iff,axiom,
! [A: list_list_a,B: list_list_a,C: list_list_a,D: list_list_a] :
( ( ( insert_list_list_a @ A @ ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a ) )
= ( insert_list_list_a @ C @ ( insert_list_list_a @ D @ bot_bo1875519244922727510list_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_156_singleton__iff,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_157_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_158_singleton__iff,axiom,
! [B: list_list_a,A: list_list_a] :
( ( member_list_list_a @ B @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_159_singletonD,axiom,
! [B: list_a,A: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_160_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_161_singletonD,axiom,
! [B: list_list_a,A: list_list_a] :
( ( member_list_list_a @ B @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_162_insert__Diff__if,axiom,
! [X2: list_a,B2: set_list_a,A2: set_list_a] :
( ( ( member_list_a @ X2 @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X2 @ A2 ) @ B2 )
= ( minus_646659088055828811list_a @ A2 @ B2 ) ) )
& ( ~ ( member_list_a @ X2 @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X2 @ A2 ) @ B2 )
= ( insert_list_a @ X2 @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_163_insert__Diff__if,axiom,
! [X2: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X2 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( insert_a @ X2 @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_164_insert__Diff__if,axiom,
! [X2: list_list_a,B2: set_list_list_a,A2: set_list_list_a] :
( ( ( member_list_list_a @ X2 @ B2 )
=> ( ( minus_5335179877275218001list_a @ ( insert_list_list_a @ X2 @ A2 ) @ B2 )
= ( minus_5335179877275218001list_a @ A2 @ B2 ) ) )
& ( ~ ( member_list_list_a @ X2 @ B2 )
=> ( ( minus_5335179877275218001list_a @ ( insert_list_list_a @ X2 @ A2 ) @ B2 )
= ( insert_list_list_a @ X2 @ ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_165_Diff__insert__absorb,axiom,
! [X2: list_a,A2: set_list_a] :
( ~ ( member_list_a @ X2 @ A2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X2 @ A2 ) @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_166_Diff__insert__absorb,axiom,
! [X2: a,A2: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X2 @ A2 ) @ ( insert_a @ X2 @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_167_Diff__insert__absorb,axiom,
! [X2: list_list_a,A2: set_list_list_a] :
( ~ ( member_list_list_a @ X2 @ A2 )
=> ( ( minus_5335179877275218001list_a @ ( insert_list_list_a @ X2 @ A2 ) @ ( insert_list_list_a @ X2 @ bot_bo1875519244922727510list_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_168_Diff__insert2,axiom,
! [A2: set_list_a,A: list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_169_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_170_Diff__insert2,axiom,
! [A2: set_list_list_a,A: list_list_a,B2: set_list_list_a] :
( ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ A @ B2 ) )
= ( minus_5335179877275218001list_a @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_171_insert__Diff,axiom,
! [A: list_a,A2: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( insert_list_a @ A @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_172_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_173_insert__Diff,axiom,
! [A: list_list_a,A2: set_list_list_a] :
( ( member_list_list_a @ A @ A2 )
=> ( ( insert_list_list_a @ A @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_174_Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B2 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_175_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_176_Diff__insert,axiom,
! [A2: set_list_list_a,A: list_list_a,B2: set_list_list_a] :
( ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ A @ B2 ) )
= ( minus_5335179877275218001list_a @ ( minus_5335179877275218001list_a @ A2 @ B2 ) @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ) ).
% Diff_insert
thf(fact_177_Ring__Divisibility_Ocarrier__mult__of,axiom,
! [R: partia2956882679547061052t_unit] :
( ( partia7172811403827572716t_unit @ ( ring_m2698952956457993885t_unit @ R ) )
= ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) ) ).
% Ring_Divisibility.carrier_mult_of
thf(fact_178_Ring__Divisibility_Ocarrier__mult__of,axiom,
! [R: partia2670972154091845814t_unit] :
( ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ R ) )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ).
% Ring_Divisibility.carrier_mult_of
thf(fact_179_Ring__Divisibility_Ocarrier__mult__of,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ R ) )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% Ring_Divisibility.carrier_mult_of
thf(fact_180_p_Omaximalideal__prime,axiom,
! [I2: set_list_a] :
( ( maxima6585700282301356660t_unit @ I2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( primei6309817859076077608t_unit @ I2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.maximalideal_prime
thf(fact_181_p_Oring__primeI_H,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( 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 ) ) )
=> ( ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.ring_primeI'
thf(fact_182_p_Omult__of_Ogcdof__exists,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ? [C2: list_a] :
( ( member_list_a @ C2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ C2 @ A @ B ) ) ) ) ).
% p.mult_of.gcdof_exists
thf(fact_183_p_Oprime__eq__prime__mult,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P ) ) ) ).
% p.prime_eq_prime_mult
thf(fact_184_p_Oring__primeE_I2_J,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P ) ) ) ).
% p.ring_primeE(2)
thf(fact_185_p_Omult__of_Ogcd__condition__monoid__axioms,axiom,
gcd_co7549656856577097917t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.mult_of.gcd_condition_monoid_axioms
thf(fact_186_p_Oprimeideal__iff__prime,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( 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 ) ) )
=> ( ( primei6309817859076077608t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.primeideal_iff_prime
thf(fact_187_p_OIdl__subset__ideal_H,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ) ) ).
% p.Idl_subset_ideal'
thf(fact_188_p_Ozero__is__prime_I2_J,axiom,
prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_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(2)
thf(fact_189_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_190_mult__of_Ocarrier__not__empty,axiom,
( ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) )
!= bot_bot_set_a ) ).
% mult_of.carrier_not_empty
thf(fact_191_monic__poly__var,axiom,
monic_3145109188698636716ly_a_b @ r @ ( var_a_b @ r ) ).
% monic_poly_var
thf(fact_192_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_193_subsetI,axiom,
! [A2: set_list_list_a,B2: set_list_list_a] :
( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A2 )
=> ( member_list_list_a @ X3 @ B2 ) )
=> ( ord_le8488217952732425610list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_194_subsetI,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( member_list_a @ X3 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_195_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_196_subset__antisym,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_197_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_198_p_Ocgenideal__self,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I ) ) ) ).
% p.cgenideal_self
thf(fact_199_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_200_mult__of_Olcmof__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( islcm_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C2 @ A @ B ) ) ) ) ).
% mult_of.lcmof_exists
thf(fact_201_p_Ogenideal__self,axiom,
! [S: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ S @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S ) ) ) ).
% p.genideal_self
thf(fact_202_p_Osubset__Idl__subset,axiom,
! [I2: set_list_a,H: set_list_a] :
( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H @ I2 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 ) ) ) ) ).
% p.subset_Idl_subset
thf(fact_203_p_Ocgenideal__is__principalideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.cgenideal_is_principalideal
thf(fact_204_subset__empty,axiom,
! [A2: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ bot_bo1875519244922727510list_a )
= ( A2 = bot_bo1875519244922727510list_a ) ) ).
% subset_empty
thf(fact_205_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_206_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_207_empty__subsetI,axiom,
! [A2: set_list_list_a] : ( ord_le8488217952732425610list_a @ bot_bo1875519244922727510list_a @ A2 ) ).
% empty_subsetI
thf(fact_208_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_209_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_210_insert__subset,axiom,
! [X2: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( insert_list_list_a @ X2 @ A2 ) @ B2 )
= ( ( member_list_list_a @ X2 @ B2 )
& ( ord_le8488217952732425610list_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_211_insert__subset,axiom,
! [X2: list_a,A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X2 @ A2 ) @ B2 )
= ( ( member_list_a @ X2 @ B2 )
& ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_212_insert__subset,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A2 ) @ B2 )
= ( ( member_a @ X2 @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_213_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_214_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_215_p_Ocgenideal__eq__genideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I )
= ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% p.cgenideal_eq_genideal
thf(fact_216_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_217_singleton__insert__inj__eq,axiom,
! [B: list_list_a,A: list_list_a,A2: set_list_list_a] :
( ( ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a )
= ( insert_list_list_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_218_singleton__insert__inj__eq,axiom,
! [B: list_a,A: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_219_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_220_singleton__insert__inj__eq_H,axiom,
! [A: list_list_a,A2: set_list_list_a,B: list_list_a] :
( ( ( insert_list_list_a @ A @ A2 )
= ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a ) )
= ( ( A = B )
& ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ B @ bot_bo1875519244922727510list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_221_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A2: set_list_a,B: list_a] :
( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B @ bot_bot_set_list_a ) )
= ( ( A = B )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_222_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_223_Diff__eq__empty__iff,axiom,
! [A2: set_list_list_a,B2: set_list_list_a] :
( ( ( minus_5335179877275218001list_a @ A2 @ B2 )
= bot_bo1875519244922727510list_a )
= ( ord_le8488217952732425610list_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_224_Diff__eq__empty__iff,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( minus_646659088055828811list_a @ A2 @ B2 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_225_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_226_mult__of_Odivisor__chain__condition__monoid__axioms,axiom,
diviso6259607970152342594t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.divisor_chain_condition_monoid_axioms
thf(fact_227_mult__of_Oprimeness__condition__monoid__axioms,axiom,
primen965786292471834261t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.primeness_condition_monoid_axioms
thf(fact_228_in__mono,axiom,
! [A2: set_list_list_a,B2: set_list_list_a,X2: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ B2 )
=> ( ( member_list_list_a @ X2 @ A2 )
=> ( member_list_list_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_229_in__mono,axiom,
! [A2: set_list_a,B2: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_230_in__mono,axiom,
! [A2: set_a,B2: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_231_subsetD,axiom,
! [A2: set_list_list_a,B2: set_list_list_a,C: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ B2 )
=> ( ( member_list_list_a @ C @ A2 )
=> ( member_list_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_232_subsetD,axiom,
! [A2: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_233_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_234_equalityE,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ~ ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_235_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_236_subset__eq,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A3: set_list_list_a,B4: set_list_list_a] :
! [X: list_list_a] :
( ( member_list_list_a @ X @ A3 )
=> ( member_list_list_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_237_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B4: set_list_a] :
! [X: list_a] :
( ( member_list_a @ X @ A3 )
=> ( member_list_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_238_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B4: set_a] :
! [X: a] :
( ( member_a @ X @ A3 )
=> ( member_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_239_equalityD1,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_240_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_241_equalityD2,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( A2 = B2 )
=> ( ord_le8861187494160871172list_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_242_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_243_subset__iff,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A3: set_list_list_a,B4: set_list_list_a] :
! [T: list_list_a] :
( ( member_list_list_a @ T @ A3 )
=> ( member_list_list_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_244_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A3: set_list_a,B4: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A3 )
=> ( member_list_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_245_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A3 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_246_subset__refl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_247_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_248_Collect__mono,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_249_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_250_subset__trans,axiom,
! [A2: set_list_a,B2: set_list_a,C4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C4 )
=> ( ord_le8861187494160871172list_a @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_251_subset__trans,axiom,
! [A2: set_a,B2: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C4 )
=> ( ord_less_eq_set_a @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_252_set__eq__subset,axiom,
( ( ^ [Y3: set_list_a,Z: set_list_a] : ( Y3 = Z ) )
= ( ^ [A3: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A3 @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_253_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : ( Y3 = Z ) )
= ( ^ [A3: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A3 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_254_Collect__mono__iff,axiom,
! [P2: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q ) )
= ( ! [X: list_a] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_255_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_256_insert__mono,axiom,
! [C4: set_list_list_a,D2: set_list_list_a,A: list_list_a] :
( ( ord_le8488217952732425610list_a @ C4 @ D2 )
=> ( ord_le8488217952732425610list_a @ ( insert_list_list_a @ A @ C4 ) @ ( insert_list_list_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_257_insert__mono,axiom,
! [C4: set_list_a,D2: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ C4 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ A @ C4 ) @ ( insert_list_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_258_insert__mono,axiom,
! [C4: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C4 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C4 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_259_subset__insert,axiom,
! [X2: list_list_a,A2: set_list_list_a,B2: set_list_list_a] :
( ~ ( member_list_list_a @ X2 @ A2 )
=> ( ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ X2 @ B2 ) )
= ( ord_le8488217952732425610list_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_260_subset__insert,axiom,
! [X2: list_a,A2: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ B2 ) )
= ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_261_subset__insert,axiom,
! [X2: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_262_subset__insertI,axiom,
! [B2: set_list_list_a,A: list_list_a] : ( ord_le8488217952732425610list_a @ B2 @ ( insert_list_list_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_263_subset__insertI,axiom,
! [B2: set_list_a,A: list_a] : ( ord_le8861187494160871172list_a @ B2 @ ( insert_list_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_264_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_265_subset__insertI2,axiom,
! [A2: set_list_list_a,B2: set_list_list_a,B: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ B2 )
=> ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_266_subset__insertI2,axiom,
! [A2: set_list_a,B2: set_list_a,B: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_267_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_268_Diff__mono,axiom,
! [A2: set_list_list_a,C4: set_list_list_a,D2: set_list_list_a,B2: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ C4 )
=> ( ( ord_le8488217952732425610list_a @ D2 @ B2 )
=> ( ord_le8488217952732425610list_a @ ( minus_5335179877275218001list_a @ A2 @ B2 ) @ ( minus_5335179877275218001list_a @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_269_Diff__mono,axiom,
! [A2: set_list_a,C4: set_list_a,D2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C4 )
=> ( ( ord_le8861187494160871172list_a @ D2 @ B2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ ( minus_646659088055828811list_a @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_270_Diff__mono,axiom,
! [A2: set_a,C4: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C4 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C4 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_271_Diff__subset,axiom,
! [A2: set_list_list_a,B2: set_list_list_a] : ( ord_le8488217952732425610list_a @ ( minus_5335179877275218001list_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_272_Diff__subset,axiom,
! [A2: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_273_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_274_double__diff,axiom,
! [A2: set_list_list_a,B2: set_list_list_a,C4: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ B2 )
=> ( ( ord_le8488217952732425610list_a @ B2 @ C4 )
=> ( ( minus_5335179877275218001list_a @ B2 @ ( minus_5335179877275218001list_a @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_275_double__diff,axiom,
! [A2: set_list_a,B2: set_list_a,C4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ C4 )
=> ( ( minus_646659088055828811list_a @ B2 @ ( minus_646659088055828811list_a @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_276_double__diff,axiom,
! [A2: set_a,B2: set_a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C4 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C4 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_277_subset__singletonD,axiom,
! [A2: set_list_list_a,X2: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ X2 @ bot_bo1875519244922727510list_a ) )
=> ( ( A2 = bot_bo1875519244922727510list_a )
| ( A2
= ( insert_list_list_a @ X2 @ bot_bo1875519244922727510list_a ) ) ) ) ).
% subset_singletonD
thf(fact_278_subset__singletonD,axiom,
! [A2: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) )
=> ( ( A2 = bot_bot_set_list_a )
| ( A2
= ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) ) ) ).
% subset_singletonD
thf(fact_279_subset__singletonD,axiom,
! [A2: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_280_subset__singleton__iff,axiom,
! [X4: set_list_list_a,A: list_list_a] :
( ( ord_le8488217952732425610list_a @ X4 @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
= ( ( X4 = bot_bo1875519244922727510list_a )
| ( X4
= ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_281_subset__singleton__iff,axiom,
! [X4: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ X4 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( ( X4 = bot_bot_set_list_a )
| ( X4
= ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_282_subset__singleton__iff,axiom,
! [X4: set_a,A: a] :
( ( ord_less_eq_set_a @ X4 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X4 = bot_bot_set_a )
| ( X4
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_283_subset__Diff__insert,axiom,
! [A2: set_list_list_a,B2: set_list_list_a,X2: list_list_a,C4: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( minus_5335179877275218001list_a @ B2 @ ( insert_list_list_a @ X2 @ C4 ) ) )
= ( ( ord_le8488217952732425610list_a @ A2 @ ( minus_5335179877275218001list_a @ B2 @ C4 ) )
& ~ ( member_list_list_a @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_284_subset__Diff__insert,axiom,
! [A2: set_list_a,B2: set_list_a,X2: list_a,C4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B2 @ ( insert_list_a @ X2 @ C4 ) ) )
= ( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B2 @ C4 ) )
& ~ ( member_list_a @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_285_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X2: a,C4: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X2 @ C4 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C4 ) )
& ~ ( member_a @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_286_field_Omonic__poly__var,axiom,
! [R: partia4960592913263135132t_unit] :
( ( field_1540243473349940225t_unit @ R )
=> ( monic_8143709425247463374t_unit @ R @ ( var_se2996050386653789495t_unit @ R ) ) ) ).
% field.monic_poly_var
thf(fact_287_field_Omonic__poly__var,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( monic_3395465470813675732t_unit @ R @ ( var_se6008125447796440765t_unit @ R ) ) ) ).
% field.monic_poly_var
thf(fact_288_field_Omonic__poly__var,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( monic_6852076386464302682t_unit @ R @ ( var_se2415970144172829891t_unit @ R ) ) ) ).
% field.monic_poly_var
thf(fact_289_field_Omonic__poly__var,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( monic_5008461317928820916t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) ) ) ).
% field.monic_poly_var
thf(fact_290_field_Omonic__poly__var,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( monic_3145109188698636716ly_a_b @ R @ ( var_a_b @ R ) ) ) ).
% field.monic_poly_var
thf(fact_291_subset__insert__iff,axiom,
! [A2: set_list_list_a,X2: list_list_a,B2: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ X2 @ B2 ) )
= ( ( ( member_list_list_a @ X2 @ A2 )
=> ( ord_le8488217952732425610list_a @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ X2 @ bot_bo1875519244922727510list_a ) ) @ B2 ) )
& ( ~ ( member_list_list_a @ X2 @ A2 )
=> ( ord_le8488217952732425610list_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_292_subset__insert__iff,axiom,
! [A2: set_list_a,X2: list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ B2 ) )
= ( ( ( member_list_a @ X2 @ A2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) @ B2 ) )
& ( ~ ( member_list_a @ X2 @ A2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_293_subset__insert__iff,axiom,
! [A2: set_a,X2: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B2 ) )
= ( ( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_294_Diff__single__insert,axiom,
! [A2: set_list_list_a,X2: list_list_a,B2: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ X2 @ bot_bo1875519244922727510list_a ) ) @ B2 )
=> ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_295_Diff__single__insert,axiom,
! [A2: set_list_a,X2: list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) @ B2 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_296_Diff__single__insert,axiom,
! [A2: set_a,X2: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_297_noetherian__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R )
=> ( ring_n3639167112692572309ng_a_b @ R ) ) ).
% noetherian_domain.axioms(1)
thf(fact_298_noetherian__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_n4705423059119889713t_unit @ R )
=> ( ring_n5188127996776581661t_unit @ R ) ) ).
% noetherian_domain.axioms(1)
thf(fact_299_ring__prime__def,axiom,
( ring_r6430282645014804837t_unit
= ( ^ [R2: partia2670972154091845814t_unit,A4: list_a] :
( ( A4
!= ( zero_l4142658623432671053t_unit @ R2 ) )
& ( prime_2011924034616061926t_unit @ R2 @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_300_ring__prime__def,axiom,
( ring_ring_prime_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,A4: a] :
( ( A4
!= ( zero_a_b @ R2 ) )
& ( prime_a_ring_ext_a_b @ R2 @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_301_ring__prime__def,axiom,
( ring_r5437400583859147359t_unit
= ( ^ [R2: partia2956882679547061052t_unit,A4: list_list_a] :
( ( A4
!= ( zero_l347298301471573063t_unit @ R2 ) )
& ( prime_1232919612140715622t_unit @ R2 @ A4 ) ) ) ) ).
% ring_prime_def
thf(fact_302_p_Ofield__iff__prime,axiom,
! [A: list_a] :
( ( 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 ) ) )
=> ( ( field_26233345952514695t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) ) )
= ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) ) ) ).
% p.field_iff_prime
thf(fact_303_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_304_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_305_p_Oa__lcos__mult__one,axiom,
! [M: set_list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ M )
= M ) ) ).
% p.a_lcos_mult_one
thf(fact_306_ring__primeE_I1_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( P
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_307_p_Omult__of_Ogcd__isgcd,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ A @ B ) ) ) ).
% p.mult_of.gcd_isgcd
thf(fact_308_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_309_p_Ocarrier__is__subalgebra,axiom,
! [K: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.carrier_is_subalgebra
thf(fact_310_p_Osubalgebra__in__carrier,axiom,
! [K: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.subalgebra_in_carrier
thf(fact_311_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_312_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_313_ring__primeE_I3_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_ring_ext_a_b @ r @ P ) ) ) ).
% ring_primeE(3)
thf(fact_314_ring__primeI,axiom,
! [P: a] :
( ( P
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI
thf(fact_315_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_316_p_Opow__non__zero,axiom,
! [X2: list_a,N: nat] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.pow_non_zero
thf(fact_317_p_Omult__of_Ogcd__exists,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.mult_of.gcd_exists
thf(fact_318_p_Omult__of_Ogcd__closed,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.mult_of.gcd_closed
thf(fact_319_p_Oa__l__coset__subset__G,axiom,
! [H: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ H ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.a_l_coset_subset_G
thf(fact_320_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_321_p_Onat__pow__closed,axiom,
! [X2: list_a,N: nat] :
( ( member_list_a @ X2 @ ( 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 ) ) @ X2 @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.nat_pow_closed
thf(fact_322_p_OFactRing__zeroideal_I2_J,axiom,
is_rin2993610189962786360t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% p.FactRing_zeroideal(2)
thf(fact_323_p_OFactRing__zeroideal_I1_J,axiom,
is_rin4843644836746533432t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.FactRing_zeroideal(1)
thf(fact_324_field_Omonic__poly__pow,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a,N: nat] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_8143709425247463374t_unit @ R @ F )
=> ( monic_8143709425247463374t_unit @ R @ ( pow_li5711373720449609902it_nat @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_325_field_Omonic__poly__pow,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a,N: nat] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_3395465470813675732t_unit @ R @ F )
=> ( monic_3395465470813675732t_unit @ R @ ( pow_li690586108738686766it_nat @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_326_field_Omonic__poly__pow,axiom,
! [R: partia6043505979758434576t_unit,F: list_set_a,N: nat] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( monic_6852076386464302682t_unit @ R @ F )
=> ( monic_6852076386464302682t_unit @ R @ ( pow_li8158544010604435374it_nat @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_327_field_Omonic__poly__pow,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a,N: nat] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_5008461317928820916t_unit @ R @ F )
=> ( monic_5008461317928820916t_unit @ R @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_328_field_Omonic__poly__pow,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a,N: nat] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_5986596350207772206t_unit @ R @ F )
=> ( monic_5986596350207772206t_unit @ R @ ( pow_li6759500793967859886it_nat @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_329_field_Omonic__poly__pow,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a,N: nat] :
( ( field_a_b @ R )
=> ( ( monic_3145109188698636716ly_a_b @ R @ F )
=> ( monic_3145109188698636716ly_a_b @ R @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ N ) ) ) ) ).
% field.monic_poly_pow
thf(fact_330_principal__domain_Ofield__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) ) )
= ( ring_ring_prime_a_b @ R @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_331_principal__domain_Ofield__iff__prime,axiom,
! [R: partia2670972154091845814t_unit,A: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( field_26233345952514695t_unit @ ( factRi3329376332477095402t_unit @ R @ ( cgenid9131348535277946915t_unit @ R @ A ) ) )
= ( ring_r6430282645014804837t_unit @ R @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_332_principal__domain_Ofield__iff__prime,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ A @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( field_1540243473349940225t_unit @ ( factRi7259693425559269476t_unit @ R @ ( cgenid24865672677839267t_unit @ R @ A ) ) )
= ( ring_r5437400583859147359t_unit @ R @ A ) ) ) ) ).
