TPTP Problem File: SLH0046^1.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 : Interpolation_Polynomials_HOL_Algebra/0001_Lagrange_Interpolation/prob_00272_010702__17391280_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1618 ( 422 unt; 337 typ; 0 def)
% Number of atoms : 3618 (1117 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 15661 ( 207 ~; 29 |; 109 &;13554 @)
% ( 0 <=>;1762 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 36 ( 35 usr)
% Number of type conns : 681 ( 681 >; 0 *; 0 +; 0 <<)
% Number of symbols : 305 ( 302 usr; 18 con; 0-4 aty)
% Number of variables : 2966 ( 92 ^;2830 !; 44 ?;2966 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:38:33.121
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
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thf(sy_c_Ring__Divisibility_Ofactorial__domain_001tf__a_001tf__b,type,
ring_f5272581269873410839in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n4705423059119889713t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n3212398840814694743t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001tf__a_001tf__b,type,
ring_n4045954140777738665in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n5188127996776581661t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_n5014428767265248323t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001tf__a_001tf__b,type,
ring_n3639167112692572309ng_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_p715737262848045090t_unit: partia2956882679547061052t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p2862007038493914190t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r360171070648044744t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__Set__Oset_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r8677422918745460462t_unit: partia3925755165846298134t_unit > list_set_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r7790391342995787508t_unit: partia6043505979758434576t_unit > set_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
ring_r5437400583859147359t_unit: partia2956882679547061052t_unit > list_list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_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_It__Set__Oset_Itf__a_J_J,type,
insert_list_set_a: list_set_a > set_list_set_a > set_list_set_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_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_Osubcring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subcri8676831449680469861t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_It__Set__Oset_Itf__a_J_J_001t__Product____Type__Ounit,type,
subcri2121798867598172171t_unit: set_list_set_a > partia3925755165846298134t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subcri7763218559781929323t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
subcri4445174380595745425t_unit: set_set_a > partia6043505979758434576t_unit > $o ).
thf(sy_c_Subrings_Osubcring_001tf__a_001tf__b,type,
subcring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subdom561091866123308472t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subdom7821232466298058046t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
subdom4943114742163587044t_unit: set_set_a > partia6043505979758434576t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001tf__a_001tf__b,type,
subdomain_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
subrin3541368690557094692t_unit: set_list_list_a > partia2956882679547061052t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_It__Set__Oset_Itf__a_J_J_001t__Product____Type__Ounit,type,
subrin9205269123103177546t_unit: set_list_set_a > partia3925755165846298134t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
subrin6918843898125473962t_unit: set_list_a > partia2670972154091845814t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
subrin1511138061850335568t_unit: set_set_a > partia6043505979758434576t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_UnivPoly_Obound_001t__List__Olist_Itf__a_J,type,
bound_list_a: list_a > nat > ( nat > list_a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__a,type,
bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member8231385768148312316list_a: ( list_list_a > list_list_a ) > set_li5608457238520824219list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_Itf__a_J_J,type,
member7168557129179038582list_a: ( list_list_a > list_a ) > set_li3422455791611400469list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__List__Olist_Itf__a_J_J_Mtf__a_J,type,
member_list_list_a_a: ( list_list_a > a ) > set_list_list_a_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member6714375691612171394list_a: ( list_a > list_list_a ) > set_li6773872926390105121list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member_list_a_list_a: ( list_a > list_a ) > set_list_a_list_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member_a_list_list_a: ( a > list_list_a ) > set_a_list_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
member_a_list_a: ( a > list_a ) > set_a_list_a > $o ).
thf(sy_c_member_001_062_Itf__a_Mtf__a_J,type,
member_a_a: ( a > a ) > set_a_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
member6842060177613954879list_a: list_l7815035709764258326list_a > set_li3407770045201608054list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
member6735289990665601811_set_a: list_list_set_a > set_list_list_set_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_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__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_S,type,
s: set_a ).
thf(sy_v_p____,type,
p: list_a ).