% principal_domain.field_iff_prime
thf(fact_333_mult__of_Otrivial__group__alt,axiom,
( ( elemen1145482699608675729t_unit @ ( ring_mult_of_a_b @ r ) )
= ( ? [A4: a] : ( ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% mult_of.trivial_group_alt
thf(fact_334_mult__of_Otrivial__group,axiom,
( ( elemen1145482699608675729t_unit @ ( ring_mult_of_a_b @ r ) )
= ( ? [A4: a] :
( ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) )
= ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ).
% mult_of.trivial_group
thf(fact_335_ring__primeI_H,axiom,
! [P: a] :
( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% ring_primeI'
thf(fact_336_primeness__condition,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeness_condition
thf(fact_337_mult__of_Ogcdof__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C2: a] :
( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ C2 @ A @ B ) ) ) ) ).
% mult_of.gcdof_exists
thf(fact_338_p_Odomain__iff__prime,axiom,
! [A: list_a] :
( ( 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 ) ) )
=> ( ( domain1617769409708967785t_unit @ ( factRi3329376332477095402t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) ) )
= ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) ) ) ).
% p.domain_iff_prime
thf(fact_339_ring__irreducibleE_I1_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( R3
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_340_ring__primeE_I2_J,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% ring_primeE(2)
thf(fact_341_p_Ocring__fieldI,axiom,
( ( ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.cring_fieldI
thf(fact_342_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_343_cgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_344_zero__is__prime_I2_J,axiom,
prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_prime(2)
thf(fact_345_p_OUnits__closed,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.Units_closed
thf(fact_346_irreducible__imp__maximalideal,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r ) ) ) ).
% irreducible_imp_maximalideal
thf(fact_347_p_OUnits__pow__closed,axiom,
! [X2: list_a,D: nat] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ D ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.Units_pow_closed
thf(fact_348_prime__eq__prime__mult,axiom,
! [P: a] :
( ( member_a @ P @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P )
= ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% prime_eq_prime_mult
thf(fact_349_p_Oideal__eq__carrier__iff,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ideal_eq_carrier_iff
thf(fact_350_field__iff__prime,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
= ( ring_ring_prime_a_b @ r @ A ) ) ) ).
% field_iff_prime
thf(fact_351_FactRing__zeroideal_I2_J,axiom,
is_rin9099215527551818550t_unit @ r @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% FactRing_zeroideal(2)
thf(fact_352_FactRing__zeroideal_I1_J,axiom,
is_rin6001486760346555702it_a_b @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) @ r ).
% FactRing_zeroideal(1)
thf(fact_353_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia7496981018696276118t_unit,R3: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ R3 )
=> ~ ( member_set_list_a @ R3 @ ( units_5837875185506529638t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_354_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia6043505979758434576t_unit,R3: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R3 )
=> ~ ( member_set_a @ R3 @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_355_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R3 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_356_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R3 )
=> ~ ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_357_domain_Oring__irreducibleE_I4_J,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R3 )
=> ~ ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(4)
thf(fact_358_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,R3: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ R3 )
=> ( R3
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_359_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia6043505979758434576t_unit,R3: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R3 )
=> ( R3
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_360_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R3 )
=> ( R3
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_361_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R3 )
=> ( R3
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_362_domain_Oring__irreducibleE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R3 )
=> ( R3
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.ring_irreducibleE(1)
thf(fact_363_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_p2468016639901664456t_unit @ R )
=> ( domain1617769409708967785t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_364_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_365_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_366_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_367_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_p715737262848045090t_unit @ R )
=> ( domain7810152921033798211t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_368_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_n1398569921632463185t_unit @ R )
=> ( domain1617769409708967785t_unit @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_369_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_n3212398840814694743t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_370_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_n8900817365880610859t_unit @ R )
=> ( domain7810152921033798211t_unit @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_371_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n4045954140777738665in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_372_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_n4705423059119889713t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_373_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_f4451355594461056447t_unit @ R )
=> ( domain1617769409708967785t_unit @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_374_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_f6820247627256571077t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_375_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_f5761620020419587481t_unit @ R )
=> ( domain7810152921033798211t_unit @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_376_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_f5272581269873410839in_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_377_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_f796907574329358751t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_378_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( prime_5738381090551951334t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_379_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( prime_4522187476880896870t_unit @ R @ ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_380_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R )
=> ( prime_1232919612140715622t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_381_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( prime_2011924034616061926t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_382_domain_Ozero__is__prime_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(1)
thf(fact_383_noetherian__domain__def,axiom,
( ring_n1398569921632463185t_unit
= ( ^ [R2: partia7496981018696276118t_unit] :
( ( ring_n7704429503468267069t_unit @ R2 )
& ( domain1617769409708967785t_unit @ R2 ) ) ) ) ).
% noetherian_domain_def
thf(fact_384_noetherian__domain__def,axiom,
( ring_n3212398840814694743t_unit
= ( ^ [R2: partia6043505979758434576t_unit] :
( ( ring_n5014428767265248323t_unit @ R2 )
& ( domain4236798911309298543t_unit @ R2 ) ) ) ) ).
% noetherian_domain_def
thf(fact_385_noetherian__domain__def,axiom,
( ring_n8900817365880610859t_unit
= ( ^ [R2: partia2956882679547061052t_unit] :
( ( ring_n1719824158142654231t_unit @ R2 )
& ( domain7810152921033798211t_unit @ R2 ) ) ) ) ).
% noetherian_domain_def
thf(fact_386_noetherian__domain__def,axiom,
( ring_n4045954140777738665in_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b] :
( ( ring_n3639167112692572309ng_a_b @ R2 )
& ( domain_a_b @ R2 ) ) ) ) ).
% noetherian_domain_def
thf(fact_387_noetherian__domain__def,axiom,
( ring_n4705423059119889713t_unit
= ( ^ [R2: partia2670972154091845814t_unit] :
( ( ring_n5188127996776581661t_unit @ R2 )
& ( domain6553523120543210313t_unit @ R2 ) ) ) ) ).
% noetherian_domain_def
thf(fact_388_noetherian__domain_Ointro,axiom,
! [R: partia7496981018696276118t_unit] :
( ( ring_n7704429503468267069t_unit @ R )
=> ( ( domain1617769409708967785t_unit @ R )
=> ( ring_n1398569921632463185t_unit @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_389_noetherian__domain_Ointro,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_n5014428767265248323t_unit @ R )
=> ( ( domain4236798911309298543t_unit @ R )
=> ( ring_n3212398840814694743t_unit @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_390_noetherian__domain_Ointro,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_n1719824158142654231t_unit @ R )
=> ( ( domain7810152921033798211t_unit @ R )
=> ( ring_n8900817365880610859t_unit @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_391_noetherian__domain_Ointro,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n3639167112692572309ng_a_b @ R )
=> ( ( domain_a_b @ R )
=> ( ring_n4045954140777738665in_a_b @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_392_noetherian__domain_Ointro,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_n5188127996776581661t_unit @ R )
=> ( ( domain6553523120543210313t_unit @ R )
=> ( ring_n4705423059119889713t_unit @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_393_domain_Oring__primeE_I1_J,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r1091214237498979717t_unit @ R @ P )
=> ( P
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_394_domain_Oring__primeE_I1_J,axiom,
! [R: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r6795642478576035723t_unit @ R @ P )
=> ( P
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_395_domain_Oring__primeE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P )
=> ( P
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_396_domain_Oring__primeE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( P
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_397_domain_Oring__primeE_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P )
=> ( P
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.ring_primeE(1)
thf(fact_398_domain_Ozero__is__prime_I2_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( prime_7398342416637585285t_unit @ ( ring_m294936264769644739t_unit @ R ) @ ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% domain.zero_is_prime(2)
thf(fact_399_domain_Ozero__is__prime_I2_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% domain.zero_is_prime(2)
thf(fact_400_domain_Ozero__is__prime_I2_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R )
=> ( prime_7052497190834259039t_unit @ ( ring_m2698952956457993885t_unit @ R ) @ ( zero_l347298301471573063t_unit @ R ) ) ) ).
% domain.zero_is_prime(2)
thf(fact_401_domain_Ozero__is__prime_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ R ) @ ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.zero_is_prime(2)
thf(fact_402_domain_Ozero__is__prime_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R ) @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_prime(2)
thf(fact_403_domain_Oring__primeE_I3_J,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r1091214237498979717t_unit @ R @ P )
=> ( prime_5738381090551951334t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_404_domain_Oring__primeE_I3_J,axiom,
! [R: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r6795642478576035723t_unit @ R @ P )
=> ( prime_4522187476880896870t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_405_domain_Oring__primeE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P )
=> ( prime_2011924034616061926t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_406_domain_Oring__primeE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( prime_a_ring_ext_a_b @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_407_domain_Oring__primeE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P )
=> ( prime_1232919612140715622t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeE(3)
thf(fact_408_domain_Oring__primeE_I2_J,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r1091214237498979717t_unit @ R @ P )
=> ( prime_7398342416637585285t_unit @ ( ring_m294936264769644739t_unit @ R ) @ P ) ) ) ) ).
% domain.ring_primeE(2)
thf(fact_409_domain_Oring__primeE_I2_J,axiom,
! [R: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r6795642478576035723t_unit @ R @ P )
=> ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ P ) ) ) ) ).
% domain.ring_primeE(2)
thf(fact_410_domain_Oring__primeE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r6430282645014804837t_unit @ R @ P )
=> ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ R ) @ P ) ) ) ) ).
% domain.ring_primeE(2)
thf(fact_411_domain_Oring__primeE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_ring_prime_a_b @ R @ P )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R ) @ P ) ) ) ) ).
% domain.ring_primeE(2)
thf(fact_412_domain_Oring__primeE_I2_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r5437400583859147359t_unit @ R @ P )
=> ( prime_7052497190834259039t_unit @ ( ring_m2698952956457993885t_unit @ R ) @ P ) ) ) ) ).
% domain.ring_primeE(2)
thf(fact_413_domain_Oprime__eq__prime__mult,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P @ ( partia141011252114345353t_unit @ R ) )
=> ( ( prime_5738381090551951334t_unit @ R @ P )
= ( prime_7398342416637585285t_unit @ ( ring_m294936264769644739t_unit @ R ) @ P ) ) ) ) ).
% domain.prime_eq_prime_mult
thf(fact_414_domain_Oprime__eq__prime__mult,axiom,
! [R: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( prime_4522187476880896870t_unit @ R @ P )
= ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ P ) ) ) ) ).
% domain.prime_eq_prime_mult
thf(fact_415_domain_Oprime__eq__prime__mult,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( prime_2011924034616061926t_unit @ R @ P )
= ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ R ) @ P ) ) ) ) ).
% domain.prime_eq_prime_mult
thf(fact_416_domain_Oprime__eq__prime__mult,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( prime_a_ring_ext_a_b @ R @ P )
= ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R ) @ P ) ) ) ) ).
% domain.prime_eq_prime_mult
thf(fact_417_domain_Oprime__eq__prime__mult,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( prime_1232919612140715622t_unit @ R @ P )
= ( prime_7052497190834259039t_unit @ ( ring_m2698952956457993885t_unit @ R ) @ P ) ) ) ) ).
% domain.prime_eq_prime_mult
thf(fact_418_principal__domain_Oprimeness__condition,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P )
= ( ring_ring_prime_a_b @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_419_principal__domain_Oprimeness__condition,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P )
= ( ring_r6430282645014804837t_unit @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_420_principal__domain_Oprimeness__condition,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ P )
= ( ring_r5437400583859147359t_unit @ R @ P ) ) ) ) ).
% principal_domain.primeness_condition
thf(fact_421_domain_Oring__primeI_H,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P @ ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
=> ( ( prime_7398342416637585285t_unit @ ( ring_m294936264769644739t_unit @ R ) @ P )
=> ( ring_r1091214237498979717t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeI'
thf(fact_422_domain_Oring__primeI_H,axiom,
! [R: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( prime_8576247383786985867t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ P )
=> ( ring_r6795642478576035723t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeI'
thf(fact_423_domain_Oring__primeI_H,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ R ) @ P )
=> ( ring_r6430282645014804837t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeI'
thf(fact_424_domain_Oring__primeI_H,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ R ) @ P )
=> ( ring_ring_prime_a_b @ R @ P ) ) ) ) ).
% domain.ring_primeI'
thf(fact_425_domain_Oring__primeI_H,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( prime_7052497190834259039t_unit @ ( ring_m2698952956457993885t_unit @ R ) @ P )
=> ( ring_r5437400583859147359t_unit @ R @ P ) ) ) ) ).
% domain.ring_primeI'
thf(fact_426_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ P @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ P )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ P ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_427_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ P )
=> ( maxima6585700282301356660t_unit @ ( cgenid9131348535277946915t_unit @ R @ P ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_428_principal__domain_Oirreducible__imp__maximalideal,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ P )
=> ( maxima7552488817642790894t_unit @ ( cgenid24865672677839267t_unit @ R @ P ) @ R ) ) ) ) ).
% principal_domain.irreducible_imp_maximalideal
thf(fact_429_domain_Oprimeideal__iff__prime,axiom,
! [R: partia7496981018696276118t_unit,P: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ P @ ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
=> ( ( primei7796083425553868872t_unit @ ( cgenid9032708300698165283t_unit @ R @ P ) @ R )
= ( ring_r1091214237498979717t_unit @ R @ P ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_430_domain_Oprimeideal__iff__prime,axiom,
! [R: partia6043505979758434576t_unit,P: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ P @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( primei7645216761534224334t_unit @ ( cgenid6682780793756002467t_unit @ R @ P ) @ R )
= ( ring_r6795642478576035723t_unit @ R @ P ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_431_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ P @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( primei6309817859076077608t_unit @ ( cgenid9131348535277946915t_unit @ R @ P ) @ R )
= ( ring_r6430282645014804837t_unit @ R @ P ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_432_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ P ) @ R )
= ( ring_ring_prime_a_b @ R @ P ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_433_domain_Oprimeideal__iff__prime,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ P @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( primei2288432046033540002t_unit @ ( cgenid24865672677839267t_unit @ R @ P ) @ R )
= ( ring_r5437400583859147359t_unit @ R @ P ) ) ) ) ).
% domain.primeideal_iff_prime
thf(fact_434_principal__domain_Odomain__iff__prime,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) ) )
= ( ring_ring_prime_a_b @ R @ A ) ) ) ) ).
% principal_domain.domain_iff_prime
thf(fact_435_principal__domain_Odomain__iff__prime,axiom,
! [R: partia2670972154091845814t_unit,A: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( domain1617769409708967785t_unit @ ( factRi3329376332477095402t_unit @ R @ ( cgenid9131348535277946915t_unit @ R @ A ) ) )
= ( ring_r6430282645014804837t_unit @ R @ A ) ) ) ) ).
% principal_domain.domain_iff_prime
thf(fact_436_principal__domain_Odomain__iff__prime,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ A @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( domain7421296078544666595t_unit @ ( factRi7259693425559269476t_unit @ R @ ( cgenid24865672677839267t_unit @ R @ A ) ) )
= ( ring_r5437400583859147359t_unit @ R @ A ) ) ) ) ).
% principal_domain.domain_iff_prime
thf(fact_437_primeideal__iff__prime,axiom,
! [P: a] :
( ( member_a @ P @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P ) @ r )
= ( ring_ring_prime_a_b @ r @ P ) ) ) ).
% primeideal_iff_prime
thf(fact_438_p_Ofield__intro2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( 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 @ X3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.field_intro2
thf(fact_439_p_Ofinite__domain__units,axiom,
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ).
% p.finite_domain_units
thf(fact_440_Ring_Ofield__Units,axiom,
! [R: partia4960592913263135132t_unit] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( units_6855253132659218406t_unit @ R )
= ( minus_1475669189178701489list_a @ ( partia3317168157747563407t_unit @ R ) @ ( insert8656068109252948922list_a @ ( zero_s2920163772466840039t_unit @ R ) @ bot_bo6516733846734216886list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_441_Ring_Ofield__Units,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( units_5837875185506529638t_unit @ R )
= ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_442_Ring_Ofield__Units,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( units_2471184348132832486t_unit @ R )
= ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_443_Ring_Ofield__Units,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( units_2932844235741507942t_unit @ R )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_444_Ring_Ofield__Units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( units_a_ring_ext_a_b @ R )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ).
% Ring.field_Units
thf(fact_445_Ring_Ofield__Units,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( units_4903515905731149798t_unit @ R )
= ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% Ring.field_Units
thf(fact_446_p_Oset__add__zero,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( set_ad92425877771022410t_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 ) @ A2 )
= A2 ) ) ).
% p.set_add_zero
thf(fact_447_p_Omult__of_Omonoid__cancel__axioms,axiom,
monoid5117334421817186628t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.mult_of.monoid_cancel_axioms
thf(fact_448_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_449_ring__irreducibleI_H,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ R3 )
=> ( ring_r999134135267193926le_a_b @ r @ R3 ) ) ) ).
% ring_irreducibleI'
thf(fact_450_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_451_Units__closed,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_452_p_Odomain__axioms,axiom,
domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% p.domain_axioms
thf(fact_453_ideal__eq__carrier__iff,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_454_ring__irreducibleE_I4_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_455_maximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ r )
=> ( primeideal_a_b @ I2 @ r ) ) ).
% maximalideal_prime
thf(fact_456_zero__is__irreducible__mult,axiom,
irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( zero_a_b @ r ) ).
% zero_is_irreducible_mult
thf(fact_457_mult__of_Oprime__irreducible,axiom,
! [P: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ).
% mult_of.prime_irreducible
thf(fact_458_p_Oring__irreducibleE_I1_J,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 )
=> ( R3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_irreducibleE(1)
thf(fact_459_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_460_p_Oring__irreducibleE_I4_J,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 )
=> ~ ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.ring_irreducibleE(4)
thf(fact_461_p_Oprimeness__condition,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% p.primeness_condition
thf(fact_462_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_463_ring__irreducibleE_I3_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ R3 ) ) ) ).
% ring_irreducibleE(3)
thf(fact_464_mult__of_Oirreducible__prime,axiom,
! [P: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P ) ) ) ).
% mult_of.irreducible_prime
thf(fact_465_domain__eq__zeroprimeideal,axiom,
( ( domain_a_b @ r )
= ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ) ) ).
% domain_eq_zeroprimeideal
thf(fact_466_zeroprimeideal__domainI,axiom,
( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( domain_a_b @ r ) ) ).
% zeroprimeideal_domainI
thf(fact_467_p_Osetadd__subset__G,axiom,
! [H: set_list_a,K: set_list_a] :
( ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.setadd_subset_G
thf(fact_468_p_Oset__add__comm,axiom,
! [I2: set_list_a,J: set_list_a] :
( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ J @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I2 @ J )
= ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ J @ I2 ) ) ) ) ).
% p.set_add_comm
thf(fact_469_p_Oset__add__closed,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A2 @ B2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.set_add_closed
thf(fact_470_local_Ofield__Units,axiom,
( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% local.field_Units
thf(fact_471_p_Oirreducible__imp__maximalideal,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( maxima6585700282301356660t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.irreducible_imp_maximalideal
thf(fact_472_p_Ozeroprimeideal__domainI,axiom,
( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.zeroprimeideal_domainI
thf(fact_473_p_Odomain__eq__zeroprimeideal,axiom,
( ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.domain_eq_zeroprimeideal
thf(fact_474_cring__fieldI,axiom,
( ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ r ) ) ).
% cring_fieldI
thf(fact_475_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_476_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_477_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_478_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_479_p_Ogenideal__one,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.genideal_one
thf(fact_480_domain__iff__prime,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
= ( ring_ring_prime_a_b @ r @ A ) ) ) ).
% domain_iff_prime
thf(fact_481_p_Omult__of_OSomeGcd__ex,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( A2 != bot_bot_set_list_a )
=> ( member_list_a @ ( someGc1203596038196317370t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A2 ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.mult_of.SomeGcd_ex
thf(fact_482_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_483_p_OUnits__one__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.Units_one_closed
thf(fact_484_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_485_p_Omult__of_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.mult_of.one_closed
thf(fact_486_p_Ofinite__ring__finite__units,axiom,
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( finite_finite_list_a @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.finite_ring_finite_units
thf(fact_487_domain_Oone__not__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_488_domain_Oone__not__zero,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( one_se211549098623999037t_unit @ R )
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_489_domain_Oone__not__zero,axiom,
! [R: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( one_li8234411390022467901t_unit @ R )
!= ( zero_l347298301471573063t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_490_domain_Oone__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.one_not_zero
thf(fact_491_domain_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% domain.one_not_zero
thf(fact_492_domain_Ozero__not__one,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( zero_s2910681146719230829t_unit @ R )
!= ( one_se1127990129394575805t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_493_domain_Ozero__not__one,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( zero_s2174465271003423091t_unit @ R )
!= ( one_se211549098623999037t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_494_domain_Ozero__not__one,axiom,
! [R: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( zero_l347298301471573063t_unit @ R )
!= ( one_li8234411390022467901t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_495_domain_Ozero__not__one,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( zero_l4142658623432671053t_unit @ R )
!= ( one_li8328186300101108157t_unit @ R ) ) ) ).
% domain.zero_not_one
thf(fact_496_domain_Ozero__not__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( zero_a_b @ R )
!= ( one_a_ring_ext_a_b @ R ) ) ) ).
% domain.zero_not_one
thf(fact_497_Ring_Oone__not__zero,axiom,
! [R: partia4960592913263135132t_unit] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( one_se2489417650821308733t_unit @ R )
!= ( zero_s2920163772466840039t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_498_Ring_Oone__not__zero,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( ( one_se1127990129394575805t_unit @ R )
!= ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_499_Ring_Oone__not__zero,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( one_se211549098623999037t_unit @ R )
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_500_Ring_Oone__not__zero,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( one_li8234411390022467901t_unit @ R )
!= ( zero_l347298301471573063t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_501_Ring_Oone__not__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( one_li8328186300101108157t_unit @ R )
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_502_Ring_Oone__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( ( one_a_ring_ext_a_b @ R )
!= ( zero_a_b @ R ) ) ) ).
% Ring.one_not_zero
thf(fact_503_domain_Ozero__is__irreducible__mult,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( irredu5949096822098593390t_unit @ ( ring_m294936264769644739t_unit @ R ) @ ( zero_s2910681146719230829t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_mult
thf(fact_504_domain_Ozero__is__irreducible__mult,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ ( zero_s2174465271003423091t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_mult
thf(fact_505_domain_Ozero__is__irreducible__mult,axiom,
! [R: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R )
=> ( irredu7159092911615704776t_unit @ ( ring_m2698952956457993885t_unit @ R ) @ ( zero_l347298301471573063t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_mult
thf(fact_506_domain_Ozero__is__irreducible__mult,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ R ) @ ( zero_l4142658623432671053t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_mult
thf(fact_507_domain_Ozero__is__irreducible__mult,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ ( zero_a_b @ R ) ) ) ).
% domain.zero_is_irreducible_mult
thf(fact_508_field_Omonic__poly__one,axiom,
! [R: partia4960592913263135132t_unit] :
( ( field_1540243473349940225t_unit @ R )
=> ( monic_8143709425247463374t_unit @ R @ ( one_li3223383766543049405t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_509_field_Omonic__poly__one,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( monic_3395465470813675732t_unit @ R @ ( one_li1622763072977731901t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_510_field_Omonic__poly__one,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( monic_6852076386464302682t_unit @ R @ ( one_li6812200495491418557t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_511_field_Omonic__poly__one,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( monic_5008461317928820916t_unit @ R @ ( one_li8234411390022467901t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_512_field_Omonic__poly__one,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( monic_5986596350207772206t_unit @ R @ ( one_li8923720976704309949t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_513_field_Omonic__poly__one,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( monic_3145109188698636716ly_a_b @ R @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% field.monic_poly_one
thf(fact_514_domain_Oring__irreducibleE_I3_J,axiom,
! [R: partia7496981018696276118t_unit,R3: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ R3 )
=> ( irredu5949096822098593390t_unit @ ( ring_m294936264769644739t_unit @ R ) @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(3)
thf(fact_515_domain_Oring__irreducibleE_I3_J,axiom,
! [R: partia6043505979758434576t_unit,R3: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R3 )
=> ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(3)
thf(fact_516_domain_Oring__irreducibleE_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R3 )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(3)
thf(fact_517_domain_Oring__irreducibleE_I3_J,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R3 )
=> ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ R ) @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(3)
thf(fact_518_domain_Oring__irreducibleE_I3_J,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R3 )
=> ( irredu7159092911615704776t_unit @ ( ring_m2698952956457993885t_unit @ R ) @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(3)
thf(fact_519_monic__irreducible__poly__def,axiom,
( monic_868474719114584568t_unit
= ( ^ [R2: partia2956882679547061052t_unit,F2: list_list_list_a] :
( ( monic_5986596350207772206t_unit @ R2 @ F2 )
& ( ring_r5224476855413033410t_unit @ ( univ_p2250591967980070728t_unit @ R2 @ ( partia2464479390973590831t_unit @ R2 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_520_monic__irreducible__poly__def,axiom,
( monic_104106837769529726t_unit
= ( ^ [R2: partia2670972154091845814t_unit,F2: list_list_a] :
( ( monic_5008461317928820916t_unit @ R2 @ F2 )
& ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R2 @ ( partia5361259788508890537t_unit @ R2 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_521_monic__irreducible__poly__def,axiom,
( monic_4919232885364369782ly_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,F2: list_a] :
( ( monic_3145109188698636716ly_a_b @ R2 @ F2 )
& ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R2 @ ( partia707051561876973205xt_a_b @ R2 ) ) @ F2 ) ) ) ) ).
% monic_irreducible_poly_def
thf(fact_522_field_Oaxioms_I1_J,axiom,
! [R: partia4960592913263135132t_unit] :
( ( field_1540243473349940225t_unit @ R )
=> ( domain7421296078544666595t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_523_field_Oaxioms_I1_J,axiom,
! [R: partia7496981018696276118t_unit] :
( ( field_26233345952514695t_unit @ R )
=> ( domain1617769409708967785t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_524_field_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( domain_a_b @ R ) ) ).
% field.axioms(1)
thf(fact_525_field_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_526_field_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_527_field_Oaxioms_I1_J,axiom,
! [R: partia2956882679547061052t_unit] :
( ( field_1861437471013600865t_unit @ R )
=> ( domain7810152921033798211t_unit @ R ) ) ).
% field.axioms(1)
thf(fact_528_domain_Oring__irreducibleI_H,axiom,
! [R: partia7496981018696276118t_unit,R3: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R3 @ ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
=> ( ( irredu5949096822098593390t_unit @ ( ring_m294936264769644739t_unit @ R ) @ R3 )
=> ( ring_r5115406448772830318t_unit @ R @ R3 ) ) ) ) ).
% domain.ring_irreducibleI'
thf(fact_529_domain_Oring__irreducibleI_H,axiom,
! [R: partia6043505979758434576t_unit,R3: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R3 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ R3 )
=> ( ring_r7790391342995787508t_unit @ R @ R3 ) ) ) ) ).
% domain.ring_irreducibleI'
thf(fact_530_domain_Oring__irreducibleI_H,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ R3 )
=> ( ring_r999134135267193926le_a_b @ R @ R3 ) ) ) ) ).
% domain.ring_irreducibleI'
thf(fact_531_domain_Oring__irreducibleI_H,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ R ) @ R3 )
=> ( ring_r932985474545269838t_unit @ R @ R3 ) ) ) ) ).
% domain.ring_irreducibleI'
thf(fact_532_domain_Oring__irreducibleI_H,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( irredu7159092911615704776t_unit @ ( ring_m2698952956457993885t_unit @ R ) @ R3 )
=> ( ring_r360171070648044744t_unit @ R @ R3 ) ) ) ) ).
% domain.ring_irreducibleI'
thf(fact_533_p_Oideal__sum__iff__gcd,axiom,
! [A: list_a,B: list_a,D: 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 @ D @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ D )
= ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) ) )
= ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ D @ A @ B ) ) ) ) ) ).
% p.ideal_sum_iff_gcd
thf(fact_534_p_Obezout__identity,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 ) ) ) )
=> ( ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( somegc2556875419254249873t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% p.bezout_identity
thf(fact_535_domain_Ofinite__domain__units,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( finite5282473924520328461list_a @ ( partia141011252114345353t_unit @ R ) )
=> ( ( units_5837875185506529638t_unit @ R )
= ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) ) ) ) ).
% domain.finite_domain_units
thf(fact_536_domain_Ofinite__domain__units,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( finite_finite_set_a @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( units_2471184348132832486t_unit @ R )
= ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) ) ) ) ).
% domain.finite_domain_units
thf(fact_537_domain_Ofinite__domain__units,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( units_2932844235741507942t_unit @ R )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ) ) ).
% domain.finite_domain_units
thf(fact_538_domain_Ofinite__domain__units,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( units_a_ring_ext_a_b @ R )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ) ) ).
% domain.finite_domain_units
thf(fact_539_domain_Ofinite__domain__units,axiom,
! [R: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite1660835950917165235list_a @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( units_4903515905731149798t_unit @ R )
= ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) ) ) ) ).
% domain.finite_domain_units
thf(fact_540_rupture__is__field__iff__pirreducible,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) @ P ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_541_finite__Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B2: set_list_a] :
( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B2 ) ) )
= ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_542_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_543_finite__Diff__insert,axiom,
! [A2: set_list_list_a,A: list_list_a,B2: set_list_list_a] :
( ( finite1660835950917165235list_a @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ A @ B2 ) ) )
= ( finite1660835950917165235list_a @ ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_544_p_Oring__irreducibleI,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( 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 @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A5: list_a,B5: list_a] :
( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ B5 ) )
=> ( ( member_list_a @ A5 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( member_list_a @ B5 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 ) ) ) ) ).
% p.ring_irreducibleI
thf(fact_545_irreducible__mult__imp__irreducible,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( irredu6211895646901577903xt_a_b @ r @ A ) ) ) ).
% irreducible_mult_imp_irreducible
thf(fact_546_p_Oadd__additive__subgroups,axiom,
! [H: set_list_a,K: set_list_a] :
( ( additi4714453376129182166t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( additi4714453376129182166t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( additi4714453376129182166t_unit @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ K ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.add_additive_subgroups
thf(fact_547_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_548_ring__irreducibleE_I2_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( irredu6211895646901577903xt_a_b @ r @ R3 ) ) ) ).
% ring_irreducibleE(2)
thf(fact_549_zero__is__irreducible__iff__field,axiom,
( ( irredu6211895646901577903xt_a_b @ r @ ( zero_a_b @ r ) )
= ( field_a_b @ r ) ) ).
% zero_is_irreducible_iff_field
thf(fact_550_p_Ozero__is__irreducible__mult,axiom,
irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_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_irreducible_mult
thf(fact_551_p_Om__assoc,axiom,
! [X2: list_a,Y2: list_a,Z2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( member_list_a @ Z2 @ ( 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 ) ) @ X2 @ Y2 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ Z2 ) ) ) ) ) ) ).
% p.m_assoc
thf(fact_552_p_Om__comm,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ X2 ) ) ) ) ).
% p.m_comm
thf(fact_553_p_Om__lcomm,axiom,
! [X2: list_a,Y2: list_a,Z2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ Z2 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Z2 ) ) ) ) ) ) ).
% p.m_lcomm
thf(fact_554_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_555_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_556_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_557_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_558_p_Omult__of_Oprime__irreducible,axiom,
! [P: list_a] :
( ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P )
=> ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P ) ) ).
% p.mult_of.prime_irreducible
thf(fact_559_monic__poly__mult,axiom,
! [F: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G )
=> ( monic_3145109188698636716ly_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ G ) ) ) ) ).
% monic_poly_mult
thf(fact_560_irreducible__imp__irreducible__mult,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A ) ) ) ).
% irreducible_imp_irreducible_mult
thf(fact_561_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_562_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_563_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_564_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_565_p_Omult__of_Ol__cancel,axiom,
! [C: list_a,A: list_a,B: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( A = B ) ) ) ) ) ).
% p.mult_of.l_cancel
thf(fact_566_p_Omult__of_Om__assoc,axiom,
! [X2: list_a,Y2: list_a,Z2: list_a] :
( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_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 ) ) @ X2 @ Y2 ) @ Z2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ Z2 ) ) ) ) ) ) ).
% p.mult_of.m_assoc
thf(fact_567_p_Omult__of_Om__comm,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ X2 ) ) ) ) ).
% p.mult_of.m_comm
thf(fact_568_p_Omult__of_Om__lcomm,axiom,
! [X2: list_a,Y2: list_a,Z2: list_a] :
( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ Z2 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Z2 ) ) ) ) ) ) ).
% p.mult_of.m_lcomm
thf(fact_569_p_Omult__of_Or__cancel,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( A = B ) ) ) ) ) ).
% p.mult_of.r_cancel
thf(fact_570_p_Oinv__unique,axiom,
! [Y2: list_a,X2: list_a,Y4: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( member_list_a @ Y4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( Y2 = Y4 ) ) ) ) ) ) ).
% p.inv_unique
thf(fact_571_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_572_p_Oprod__unit__l,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.prod_unit_l
thf(fact_573_p_Oprod__unit__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.prod_unit_r
thf(fact_574_p_Ounit__factor,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.unit_factor
thf(fact_575_p_Ogroup__commutes__pow,axiom,
! [X2: list_a,Y2: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ X2 ) )
=> ( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N ) @ Y2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N ) ) ) ) ) ) ).
% p.group_commutes_pow
thf(fact_576_p_Onat__pow__comm,axiom,
! [X2: list_a,N: nat,M2: nat] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ M2 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ M2 ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N ) ) ) ) ).
% p.nat_pow_comm
thf(fact_577_p_Onat__pow__distrib,axiom,
! [X2: list_a,Y2: list_a,N: nat] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 ) @ 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 ) ) @ X2 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ N ) ) ) ) ) ).
% p.nat_pow_distrib
thf(fact_578_p_Opow__mult__distrib,axiom,
! [X2: list_a,Y2: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ X2 ) )
=> ( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 ) @ 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 ) ) @ X2 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ N ) ) ) ) ) ) ).
% p.pow_mult_distrib
thf(fact_579_p_Oring__irreducibleE_I3_J,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 )
=> ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ R3 ) ) ) ).
% p.ring_irreducibleE(3)
thf(fact_580_p_OUnits__inv__comm,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.Units_inv_comm
thf(fact_581_p_Omult__of_Oirreducible__prime,axiom,
! [P: list_a] :
( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P )
=> ( ( member_list_a @ P @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P ) ) ) ).
% p.mult_of.irreducible_prime
thf(fact_582_p_Omult__of_Oinv__unique,axiom,
! [Y2: list_a,X2: list_a,Y4: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y4 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( Y2 = Y4 ) ) ) ) ) ) ).
% p.mult_of.inv_unique
thf(fact_583_p_Omult__of_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_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.mult_of.one_unique
thf(fact_584_p_OUnits__l__inv__ex,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_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 ) ) @ X3 @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.Units_l_inv_ex
thf(fact_585_p_OUnits__r__inv__ex,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_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 ) ) @ X2 @ X3 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.Units_r_inv_ex
thf(fact_586_p_Oring__irreducibleE_I5_J,axiom,
! [R3: list_a,A: list_a,B: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 )
=> ( ( 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 ) ) ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ) ).
% p.ring_irreducibleE(5)
thf(fact_587_finite__domain__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% finite_domain_units
thf(fact_588_p_Omult__of_Omonoid__cancelI,axiom,
( ! [A5: list_a,B5: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A5 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B5 ) )
=> ( ( member_list_a @ A5 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B5 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( A5 = B5 ) ) ) ) )
=> ( ! [A5: list_a,B5: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B5 @ C2 ) )
=> ( ( member_list_a @ A5 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B5 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( A5 = B5 ) ) ) ) )
=> ( monoid5117334421817186628t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.mult_of.monoid_cancelI
thf(fact_589_mult__of_Otrivial__group__subset,axiom,
( ( elemen1145482699608675729t_unit @ ( ring_mult_of_a_b @ r ) )
= ( ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) @ ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% mult_of.trivial_group_subset
thf(fact_590_finite__insert,axiom,
! [A: list_list_a,A2: set_list_list_a] :
( ( finite1660835950917165235list_a @ ( insert_list_list_a @ A @ A2 ) )
= ( finite1660835950917165235list_a @ A2 ) ) ).
% finite_insert
thf(fact_591_finite__insert,axiom,
! [A: list_a,A2: set_list_a] :
( ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) )
= ( finite_finite_list_a @ A2 ) ) ).
% finite_insert
thf(fact_592_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_593_finite__Diff,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_594_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_595_finite__Diff,axiom,
! [A2: set_list_list_a,B2: set_list_list_a] :
( ( finite1660835950917165235list_a @ A2 )
=> ( finite1660835950917165235list_a @ ( minus_5335179877275218001list_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_596_finite__Diff2,axiom,
! [B2: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A2 @ B2 ) )
= ( finite_finite_list_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_597_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_598_finite__Diff2,axiom,
! [B2: set_list_list_a,A2: set_list_list_a] :
( ( finite1660835950917165235list_a @ B2 )
=> ( ( finite1660835950917165235list_a @ ( minus_5335179877275218001list_a @ A2 @ B2 ) )
= ( finite1660835950917165235list_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_599_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 ) ) ) )
=> ( ! [A5: list_a] :
( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A5
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ X5 )
= ( 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_600_field__intro2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_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 ) ) )
=> ( member_a @ X3 @ ( units_a_ring_ext_a_b @ r ) ) )
=> ( field_a_b @ r ) ) ) ).
% field_intro2
thf(fact_601_one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% one_closed
thf(fact_602_Units__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_one_closed
thf(fact_603_p_Oring__irreducibleI_H,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( 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 ) ) )
=> ( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ R3 )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 ) ) ) ).
% p.ring_irreducibleI'
thf(fact_604_mult__of_OSomeGcd__ex,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( A2 != bot_bot_set_a )
=> ( member_a @ ( someGc8133249837406473920t_unit @ ( ring_mult_of_a_b @ r ) @ A2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.SomeGcd_ex
thf(fact_605_mult__of_Oone__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.one_closed
thf(fact_606_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_607_p_Om__closed,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.m_closed
thf(fact_608_p_OUnits__m__closed,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.Units_m_closed
thf(fact_609_p_Ol__null,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.l_null
thf(fact_610_p_Or__null,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( 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_611_p_Omult__of_Om__closed,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_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 ) ) @ X2 @ Y2 ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.mult_of.m_closed
thf(fact_612_p_Ol__one,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 )
= X2 ) ) ).
% p.l_one
thf(fact_613_p_Or__one,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X2 ) ) ).
% p.r_one
thf(fact_614_p_OUnits__l__cancel,axiom,
! [X2: list_a,Y2: list_a,Z2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_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 ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Z2 ) )
= ( Y2 = Z2 ) ) ) ) ) ).
% p.Units_l_cancel
thf(fact_615_p_Omult__of_Ol__one,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_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 ) ) ) @ X2 )
= X2 ) ) ).
% p.mult_of.l_one
thf(fact_616_p_Omult__of_Or__one,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X2 ) ) ).
% p.mult_of.r_one
thf(fact_617_ring__irreducible__def,axiom,
( ring_r999134135267193926le_a_b
= ( ^ [R2: partia2175431115845679010xt_a_b,A4: a] :
( ( A4
!= ( zero_a_b @ R2 ) )
& ( irredu6211895646901577903xt_a_b @ R2 @ A4 ) ) ) ) ).
% ring_irreducible_def
thf(fact_618_ring__irreducible__def,axiom,
( ring_r932985474545269838t_unit
= ( ^ [R2: partia2670972154091845814t_unit,A4: list_a] :
( ( A4
!= ( zero_l4142658623432671053t_unit @ R2 ) )
& ( irredu4230924414530676029t_unit @ R2 @ A4 ) ) ) ) ).
% ring_irreducible_def
thf(fact_619_ring__irreducible__def,axiom,
( ring_r360171070648044744t_unit
= ( ^ [R2: partia2956882679547061052t_unit,A4: list_list_a] :
( ( A4
!= ( zero_l347298301471573063t_unit @ R2 ) )
& ( irredu4439051761327310013t_unit @ R2 @ A4 ) ) ) ) ).
% ring_irreducible_def
thf(fact_620_domain_Ointegral__iff,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
= ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_621_domain_Ointegral__iff,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
= ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_622_domain_Ointegral__iff,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
= ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_623_domain_Ointegral__iff,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
= ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_624_domain_Ointegral__iff,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( zero_l347298301471573063t_unit @ R ) )
= ( ( A
= ( zero_l347298301471573063t_unit @ R ) )
| ( B
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral_iff
thf(fact_625_domain_Om__rcancel,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ B @ A )
= ( mult_s7802724872828879953t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_626_domain_Om__rcancel,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( A
!= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ B @ A )
= ( mult_s7930653359683758801t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_627_domain_Om__rcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ B @ A )
= ( mult_l7073676228092353617t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_628_domain_Om__rcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ B @ A )
= ( mult_a_ring_ext_a_b @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_629_domain_Om__rcancel,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( A
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ B @ A )
= ( mult_l4853965630390486993t_unit @ R @ C @ A ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_rcancel
thf(fact_630_domain_Om__lcancel,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a,C: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( A
!= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ C @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( mult_s7802724872828879953t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_631_domain_Om__lcancel,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a,C: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( A
!= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ C @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( mult_s7930653359683758801t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_632_domain_Om__lcancel,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,C: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( A
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( mult_l7073676228092353617t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_633_domain_Om__lcancel,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,C: a] :
( ( domain_a_b @ R )
=> ( ( A
!= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( mult_a_ring_ext_a_b @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_634_domain_Om__lcancel,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,C: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( A
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ C @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( mult_l4853965630390486993t_unit @ R @ A @ C ) )
= ( B = C ) ) ) ) ) ) ) ).
% domain.m_lcancel
thf(fact_635_domain_Ointegral,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_636_domain_Ointegral,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_637_domain_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_638_domain_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_639_domain_Ointegral,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( A
= ( zero_l347298301471573063t_unit @ R ) )
| ( B
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% domain.integral
thf(fact_640_Ring_Ointegral,axiom,
! [R: partia4960592913263135132t_unit,A: set_list_list_a,B: set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( ( mult_s6197134818278049745t_unit @ R @ A @ B )
= ( zero_s2920163772466840039t_unit @ R ) )
=> ( ( member334759470184282131list_a @ A @ ( partia3317168157747563407t_unit @ R ) )
=> ( ( member334759470184282131list_a @ B @ ( partia3317168157747563407t_unit @ R ) )
=> ( ( A
= ( zero_s2920163772466840039t_unit @ R ) )
| ( B
= ( zero_s2920163772466840039t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_641_Ring_Ointegral,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a,B: set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( ( mult_s7802724872828879953t_unit @ R @ A @ B )
= ( zero_s2910681146719230829t_unit @ R ) )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( A
= ( zero_s2910681146719230829t_unit @ R ) )
| ( B
= ( zero_s2910681146719230829t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_642_Ring_Ointegral,axiom,
! [R: partia6043505979758434576t_unit,A: set_a,B: set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( ( mult_s7930653359683758801t_unit @ R @ A @ B )
= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( A
= ( zero_s2174465271003423091t_unit @ R ) )
| ( B
= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_643_Ring_Ointegral,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ A @ B )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( A
= ( zero_l4142658623432671053t_unit @ R ) )
| ( B
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_644_Ring_Ointegral,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( field_a_b @ R )
=> ( ( ( mult_a_ring_ext_a_b @ R @ A @ B )
= ( zero_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( A
= ( zero_a_b @ R ) )
| ( B
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_645_Ring_Ointegral,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( ( mult_l4853965630390486993t_unit @ R @ A @ B )
= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( A
= ( zero_l347298301471573063t_unit @ R ) )
| ( B
= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ) ) ).
% Ring.integral
thf(fact_646_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia7496981018696276118t_unit,R3: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ R3 )
=> ( irredu943254396193320253t_unit @ R @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_647_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia6043505979758434576t_unit,R3: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R3 )
=> ( irredu5346329325703585725t_unit @ R @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_648_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R3 )
=> ( irredu6211895646901577903xt_a_b @ R @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_649_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R3 )
=> ( irredu4230924414530676029t_unit @ R @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_650_domain_Oring__irreducibleE_I2_J,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R3 )
=> ( irredu4439051761327310013t_unit @ R @ R3 ) ) ) ) ).
% domain.ring_irreducibleE(2)
thf(fact_651_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia4960592913263135132t_unit] :
( ( domain7421296078544666595t_unit @ R )
=> ( ( irredu5077821220250117309t_unit @ R @ ( zero_s2920163772466840039t_unit @ R ) )
= ( field_1540243473349940225t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_652_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia7496981018696276118t_unit] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( irredu943254396193320253t_unit @ R @ ( zero_s2910681146719230829t_unit @ R ) )
= ( field_26233345952514695t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_653_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( irredu5346329325703585725t_unit @ R @ ( zero_s2174465271003423091t_unit @ R ) )
= ( field_6045675692312731021t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_654_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia2956882679547061052t_unit] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( irredu4439051761327310013t_unit @ R @ ( zero_l347298301471573063t_unit @ R ) )
= ( field_1861437471013600865t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_655_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( ( irredu6211895646901577903xt_a_b @ R @ ( zero_a_b @ R ) )
= ( field_a_b @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_656_domain_Ozero__is__irreducible__iff__field,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( irredu4230924414530676029t_unit @ R @ ( zero_l4142658623432671053t_unit @ R ) )
= ( field_6388047844668329575t_unit @ R ) ) ) ).