% Relevant facts (1277)
thf(fact_0_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_1_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_2_domain_Ouniv__poly__is__cring,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( cring_7032029500657136448t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_cring
thf(fact_3_domain_Ouniv__poly__is__cring,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_cring
thf(fact_4_domain_Ouniv__poly__is__cring,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( cring_5991999922451032090t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_cring
thf(fact_5_x_Ois__cring,axiom,
cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.is_cring
thf(fact_6_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_7_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_8_univ__poly__is__cring,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_cring
thf(fact_9_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_10_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_11_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_12_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_13_x_Osemiring__axioms,axiom,
semiri2871908745932252451t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.semiring_axioms
thf(fact_14_x_Oabelian__monoid__axioms,axiom,
abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.abelian_monoid_axioms
thf(fact_15_carrier__is__subcring,axiom,
subcring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subcring
thf(fact_16_x_Oring__axioms,axiom,
ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.ring_axioms
thf(fact_17_x_Ois__abelian__group,axiom,
abelia3891852623213500406t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.is_abelian_group
thf(fact_18_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_19_local_Osemiring__axioms,axiom,
semiring_a_b @ r ).
% local.semiring_axioms
thf(fact_20_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_21_subcringI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subcring_a_b @ H @ r ) ) ).
% subcringI'
thf(fact_22_univ__poly__is__ring,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_ring
thf(fact_23_univ__poly__is__abelian__group,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( abelia3891852623213500406t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_abelian_group
thf(fact_24_univ__poly__is__abelian__monoid,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_abelian_monoid
thf(fact_25_domain_Ouniv__poly__is__ring,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_ring
thf(fact_26_domain_Ouniv__poly__is__ring,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ring_l1939023646219158831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_ring
thf(fact_27_domain_Ouniv__poly__is__ring,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( ring_l2585786428786346709t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_ring
thf(fact_28_domain_Ouniv__poly__is__domain,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_29_domain_Ouniv__poly__is__domain,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( domain7810152921033798211t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_30_domain_Ouniv__poly__is__domain,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( domain5179785879681597929t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_domain
thf(fact_31_domain_Ouniv__poly__is__abelian__group,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( abelia3891852623213500406t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_group
thf(fact_32_domain_Ouniv__poly__is__abelian__group,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( abelia2778853791629620336t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_group
thf(fact_33_domain_Ouniv__poly__is__abelian__group,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( abelia8866176162151713430t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_group
thf(fact_34_domain_Ouniv__poly__is__abelian__monoid,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( abelia226231641709521465t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_monoid
thf(fact_35_domain_Ouniv__poly__is__abelian__monoid,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( abelia3641329199688042803t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_monoid
thf(fact_36_domain_Ouniv__poly__is__abelian__monoid,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( abelia6884027370077999321t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_is_abelian_monoid
thf(fact_37_subdomainI_H,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( subdomain_a_b @ H @ r ) ) ).
% subdomainI'
thf(fact_38_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a,P: list_set_a] :
( ( ring_s5549108798478166619t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
=> ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_39_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,P: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_40_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_41_ring_Ocarrier__polynomial__shell,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).
% ring.carrier_polynomial_shell
thf(fact_42_noetherian__domain_Ointro,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_n5188127996776581661t_unit @ R )
=> ( ( domain6553523120543210313t_unit @ R )
=> ( ring_n4705423059119889713t_unit @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_43_noetherian__domain_Ointro,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_n5014428767265248323t_unit @ R )
=> ( ( domain4236798911309298543t_unit @ R )
=> ( ring_n3212398840814694743t_unit @ R ) ) ) ).
% noetherian_domain.intro
thf(fact_44_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_45_noetherian__domain__def,axiom,
( ring_n4705423059119889713t_unit
= ( ^ [R2: partia2670972154091845814t_unit] :
( ( ring_n5188127996776581661t_unit @ R2 )
& ( domain6553523120543210313t_unit @ R2 ) ) ) ) ).
% noetherian_domain_def
thf(fact_46_noetherian__domain__def,axiom,
( ring_n3212398840814694743t_unit
= ( ^ [R2: partia6043505979758434576t_unit] :
( ( ring_n5014428767265248323t_unit @ R2 )
& ( domain4236798911309298543t_unit @ R2 ) ) ) ) ).
% noetherian_domain_def
thf(fact_47_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_48_x_OsubcringI_H,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.subcringI'
thf(fact_49_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_50_cring_OsubcringI_H,axiom,
! [R: partia2956882679547061052t_unit,H: set_list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ H @ R )
=> ( subcri8676831449680469861t_unit @ H @ R ) ) ) ).
% cring.subcringI'
thf(fact_51_cring_OsubcringI_H,axiom,
! [R: partia3925755165846298134t_unit,H: set_list_set_a] :
( ( cring_7032029500657136448t_unit @ R )
=> ( ( subrin9205269123103177546t_unit @ H @ R )
=> ( subcri2121798867598172171t_unit @ H @ R ) ) ) ).
% cring.subcringI'
thf(fact_52_cring_OsubcringI_H,axiom,
! [R: partia6043505979758434576t_unit,H: set_set_a] :
( ( cring_5265999997378430022t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ H @ R )
=> ( subcri4445174380595745425t_unit @ H @ R ) ) ) ).
% cring.subcringI'
thf(fact_53_cring_OsubcringI_H,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( cring_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( subcring_a_b @ H @ R ) ) ) ).
% cring.subcringI'
thf(fact_54_cring_OsubcringI_H,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( subcri7763218559781929323t_unit @ H @ R ) ) ) ).
% cring.subcringI'
thf(fact_55_ring_Oonepideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_56_ring_Oonepideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_57_ring_Oonepideal,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( princi2534607884127416211t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).
% ring.onepideal
thf(fact_58_cring_Ocarrier__is__subcring,axiom,
! [R: partia3925755165846298134t_unit] :
( ( cring_7032029500657136448t_unit @ R )
=> ( subcri2121798867598172171t_unit @ ( partia3893404292425143049t_unit @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_59_cring_Ocarrier__is__subcring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( subcring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_60_cring_Ocarrier__is__subcring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_61_cring_Ocarrier__is__subcring,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( subcri8676831449680469861t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).
% cring.carrier_is_subcring
thf(fact_62_is__cring,axiom,
cring_a_b @ r ).
% is_cring
thf(fact_63_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_64_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_65_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_66_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_67_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_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_69_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_70_x_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.onepideal
thf(fact_71_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_72_x_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 ) ) ).
% x.carrier_is_subring
thf(fact_73_x_Ocarrier__is__subcring,axiom,
subcri7763218559781929323t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
% x.carrier_is_subcring
thf(fact_74_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_75_x_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 ) ) ) ) ) ) ) ) ).
% x.carrier_polynomial_shell
thf(fact_76_subdomainE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subdomainE(4)
thf(fact_77_subdomain_Oaxioms_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( subcri7763218559781929323t_unit @ H @ R ) ) ).
% subdomain.axioms(1)
thf(fact_78_subdomain_Oaxioms_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( subcring_a_b @ H @ R ) ) ).
% subdomain.axioms(1)
thf(fact_79_ring_Ocgenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_80_ring_Ocgenideal__self,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_81_ring_Ocgenideal__self,axiom,
! [R: partia2956882679547061052t_unit,I: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ I @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ I @ ( cgenid24865672677839267t_unit @ R @ I ) ) ) ) ).
% ring.cgenideal_self
thf(fact_82_domain_OsubdomainI_H,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( subdom7821232466298058046t_unit @ H @ R ) ) ) ).
% domain.subdomainI'
thf(fact_83_domain_OsubdomainI_H,axiom,
! [R: partia6043505979758434576t_unit,H: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ H @ R )
=> ( subdom4943114742163587044t_unit @ H @ R ) ) ) ).
% domain.subdomainI'
thf(fact_84_domain_OsubdomainI_H,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( subdomain_a_b @ H @ R ) ) ) ).
% domain.subdomainI'
thf(fact_85_cring_Ocgenideal__is__principalideal,axiom,
! [R: partia3925755165846298134t_unit,I: list_set_a] :
( ( cring_7032029500657136448t_unit @ R )
=> ( ( member_list_set_a @ I @ ( partia3893404292425143049t_unit @ R ) )
=> ( princi9128422467719522105t_unit @ ( cgenid5493546629940232227t_unit @ R @ I ) @ R ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_86_cring_Ocgenideal__is__principalideal,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ R @ I ) @ R ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_87_cring_Ocgenideal__is__principalideal,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ R @ I ) @ R ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_88_cring_Ocgenideal__is__principalideal,axiom,
! [R: partia2956882679547061052t_unit,I: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ I @ ( partia2464479390973590831t_unit @ R ) )
=> ( princi2534607884127416211t_unit @ ( cgenid24865672677839267t_unit @ R @ I ) @ R ) ) ) ).