% domain.zero_is_irreducible_iff_field
thf(fact_657_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia7496981018696276118t_unit,R3: set_list_a,A: set_list_a,B: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R3 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( ring_r5115406448772830318t_unit @ R @ R3 )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B @ ( partia141011252114345353t_unit @ R ) )
=> ( ( R3
= ( mult_s7802724872828879953t_unit @ R @ A @ B ) )
=> ( ( member_set_list_a @ A @ ( units_5837875185506529638t_unit @ R ) )
| ( member_set_list_a @ B @ ( units_5837875185506529638t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_658_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia6043505979758434576t_unit,R3: set_a,A: set_a,B: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( ring_r7790391342995787508t_unit @ R @ R3 )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( R3
= ( mult_s7930653359683758801t_unit @ R @ A @ B ) )
=> ( ( member_set_a @ A @ ( units_2471184348132832486t_unit @ R ) )
| ( member_set_a @ B @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_659_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a,A: a,B: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ring_r999134135267193926le_a_b @ R @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ R @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_660_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a,A: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ring_r932985474545269838t_unit @ R @ R3 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ R @ A @ B ) )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ R ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_661_domain_Oring__irreducibleE_I5_J,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a,A: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ring_r360171070648044744t_unit @ R @ R3 )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( R3
= ( mult_l4853965630390486993t_unit @ R @ A @ B ) )
=> ( ( member_list_list_a @ A @ ( units_4903515905731149798t_unit @ R ) )
| ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_662_field_Omonic__poly__mult,axiom,
! [R: partia4960592913263135132t_unit,F: list_set_list_list_a,G: list_set_list_list_a] :
( ( field_1540243473349940225t_unit @ R )
=> ( ( monic_8143709425247463374t_unit @ R @ F )
=> ( ( monic_8143709425247463374t_unit @ R @ G )
=> ( monic_8143709425247463374t_unit @ R @ ( mult_l7436655221470123345t_unit @ ( univ_p7077926387201515752t_unit @ R @ ( partia3317168157747563407t_unit @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_663_field_Omonic__poly__mult,axiom,
! [R: partia7496981018696276118t_unit,F: list_set_list_a,G: list_set_list_a] :
( ( field_26233345952514695t_unit @ R )
=> ( ( monic_3395465470813675732t_unit @ R @ F )
=> ( ( monic_3395465470813675732t_unit @ R @ G )
=> ( monic_3395465470813675732t_unit @ R @ ( mult_l5330480240434472913t_unit @ ( univ_p863672496597069550t_unit @ R @ ( partia141011252114345353t_unit @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_664_field_Omonic__poly__mult,axiom,
! [R: partia6043505979758434576t_unit,F: list_set_a,G: list_set_a] :
( ( field_6045675692312731021t_unit @ R )
=> ( ( monic_6852076386464302682t_unit @ R @ F )
=> ( ( monic_6852076386464302682t_unit @ R @ G )
=> ( monic_6852076386464302682t_unit @ R @ ( mult_l4263563202070946897t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_665_field_Omonic__poly__mult,axiom,
! [R: partia2670972154091845814t_unit,F: list_list_a,G: list_list_a] :
( ( field_6388047844668329575t_unit @ R )
=> ( ( monic_5008461317928820916t_unit @ R @ F )
=> ( ( monic_5008461317928820916t_unit @ R @ G )
=> ( monic_5008461317928820916t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_666_field_Omonic__poly__mult,axiom,
! [R: partia2956882679547061052t_unit,F: list_list_list_a,G: list_list_list_a] :
( ( field_1861437471013600865t_unit @ R )
=> ( ( monic_5986596350207772206t_unit @ R @ F )
=> ( ( monic_5986596350207772206t_unit @ R @ G )
=> ( monic_5986596350207772206t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_667_field_Omonic__poly__mult,axiom,
! [R: partia2175431115845679010xt_a_b,F: list_a,G: list_a] :
( ( field_a_b @ R )
=> ( ( monic_3145109188698636716ly_a_b @ R @ F )
=> ( ( monic_3145109188698636716ly_a_b @ R @ G )
=> ( monic_3145109188698636716ly_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ F @ G ) ) ) ) ) ).
% field.monic_poly_mult
thf(fact_668_domain_Oirreducible__imp__irreducible__mult,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ A @ ( partia141011252114345353t_unit @ R ) )
=> ( ( irredu943254396193320253t_unit @ R @ A )
=> ( irredu5949096822098593390t_unit @ ( ring_m294936264769644739t_unit @ R ) @ A ) ) ) ) ).
% domain.irreducible_imp_irreducible_mult
thf(fact_669_domain_Oirreducible__imp__irreducible__mult,axiom,
! [R: partia6043505979758434576t_unit,A: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( irredu5346329325703585725t_unit @ R @ A )
=> ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ A ) ) ) ) ).
% domain.irreducible_imp_irreducible_mult
thf(fact_670_domain_Oirreducible__imp__irreducible__mult,axiom,
! [R: partia2670972154091845814t_unit,A: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( irredu4230924414530676029t_unit @ R @ A )
=> ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ R ) @ A ) ) ) ) ).
% domain.irreducible_imp_irreducible_mult
thf(fact_671_domain_Oirreducible__imp__irreducible__mult,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( irredu6211895646901577903xt_a_b @ R @ A )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ A ) ) ) ) ).
% domain.irreducible_imp_irreducible_mult
thf(fact_672_domain_Oirreducible__imp__irreducible__mult,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( irredu4439051761327310013t_unit @ R @ A )
=> ( irredu7159092911615704776t_unit @ ( ring_m2698952956457993885t_unit @ R ) @ A ) ) ) ) ).
% domain.irreducible_imp_irreducible_mult
thf(fact_673_principal__domain_Obezout__identity,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( set_add_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) @ ( cgenid547466209912283029xt_a_b @ R @ B ) )
= ( cgenid547466209912283029xt_a_b @ R @ ( somegc1600592057159103747xt_a_b @ R @ A @ B ) ) ) ) ) ) ).
% principal_domain.bezout_identity
thf(fact_674_principal__domain_Obezout__identity,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( set_ad92425877771022410t_unit @ R @ ( cgenid9131348535277946915t_unit @ R @ A ) @ ( cgenid9131348535277946915t_unit @ R @ B ) )
= ( cgenid9131348535277946915t_unit @ R @ ( somegc2556875419254249873t_unit @ R @ A @ B ) ) ) ) ) ) ).
% principal_domain.bezout_identity
thf(fact_675_principal__domain_Obezout__identity,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( set_ad4979488651584656068t_unit @ R @ ( cgenid24865672677839267t_unit @ R @ A ) @ ( cgenid24865672677839267t_unit @ R @ B ) )
= ( cgenid24865672677839267t_unit @ R @ ( somegc7099985650160764177t_unit @ R @ A @ B ) ) ) ) ) ) ).
% principal_domain.bezout_identity
thf(fact_676_finite__has__minimal2,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A @ A2 )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ( ord_le8861187494160871172list_a @ X3 @ A )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_677_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ X3 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_678_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ X3 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_679_finite__has__maximal2,axiom,
! [A2: set_set_list_a,A: set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( member_set_list_a @ A @ A2 )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ( ord_le8861187494160871172list_a @ A @ X3 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_680_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ( ord_less_eq_set_a @ A @ X3 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_681_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( ord_less_eq_nat @ A @ X3 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_682_rev__finite__subset,axiom,
! [B2: set_list_a,A2: set_list_a] :
( ( finite_finite_list_a @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_683_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_684_infinite__super,axiom,
! [S: set_list_a,T2: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ T2 )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ T2 ) ) ) ).
% infinite_super
thf(fact_685_infinite__super,axiom,
! [S: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_686_finite__subset,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ( finite_finite_list_a @ B2 )
=> ( finite_finite_list_a @ A2 ) ) ) ).
% finite_subset
thf(fact_687_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_688_infinite__imp__nonempty,axiom,
! [S: set_list_a] :
( ~ ( finite_finite_list_a @ S )
=> ( S != bot_bot_set_list_a ) ) ).
% infinite_imp_nonempty
thf(fact_689_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_690_infinite__imp__nonempty,axiom,
! [S: set_list_list_a] :
( ~ ( finite1660835950917165235list_a @ S )
=> ( S != bot_bo1875519244922727510list_a ) ) ).
% infinite_imp_nonempty
thf(fact_691_finite_OemptyI,axiom,
finite_finite_list_a @ bot_bot_set_list_a ).
% finite.emptyI
thf(fact_692_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_693_finite_OemptyI,axiom,
finite1660835950917165235list_a @ bot_bo1875519244922727510list_a ).
% finite.emptyI
thf(fact_694_finite_OinsertI,axiom,
! [A2: set_list_list_a,A: list_list_a] :
( ( finite1660835950917165235list_a @ A2 )
=> ( finite1660835950917165235list_a @ ( insert_list_list_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_695_finite_OinsertI,axiom,
! [A2: set_list_a,A: list_a] :
( ( finite_finite_list_a @ A2 )
=> ( finite_finite_list_a @ ( insert_list_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_696_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_697_Diff__infinite__finite,axiom,
! [T2: set_list_a,S: set_list_a] :
( ( finite_finite_list_a @ T2 )
=> ( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_698_Diff__infinite__finite,axiom,
! [T2: set_a,S: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_699_Diff__infinite__finite,axiom,
! [T2: set_list_list_a,S: set_list_list_a] :
( ( finite1660835950917165235list_a @ T2 )
=> ( ~ ( finite1660835950917165235list_a @ S )
=> ~ ( finite1660835950917165235list_a @ ( minus_5335179877275218001list_a @ S @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_700_principal__domain_Oideal__sum__iff__gcd,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B: a,D: a] :
( ( ring_p8803135361686045600in_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ D @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ( cgenid547466209912283029xt_a_b @ R @ D )
= ( set_add_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) @ ( cgenid547466209912283029xt_a_b @ R @ B ) ) )
= ( isgcd_a_ring_ext_a_b @ R @ D @ A @ B ) ) ) ) ) ) ).
% principal_domain.ideal_sum_iff_gcd
thf(fact_701_principal__domain_Oideal__sum__iff__gcd,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B: list_a,D: list_a] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ D @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ( cgenid9131348535277946915t_unit @ R @ D )
= ( set_ad92425877771022410t_unit @ R @ ( cgenid9131348535277946915t_unit @ R @ A ) @ ( cgenid9131348535277946915t_unit @ R @ B ) ) )
= ( isgcd_1118609098697428327t_unit @ R @ D @ A @ B ) ) ) ) ) ) ).
% principal_domain.ideal_sum_iff_gcd
thf(fact_702_principal__domain_Oideal__sum__iff__gcd,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B: list_list_a,D: list_list_a] :
( ( ring_p715737262848045090t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ D @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ( cgenid24865672677839267t_unit @ R @ D )
= ( set_ad4979488651584656068t_unit @ R @ ( cgenid24865672677839267t_unit @ R @ A ) @ ( cgenid24865672677839267t_unit @ R @ B ) ) )
= ( isgcd_3804025100609598183t_unit @ R @ D @ A @ B ) ) ) ) ) ) ).
% principal_domain.ideal_sum_iff_gcd
thf(fact_703_domain_Oring__irreducibleI,axiom,
! [R: partia7496981018696276118t_unit,R3: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ R3 @ ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
=> ( ~ ( member_set_list_a @ R3 @ ( units_5837875185506529638t_unit @ R ) )
=> ( ! [A5: set_list_a,B5: set_list_a] :
( ( member_set_list_a @ A5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( member_set_list_a @ B5 @ ( partia141011252114345353t_unit @ R ) )
=> ( ( R3
= ( mult_s7802724872828879953t_unit @ R @ A5 @ B5 ) )
=> ( ( member_set_list_a @ A5 @ ( units_5837875185506529638t_unit @ R ) )
| ( member_set_list_a @ B5 @ ( units_5837875185506529638t_unit @ R ) ) ) ) ) )
=> ( ring_r5115406448772830318t_unit @ R @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_704_domain_Oring__irreducibleI,axiom,
! [R: partia6043505979758434576t_unit,R3: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ R3 @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ~ ( member_set_a @ R3 @ ( units_2471184348132832486t_unit @ R ) )
=> ( ! [A5: set_a,B5: set_a] :
( ( member_set_a @ A5 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( member_set_a @ B5 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( R3
= ( mult_s7930653359683758801t_unit @ R @ A5 @ B5 ) )
=> ( ( member_set_a @ A5 @ ( units_2471184348132832486t_unit @ R ) )
| ( member_set_a @ B5 @ ( units_2471184348132832486t_unit @ R ) ) ) ) ) )
=> ( ring_r7790391342995787508t_unit @ R @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_705_domain_Oring__irreducibleI,axiom,
! [R: partia2175431115845679010xt_a_b,R3: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ R3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ R ) )
=> ( ! [A5: a,B5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ R @ A5 @ B5 ) )
=> ( ( member_a @ A5 @ ( units_a_ring_ext_a_b @ R ) )
| ( member_a @ B5 @ ( units_a_ring_ext_a_b @ R ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ R @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_706_domain_Oring__irreducibleI,axiom,
! [R: partia2670972154091845814t_unit,R3: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ R3 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ~ ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ R ) )
=> ( ! [A5: list_a,B5: list_a] :
( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ R @ A5 @ B5 ) )
=> ( ( member_list_a @ A5 @ ( units_2932844235741507942t_unit @ R ) )
| ( member_list_a @ B5 @ ( units_2932844235741507942t_unit @ R ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ R @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_707_domain_Oring__irreducibleI,axiom,
! [R: partia2956882679547061052t_unit,R3: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ R3 @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ~ ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ R ) )
=> ( ! [A5: list_list_a,B5: list_list_a] :
( ( member_list_list_a @ A5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B5 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( R3
= ( mult_l4853965630390486993t_unit @ R @ A5 @ B5 ) )
=> ( ( member_list_list_a @ A5 @ ( units_4903515905731149798t_unit @ R ) )
| ( member_list_list_a @ B5 @ ( units_4903515905731149798t_unit @ R ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ R @ R3 ) ) ) ) ) ).
% domain.ring_irreducibleI
thf(fact_708_domain_Oirreducible__mult__imp__irreducible,axiom,
! [R: partia7496981018696276118t_unit,A: set_list_a] :
( ( domain1617769409708967785t_unit @ R )
=> ( ( member_set_list_a @ A @ ( minus_4782336368215558443list_a @ ( partia141011252114345353t_unit @ R ) @ ( insert_set_list_a @ ( zero_s2910681146719230829t_unit @ R ) @ bot_bo3186585308812441520list_a ) ) )
=> ( ( irredu5949096822098593390t_unit @ ( ring_m294936264769644739t_unit @ R ) @ A )
=> ( irredu943254396193320253t_unit @ R @ A ) ) ) ) ).
% domain.irreducible_mult_imp_irreducible
thf(fact_709_domain_Oirreducible__mult__imp__irreducible,axiom,
! [R: partia6043505979758434576t_unit,A: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ A @ ( minus_5736297505244876581_set_a @ ( partia5907974310037520643t_unit @ R ) @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) ) )
=> ( ( irredu8646402277169070324t_unit @ ( ring_m2800496791135293897t_unit @ R ) @ A )
=> ( irredu5346329325703585725t_unit @ R @ A ) ) ) ) ).
% domain.irreducible_mult_imp_irreducible
thf(fact_710_domain_Oirreducible__mult__imp__irreducible,axiom,
! [R: partia2670972154091845814t_unit,A: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ R ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) )
=> ( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ R ) @ A )
=> ( irredu4230924414530676029t_unit @ R @ A ) ) ) ) ).
% domain.irreducible_mult_imp_irreducible
thf(fact_711_domain_Oirreducible__mult__imp__irreducible,axiom,
! [R: partia2175431115845679010xt_a_b,A: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ R ) @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) )
=> ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ R ) @ A )
=> ( irredu6211895646901577903xt_a_b @ R @ A ) ) ) ) ).
% domain.irreducible_mult_imp_irreducible
thf(fact_712_domain_Oirreducible__mult__imp__irreducible,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ A @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ R ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) ) )
=> ( ( irredu7159092911615704776t_unit @ ( ring_m2698952956457993885t_unit @ R ) @ A )
=> ( irredu4439051761327310013t_unit @ R @ A ) ) ) ) ).
% domain.irreducible_mult_imp_irreducible
thf(fact_713_finite__has__maximal,axiom,
! [A2: set_set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( A2 != bot_bo3186585308812441520list_a )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_714_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_715_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_716_finite__has__minimal,axiom,
! [A2: set_set_list_a] :
( ( finite5282473924520328461list_a @ A2 )
=> ( ( A2 != bot_bo3186585308812441520list_a )
=> ? [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A2 )
& ! [Xa: set_list_a] :
( ( member_set_list_a @ Xa @ A2 )
=> ( ( ord_le8861187494160871172list_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_717_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_718_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_719_finite_Ocases,axiom,
! [A: set_list_a] :
( ( finite_finite_list_a @ A )
=> ( ( A != bot_bot_set_list_a )
=> ~ ! [A6: set_list_a] :
( ? [A5: list_a] :
( A
= ( insert_list_a @ A5 @ A6 ) )
=> ~ ( finite_finite_list_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_720_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A6: set_a] :
( ? [A5: a] :
( A
= ( insert_a @ A5 @ A6 ) )
=> ~ ( finite_finite_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_721_finite_Ocases,axiom,
! [A: set_list_list_a] :
( ( finite1660835950917165235list_a @ A )
=> ( ( A != bot_bo1875519244922727510list_a )
=> ~ ! [A6: set_list_list_a] :
( ? [A5: list_list_a] :
( A
= ( insert_list_list_a @ A5 @ A6 ) )
=> ~ ( finite1660835950917165235list_a @ A6 ) ) ) ) ).
% finite.cases
thf(fact_722_finite_Osimps,axiom,
( finite_finite_list_a
= ( ^ [A4: set_list_a] :
( ( A4 = bot_bot_set_list_a )
| ? [A3: set_list_a,B6: list_a] :
( ( A4
= ( insert_list_a @ B6 @ A3 ) )
& ( finite_finite_list_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_723_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A4: set_a] :
( ( A4 = bot_bot_set_a )
| ? [A3: set_a,B6: a] :
( ( A4
= ( insert_a @ B6 @ A3 ) )
& ( finite_finite_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_724_finite_Osimps,axiom,
( finite1660835950917165235list_a
= ( ^ [A4: set_list_list_a] :
( ( A4 = bot_bo1875519244922727510list_a )
| ? [A3: set_list_list_a,B6: list_list_a] :
( ( A4
= ( insert_list_list_a @ B6 @ A3 ) )
& ( finite1660835950917165235list_a @ A3 ) ) ) ) ) ).
% finite.simps
thf(fact_725_finite__induct,axiom,
! [F3: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [X3: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ~ ( member_list_a @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_list_a @ X3 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_726_finite__induct,axiom,
! [F3: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_a @ X3 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_727_finite__induct,axiom,
! [F3: set_list_list_a,P2: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ F3 )
=> ( ( P2 @ bot_bo1875519244922727510list_a )
=> ( ! [X3: list_list_a,F4: set_list_list_a] :
( ( finite1660835950917165235list_a @ F4 )
=> ( ~ ( member_list_list_a @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_list_list_a @ X3 @ F4 ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_induct
thf(fact_728_finite__ne__induct,axiom,
! [F3: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( F3 != bot_bot_set_list_a )
=> ( ! [X3: list_a] : ( P2 @ ( insert_list_a @ X3 @ bot_bot_set_list_a ) )
=> ( ! [X3: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ( F4 != bot_bot_set_list_a )
=> ( ~ ( member_list_a @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_list_a @ X3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_729_finite__ne__induct,axiom,
! [F3: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ! [X3: a] : ( P2 @ ( insert_a @ X3 @ bot_bot_set_a ) )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( F4 != bot_bot_set_a )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_a @ X3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_730_finite__ne__induct,axiom,
! [F3: set_list_list_a,P2: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ F3 )
=> ( ( F3 != bot_bo1875519244922727510list_a )
=> ( ! [X3: list_list_a] : ( P2 @ ( insert_list_list_a @ X3 @ bot_bo1875519244922727510list_a ) )
=> ( ! [X3: list_list_a,F4: set_list_list_a] :
( ( finite1660835950917165235list_a @ F4 )
=> ( ( F4 != bot_bo1875519244922727510list_a )
=> ( ~ ( member_list_list_a @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_list_list_a @ X3 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_731_infinite__finite__induct,axiom,
! [P2: set_list_a > $o,A2: set_list_a] :
( ! [A6: set_list_a] :
( ~ ( finite_finite_list_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [X3: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ~ ( member_list_a @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_list_a @ X3 @ F4 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_732_infinite__finite__induct,axiom,
! [P2: set_a > $o,A2: set_a] :
( ! [A6: set_a] :
( ~ ( finite_finite_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X3: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ~ ( member_a @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_a @ X3 @ F4 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_733_infinite__finite__induct,axiom,
! [P2: set_list_list_a > $o,A2: set_list_list_a] :
( ! [A6: set_list_list_a] :
( ~ ( finite1660835950917165235list_a @ A6 )
=> ( P2 @ A6 ) )
=> ( ( P2 @ bot_bo1875519244922727510list_a )
=> ( ! [X3: list_list_a,F4: set_list_list_a] :
( ( finite1660835950917165235list_a @ F4 )
=> ( ~ ( member_list_list_a @ X3 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_list_list_a @ X3 @ F4 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_734_finite__subset__induct,axiom,
! [F3: set_list_list_a,A2: set_list_list_a,P2: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ F3 )
=> ( ( ord_le8488217952732425610list_a @ F3 @ A2 )
=> ( ( P2 @ bot_bo1875519244922727510list_a )
=> ( ! [A5: list_list_a,F4: set_list_list_a] :
( ( finite1660835950917165235list_a @ F4 )
=> ( ( member_list_list_a @ A5 @ A2 )
=> ( ~ ( member_list_list_a @ A5 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_list_list_a @ A5 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_735_finite__subset__induct,axiom,
! [F3: set_list_a,A2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A2 )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [A5: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ( member_list_a @ A5 @ A2 )
=> ( ~ ( member_list_a @ A5 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_list_a @ A5 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_736_finite__subset__induct,axiom,
! [F3: set_a,A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A5: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A5 @ A2 )
=> ( ~ ( member_a @ A5 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_a @ A5 @ F4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_737_finite__subset__induct_H,axiom,
! [F3: set_list_list_a,A2: set_list_list_a,P2: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ F3 )
=> ( ( ord_le8488217952732425610list_a @ F3 @ A2 )
=> ( ( P2 @ bot_bo1875519244922727510list_a )
=> ( ! [A5: list_list_a,F4: set_list_list_a] :
( ( finite1660835950917165235list_a @ F4 )
=> ( ( member_list_list_a @ A5 @ A2 )
=> ( ( ord_le8488217952732425610list_a @ F4 @ A2 )
=> ( ~ ( member_list_list_a @ A5 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_list_list_a @ A5 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_738_finite__subset__induct_H,axiom,
! [F3: set_list_a,A2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ F3 )
=> ( ( ord_le8861187494160871172list_a @ F3 @ A2 )
=> ( ( P2 @ bot_bot_set_list_a )
=> ( ! [A5: list_a,F4: set_list_a] :
( ( finite_finite_list_a @ F4 )
=> ( ( member_list_a @ A5 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ F4 @ A2 )
=> ( ~ ( member_list_a @ A5 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_list_a @ A5 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_739_finite__subset__induct_H,axiom,
! [F3: set_a,A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F3 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A5: a,F4: set_a] :
( ( finite_finite_a @ F4 )
=> ( ( member_a @ A5 @ A2 )
=> ( ( ord_less_eq_set_a @ F4 @ A2 )
=> ( ~ ( member_a @ A5 @ F4 )
=> ( ( P2 @ F4 )
=> ( P2 @ ( insert_a @ A5 @ F4 ) ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_740_infinite__remove,axiom,
! [S: set_list_a,A: list_a] :
( ~ ( finite_finite_list_a @ S )
=> ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ S @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ) ).
% infinite_remove
thf(fact_741_infinite__remove,axiom,
! [S: set_a,A: a] :
( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_742_infinite__remove,axiom,
! [S: set_list_list_a,A: list_list_a] :
( ~ ( finite1660835950917165235list_a @ S )
=> ~ ( finite1660835950917165235list_a @ ( minus_5335179877275218001list_a @ S @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ) ) ).
% infinite_remove
thf(fact_743_infinite__coinduct,axiom,
! [X4: set_list_a > $o,A2: set_list_a] :
( ( X4 @ A2 )
=> ( ! [A6: set_list_a] :
( ( X4 @ A6 )
=> ? [X5: list_a] :
( ( member_list_a @ X5 @ A6 )
& ( ( X4 @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) )
| ~ ( finite_finite_list_a @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ X5 @ bot_bot_set_list_a ) ) ) ) ) )
=> ~ ( finite_finite_list_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_744_infinite__coinduct,axiom,
! [X4: set_a > $o,A2: set_a] :
( ( X4 @ A2 )
=> ( ! [A6: set_a] :
( ( X4 @ A6 )
=> ? [X5: a] :
( ( member_a @ X5 @ A6 )
& ( ( X4 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_745_infinite__coinduct,axiom,
! [X4: set_list_list_a > $o,A2: set_list_list_a] :
( ( X4 @ A2 )
=> ( ! [A6: set_list_list_a] :
( ( X4 @ A6 )
=> ? [X5: list_list_a] :
( ( member_list_list_a @ X5 @ A6 )
& ( ( X4 @ ( minus_5335179877275218001list_a @ A6 @ ( insert_list_list_a @ X5 @ bot_bo1875519244922727510list_a ) ) )
| ~ ( finite1660835950917165235list_a @ ( minus_5335179877275218001list_a @ A6 @ ( insert_list_list_a @ X5 @ bot_bo1875519244922727510list_a ) ) ) ) ) )
=> ~ ( finite1660835950917165235list_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_746_finite__empty__induct,axiom,
! [A2: set_list_a,P2: set_list_a > $o] :
( ( finite_finite_list_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A5: list_a,A6: set_list_a] :
( ( finite_finite_list_a @ A6 )
=> ( ( member_list_a @ A5 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_646659088055828811list_a @ A6 @ ( insert_list_a @ A5 @ bot_bot_set_list_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_747_finite__empty__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A5: a,A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( member_a @ A5 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_748_finite__empty__induct,axiom,
! [A2: set_list_list_a,P2: set_list_list_a > $o] :
( ( finite1660835950917165235list_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A5: list_list_a,A6: set_list_list_a] :
( ( finite1660835950917165235list_a @ A6 )
=> ( ( member_list_list_a @ A5 @ A6 )
=> ( ( P2 @ A6 )
=> ( P2 @ ( minus_5335179877275218001list_a @ A6 @ ( insert_list_list_a @ A5 @ bot_bo1875519244922727510list_a ) ) ) ) ) )
=> ( P2 @ bot_bo1875519244922727510list_a ) ) ) ) ).
% finite_empty_induct
thf(fact_749_remove__induct,axiom,
! [P2: set_a > $o,B2: set_a] :
( ( P2 @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ( A6 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A6 @ B2 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ A6 )
=> ( P2 @ ( minus_minus_set_a @ A6 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A6 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_750_p_Omonoid__cancelI,axiom,
( ! [A5: list_a,B5: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A5 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B5 ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B5 @ ( 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 ) ) ) )
=> ( A5 = B5 ) ) ) ) )
=> ( ! [A5: list_a,B5: list_a,C2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ C2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B5 @ C2 ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B5 @ ( 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 ) ) ) )
=> ( A5 = B5 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.monoid_cancelI
thf(fact_751_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q2: list_a,X2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) @ X2 )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
| ( polyno4133073214067823460ot_a_b @ r @ Q2 @ X2 ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_752_p_Oirreducible__mult__imp__irreducible,axiom,
! [A: list_a] :
( ( 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 ) ) )
=> ( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) ) ) ).
% p.irreducible_mult_imp_irreducible
thf(fact_753_p_Oexists__irreducible__divisor,axiom,
! [A: list_a] :
( ( 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 ) ) )
=> ( ~ ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ~ ! [B5: list_a] :
( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B5 )
=> ~ ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B5 @ A ) ) ) ) ) ).
% p.exists_irreducible_divisor
thf(fact_754_ring__irreducibleI,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R3 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A5: a,B5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ r @ A5 @ B5 ) )
=> ( ( member_a @ A5 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B5 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R3 ) ) ) ) ).
% ring_irreducibleI
thf(fact_755_mult__of_Ogcd__condition__monoid__axioms,axiom,
gcd_co701944698663231555t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.gcd_condition_monoid_axioms
thf(fact_756_setadd__subset__G,axiom,
! [H: set_a,K: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_757_set__add__comm,axiom,
! [I2: set_a,J: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I2 @ J )
= ( set_add_a_b @ r @ J @ I2 ) ) ) ) ).
% set_add_comm
thf(fact_758_set__add__closed,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_759_m__lcomm,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z2 ) )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( mult_a_ring_ext_a_b @ r @ X2 @ Z2 ) ) ) ) ) ) ).
% m_lcomm
thf(fact_760_m__comm,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X2 ) ) ) ) ).
% m_comm
thf(fact_761_m__assoc,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% m_assoc
thf(fact_762_bezout__identity,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B ) )
= ( cgenid547466209912283029xt_a_b @ r @ ( somegc1600592057159103747xt_a_b @ r @ A @ B ) ) ) ) ) ).
% bezout_identity
thf(fact_763_ideal__sum__iff__gcd,axiom,
! [A: a,B: a,D: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ D @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( cgenid547466209912283029xt_a_b @ r @ D )
= ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B ) ) )
= ( isgcd_a_ring_ext_a_b @ r @ D @ A @ B ) ) ) ) ) ).
% ideal_sum_iff_gcd
thf(fact_764_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_765_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_766_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_767_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_768_f__comm__group__1,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X2
!= ( zero_a_b @ r ) )
=> ( ( Y2
!= ( zero_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 )
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% f_comm_group_1
thf(fact_769_mult__of_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ C )
= ( mult_a_ring_ext_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B ) ) ) ) ) ).
% mult_of.r_cancel
thf(fact_770_mult__of_Om__lcomm,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z2 ) )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( mult_a_ring_ext_a_b @ r @ X2 @ Z2 ) ) ) ) ) ) ).
% mult_of.m_lcomm
thf(fact_771_mult__of_Om__comm,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X2 ) ) ) ) ).
% mult_of.m_comm
thf(fact_772_mult__of_Om__assoc,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 ) @ Z2 )
= ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y2 @ Z2 ) ) ) ) ) ) ).
% mult_of.m_assoc
thf(fact_773_mult__of_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A = B ) ) ) ) ) ).
% mult_of.l_cancel
thf(fact_774_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_775_inv__unique,axiom,
! [Y2: a,X2: a,Y4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y2 = Y4 ) ) ) ) ) ) ).
% inv_unique
thf(fact_776_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_777_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_778_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_779_Units__inv__comm,axiom,
! [X2: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_inv_comm
thf(fact_780_p_Odivides__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( factor1757716651909850160t_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 ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ).
% p.divides_trans
thf(fact_781_p_Ozero__divides,axiom,
! [A: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.zero_divides
thf(fact_782_mult__of_Or__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.r_inv_ex
thf(fact_783_mult__of_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.one_unique
thf(fact_784_mult__of_Ol__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.l_inv_ex
thf(fact_785_mult__of_Oinv__unique,axiom,
! [Y2: a,X2: a,Y4: a] :
( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y4 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( Y2 = Y4 ) ) ) ) ) ) ).
% mult_of.inv_unique
thf(fact_786_mult__of_Oinv__comm,axiom,
! [X2: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.inv_comm
thf(fact_787_Units__r__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_r_inv_ex
thf(fact_788_Units__l__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_l_inv_ex
thf(fact_789_irreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_rI
thf(fact_790_irreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irredu6211895646901577903xt_a_b @ r @ B )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( irredu6211895646901577903xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% irreducible_prod_lI
thf(fact_791_ring__irreducibleE_I5_J,axiom,
! [R3: a,A: a,B: a] :
( ( member_a @ R3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R3 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_792_p_Odivides__zero,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.divides_zero
thf(fact_793_p_Odivides__prod__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( 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 ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ).
% p.divides_prod_r
thf(fact_794_p_Odivides__prod__l,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 ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ) ).
% p.divides_prod_l
thf(fact_795_p_Odivides__mult,axiom,
! [A: list_a,C: list_a,B: list_a] :
( ( member_list_a @ A @ ( 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 ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ).
% p.divides_mult
thf(fact_796_set__add__zero,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ A2 )
= A2 ) ) ).
% set_add_zero
thf(fact_797_p_Oone__divides,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A ) ) ).
% p.one_divides
thf(fact_798_p_Ounit__divides,axiom,
! [U: list_a,A: list_a] :
( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ A ) ) ) ).
% p.unit_divides
thf(fact_799_p_Odivides__unit,axiom,
! [A: list_a,U: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ U )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.divides_unit
thf(fact_800_cring__fieldI2,axiom,
( ( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) )
=> ( ! [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A5
!= ( zero_a_b @ r ) )
=> ? [X5: a] :
( ( member_a @ X5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( mult_a_ring_ext_a_b @ r @ A5 @ X5 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) )
=> ( field_a_b @ r ) ) ) ).
% cring_fieldI2
thf(fact_801_p_Oring__irreducibleE_I2_J,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ R3 ) ) ) ).
% p.ring_irreducibleE(2)
thf(fact_802_p_Ozero__is__irreducible__iff__field,axiom,
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.zero_is_irreducible_iff_field
thf(fact_803_p_Oisgcd__divides__r,axiom,
! [B: list_a,A: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ 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 ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A @ B ) ) ) ) ).
% p.isgcd_divides_r
thf(fact_804_p_Oisgcd__divides__l,axiom,
! [A: list_a,B: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ 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 ) ) ) )
=> ( isgcd_1118609098697428327t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A @ B ) ) ) ) ).
% p.isgcd_divides_l
thf(fact_805_p_Odivides__one,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.divides_one
thf(fact_806_p_OUnit__eq__dividesone,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.Unit_eq_dividesone
thf(fact_807_p_Oto__contain__is__to__divide,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A ) )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ).
% p.to_contain_is_to_divide
thf(fact_808_p_Oirreducible__prod__rI,axiom,
! [A: list_a,B: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ).
% p.irreducible_prod_rI
thf(fact_809_p_Oirreducible__prod__lI,axiom,
! [B: list_a,A: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B )
=> ( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ).
% p.irreducible_prod_lI
thf(fact_810_p_Oirreducible__imp__irreducible__mult,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A )
=> ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A ) ) ) ).
% p.irreducible_imp_irreducible_mult
thf(fact_811_f__comm__group__2,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X2
!= ( 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 @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% f_comm_group_2
thf(fact_812_p_Omult__divides,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 @ ( 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 ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% p.mult_divides
thf(fact_813_m__closed,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_814_Units__m__closed,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_815_r__null,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_816_l__null,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X2 )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_817_mult__of_Oright__cancel,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X2 )
= ( mult_a_ring_ext_a_b @ r @ Z2 @ X2 ) )
= ( Y2 = Z2 ) ) ) ) ) ).
% mult_of.right_cancel
thf(fact_818_mult__of_Om__closed,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.m_closed
thf(fact_819_r__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( one_a_ring_ext_a_b @ r ) )
= X2 ) ) ).
% r_one
thf(fact_820_l__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X2 )
= X2 ) ) ).
% l_one
thf(fact_821_Units__l__cancel,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ X2 @ Z2 ) )
= ( Y2 = Z2 ) ) ) ) ) ).
% Units_l_cancel
thf(fact_822_p_Odivides__refl,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A ) ) ).
% p.divides_refl
thf(fact_823_mult__of_Or__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( one_a_ring_ext_a_b @ r ) )
= X2 ) ) ).
% mult_of.r_one
thf(fact_824_mult__of_Or__cancel__one_H,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( X2
= ( mult_a_ring_ext_a_b @ r @ A @ X2 ) )
= ( A
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.r_cancel_one'
thf(fact_825_mult__of_Or__cancel__one,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ X2 )
= X2 )
= ( A
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.r_cancel_one
thf(fact_826_mult__of_Ol__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ X2 )
= X2 ) ) ).
% mult_of.l_one
thf(fact_827_mult__of_Ol__cancel__one_H,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( X2
= ( mult_a_ring_ext_a_b @ r @ X2 @ A ) )
= ( A
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.l_cancel_one'
thf(fact_828_mult__of_Ol__cancel__one,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ A )
= X2 )
= ( A
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.l_cancel_one
thf(fact_829_p_Odivides__mult__rI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ 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 ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ) ).
% p.divides_mult_rI
thf(fact_830_p_Odivides__mult__lI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( 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 ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ).
% p.divides_mult_lI
thf(fact_831_p_Odivides__imp__divides__mult,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 @ ( 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 ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ) ).
% p.divides_imp_divides_mult
thf(fact_832_mult__of_Omonoid__cancelI,axiom,
( ! [A5: a,B5: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A5 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B5 ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A5 = B5 ) ) ) ) )
=> ( ! [A5: a,B5: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A5 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B5 @ C2 ) )
=> ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( A5 = B5 ) ) ) ) )
=> ( monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.monoid_cancelI
thf(fact_833_mult__of_Oderived__eq__singleton,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ H )
= ( insert_a @ ( one_a_ring_ext_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% mult_of.derived_eq_singleton
thf(fact_834_p_Osubfield__m__inv__simprule,axiom,
! [K: set_list_a,K2: list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A ) @ K )
=> ( member_list_a @ A @ K ) ) ) ) ) ).
% p.subfield_m_inv_simprule
thf(fact_835_p_Ois__root__poly__mult__imp__is__root,axiom,
! [P: list_list_a,Q2: list_list_a,X2: list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ Q2 ) @ X2 )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X2 )
| ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ X2 ) ) ) ) ) ).
% p.is_root_poly_mult_imp_is_root
thf(fact_836_divides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ).
% divides_trans
thf(fact_837_zero__divides,axiom,
! [A: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A )
= ( A
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_838_Units__pow__closed,axiom,
! [X2: a,D: nat] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ D ) @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% Units_pow_closed
thf(fact_839_pow__non__zero,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X2
!= ( zero_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N )
!= ( zero_a_b @ r ) ) ) ) ).
% pow_non_zero
thf(fact_840_divides__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( zero_a_b @ r ) ) ) ).
% divides_zero
thf(fact_841_group__commutes__pow,axiom,
! [X2: a,Y2: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X2 ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_842_nat__pow__comm,axiom,
! [X2: a,N: nat,M2: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ M2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ M2 ) @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_843_nat__pow__distrib,axiom,
! [X2: a,Y2: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_844_pow__mult__distrib,axiom,
! [X2: a,Y2: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ Y2 @ X2 ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y2 @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_845_local_Odivides__mult,axiom,
! [A: a,C: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% local.divides_mult
thf(fact_846_divides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_847_divides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_848_monoid__cancelI,axiom,
( ! [A5: a,B5: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C2 @ A5 )
= ( mult_a_ring_ext_a_b @ r @ C2 @ B5 ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A5 = B5 ) ) ) ) )
=> ( ! [A5: a,B5: a,C2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A5 @ C2 )
= ( mult_a_ring_ext_a_b @ r @ B5 @ C2 ) )
=> ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A5 = B5 ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_849_mult__of_Odivides__unit,axiom,
! [A: a,U: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ U )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.divides_unit
thf(fact_850_mult__of_Ounit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ U @ A ) ) ) ).
% mult_of.unit_divides
thf(fact_851_mult__of_Ogcd__divides,axiom,
! [Z2: a,X2: a,Y2: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z2 @ X2 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z2 @ Y2 )
=> ( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Z2 @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X2 @ Y2 ) ) ) ) ) ) ) ).
% mult_of.gcd_divides
thf(fact_852_mult__of_Ogcd__divides__l,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ A ) ) ) ).
% mult_of.gcd_divides_l
thf(fact_853_mult__of_Ogcd__divides__r,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ B ) ) ) ).
% mult_of.gcd_divides_r
thf(fact_854_mult__of_OUnits__closed,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.Units_closed
thf(fact_855_mult__of_Odivides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C ) ) ) ) ).
% mult_of.divides_trans
thf(fact_856_mult__of_Ogcd__closed,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.gcd_closed
thf(fact_857_mult__of_Ogcd__exists,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.gcd_exists
thf(fact_858_one__divides,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ A ) ) ).
% one_divides
thf(fact_859_divides__unit,axiom,
! [A: a,U: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ U )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% divides_unit
thf(fact_860_unit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ U @ A ) ) ) ).
% unit_divides
thf(fact_861_isgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ A @ A @ B ) ) ) ) ).
% isgcd_divides_l
thf(fact_862_isgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor8216151070175719842xt_a_b @ r @ B @ A )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ B @ A @ B ) ) ) ) ).
% isgcd_divides_r
thf(fact_863_divides__mult__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( zero_a_b @ r ) )
=> ( A
= ( zero_a_b @ r ) ) ) ) ).
% divides_mult_zero
thf(fact_864_mult__of_OUnits,axiom,
ord_less_eq_set_a @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.Units
thf(fact_865_mult__of_Oprod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_l
thf(fact_866_mult__of_Oprod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.prod_unit_r
thf(fact_867_mult__of_Ounit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.unit_factor
thf(fact_868_mult__of_Odivides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% mult_of.divides_prod_l
thf(fact_869_mult__of_Odivides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% mult_of.divides_prod_r
thf(fact_870_mult__of_OUnit__eq__dividesone,axiom,
! [U: a] :
( ( member_a @ U @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ U @ ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Unit_eq_dividesone
thf(fact_871_mult__of_OUnits__inv__comm,axiom,
! [X2: a,Y2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Y2 @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ) ).
% mult_of.Units_inv_comm
thf(fact_872_Unit__eq__dividesone,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
= ( factor8216151070175719842xt_a_b @ r @ U @ ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% Unit_eq_dividesone
thf(fact_873_divides__one,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ ( one_a_ring_ext_a_b @ r ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% divides_one
thf(fact_874_mult__of_Omonoid__cancel__axioms,axiom,
monoid1999574367301118026t_unit @ ( ring_mult_of_a_b @ r ) ).
% mult_of.monoid_cancel_axioms
thf(fact_875_to__contain__is__to__divide,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ B ) @ ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( factor8216151070175719842xt_a_b @ r @ A @ B ) ) ) ) ).
% to_contain_is_to_divide
thf(fact_876_mult__of_Omono__derived,axiom,
! [K: set_a,H: set_a] :
( ( ord_less_eq_set_a @ K @ H )
=> ( ord_less_eq_set_a @ ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ K ) @ ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ H ) ) ) ).
% mult_of.mono_derived
thf(fact_877_mult__of_Oisgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A @ A @ B ) ) ) ) ).
% mult_of.isgcd_divides_l
thf(fact_878_mult__of_Oisgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ B @ A )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ B @ A @ B ) ) ) ) ).
% mult_of.isgcd_divides_r
thf(fact_879_mult__of_Ogcd__isgcd,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ A @ B ) ) ) ).
% mult_of.gcd_isgcd
thf(fact_880_p_Omult__of_OUnits__closed,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.mult_of.Units_closed
thf(fact_881_p_Osubring__props_I2_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).
% p.subring_props(2)
thf(fact_882_p_Ouniv__poly__is__principal,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ring_p715737262848045090t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).
% p.univ_poly_is_principal
thf(fact_883_p_Osubring__props_I6_J,axiom,
! [K: set_list_a,H1: list_a,H2: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H1 @ K )
=> ( ( member_list_a @ H2 @ K )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H2 ) @ K ) ) ) ) ).
% p.subring_props(6)
thf(fact_884_p_Osubring__props_I4_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( K != bot_bot_set_list_a ) ) ).
% p.subring_props(4)
thf(fact_885_p_Osubring__props_I3_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ K ) ) ).
% p.subring_props(3)
thf(fact_886_mult__of_OUnits__l__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X3 @ X2 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_l_inv_ex
thf(fact_887_mult__of_OUnits__r__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( ( mult_a_ring_ext_a_b @ r @ X2 @ X3 )
= ( one_a_ring_ext_a_b @ r ) ) ) ) ).
% mult_of.Units_r_inv_ex
thf(fact_888_mult__of_Oirreducible__prod__rI,axiom,
! [A: a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prod_rI
thf(fact_889_mult__of_Oirreducible__prod__lI,axiom,
! [B: a,A: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ B )
=> ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prod_lI
thf(fact_890_mult__of_Oirreducible__prodE,axiom,
! [A: a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ~ ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) )
=> ~ ( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ~ ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ) ).
% mult_of.irreducible_prodE
thf(fact_891_mult__of_Oderived__in__carrier,axiom,
! [H: set_a] :
( ( ord_less_eq_set_a @ H @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_set_a @ ( genera353947490595344117t_unit @ ( ring_mult_of_a_b @ r ) @ H ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ).
% mult_of.derived_in_carrier
thf(fact_892_mult__of_Oprime__divides,axiom,
! [A: a,B: a,P: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ A )
| ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ P @ B ) ) ) ) ) ) ).
% mult_of.prime_divides
thf(fact_893_p_Omult__of_Oprod__unit__l,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ B @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% p.mult_of.prod_unit_l
thf(fact_894_p_Omult__of_Oprod__unit__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% p.mult_of.prod_unit_r
thf(fact_895_p_Omult__of_Ounit__factor,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.mult_of.unit_factor
thf(fact_896_p_Osubring__props_I1_J,axiom,
! [K: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.subring_props(1)
thf(fact_897_p_Omult__of_Odivides__unit,axiom,
! [A: list_a,U: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ U )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ U @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.mult_of.divides_unit
thf(fact_898_p_Omult__of_Ounit__divides,axiom,
! [U: list_a,A: list_a] :
( ( member_list_a @ U @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ U @ A ) ) ) ).
% p.mult_of.unit_divides
thf(fact_899_p_Omult__of_Odivides__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ C )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ C ) ) ) ) ).
% p.mult_of.divides_trans
thf(fact_900_p_Omult__of_OUnits__inv__comm,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ X2 @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.mult_of.Units_inv_comm
thf(fact_901_p_Opprime__iff__pirreducible,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) ) ) ) ).
% p.pprime_iff_pirreducible
thf(fact_902_p_OpprimeE_I2_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ) ) ).
% p.pprimeE(2)
thf(fact_903_p_Odivides__mult__zero,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ ( 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 ) ) ) ) ) ) ).
% p.divides_mult_zero
thf(fact_904_p_Omult__of_OUnits__l__inv__ex,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.mult_of.Units_l_inv_ex
thf(fact_905_p_Omult__of_OUnits__r__inv__ex,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X3 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.mult_of.Units_r_inv_ex
thf(fact_906_p_Omult__of_Odivides__prod__l,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ) ).
% p.mult_of.divides_prod_l
thf(fact_907_p_Omult__of_Odivides__prod__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ).
% p.mult_of.divides_prod_r
thf(fact_908_p_Omult__of_OUnit__eq__dividesone,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ U @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
= ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ U @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.mult_of.Unit_eq_dividesone
thf(fact_909_p_Omult__of_Oirreducible__prodE,axiom,
! [A: list_a,B: list_a] :
( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A )
=> ~ ( member_list_a @ B @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ~ ( ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ~ ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B ) ) ) ) ) ) ).
% p.mult_of.irreducible_prodE
thf(fact_910_p_Omult__of_Oirreducible__prod__lI,axiom,
! [B: list_a,A: list_a] :
( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B )
=> ( ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ).
% p.mult_of.irreducible_prod_lI
thf(fact_911_p_Omult__of_Oirreducible__prod__rI,axiom,
! [A: list_a,B: list_a] :
( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A )
=> ( ( member_list_a @ B @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ).