% cring.cgenideal_is_principalideal
thf(fact_89_principalideal_Ois__principalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ( principalideal_a_b @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_90_principalideal_Ois__principalideal,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( princi8786919440553033881t_unit @ I2 @ R )
=> ( princi8786919440553033881t_unit @ I2 @ R ) ) ).
% principalideal.is_principalideal
thf(fact_91_subringE_I4_J,axiom,
! [H: set_set_a,R: partia6043505979758434576t_unit] :
( ( subrin1511138061850335568t_unit @ H @ R )
=> ( H != bot_bot_set_set_a ) ) ).
% subringE(4)
thf(fact_92_subringE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subringE(4)
thf(fact_93_subringE_I4_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( H != bot_bot_set_list_a ) ) ).
% subringE(4)
thf(fact_94_subcringE_I4_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( H != bot_bot_set_a ) ) ).
% subcringE(4)
thf(fact_95_subcringE_I4_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( H != bot_bot_set_list_a ) ) ).
% subcringE(4)
thf(fact_96_ring__iso__memE_I1_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( H2 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_97_ring__iso__memE_I1_J,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( H2 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_98_ring__iso__memE_I1_J,axiom,
! [H2: a > list_list_a,R: partia2175431115845679010xt_a_b,S: partia2956882679547061052t_unit,X2: a] :
( ( member_a_list_list_a @ H2 @ ( ring_i4464730343205239444t_unit @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_list_a @ ( H2 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_99_ring__iso__memE_I1_J,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_a @ ( H2 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_100_ring__iso__memE_I1_J,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_101_ring__iso__memE_I1_J,axiom,
! [H2: list_a > list_list_a,R: partia2670972154091845814t_unit,S: partia2956882679547061052t_unit,X2: list_a] :
( ( member6714375691612171394list_a @ H2 @ ( ring_i7582117978422105628t_unit @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( H2 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_102_ring__iso__memE_I1_J,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_a @ ( H2 @ X2 ) @ ( partia707051561876973205xt_a_b @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_103_ring__iso__memE_I1_J,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_a @ ( H2 @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_104_ring__iso__memE_I1_J,axiom,
! [H2: list_list_a > list_list_a,R: partia2956882679547061052t_unit,S: partia2956882679547061052t_unit,X2: list_list_a] :
( ( member8231385768148312316list_a @ H2 @ ( ring_i6186174840089424918t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( H2 @ X2 ) @ ( partia2464479390973590831t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_105_subcring_Oaxioms_I1_J,axiom,
! [H: set_set_a,R: partia6043505979758434576t_unit] :
( ( subcri4445174380595745425t_unit @ H @ R )
=> ( subrin1511138061850335568t_unit @ H @ R ) ) ).
% subcring.axioms(1)
thf(fact_106_subcring_Oaxioms_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( subring_a_b @ H @ R ) ) ).
% subcring.axioms(1)
thf(fact_107_subcring_Oaxioms_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( subrin6918843898125473962t_unit @ H @ R ) ) ).
% subcring.axioms(1)
thf(fact_108_noetherian__ring_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_n5188127996776581661t_unit @ R )
=> ( ring_l6212528067271185461t_unit @ R ) ) ).
% noetherian_ring.axioms(1)
thf(fact_109_noetherian__ring_Oaxioms_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_n3639167112692572309ng_a_b @ R )
=> ( ring_a_b @ R ) ) ).
% noetherian_ring.axioms(1)
thf(fact_110_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_p8098905331641078952t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_111_principal__domain_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_p2862007038493914190t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% principal_domain.axioms(1)
thf(fact_112_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_113_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_n4705423059119889713t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_114_noetherian__domain_Oaxioms_I2_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_n3212398840814694743t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% noetherian_domain.axioms(2)
thf(fact_115_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_116_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_f796907574329358751t_unit @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_117_factorial__domain_Oaxioms_I1_J,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_f6820247627256571077t_unit @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ).