% p.mult_of.irreducible_prod_rI
thf(fact_912_divides__imp__divides__mult,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% divides_imp_divides_mult
thf(fact_913_mult__divides,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ B ) ) ) ) ) ).
% mult_divides
thf(fact_914_p_Omult__of_Ogcd__divides,axiom,
! [Z2: list_a,X2: list_a,Y2: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Z2 @ X2 )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Z2 @ Y2 )
=> ( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Z2 @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 @ Y2 ) ) ) ) ) ) ) ).
% p.mult_of.gcd_divides
thf(fact_915_p_Omult__of_Ogcd__divides__l,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ A ) ) ) ).
% p.mult_of.gcd_divides_l
thf(fact_916_p_Omult__of_Ogcd__divides__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ B ) ) ) ).
% p.mult_of.gcd_divides_r
thf(fact_917_p_Omult__of_Oisgcd__divides__l,axiom,
! [A: list_a,B: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ A @ B ) ) ) ) ).
% p.mult_of.isgcd_divides_l
thf(fact_918_p_Omult__of_Oisgcd__divides__r,axiom,
! [B: list_a,A: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ A )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ A @ B ) ) ) ) ).
% p.mult_of.isgcd_divides_r
thf(fact_919_p_Omult__of_Oprime__divides,axiom,
! [A: list_a,B: list_a,P: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P @ A )
| ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P @ B ) ) ) ) ) ) ).
% p.mult_of.prime_divides
thf(fact_920_exists__irreducible__divisor,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ~ ! [B5: a] :
( ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ B5 )
=> ~ ( factor8216151070175719842xt_a_b @ r @ B5 @ A ) ) ) ) ) ).
% exists_irreducible_divisor
thf(fact_921_nat__pow__closed,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% nat_pow_closed
thf(fact_922_divides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ A ) ) ).
% divides_refl
thf(fact_923_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_924_divides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% divides_mult_lI
thf(fact_925_divides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_926_mult__of_OUnits__eq,axiom,
( ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) )
= ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ).
% mult_of.Units_eq
thf(fact_927_mult__of_Odivides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A ) ) ).
% mult_of.divides_refl
thf(fact_928_mult__of_OUnits__m__closed,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ).
% mult_of.Units_m_closed
thf(fact_929_mult__of_OUnits__one__closed,axiom,
member_a @ ( one_a_ring_ext_a_b @ r ) @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ).
% mult_of.Units_one_closed
thf(fact_930_Units__mult__eq__Units,axiom,
( ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) )
= ( units_a_ring_ext_a_b @ r ) ) ).
% Units_mult_eq_Units
thf(fact_931_mult__of_OUnits__l__cancel,axiom,
! [X2: a,Y2: a,Z2: a] :
( ( member_a @ X2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Z2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 )
= ( mult_a_ring_ext_a_b @ r @ X2 @ Z2 ) )
= ( Y2 = Z2 ) ) ) ) ) ).
% mult_of.Units_l_cancel
thf(fact_932_mult__of_Odivides__mult__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ) ).
% mult_of.divides_mult_l
thf(fact_933_mult__of_Odivides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% mult_of.divides_mult_lI
thf(fact_934_mult__of_Odivides__mult__r,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) )
= ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ) ).
% mult_of.divides_mult_r
thf(fact_935_mult__of_Odivides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% mult_of.divides_mult_rI
thf(fact_936_p_Omult__of_OUnits__m__closed,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_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 ) ) @ X2 @ Y2 ) @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.mult_of.Units_m_closed
thf(fact_937_p_Omult__of_OUnits__one__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.mult_of.Units_one_closed
thf(fact_938_p_OUnits__mult__eq__Units,axiom,
( ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.Units_mult_eq_Units
thf(fact_939_p_Omult__of_OUnits__l__cancel,axiom,
! [X2: list_a,Y2: list_a,Z2: list_a] :
( ( member_list_a @ X2 @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Z2 ) )
= ( Y2 = Z2 ) ) ) ) ) ).
% p.mult_of.Units_l_cancel
thf(fact_940_p_Omult__of_Odivides__refl,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ A ) ) ).
% p.mult_of.divides_refl
thf(fact_941_p_Omult__of_Odivides__mult__l,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
= ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ) ) ).
% p.mult_of.divides_mult_l
thf(fact_942_p_Omult__of_Odivides__mult__lI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ).
% p.mult_of.divides_mult_lI
thf(fact_943_p_Omult__of_Odivides__mult__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
= ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ) ) ).
% p.mult_of.divides_mult_r
thf(fact_944_p_Omult__of_Odivides__mult__rI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) ) ) ) ) ).
% p.mult_of.divides_mult_rI
thf(fact_945_p_Omult__of_Odivides__fcount,axiom,
! [A: list_a,B: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ord_less_eq_nat @ ( factor8927536889056116518t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A ) @ ( factor8927536889056116518t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B ) ) ) ) ) ).
% p.mult_of.divides_fcount
thf(fact_946_mult__of_Odivides__fcount,axiom,
! [A: a,B: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ord_less_eq_nat @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A ) @ ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ).
% mult_of.divides_fcount
thf(fact_947_p_Olong__division__closed_I1_J,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ) ) ).
% p.long_division_closed(1)
thf(fact_948_p_Orupture__is__field__iff__pirreducible,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( field_1540243473349940225t_unit @ ( polyno859807163042199155t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ P ) )
= ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) ) ) ) ).
% p.rupture_is_field_iff_pirreducible
thf(fact_949_p_Oline__extension__smult__closed,axiom,
! [K: set_list_a,E: set_list_a,A: list_a,K2: list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [K3: list_a,V2: list_a] :
( ( member_list_a @ K3 @ K )
=> ( ( member_list_a @ V2 @ E )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K3 @ V2 ) @ E ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ U ) @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) ) ) ) ) ) ) ) ).
% p.line_extension_smult_closed
thf(fact_950_p_OpprimeE_I1_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ).
% p.pprimeE(1)
thf(fact_951_polynomial__ring__assms,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% polynomial_ring_assms
thf(fact_952_subring__props_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K ) ) ).
% subring_props(2)
thf(fact_953_subring__props_I6_J,axiom,
! [K: set_a,H1: a,H2: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H2 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H2 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_954_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_955_subring__props_I3_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K ) ) ).
% subring_props(3)
thf(fact_956_subring__props_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_957_univ__poly__is__principal,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_principal
thf(fact_958_pprimeE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_959_pprime__iff__pirreducible,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_960_p_Oline__extension__in__carrier,axiom,
! [K: set_list_a,A: list_a,E: set_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ E @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.line_extension_in_carrier
thf(fact_961_p_Opolynomial__pow__not__zero,axiom,
! [P: list_list_a,N: nat] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ N )
!= nil_list_a ) ) ) ).
% p.polynomial_pow_not_zero
thf(fact_962_subfield__m__inv__simprule,axiom,
! [K: set_a,K2: a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K2 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K2 @ A ) @ K )
=> ( member_a @ A @ K ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_963_p_Olong__division__zero_I1_J,axiom,
! [K: set_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Q2 )
= nil_list_a ) ) ) ).
% p.long_division_zero(1)
thf(fact_964_p_Oexists__unique__long__division,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( Q2 != nil_list_a )
=> ? [X3: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 @ X3 )
& ! [Y5: produc7709606177366032167list_a] :
( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 @ Y5 )
=> ( Y5 = X3 ) ) ) ) ) ) ) ).
% p.exists_unique_long_division
thf(fact_965_p_OpprimeI,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ! [Q3: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q3 @ R4 ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q3 )
| ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ R4 ) ) ) ) )
=> ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) ) ) ) ) ) ).
% p.pprimeI
thf(fact_966_long__division__zero_I1_J,axiom,
! [K: set_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q2 )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_967_long__division__closed_I1_J,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_968_polynomial__pow__not__zero,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N )
!= nil_a ) ) ) ).
% polynomial_pow_not_zero
thf(fact_969_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_970_p_Ozero__pdivides,axiom,
! [P: list_list_a] :
( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= ( P = nil_list_a ) ) ).
% p.zero_pdivides
thf(fact_971_pprimeE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_972_p_Opdivides__iff__shell,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 )
= ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q2 ) ) ) ) ) ).
% p.pdivides_iff_shell
thf(fact_973_p_Opolynomial__pow__division,axiom,
! [P: list_list_a,N: nat,M2: nat] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ M2 ) ) ) ) ).
% p.polynomial_pow_division
thf(fact_974_p_OpprimeE_I3_J,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a,R3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r5437400583859147359t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q2 @ R3 ) )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 )
| ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ R3 ) ) ) ) ) ) ) ) ).
% p.pprimeE(3)
thf(fact_975_p_Opirreducible__pow__pdivides__iff,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a,R3: list_list_a,N: nat] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( ~ ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ N ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q2 @ R3 ) )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ).
% p.pirreducible_pow_pdivides_iff
thf(fact_976_exists__unique__long__division,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ? [X3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q2 @ X3 )
& ! [Y5: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P @ Q2 @ Y5 )
=> ( Y5 = X3 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_977_p_Opmod__zero__iff__pdivides,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 )
= nil_list_a )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ P ) ) ) ) ) ).
% p.pmod_zero_iff_pdivides
thf(fact_978_p_OpirreducibleI,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ! [Q3: list_list_a,R4: list_list_a] :
( ( member_list_list_a @ Q3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q3 @ R4 ) )
=> ( ( member_list_list_a @ Q3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
| ( member_list_list_a @ R4 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ) ) )
=> ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) ) ) ) ) ) ).
% p.pirreducibleI
thf(fact_979_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_980_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_981_pdivides__zero,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ).
% pdivides_zero
thf(fact_982_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_983_p_Ouniv__poly__not__field,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ~ ( field_1861437471013600865t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).
% p.univ_poly_not_field
thf(fact_984_p_Ouniv__poly__is__domain,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ).
% p.univ_poly_is_domain
thf(fact_985_pdivides__iff__shell,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 ) ) ) ) ) ).
% pdivides_iff_shell
thf(fact_986_pdivides__mult__r,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
= ( polyno5814909790663948098es_a_b @ r @ A @ B ) ) ) ) ) ).
% pdivides_mult_r
thf(fact_987_polynomial__pow__division,axiom,
! [P: list_a,N: nat,M2: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ M2 ) ) ) ) ).
% polynomial_pow_division
thf(fact_988_p_Ovar__carr,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% p.var_carr
thf(fact_989_p_Ovar__closed_I1_J,axiom,
! [K: set_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ).
% p.var_closed(1)
thf(fact_990_pprimeE_I3_J,axiom,
! [K: set_a,P: list_a,Q2: list_a,R3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R3 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
| ( polyno5814909790663948098es_a_b @ r @ P @ R3 ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_991_p_Olong__division__closed_I2_J,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_list_a @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ) ) ).
% p.long_division_closed(2)
thf(fact_992_p_Ovar__pow__carr,axiom,
! [K: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) @ ( minus_5335179877275218001list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) @ bot_bo1875519244922727510list_a ) ) ) ) ).
% p.var_pow_carr
thf(fact_993_p_Ovar__pow__closed,axiom,
! [K: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( member_list_list_a @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ).
% p.var_pow_closed
thf(fact_994_pirreducible__pow__pdivides__iff,axiom,
! [K: set_a,P: list_a,Q2: list_a,R3: list_a,N: nat] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ~ ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R3 ) )
= ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ R3 ) ) ) ) ) ) ) ) ).
% pirreducible_pow_pdivides_iff
thf(fact_995_p_Opdivides__zero,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a ) ) ) ).
% p.pdivides_zero
thf(fact_996_p_Olong__division__zero_I2_J,axiom,
! [K: set_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Q2 )
= nil_list_a ) ) ) ).
% p.long_division_zero(2)
thf(fact_997_p_Osubring__polynomial__pow__not__zero,axiom,
! [K: set_list_a,P: list_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( P != nil_list_a )
=> ( ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ N )
!= nil_list_a ) ) ) ) ).
% p.subring_polynomial_pow_not_zero
thf(fact_998_p_OpirreducibleE_I1_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ).
% p.pirreducibleE(1)
thf(fact_999_pprimeI,axiom,
! [K: set_a,P: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R4 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q3 )
| ( polyno5814909790663948098es_a_b @ r @ P @ R4 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).
% pprimeI
thf(fact_1000_p_OpirreducibleE_I2_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ~ ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ) ) ).
% p.pirreducibleE(2)
thf(fact_1001_p_OpirreducibleE_I3_J,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a,R3: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q2 @ R3 ) )
=> ( ( member_list_list_a @ Q2 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
| ( member_list_list_a @ R3 @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) ) ) ) ) ) ) ) ) ).
% p.pirreducibleE(3)
thf(fact_1002_p_Osubring__polynomial__pow__division,axiom,
! [K: set_list_a,P: list_list_a,N: nat,M2: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ N ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ M2 ) ) ) ) ) ).
% p.subring_polynomial_pow_division
thf(fact_1003_p_Ocarrier__polynomial__shell,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.carrier_polynomial_shell
thf(fact_1004_pdivides__imp__splitted,axiom,
! [P: list_a,Q2: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q2 )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q2 )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_1005_p_Osame__pmod__iff__pdivides,axiom,
! [K: set_list_a,A: list_list_a,B: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ Q2 )
= ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ Q2 ) )
= ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ A @ B ) ) ) ) ) ) ) ).
% p.same_pmod_iff_pdivides
thf(fact_1006_p_Ouniv__poly__a__minus__consistent,axiom,
! [K: set_list_a,Q2: list_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q2 )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P @ Q2 ) ) ) ) ).
% p.univ_poly_a_minus_consistent
thf(fact_1007_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_1008_univ__poly__not__field,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_not_field
thf(fact_1009_univ__poly__is__domain,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_domain
thf(fact_1010_long__division__closed_I2_J,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P @ Q2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_1011_long__division__zero_I2_J,axiom,
! [K: set_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q2 )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_1012_subring__polynomial__pow__not__zero,axiom,
! [K: set_a,P: list_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N )
!= nil_a ) ) ) ) ).
% subring_polynomial_pow_not_zero
thf(fact_1013_pirreducibleE_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( P != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_1014_pirreducibleE_I2_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_1015_subring__polynomial__pow__division,axiom,
! [K: set_a,P: list_a,N: nat,M2: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ M2 ) ) ) ) ) ).
% subring_polynomial_pow_division
thf(fact_1016_pmod__zero__iff__pdivides,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P @ Q2 )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q2 @ P ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_1017_pirreducibleE_I3_J,axiom,
! [K: set_a,P: list_a,Q2: list_a,R3: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 @ R3 ) )
=> ( ( member_list_a @ Q2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ) ) ) ).
% pirreducibleE(3)
thf(fact_1018_pirreducibleI,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P != nil_a )
=> ( ~ ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R4: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R4 ) )
=> ( ( member_list_a @ Q3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_1019_carrier__polynomial__shell,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_1020_p_Ounitary__monom__eq__var__pow,axiom,
! [K: set_list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N )
= ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) ) ) ).
% p.unitary_monom_eq_var_pow
thf(fact_1021_p_Olong__divisionI,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a,B: list_list_a,R3: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( Q2 != nil_list_a )
=> ( ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 @ ( produc8696003437204565271list_a @ B @ R3 ) )
=> ( ( produc8696003437204565271list_a @ B @ R3 )
= ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) ) ) ) ) ) ) ) ).
% p.long_divisionI
thf(fact_1022_p_Olong__divisionE,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( Q2 != nil_list_a )
=> ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 @ ( produc8696003437204565271list_a @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) ) ) ) ) ) ) ).
% p.long_divisionE
thf(fact_1023_p_Opoly__add_Ocases,axiom,
! [X2: produc7709606177366032167list_a] :
~ ! [P1: list_list_a,P22: list_list_a] :
( X2
!= ( produc8696003437204565271list_a @ P1 @ P22 ) ) ).
% p.poly_add.cases
thf(fact_1024_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q2: list_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_1025_same__pmod__iff__pdivides,axiom,
! [K: set_a,A: list_a,B: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q2 )
= ( polynomial_pmod_a_b @ r @ B @ Q2 ) )
= ( polyno5814909790663948098es_a_b @ r @ Q2 @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_1026_p_Oexists__long__division,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( Q2 != nil_list_a )
=> ~ ! [B5: list_list_a] :
( ( member_list_list_a @ B5 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ! [R4: list_list_a] :
( ( member_list_list_a @ R4 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ~ ( polyno6947042923167803568t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 @ ( produc8696003437204565271list_a @ B5 @ R4 ) ) ) ) ) ) ) ) ).
% p.exists_long_division
thf(fact_1027_p_Ominus__closed,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.minus_closed
thf(fact_1028_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_1029_p_Omonom__eq__var__pow,axiom,
! [K: set_list_a,A: list_a,N: nat] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ A @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N )
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( cons_list_a @ A @ nil_list_a ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N ) ) ) ) ) ).
% p.monom_eq_var_pow
thf(fact_1030_unitary__monom__eq__var__pow,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( monom_a_b @ r @ ( one_a_ring_ext_a_b @ r ) @ N )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ).
% unitary_monom_eq_var_pow
thf(fact_1031_p_Oconst__term__simprules__shell_I2_J,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q2 ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 ) ) ) ) ) ) ).
% p.const_term_simprules_shell(2)
thf(fact_1032_p_Onormalize_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ~ ! [V2: list_a,Va: list_list_a] :
( X2
!= ( cons_list_a @ V2 @ Va ) ) ) ).
% p.normalize.cases
thf(fact_1033_p_Opoly__mult_Ocases,axiom,
! [X2: produc7709606177366032167list_a] :
( ! [P22: list_list_a] :
( X2
!= ( produc8696003437204565271list_a @ nil_list_a @ P22 ) )
=> ~ ! [V2: list_a,Va: list_list_a,P22: list_list_a] :
( X2
!= ( produc8696003437204565271list_a @ ( cons_list_a @ V2 @ Va ) @ P22 ) ) ) ).
% p.poly_mult.cases
thf(fact_1034_p_Ocombine_Ocases,axiom,
! [X2: produc7709606177366032167list_a] :
( ! [K3: list_a,Ks: list_list_a,U2: list_a,Us: list_list_a] :
( X2
!= ( produc8696003437204565271list_a @ ( cons_list_a @ K3 @ Ks ) @ ( cons_list_a @ U2 @ Us ) ) )
=> ( ! [Us: list_list_a] :
( X2
!= ( produc8696003437204565271list_a @ nil_list_a @ Us ) )
=> ~ ! [Ks: list_list_a] :
( X2
!= ( produc8696003437204565271list_a @ Ks @ nil_list_a ) ) ) ) ).
% p.combine.cases
thf(fact_1035_p_Oconst__term__not__zero,axiom,
! [P: list_list_a] :
( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P != nil_list_a ) ) ).
% p.const_term_not_zero
thf(fact_1036_exists__long__division,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ~ ! [B5: list_a] :
( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ! [R4: list_a] :
( ( member_list_a @ R4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P @ Q2 @ ( produc6837034575241423639list_a @ B5 @ R4 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_1037_p_Oconst__term__simprules__shell_I1_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ K ) ) ) ).
% p.const_term_simprules_shell(1)
thf(fact_1038_long__divisionE,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P @ Q2 @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P @ Q2 ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_1039_long__divisionI,axiom,
! [K: set_a,P: list_a,Q2: list_a,B: list_a,R3: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q2 != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P @ Q2 @ ( produc6837034575241423639list_a @ B @ R3 ) )
=> ( ( produc6837034575241423639list_a @ B @ R3 )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P @ Q2 ) @ ( polynomial_pmod_a_b @ r @ P @ Q2 ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_1040_monom__eq__var__pow,axiom,
! [K: set_a,A: a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( monom_a_b @ r @ A @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A @ nil_a ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) ) ) ) ) ).
% monom_eq_var_pow
thf(fact_1041_p_Oassociated__polynomials__iff,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q2 )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
& ( P
= ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( cons_list_a @ X @ nil_list_a ) @ Q2 ) ) ) ) ) ) ) ) ).
% p.associated_polynomials_iff
thf(fact_1042_p_Opoly__mult__var,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( P = nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= nil_list_a ) )
& ( ( P != nil_list_a )
=> ( ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( append_list_a @ P @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ) ).
% p.poly_mult_var
thf(fact_1043_poly__add_Ocases,axiom,
! [X2: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X2
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_1044_normalize_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ~ ! [V2: a,Va: list_a] :
( X2
!= ( cons_a @ V2 @ Va ) ) ) ).
% normalize.cases
thf(fact_1045_poly__mult_Ocases,axiom,
! [X2: produc9164743771328383783list_a] :
( ! [P22: list_a] :
( X2
!= ( produc6837034575241423639list_a @ nil_a @ P22 ) )
=> ~ ! [V2: a,Va: list_a,P22: list_a] :
( X2
!= ( produc6837034575241423639list_a @ ( cons_a @ V2 @ Va ) @ P22 ) ) ) ).
% poly_mult.cases
thf(fact_1046_combine_Ocases,axiom,
! [X2: produc9164743771328383783list_a] :
( ! [K3: a,Ks: list_a,U2: a,Us: list_a] :
( X2
!= ( produc6837034575241423639list_a @ ( cons_a @ K3 @ Ks ) @ ( cons_a @ U2 @ Us ) ) )
=> ( ! [Us: list_a] :
( X2
!= ( produc6837034575241423639list_a @ nil_a @ Us ) )
=> ~ ! [Ks: list_a] :
( X2
!= ( produc6837034575241423639list_a @ Ks @ nil_a ) ) ) ) ).
% combine.cases
thf(fact_1047_const__term__not__zero,axiom,
! [P: list_a] :
( ( ( const_term_a_b @ r @ P )
!= ( zero_a_b @ r ) )
=> ( P != nil_a ) ) ).
% const_term_not_zero
thf(fact_1048_const__term__simprules__shell_I1_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ K ) ) ) ).
% const_term_simprules_shell(1)
thf(fact_1049_const__term__simprules__shell_I2_J,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q2 ) ) ) ) ) ) ).
% const_term_simprules_shell(2)
thf(fact_1050_p_Oassociated__polynomials__imp__same__is__root,axiom,
! [P: list_list_a,Q2: list_list_a,X2: list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( 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 ) ) ) ) ) )
=> ( ( associ5603075271488036121t_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 @ Q2 )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X2 )
= ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q2 @ X2 ) ) ) ) ) ).
% p.associated_polynomials_imp_same_is_root
thf(fact_1051_p_Ocgenideal__pirreducible,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q2 )
=> ( ( member_list_list_a @ Q2 @ ( cgenid24865672677839267t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) )
=> ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q2 ) ) ) ) ) ) ).
% p.cgenideal_pirreducible
thf(fact_1052_p_Osubring__degree__one__associatedI,axiom,
! [K: set_list_a,A: list_a,A7: list_a,B: list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ A @ K )
=> ( ( member_list_a @ A7 @ K )
=> ( ( member_list_a @ B @ K )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A7 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A7 @ B ) @ nil_list_a ) ) ) ) ) ) ) ) ).
% p.subring_degree_one_associatedI
thf(fact_1053_p_Odivides__pirreducible__condition,axiom,
! [K: set_list_a,Q2: list_list_a,P: list_list_a] :
( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ Q2 )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( factor6954119973539764400t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q2 )
=> ( ( member_list_list_a @ P @ ( units_4903515905731149798t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
| ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q2 ) ) ) ) ) ).
% p.divides_pirreducible_condition
thf(fact_1054_p_Ois__root__imp__pdivides,axiom,
! [P: list_list_a,X2: list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X2 )
=> ( 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 ) ) @ X2 ) @ nil_list_a ) ) @ P ) ) ) ).