% factorial_domain.axioms(1)
thf(fact_118_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_119_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_120_ring_Ocarrier__is__subring,axiom,
! [R: partia6043505979758434576t_unit] :
( ( ring_s5549108798478166619t_unit @ R )
=> ( subrin1511138061850335568t_unit @ ( partia5907974310037520643t_unit @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_121_ring_Ocarrier__is__subring,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( subring_a_b @ ( partia707051561876973205xt_a_b @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_122_ring_Ocarrier__is__subring,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( subrin6918843898125473962t_unit @ ( partia5361259788508890537t_unit @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_123_ring_Ocarrier__is__subring,axiom,
! [R: partia2956882679547061052t_unit] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( subrin3541368690557094692t_unit @ ( partia2464479390973590831t_unit @ R ) @ R ) ) ).
% ring.carrier_is_subring
thf(fact_124_b,axiom,
member_list_a @ p @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% b
thf(fact_125_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q: list_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_126_x_OsubcringI,axiom,
! [H: set_list_a] :
( ( subrin6918843898125473962t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H1 @ H22 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H22 @ H1 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subcringI
thf(fact_127_x_Opoly__of__const__in__carrier,axiom,
! [S2: list_a] :
( ( member_list_a @ S2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_list_a @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ S2 ) @ ( 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 ) ) ) ) ) ) ) ).
% x.poly_of_const_in_carrier
thf(fact_128_var__closed_I1_J,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( member_list_a @ ( var_a_b @ r ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% var_closed(1)
thf(fact_129_x_Osubcring__inter,axiom,
! [I2: set_list_a,J: set_list_a] :
( ( subcri7763218559781929323t_unit @ I2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subcri7763218559781929323t_unit @ J @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subcri7763218559781929323t_unit @ ( inf_inf_set_list_a @ I2 @ J ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subcring_inter
thf(fact_130_x_Osubring__inter,axiom,
! [I2: set_list_a,J: set_list_a] :
( ( subrin6918843898125473962t_unit @ I2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( subrin6918843898125473962t_unit @ J @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( subrin6918843898125473962t_unit @ ( inf_inf_set_list_a @ I2 @ J ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subring_inter
thf(fact_131_x_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
!= bot_bot_set_list_a ) ).
% x.carrier_not_empty
thf(fact_132_subcringI,axiom,
! [H: set_a] :
( ( subring_a_b @ H @ r )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ r @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ r @ H22 @ H1 ) ) ) )
=> ( subcring_a_b @ H @ r ) ) ) ).
% subcringI
thf(fact_133_poly__of__const__in__carrier,axiom,
! [S2: a] :
( ( member_a @ S2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( poly_of_const_a_b @ r @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% poly_of_const_in_carrier
thf(fact_134_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_135_m__lcomm,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_136_m__comm,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X2 ) ) ) ) ).
% m_comm
thf(fact_137_m__assoc,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X2 @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_138_x_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 ) ) ) ).
% x.cgenideal_self
thf(fact_139_x_Om__lcomm,axiom,
! [X2: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Z ) ) ) ) ) ) ).
% x.m_lcomm
thf(fact_140_x_Om__comm,axiom,
! [X2: list_a,Y: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X2 ) ) ) ) ).
% x.m_comm
thf(fact_141_x_Om__assoc,axiom,
! [X2: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) @ Z )
= ( 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 ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.m_assoc
thf(fact_142_x_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 ) ) ) ) ).
% x.cgenideal_is_principalideal
thf(fact_143_const__term__simprules__shell_I2_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(2)
thf(fact_144_m__closed,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_145_x_Om__closed,axiom,
! [X2: list_a,Y: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.m_closed
thf(fact_146_x_Ominus__closed,axiom,
! [X2: list_a,Y: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.minus_closed
thf(fact_147_ring_Opoly__of__const_Ocong,axiom,
poly_o8716471131768098070t_unit = poly_o8716471131768098070t_unit ).
% ring.poly_of_const.cong
thf(fact_148_ring_Opoly__of__const_Ocong,axiom,
poly_of_const_a_b = poly_of_const_a_b ).