% p.is_root_imp_pdivides
thf(fact_1055_p_Ofactors__mult__single,axiom,
! [A: list_a,Fb: list_list_a,B: list_a] :
( ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fb @ B )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ A @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ).
% p.factors_mult_single
thf(fact_1056_associated__polynomials__iff,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 )
= ( ? [X: a] :
( ( member_a @ X @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
& ( P
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ X @ nil_a ) @ Q2 ) ) ) ) ) ) ) ) ).
% associated_polynomials_iff
thf(fact_1057_associated__sym,axiom,
! [A: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( associ5860276527279195403xt_a_b @ r @ B @ A ) ) ).
% associated_sym
thf(fact_1058_associated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( associ5860276527279195403xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ C ) ) ) ) ) ).
% associated_trans
thf(fact_1059_assoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ! [A5: a,B5: a] :
( ( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ B5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ A5 @ B5 ) )
=> ( ( member_a @ ( F @ A5 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ ( F @ B5 ) @ ( partia707051561876973205xt_a_b @ r ) )
& ( associ5860276527279195403xt_a_b @ r @ ( F @ A5 ) @ ( F @ B5 ) ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% assoc_subst
thf(fact_1060_mult__of_Oassociated__sym,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ A ) ) ).
% mult_of.associated_sym
thf(fact_1061_Units__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ).
% Units_assoc
thf(fact_1062_assoc__iff__assoc__mult,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% assoc_iff_assoc_mult
thf(fact_1063_mult__cong__r,axiom,
! [B: a,B7: a,A: a] :
( ( associ5860276527279195403xt_a_b @ r @ B @ B7 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B7 ) ) ) ) ) ) ).
% mult_cong_r
thf(fact_1064_mult__cong__l,axiom,
! [A: a,A7: a,B: a] :
( ( associ5860276527279195403xt_a_b @ r @ A @ A7 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A7 @ B ) ) ) ) ) ) ).
% mult_cong_l
thf(fact_1065_mult__of_Oassociated__trans,axiom,
! [A: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C ) ) ) ) ) ).
% mult_of.associated_trans
thf(fact_1066_mult__of_Oassoc__subst,axiom,
! [A: a,B: a,F: a > a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ! [A5: a,B5: a] :
( ( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ B5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A5 @ B5 ) )
=> ( ( member_a @ ( F @ A5 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( member_a @ ( F @ B5 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F @ A5 ) @ ( F @ B5 ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% mult_of.assoc_subst
thf(fact_1067_Units__cong,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% Units_cong
thf(fact_1068_divides__cong__r,axiom,
! [X2: a,Y2: a,Y4: a] :
( ( factor8216151070175719842xt_a_b @ r @ X2 @ Y2 )
=> ( ( associ5860276527279195403xt_a_b @ r @ Y2 @ Y4 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X2 @ Y4 ) ) ) ) ).
% divides_cong_r
thf(fact_1069_divides__cong__l,axiom,
! [X2: a,X6: a,Y2: a] :
( ( associ5860276527279195403xt_a_b @ r @ X2 @ X6 )
=> ( ( factor8216151070175719842xt_a_b @ r @ X6 @ Y2 )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ X2 @ Y2 ) ) ) ) ).
% divides_cong_l
thf(fact_1070_mult__of_OUnits__assoc,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ).
% mult_of.Units_assoc
thf(fact_1071_associated__iff__same__ideal,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ( cgenid547466209912283029xt_a_b @ r @ A )
= ( cgenid547466209912283029xt_a_b @ r @ B ) ) ) ) ) ).
% associated_iff_same_ideal
thf(fact_1072_p_Omult__of_Oassociated__sym,axiom,
! [A: list_a,B: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ A ) ) ).
% p.mult_of.associated_sym
thf(fact_1073_mult__of_Omult__cong__r,axiom,
! [B: a,B7: a,A: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ B7 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B7 ) ) ) ) ) ) ).
% mult_of.mult_cong_r
thf(fact_1074_mult__of_Omult__cong__l,axiom,
! [A: a,A7: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A7 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A7 @ B ) ) ) ) ) ) ).
% mult_of.mult_cong_l
thf(fact_1075_mult__of_Oassoc__r__cancel,axiom,
! [A: a,B: a,A7: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A7 @ B ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A7 ) ) ) ) ) ).
% mult_of.assoc_r_cancel
thf(fact_1076_mult__of_Oassoc__l__cancel,axiom,
! [A: a,B: a,B7: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( mult_a_ring_ext_a_b @ r @ A @ B7 ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ B @ B7 ) ) ) ) ) ).
% mult_of.assoc_l_cancel
thf(fact_1077_p_Oassociated__sym,axiom,
! [A: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ A ) ) ).
% p.associated_sym
thf(fact_1078_ring__associated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ B )
= ( ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ X @ B ) ) ) ) ) ) ) ).
% ring_associated_iff
thf(fact_1079_associatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2'
thf(fact_1080_associatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ B ) ) ) ) ).
% associatedI2
thf(fact_1081_mult__of_Odivides__cong__r,axiom,
! [X2: a,Y2: a,Y4: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X2 @ Y2 )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y2 @ Y4 )
=> ( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X2 @ Y4 ) ) ) ) ).
% mult_of.divides_cong_r
thf(fact_1082_mult__of_Odivides__cong__l,axiom,
! [X2: a,X6: a,Y2: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X2 @ X6 )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X6 @ Y2 )
=> ( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ X2 @ Y2 ) ) ) ) ).
% mult_of.divides_cong_l
thf(fact_1083_mult__of_Oassoc__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_r
thf(fact_1084_mult__of_Oassoc__unit__l,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ B @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.assoc_unit_l
thf(fact_1085_mult__of_Oirreducible__cong,axiom,
! [A: a,A7: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A7 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A7 ) ) ) ) ) ).
% mult_of.irreducible_cong
thf(fact_1086_mult__of_Ogcd__cong__r,axiom,
! [Y2: a,Y4: a,X2: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ Y2 @ Y4 )
=> ( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y4 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X2 @ Y2 ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X2 @ Y4 ) ) ) ) ) ) ).
% mult_of.gcd_cong_r
thf(fact_1087_mult__of_Ogcd__cong__l,axiom,
! [X2: a,X6: a,Y2: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ X2 @ X6 )
=> ( ( member_a @ X2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ X6 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ Y2 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X2 @ Y2 ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ X6 @ Y2 ) ) ) ) ) ) ).
% mult_of.gcd_cong_l
thf(fact_1088_mult__of_Ogcd__assoc,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ C ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C ) ) ) ) ) ) ).
% mult_of.gcd_assoc
thf(fact_1089_mult__of_Oprime__cong,axiom,
! [P: a,P3: a] :
( ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ P @ P3 )
=> ( ( member_a @ P @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ P3 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( prime_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ P3 ) ) ) ) ) ).
% mult_of.prime_cong
thf(fact_1090_mult__of_Oassociated__fcount,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A )
= ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ B ) ) ) ) ) ).
% mult_of.associated_fcount
thf(fact_1091_mult__of_Ogcdof__cong__l,axiom,
! [A7: a,A: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A7 @ A )
=> ( ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A @ B @ C )
=> ( ( member_a @ A7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A7 @ B @ C ) ) ) ) ) ) ) ).
% mult_of.gcdof_cong_l
thf(fact_1092_p_Omult__of_Oassociated__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ C )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ C ) ) ) ) ) ).
% p.mult_of.associated_trans
thf(fact_1093_p_Omult__of_Oassoc__subst,axiom,
! [A: list_a,B: list_a,F: list_a > list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ! [A5: list_a,B5: list_a] :
( ( ( member_list_a @ A5 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( member_list_a @ B5 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A5 @ B5 ) )
=> ( ( member_list_a @ ( F @ A5 ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( member_list_a @ ( F @ B5 ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( F @ A5 ) @ ( F @ B5 ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% p.mult_of.assoc_subst
thf(fact_1094_p_Omult__of_OUnits__assoc,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ).
% p.mult_of.Units_assoc
thf(fact_1095_p_Oassociated__trans,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( associ8407585678920448409t_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 @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C ) ) ) ) ) ).
% p.associated_trans
thf(fact_1096_p_Oassoc__subst,axiom,
! [A: list_a,B: list_a,F: list_a > list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ! [A5: list_a,B5: list_a] :
( ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A5 @ B5 ) )
=> ( ( member_list_a @ ( F @ A5 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( member_list_a @ ( F @ B5 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ A5 ) @ ( F @ B5 ) ) ) )
=> ( ( 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 ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( F @ A ) @ ( F @ B ) ) ) ) ) ) ).
% p.assoc_subst
thf(fact_1097_p_Osubring__props_I5_J,axiom,
! [K: set_list_a,H3: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_a @ H3 @ K )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H3 ) @ K ) ) ) ).
% p.subring_props(5)
thf(fact_1098_p_OUnits__assoc,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ).
% p.Units_assoc
thf(fact_1099_mult__of_OassociatedD2,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ B @ X3 ) ) ) ) ) ) ).
% mult_of.associatedD2
thf(fact_1100_mult__of_OassociatedE2,axiom,
! [A: a,B: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ! [U2: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U2 ) )
=> ~ ( member_a @ U2 @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ~ ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) ) ) ) ) ).
% mult_of.associatedE2
thf(fact_1101_mult__of_OassociatedI2,axiom,
! [U: a,A: a,B: a] :
( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% mult_of.associatedI2
thf(fact_1102_mult__of_OassociatedI2_H,axiom,
! [A: a,B: a,U: a] :
( ( A
= ( mult_a_ring_ext_a_b @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) ) ) ).
% mult_of.associatedI2'
thf(fact_1103_mult__of_Oassociated__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
= ( ? [X: a] :
( ( member_a @ X @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( A
= ( mult_a_ring_ext_a_b @ r @ B @ X ) ) ) ) ) ) ) ).
% mult_of.associated_iff
thf(fact_1104_monic__poly__not__assoc,axiom,
! [F: list_a,G: list_a] :
( ( monic_3145109188698636716ly_a_b @ r @ F )
=> ( ( monic_3145109188698636716ly_a_b @ r @ G )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ F @ G )
=> ( F = G ) ) ) ) ).
% monic_poly_not_assoc
thf(fact_1105_mult__of_Ogcd__mult,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% mult_of.gcd_mult
thf(fact_1106_mult__of_OgcdI,axiom,
! [A: a,B: a,C: a] :
( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C )
=> ( ! [Y: a] :
( ( member_a @ Y @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ B )
=> ( ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ C )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ Y @ A ) ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C ) ) ) ) ) ) ) ) ).
% mult_of.gcdI
thf(fact_1107_mult__of_OgcdI2,axiom,
! [A: a,B: a,C: a] :
( ( isgcd_a_Product_unit @ ( ring_mult_of_a_b @ r ) @ A @ B @ C )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ B @ C ) ) ) ) ) ) ).
% mult_of.gcdI2
thf(fact_1108_p_Omult__of_Omult__cong__r,axiom,
! [B: list_a,B7: list_a,A: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ B7 )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B7 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_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 @ B7 ) ) ) ) ) ) ).
% p.mult_of.mult_cong_r
thf(fact_1109_p_Omult__of_Omult__cong__l,axiom,
! [A: list_a,A7: list_a,B: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ A7 )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A7 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_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 ) ) @ A7 @ B ) ) ) ) ) ) ).
% p.mult_of.mult_cong_l
thf(fact_1110_p_Omult__of_Oassoc__r__cancel,axiom,
! [A: list_a,B: list_a,A7: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_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 ) ) @ A7 @ B ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A7 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ A7 ) ) ) ) ) ).
% p.mult_of.assoc_r_cancel
thf(fact_1111_p_Omult__of_Oassoc__l__cancel,axiom,
! [A: list_a,B: list_a,B7: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B7 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_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 @ B7 ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ B7 ) ) ) ) ) ).
% p.mult_of.assoc_l_cancel
thf(fact_1112_p_Omult__of_Odivides__cong__r,axiom,
! [X2: list_a,Y2: list_a,Y4: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 @ Y2 )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Y2 @ Y4 )
=> ( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 @ Y4 ) ) ) ) ).
% p.mult_of.divides_cong_r
thf(fact_1113_p_Omult__of_Odivides__cong__l,axiom,
! [X2: list_a,X6: list_a,Y2: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 @ X6 )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X6 @ Y2 )
=> ( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 @ Y2 ) ) ) ) ).
% p.mult_of.divides_cong_l
thf(fact_1114_p_Omult__of_Oassoc__unit__r,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ B @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.mult_of.assoc_unit_r
thf(fact_1115_p_Omult__of_Oassoc__unit__l,axiom,
! [A: list_a,B: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ B @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.mult_of.assoc_unit_l
thf(fact_1116_p_Oassoc__iff__assoc__mult,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 ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ) ).
% p.assoc_iff_assoc_mult
thf(fact_1117_p_Or__minus,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 ) )
= ( 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 ) ) @ X2 @ Y2 ) ) ) ) ) ).
% p.r_minus
thf(fact_1118_p_Ol__minus,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( 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 ) ) @ X2 ) @ Y2 )
= ( 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 ) ) @ X2 @ Y2 ) ) ) ) ) ).
% p.l_minus
thf(fact_1119_p_Omult__cong__r,axiom,
! [B: list_a,B7: list_a,A: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ B7 )
=> ( ( 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 @ B7 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_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 @ B7 ) ) ) ) ) ) ).
% p.mult_cong_r
thf(fact_1120_p_Omult__cong__l,axiom,
! [A: list_a,A7: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A7 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A7 @ ( 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 ) ) ) )
=> ( associ8407585678920448409t_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 ) ) @ A7 @ B ) ) ) ) ) ) ).
% p.mult_cong_l
thf(fact_1121_p_Omult__of_Oirreducible__cong,axiom,
! [A: list_a,A7: list_a] :
( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ A7 )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A7 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A7 ) ) ) ) ) ).
% p.mult_of.irreducible_cong
thf(fact_1122_univ__poly__a__inv__consistent,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_1123_p_Odiv__neg,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 ) ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) ) ) ) ) ).
% p.div_neg
thf(fact_1124_p_OUnits__cong,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% p.Units_cong
thf(fact_1125_p_Odivides__cong__r,axiom,
! [X2: list_a,Y2: list_a,Y4: list_a] :
( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ Y4 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y4 ) ) ) ) ).
% p.divides_cong_r
thf(fact_1126_p_Odivides__cong__l,axiom,
! [X2: list_a,X6: list_a,Y2: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X6 )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X6 @ Y2 )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 ) ) ) ) ).
% p.divides_cong_l
thf(fact_1127_p_Omult__of_Ogcd__cong__r,axiom,
! [Y2: list_a,Y4: list_a,X2: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Y2 @ Y4 )
=> ( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y4 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 @ Y2 ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 @ Y4 ) ) ) ) ) ) ).
% p.mult_of.gcd_cong_r
thf(fact_1128_p_Omult__of_Ogcd__cong__l,axiom,
! [X2: list_a,X6: list_a,Y2: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 @ X6 )
=> ( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ X6 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 @ Y2 ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X6 @ Y2 ) ) ) ) ) ) ).
% p.mult_of.gcd_cong_l
thf(fact_1129_p_Omult__of_Ogcd__assoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ C ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ C ) ) ) ) ) ) ).
% p.mult_of.gcd_assoc
thf(fact_1130_p_Oassociated__iff__same__ideal,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 ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B ) ) ) ) ) ).
% p.associated_iff_same_ideal
thf(fact_1131_p_Omult__of_Oprime__cong,axiom,
! [P: list_a,P3: list_a] :
( ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P @ P3 )
=> ( ( member_list_a @ P @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ P3 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( prime_769485436051073893t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ P3 ) ) ) ) ) ).
% p.mult_of.prime_cong
thf(fact_1132_p_Omult__of_Ogcdof__cong__l,axiom,
! [A7: list_a,A: list_a,B: list_a,C: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A7 @ A )
=> ( ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B @ C )
=> ( ( member_list_a @ A7 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A7 @ B @ C ) ) ) ) ) ) ) ).
% p.mult_of.gcdof_cong_l
thf(fact_1133_associated__polynomials__imp__same__is__root,axiom,
! [P: list_a,Q2: list_a,X2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
= ( polyno4133073214067823460ot_a_b @ r @ Q2 @ X2 ) ) ) ) ) ).
% associated_polynomials_imp_same_is_root
thf(fact_1134_long__division__a__inv_I2_J,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P @ Q2 ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_1135_mult__of_Orelprime__mult,axiom,
! [A: a,B: a,C: a] :
( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ B ) @ ( one_a_ring_ext_a_b @ r ) )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ C ) @ ( one_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ C @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ ( somegc8962790057355718400t_unit @ ( ring_mult_of_a_b @ r ) @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ).
% mult_of.relprime_mult
thf(fact_1136_p_Omult__of_Oassociated__fcount,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( factor8927536889056116518t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A )
= ( factor8927536889056116518t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B ) ) ) ) ) ).
% p.mult_of.associated_fcount
thf(fact_1137_long__division__a__inv_I1_J,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) @ Q2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P @ Q2 ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_1138_p_Omult__of_Oassociated__iff,axiom,
! [A: list_a,B: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( A
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ X ) ) ) ) ) ) ) ).
% p.mult_of.associated_iff
thf(fact_1139_p_Omult__of_OassociatedI2_H,axiom,
! [A: list_a,B: list_a,U: list_a] :
( ( A
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ U ) )
=> ( ( member_list_a @ U @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ) ).
% p.mult_of.associatedI2'
thf(fact_1140_p_Omult__of_OassociatedI2,axiom,
! [U: list_a,A: list_a,B: list_a] :
( ( member_list_a @ U @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( A
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ U ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) ) ) ).
% p.mult_of.associatedI2
thf(fact_1141_p_Omult__of_OassociatedE2,axiom,
! [A: list_a,B: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ! [U2: list_a] :
( ( A
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ U2 ) )
=> ~ ( member_list_a @ U2 @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ~ ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% p.mult_of.associatedE2
thf(fact_1142_p_Omult__of_OassociatedD2,axiom,
! [A: list_a,B: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( A
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ X3 ) ) ) ) ) ) ).
% p.mult_of.associatedD2
thf(fact_1143_p_Osquare__eq__one,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( X2
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
| ( X2
= ( 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_1144_p_Oring__associated__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 ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( A
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ B ) ) ) ) ) ) ) ).
% p.ring_associated_iff
thf(fact_1145_p_OassociatedI2_H,axiom,
! [A: list_a,B: list_a,U: list_a] :
( ( A
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ U ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_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 ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ).
% p.associatedI2'
thf(fact_1146_p_OassociatedI2,axiom,
! [U: list_a,A: list_a,B: list_a] :
( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( A
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ U ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ).
% p.associatedI2
thf(fact_1147_p_Omult__of_Ogcd__mult,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) ) ) ) ) ) ).
% p.mult_of.gcd_mult
thf(fact_1148_p_Omult__of_OgcdI,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ C )
=> ( ! [Y: list_a] :
( ( member_list_a @ Y @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Y @ B )
=> ( ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Y @ C )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Y @ A ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ C ) ) ) ) ) ) ) ) ).
% p.mult_of.gcdI
thf(fact_1149_divides__pirreducible__condition,axiom,
! [K: set_a,Q2: list_a,P: list_a] :
( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 )
=> ( ( member_list_a @ P @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 ) ) ) ) ) ).
% divides_pirreducible_condition
thf(fact_1150_cgenideal__pirreducible,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ Q2 )
=> ( ( member_list_a @ Q2 @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 ) ) ) ) ) ) ).
% cgenideal_pirreducible
thf(fact_1151_p_Omult__of_OgcdI2,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( isgcd_454116541068447652t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B @ C )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ B @ C ) ) ) ) ) ) ).
% p.mult_of.gcdI2
thf(fact_1152_monic__poly__span,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 )
=> ? [Y: list_a] :
( ( monic_4919232885364369782ly_a_b @ r @ Y )
& ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 @ Y ) ) ) ) ).
% monic_poly_span
thf(fact_1153_p_Omult__of_Orelprime__mult,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ B ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ C ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ C @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( somegc4251160090145361146t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% p.mult_of.relprime_mult
thf(fact_1154_subring__degree__one__associatedI,axiom,
! [K: set_a,A: a,A7: a,B: a] :
( ( subring_a_b @ K @ r )
=> ( ( member_a @ A @ K )
=> ( ( member_a @ A7 @ K )
=> ( ( member_a @ B @ K )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ A7 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( mult_a_ring_ext_a_b @ r @ A7 @ B ) @ nil_a ) ) ) ) ) ) ) ) ).
% subring_degree_one_associatedI
thf(fact_1155_associated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( associ5860276527279195403xt_a_b @ r @ A @ A ) ) ).
% associated_refl
thf(fact_1156_poly__mult__var,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( P = nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
= nil_a ) )
& ( ( P != nil_a )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ ( var_a_b @ r ) )
= ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ) ).
% poly_mult_var
thf(fact_1157_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_1158_p_Opdivides__imp__is__root,axiom,
! [P: list_list_a,X2: list_a] :
( ( P != nil_list_a )
=> ( ( member_list_a @ X2 @ ( 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 ) ) @ X2 ) @ nil_list_a ) ) @ P )
=> ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X2 ) ) ) ) ).
% p.pdivides_imp_is_root
thf(fact_1159_mult__of_Oassociated__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A ) ) ).
% mult_of.associated_refl
thf(fact_1160_p_Omult__of_Oassociated__refl,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ A ) ) ).
% p.mult_of.associated_refl
thf(fact_1161_p_Ominus__minus,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) @ X2 ) )
= X2 ) ) ).
% p.minus_minus
thf(fact_1162_p_Oadd_Oinv__closed,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) @ X2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.add.inv_closed
thf(fact_1163_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_1164_p_Oassociated__refl,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A ) ) ).
% p.associated_refl
thf(fact_1165_p_Oadd_Oinv__eq__1__iff,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( X2
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.add.inv_eq_1_iff
thf(fact_1166_p_OUnits__minus__one__closed,axiom,
member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% p.Units_minus_one_closed
thf(fact_1167_p_Oalg__multE_I2_J,axiom,
! [X2: list_a,P: list_list_a,N: nat] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 ) @ nil_list_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X2 ) ) ) ) ) ) ).
% p.alg_multE(2)
thf(fact_1168_p_Ole__alg__mult__imp__pdivides,axiom,
! [X2: list_a,P: list_list_a,N: nat] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X2 ) )
=> ( 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 ) ) @ X2 ) @ nil_list_a ) ) @ N ) @ P ) ) ) ) ).
% p.le_alg_mult_imp_pdivides
thf(fact_1169_subring__props_I5_J,axiom,
! [K: set_a,H3: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H3 @ K )
=> ( member_a @ ( a_inv_a_b @ r @ H3 ) @ K ) ) ) ).
% subring_props(5)
thf(fact_1170_l__minus,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X2 ) @ Y2 )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 ) ) ) ) ) ).
% l_minus
thf(fact_1171_r__minus,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( a_inv_a_b @ r @ Y2 ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y2 ) ) ) ) ) ).
% r_minus
thf(fact_1172_div__neg,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( a_inv_a_b @ r @ B ) ) ) ) ) ).
% div_neg
thf(fact_1173_square__eq__one,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X2 @ X2 )
= ( one_a_ring_ext_a_b @ r ) )
=> ( ( X2
= ( one_a_ring_ext_a_b @ r ) )
| ( X2
= ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% square_eq_one
thf(fact_1174_const__term__simprules__shell_I4_J,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
= ( a_inv_a_b @ r @ ( const_term_a_b @ r @ P ) ) ) ) ) ).
% const_term_simprules_shell(4)
thf(fact_1175_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_1176_p_Ouniv__poly__a__inv__consistent,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ P ) ) ) ) ).
% p.univ_poly_a_inv_consistent
thf(fact_1177_p_Olong__division__a__inv_I2_J,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) @ Q2 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( polyno1727750685288865234t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) ) ) ) ) ) ).
% p.long_division_a_inv(2)
thf(fact_1178_p_Olong__division__a__inv_I1_J,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) @ Q2 )
= ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ ( polyno5893782122288709345t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q2 ) ) ) ) ) ) ).
% p.long_division_a_inv(1)
thf(fact_1179_pdivides__imp__is__root,axiom,
! [P: list_a,X2: a] :
( ( P != nil_a )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X2 ) @ nil_a ) ) @ P )
=> ( polyno4133073214067823460ot_a_b @ r @ P @ X2 ) ) ) ) ).
% pdivides_imp_is_root
thf(fact_1180_p_Oconst__term__simprules__shell_I4_J,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ) ) ).
% p.const_term_simprules_shell(4)
thf(fact_1181_a__inv__closed,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_1182_local_Ominus__minus,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X2 ) )
= X2 ) ) ).
% local.minus_minus
thf(fact_1183_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_1184_is__root__imp__pdivides,axiom,
! [P: list_a,X2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
=> ( polyno5814909790663948098es_a_b @ r @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X2 ) @ nil_a ) ) @ P ) ) ) ).
% is_root_imp_pdivides
thf(fact_1185_add_Oinv__eq__1__iff,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X2 )
= ( zero_a_b @ r ) )
= ( X2
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_1186_Units__minus__one__closed,axiom,
member_a @ ( a_inv_a_b @ r @ ( one_a_ring_ext_a_b @ r ) ) @ ( units_a_ring_ext_a_b @ r ) ).
% Units_minus_one_closed
thf(fact_1187_p_Oalg__multE_I1_J,axiom,
! [X2: list_a,P: list_list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( P != nil_list_a )
=> ( polyno8016796738000020810t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li488931774710091566it_nat @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) @ ( cons_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 ) @ nil_list_a ) ) @ ( polyno4259638811958763678t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X2 ) ) @ P ) ) ) ) ).
% p.alg_multE(1)
thf(fact_1188_alg__multE_I2_J,axiom,
! [X2: a,P: list_a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X2 ) @ nil_a ) ) @ N ) @ P )
=> ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X2 ) ) ) ) ) ) ).
% alg_multE(2)
thf(fact_1189_le__alg__mult__imp__pdivides,axiom,
! [X2: a,P: list_a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ ( polyno4422430861927485590lt_a_b @ r @ P @ X2 ) )
=> ( 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 @ X2 ) @ nil_a ) ) @ N ) @ P ) ) ) ) ).
% le_alg_mult_imp_pdivides
thf(fact_1190_alg__multE_I1_J,axiom,
! [X2: a,P: list_a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_a @ ( one_a_ring_ext_a_b @ r ) @ ( cons_a @ ( a_inv_a_b @ r @ X2 ) @ nil_a ) ) @ ( polyno4422430861927485590lt_a_b @ r @ P @ X2 ) ) @ P ) ) ) ) ).
% alg_multE(1)
thf(fact_1191_p_Oconst__term__eq__last,axiom,
! [P: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( 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 @ P @ ( cons_list_a @ A @ nil_list_a ) ) )
= A ) ) ) ).
% p.const_term_eq_last
thf(fact_1192_p_Oconst__term__explicit,axiom,
! [P: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_list_a )
=> ( ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= A )
=> ~ ! [P4: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P
!= ( append_list_a @ P4 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ).
% p.const_term_explicit
thf(fact_1193_p_Oconst__term__simprules_I1_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( 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 ) ) @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.const_term_simprules(1)
thf(fact_1194_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_1195_p_Ofactors__dividesI,axiom,
! [Fs: list_list_a,A: list_a,F: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A )
=> ( ( member_list_a @ F @ ( set_list_a2 @ Fs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) ) ) ).
% p.factors_dividesI
thf(fact_1196_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_1197_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_1198_p_Oexp__base__closed,axiom,
! [X2: list_a,N: nat] :
( ( member_list_a @ X2 @ ( 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 ) ) @ X2 @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.exp_base_closed
thf(fact_1199_p_Ofactorization__property,axiom,
! [A: list_a] :
( ( 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 ) ) )
=> ( ~ ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [Fs2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs2 ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fs2 @ A ) ) ) ) ).
% p.factorization_property
thf(fact_1200_p_Omult__of_Ounit__wfactors,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a @ A ) ) ).
% p.mult_of.unit_wfactors
thf(fact_1201_p_Omult__of_Owfactors__cong__r,axiom,
! [Fs: list_list_a,A: list_a,A7: list_a] :
( ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fs @ A )
=> ( ( associ6442652891116239602t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A @ A7 )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A7 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fs @ A7 ) ) ) ) ) ) ).
% p.mult_of.wfactors_cong_r
thf(fact_1202_p_Omult__of_Owfactors__dividesI,axiom,
! [Fs: list_list_a,A: list_a,F: list_a] :
( ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fs @ A )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ F @ ( set_list_a2 @ Fs ) )
=> ( factor8389468621490698395t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ F @ A ) ) ) ) ) ).
% p.mult_of.wfactors_dividesI
thf(fact_1203_p_Omult__of_Ounit__wfactors__empty,axiom,
! [A: list_a,Fs: list_list_a] :
( ( member_list_a @ A @ ( units_8735880885477018085t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fs @ A )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( Fs = nil_list_a ) ) ) ) ).
% p.mult_of.unit_wfactors_empty
thf(fact_1204_p_Omult__of_Owfactors__mult__single,axiom,
! [A: list_a,Fb: list_list_a,B: list_a] :
( ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A )
=> ( ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fb @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fb ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( cons_list_a @ A @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ) ).
% p.mult_of.wfactors_mult_single
thf(fact_1205_p_Omult__of_Owfactors__exist,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ? [Fs2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs2 ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Fs2 @ A ) ) ) ).
% p.mult_of.wfactors_exist
thf(fact_1206_p_Omult__of_Owfactors__prod__exists,axiom,
! [As: list_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ As ) )
=> ( irredu7180820467033665102t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X3 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ? [A5: list_a] :
( ( member_list_a @ A5 @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ As @ A5 ) ) ) ) ).
% p.mult_of.wfactors_prod_exists
thf(fact_1207_p_Omult__of_Owfactors__mult,axiom,
! [As: list_list_a,A: list_a,Bs: list_list_a,B: list_a] :
( ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ As @ A )
=> ( ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ Bs @ B )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( member_list_a @ B @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( append_list_a @ As @ Bs ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B ) ) ) ) ) ) ) ) ).
% p.mult_of.wfactors_mult
thf(fact_1208_p_OSpan__m__inv__simprule,axiom,
! [K: set_list_a,Us2: list_list_a,K2: list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K2 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ( member_list_a @ A @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ) ) ).
% p.Span_m_inv_simprule
thf(fact_1209_p_Omult__of_Ofactorcount__unique,axiom,
! [As: list_list_a,A: list_a] :
( ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ As @ A )
=> ( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ( factor8927536889056116518t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ A )
= ( size_s349497388124573686list_a @ As ) ) ) ) ) ).
% p.mult_of.factorcount_unique
thf(fact_1210_unit__wfactors,axiom,
! [A: a] :
( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ nil_a @ A ) ) ).
% unit_wfactors
thf(fact_1211_wfactors__dividesI,axiom,
! [Fs: list_a,A: a,F: a] :
( ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( factor8216151070175719842xt_a_b @ r @ F @ A ) ) ) ) ) ).
% wfactors_dividesI
thf(fact_1212_wfactors__cong__r,axiom,
! [Fs: list_a,A: a,A7: a] :
( ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A )
=> ( ( associ5860276527279195403xt_a_b @ r @ A @ A7 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A7 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( wfacto3557276942076956612xt_a_b @ r @ Fs @ A7 ) ) ) ) ) ) ).
% wfactors_cong_r
thf(fact_1213_mult__of_Ounit__wfactors,axiom,
! [A: a] :
( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ nil_a @ A ) ) ).
% mult_of.unit_wfactors
thf(fact_1214_const__term__simprules_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( const_term_a_b @ r @ P ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% const_term_simprules(1)
thf(fact_1215_p_Ounit__wfactors,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( wfacto3834028397835396690t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ A ) ) ).
% p.unit_wfactors
thf(fact_1216_mult__of_Owfactors__cong__r,axiom,
! [Fs: list_a,A: a,A7: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A )
=> ( ( associ6879500422977059064t_unit @ ( ring_mult_of_a_b @ r ) @ A @ A7 )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A7 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A7 ) ) ) ) ) ) ).
% mult_of.wfactors_cong_r
thf(fact_1217_mult__of_Owfactors__dividesI,axiom,
! [Fs: list_a,A: a,F: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ F @ ( set_a2 @ Fs ) )
=> ( factor3040189038382604065t_unit @ ( ring_mult_of_a_b @ r ) @ F @ A ) ) ) ) ) ).
% mult_of.wfactors_dividesI
thf(fact_1218_mult__of_Ounit__wfactors__empty,axiom,
! [A: a,Fs: list_a] :
( ( member_a @ A @ ( units_a_Product_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( Fs = nil_a ) ) ) ) ).
% mult_of.unit_wfactors_empty
thf(fact_1219_p_OSpan__in__carrier,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( ord_le8861187494160871172list_a @ K @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.Span_in_carrier
thf(fact_1220_p_Owfactors__dividesI,axiom,
! [Fs: list_list_a,A: list_a,F: list_a] :
( ( wfacto3834028397835396690t_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 ) ) ) )
=> ( ( member_list_a @ F @ ( set_list_a2 @ Fs ) )
=> ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F @ A ) ) ) ) ) ).
% p.wfactors_dividesI
thf(fact_1221_p_Owfactors__cong__r,axiom,
! [Fs: list_list_a,A: list_a,A7: list_a] :
( ( wfacto3834028397835396690t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ A7 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A7 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( wfacto3834028397835396690t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Fs @ A7 ) ) ) ) ) ) ).
% p.wfactors_cong_r
thf(fact_1222_p_Ofactors__wfactors,axiom,
! [As: list_list_a,A: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ A )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( wfacto3834028397835396690t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ A ) ) ) ).
% p.factors_wfactors
thf(fact_1223_mult__of_Owfactors__mult__single,axiom,
! [A: a,Fb: list_a,B: a] :
( ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ A )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fb @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ ( cons_a @ A @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ) ).
% mult_of.wfactors_mult_single
thf(fact_1224_const__term__eq__last,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( append_a @ P @ ( cons_a @ A @ nil_a ) ) )
= A ) ) ) ).
% const_term_eq_last
thf(fact_1225_const__term__explicit,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( P != nil_a )
=> ( ( ( const_term_a_b @ r @ P )
= A )
=> ~ ! [P4: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P4 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P4 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_1226_p_OSpan__subgroup__props_I1_J,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.Span_subgroup_props(1)
thf(fact_1227_p_OSpan__base__incl,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ).
% p.Span_base_incl
thf(fact_1228_p_OSpan__same__set,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( set_list_a2 @ Us2 )
= ( set_list_a2 @ Vs ) )
=> ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
= ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).
% p.Span_same_set
thf(fact_1229_p_Omono__Span__sublist,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( set_list_a2 @ Vs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).
% p.mono_Span_sublist
thf(fact_1230_p_Omono__Span__subset,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ).
% p.mono_Span_subset
thf(fact_1231_p_Owfactors__factors,axiom,
! [As: list_list_a,A: list_a] :
( ( wfacto3834028397835396690t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ A )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [A8: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ A8 )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A8 @ A ) ) ) ) ).
% p.wfactors_factors
thf(fact_1232_p_OSpan__subalgebraI,axiom,
! [K: set_list_a,E: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ E )
=> ( ! [V3: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ V3 )
=> ( ord_le8861187494160871172list_a @ E @ V3 ) ) )
=> ( E
= ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ) ).
% p.Span_subalgebraI
thf(fact_1233_p_Osubalgebra__Span__incl,axiom,
! [K: set_list_a,V: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ V )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ V ) ) ) ) ).
% p.subalgebra_Span_incl
thf(fact_1234_p_Ouniv__poly__a__inv__length,axiom,
! [K: set_list_a,P: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( size_s349497388124573686list_a @ ( a_inv_7033018035630854991t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P ) )
= ( size_s349497388124573686list_a @ P ) ) ) ) ).
% p.univ_poly_a_inv_length
thf(fact_1235_p_Oassociated__polynomials__imp__same__length,axiom,
! [K: set_list_a,P: list_list_a,Q2: list_list_a] :
( ( subrin6918843898125473962t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) @ P @ Q2 )
=> ( ( size_s349497388124573686list_a @ P )
= ( size_s349497388124573686list_a @ Q2 ) ) ) ) ) ) ).
% p.associated_polynomials_imp_same_length
thf(fact_1236_p_OSpan__subgroup__props_I2_J,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ).
% p.Span_subgroup_props(2)
thf(fact_1237_p_Omult__of_Ofactorcount__exists,axiom,
! [A: list_a] :
( ( member_list_a @ A @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ? [C2: nat] :
! [As2: list_list_a] :
( ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As2 ) @ ( partia7074150537345710456t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
& ( wfacto5606736950954242553t_unit @ ( ring_m2863707994090333347t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ As2 @ A ) )
=> ( C2
= ( size_s349497388124573686list_a @ As2 ) ) ) ) ).
% p.mult_of.factorcount_exists
thf(fact_1238_p_OSpan__smult__closed,axiom,
! [K: set_list_a,Us2: list_list_a,K2: list_a,V4: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ K2 @ K )
=> ( ( member_list_a @ V4 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ V4 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ) ).
% p.Span_smult_closed
thf(fact_1239_p_Omono__Span,axiom,
! [K: set_list_a,Us2: list_list_a,U: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( cons_list_a @ U @ Us2 ) ) ) ) ) ) ).
% p.mono_Span
thf(fact_1240_p_OSpan__subgroup__props_I4_J,axiom,
! [K: set_list_a,Us2: list_list_a,V4: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ V4 @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ( member_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ V4 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ) ).
% p.Span_subgroup_props(4)
thf(fact_1241_p_Omono__Span__append_I2_J,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Vs @ Us2 ) ) ) ) ) ) ).
% p.mono_Span_append(2)
thf(fact_1242_p_Omono__Span__append_I1_J,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us2 @ Vs ) ) ) ) ) ) ).
% p.mono_Span_append(1)
thf(fact_1243_p_OSpan__is__subalgebra,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.Span_is_subalgebra
thf(fact_1244_factorization__property,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ? [Fs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs2 @ A ) ) ) ) ).
% factorization_property
thf(fact_1245_p_OSpan__append__eq__set__add,axiom,
! [K: set_list_a,Us2: list_list_a,Vs: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( append_list_a @ Us2 @ Vs ) )
= ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Vs ) ) ) ) ) ) ).
% p.Span_append_eq_set_add
thf(fact_1246_mult__of_Owfactors__exist,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [Fs2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Fs2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Fs2 @ A ) ) ) ).
% mult_of.wfactors_exist
thf(fact_1247_monom__in__carrier,axiom,
! [A: a,N: nat] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( monom_a_b @ r @ A @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% monom_in_carrier
thf(fact_1248_wfactors__prod__exists,axiom,
! [As: list_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ As ) )
=> ( irredu6211895646901577903xt_a_b @ r @ X3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [A5: a] :
( ( member_a @ A5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( wfacto3557276942076956612xt_a_b @ r @ As @ A5 ) ) ) ) ).
% wfactors_prod_exists
thf(fact_1249_mult__of_Owfactors__prod__exists,axiom,
! [As: list_a] :
( ! [X3: a] :
( ( member_a @ X3 @ ( set_a2 @ As ) )
=> ( irredu4023057619401689684t_unit @ ( ring_mult_of_a_b @ r ) @ X3 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [A5: a] :
( ( member_a @ A5 @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A5 ) ) ) ) ).
% mult_of.wfactors_prod_exists
thf(fact_1250_mult__of_Owfactors__mult,axiom,
! [As: list_a,A: a,Bs: list_a,B: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A )
=> ( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ Bs @ B )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( member_a @ B @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Bs ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ ( append_a @ As @ Bs ) @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ) ) ) ).
% mult_of.wfactors_mult
thf(fact_1251_p_Owfactors__prod__exists,axiom,
! [As: list_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ As ) )
=> ( irredu4230924414530676029t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [A5: list_a] :
( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( wfacto3834028397835396690t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ A5 ) ) ) ) ).
% p.wfactors_prod_exists
thf(fact_1252_exp__base__closed,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ r @ X2 @ N ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% exp_base_closed
thf(fact_1253_p_OSpan__mem__imp__non__trivial__combine,axiom,
! [K: set_list_a,Us2: list_list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
=> ~ ! [K3: list_a] :
( ( member_list_a @ K3 @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ! [Ks: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ K )
=> ( ( ( size_s349497388124573686list_a @ Ks )
= ( size_s349497388124573686list_a @ Us2 ) )
=> ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ K3 @ Ks ) @ ( cons_list_a @ A @ Us2 ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ) ).
% p.Span_mem_imp_non_trivial_combine
thf(fact_1254_univ__poly__a__inv__length,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( size_size_list_a @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P ) )
= ( size_size_list_a @ P ) ) ) ) ).
% univ_poly_a_inv_length
thf(fact_1255_associated__polynomials__imp__same__length,axiom,
! [K: set_a,P: list_a,Q2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q2 )
=> ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q2 ) ) ) ) ) ) ).
% associated_polynomials_imp_same_length
thf(fact_1256_p_Ocombine_Osimps_I2_J,axiom,
! [Us2: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ Us2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.combine.simps(2)
thf(fact_1257_p_Ocombine_Osimps_I3_J,axiom,
! [Ks2: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ nil_list_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% p.combine.simps(3)
thf(fact_1258_mult__of_Ofactorcount__exists,axiom,
! [A: a] :
( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ? [C2: nat] :
! [As2: list_a] :
( ( ( ord_less_eq_set_a @ ( set_a2 @ As2 ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
& ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As2 @ A ) )
=> ( C2
= ( size_size_list_a @ As2 ) ) ) ) ).
% mult_of.factorcount_exists
thf(fact_1259_mult__of_Ofactorcount__unique,axiom,
! [As: list_a,A: a] :
( ( wfacto3536202916627062655t_unit @ ( ring_mult_of_a_b @ r ) @ As @ A )
=> ( ( member_a @ A @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ As ) @ ( partia6735698275553448452t_unit @ ( ring_mult_of_a_b @ r ) ) )
=> ( ( factor4067924603488134956t_unit @ ( ring_mult_of_a_b @ r ) @ A )
= ( size_size_list_a @ As ) ) ) ) ) ).
% mult_of.factorcount_unique
thf(fact_1260_p_OSpan__mem__iff__length__version,axiom,
! [K: set_list_a,Us2: list_list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
= ( ? [Ks3: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks3 ) @ K )
& ( ( size_s349497388124573686list_a @ Ks3 )
= ( size_s349497388124573686list_a @ Us2 ) )
& ( A
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks3 @ Us2 ) ) ) ) ) ) ) ).
% p.Span_mem_iff_length_version
thf(fact_1261_p_Ocombine__append__zero,axiom,
! [Us2: list_list_a,Ks2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Ks2 @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) @ Us2 )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 ) ) ) ).
% p.combine_append_zero
thf(fact_1262_p_OSpan__mem__iff,axiom,
! [K: set_list_a,Us2: list_list_a,A: list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( 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 @ A @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( minus_646659088055828811list_a @ K @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) )
& ? [Ks3: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks3 ) @ K )
& ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ X @ Ks3 ) @ ( cons_list_a @ A @ Us2 ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ) ) ).
% p.Span_mem_iff
thf(fact_1263_p_Ocombine__in__carrier,axiom,
! [Ks2: list_list_a,Us2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.combine_in_carrier
thf(fact_1264_p_Opoly__add__monom,axiom,
! [P: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( 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 ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( size_s349497388124573686list_a @ P ) ) @ P )
= ( cons_list_a @ A @ P ) ) ) ) ).
% p.poly_add_monom
thf(fact_1265_p_OSpan__finite__dimension,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( embedd4402942584324845940t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 ) ) ) ) ).
% p.Span_finite_dimension
thf(fact_1266_p_Otelescopic__base__dim_I1_J,axiom,
! [K: set_list_a,F3: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subfie1779122896746047282t_unit @ F3 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ F3 )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ F3 @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E ) ) ) ) ) ).
% p.telescopic_base_dim(1)
thf(fact_1267_p_Osum__space__dim_I1_J,axiom,
! [K: set_list_a,E: set_list_a,F3: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ F3 )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ ( set_ad92425877771022410t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ E @ F3 ) ) ) ) ) ).
% p.sum_space_dim(1)
thf(fact_1268_p_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_list_a,E: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ( embedd1768981623711841426t_unit @ K @ E @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% p.finite_dimension_imp_subalgebra
thf(fact_1269_p_Opoly__add__comm,axiom,
! [P12: list_list_a,P23: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P12 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P23 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P12 @ P23 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P23 @ P12 ) ) ) ) ).
% p.poly_add_comm
thf(fact_1270_p_Opoly__add__in__carrier,axiom,
! [P12: list_list_a,P23: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P12 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P23 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P12 @ P23 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% p.poly_add_in_carrier
thf(fact_1271_p_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_list_a,E: set_list_a,V: set_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ E )
=> ( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ V @ E )
=> ( embedd1345800358437254783t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ V ) ) ) ) ) ).
% p.subalbegra_incl_imp_finite_dimension
thf(fact_1272_p_Ocombine__append,axiom,
! [Ks2: list_list_a,Us2: list_list_a,Ks4: list_list_a,Vs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Ks2 )
= ( size_s349497388124573686list_a @ Us2 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks4 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us2 ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks4 @ Vs ) )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Ks2 @ Ks4 ) @ ( append_list_a @ Us2 @ Vs ) ) ) ) ) ) ) ) ).
% p.combine_append
thf(fact_1273_p_Odependent__imp__non__trivial__combine,axiom,
! [K: set_list_a,Us2: list_list_a] :
( ( subfie1779122896746047282t_unit @ K @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ~ ( embedd3875673156127067906t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ Us2 )
=> ~ ! [Ks: list_list_a] :
( ( ( size_s349497388124573686list_a @ Ks )
= ( size_s349497388124573686list_a @ Us2 ) )
=> ( ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ K )
=> ( ( set_list_a2 @ Ks )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ bot_bot_set_list_a ) ) ) ) ) ) ) ) ).
% p.dependent_imp_non_trivial_combine
thf(fact_1274_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_1275_p_Oadd_Om__lcomm,axiom,
! [X2: list_a,Y2: list_a,Z2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( member_list_a @ Z2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ Z2 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Z2 ) ) ) ) ) ) ).
% p.add.m_lcomm
thf(fact_1276_p_Oadd_Om__comm,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ X2 ) ) ) ) ).
% p.add.m_comm
thf(fact_1277_p_Oadd_Om__assoc,axiom,
! [X2: list_a,Y2: list_a,Z2: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( member_list_a @ Z2 @ ( 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 ) ) @ X2 @ Y2 ) @ Z2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y2 @ Z2 ) ) ) ) ) ) ).
% p.add.m_assoc
% Conjectures (1)
thf(conj_0,conjecture,
member_list_a @ d @ ( 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 ) ) ).
%------------------------------------------------------------------------------