% ring.poly_of_const.cong
thf(fact_149_ring_Oconst__term_Ocong,axiom,
const_term_a_b = const_term_a_b ).
% ring.const_term.cong
thf(fact_150_ring_Oconst__term_Ocong,axiom,
const_6738166269504826821t_unit = const_6738166269504826821t_unit ).
% ring.const_term.cong
thf(fact_151_subringE_I6_J,axiom,
! [H: set_set_a,R: partia6043505979758434576t_unit,H12: set_a,H23: set_a] :
( ( subrin1511138061850335568t_unit @ H @ R )
=> ( ( member_set_a @ H12 @ H )
=> ( ( member_set_a @ H23 @ H )
=> ( member_set_a @ ( mult_s7930653359683758801t_unit @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_152_subringE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_153_subringE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subringE(6)
thf(fact_154_subcringE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_155_subcringE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subcringE(6)
thf(fact_156_subcring_Osub__m__comm,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ R @ H23 @ H12 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_157_subcring_Osub__m__comm,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 )
= ( mult_l7073676228092353617t_unit @ R @ H23 @ H12 ) ) ) ) ) ).
% subcring.sub_m_comm
thf(fact_158_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a,P: list_set_a,Q: list_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
=> ( ( member_list_set_a @ Q @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
=> ( ( const_4974875009045744619t_unit @ R @ ( mult_l4263563202070946897t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) @ P @ Q ) )
= ( mult_s7930653359683758801t_unit @ R @ ( const_4974875009045744619t_unit @ R @ P ) @ ( const_4974875009045744619t_unit @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_159_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ R @ ( const_term_a_b @ R @ P ) @ ( const_term_a_b @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_160_domain_Oconst__term__simprules__shell_I2_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) @ ( const_6738166269504826821t_unit @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(2)
thf(fact_161_subdomainE_I8_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 )
= ( mult_l7073676228092353617t_unit @ R @ H23 @ H12 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_162_subdomainE_I8_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H12 @ H23 )
= ( mult_a_ring_ext_a_b @ R @ H23 @ H12 ) ) ) ) ) ).
% subdomainE(8)
thf(fact_163_subdomainE_I6_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_164_subdomainE_I6_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( mult_a_ring_ext_a_b @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subdomainE(6)
thf(fact_165_ring_Osubring__inter,axiom,
! [R: partia6043505979758434576t_unit,I2: set_set_a,J: set_set_a] :
( ( ring_s5549108798478166619t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ I2 @ R )
=> ( ( subrin1511138061850335568t_unit @ J @ R )
=> ( subrin1511138061850335568t_unit @ ( inf_inf_set_set_a @ I2 @ J ) @ R ) ) ) ) ).
% ring.subring_inter
thf(fact_166_ring_Osubring__inter,axiom,
! [R: partia2670972154091845814t_unit,I2: set_list_a,J: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ I2 @ R )
=> ( ( subrin6918843898125473962t_unit @ J @ R )
=> ( subrin6918843898125473962t_unit @ ( inf_inf_set_list_a @ I2 @ J ) @ R ) ) ) ) ).
% ring.subring_inter
thf(fact_167_ring_Osubring__inter,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a,J: set_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ I2 @ R )
=> ( ( subring_a_b @ J @ R )
=> ( subring_a_b @ ( inf_inf_set_a @ I2 @ J ) @ R ) ) ) ) ).
% ring.subring_inter
thf(fact_168_ring__iso__memE_I2_J,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a,Y: a] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_169_ring__iso__memE_I2_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( mult_a_ring_ext_a_b @ R @ X2 @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_170_ring__iso__memE_I2_J,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_171_ring__iso__memE_I2_J,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( mult_l7073676228092353617t_unit @ R @ X2 @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_172_ring__iso__memE_I2_J,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R @ X2 @ Y ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_173_ring__iso__memE_I2_J,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( mult_l4853965630390486993t_unit @ R @ X2 @ Y ) )
= ( mult_a_ring_ext_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_174_ring_Osubcring__inter,axiom,
! [R: partia2670972154091845814t_unit,I2: set_list_a,J: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subcri7763218559781929323t_unit @ I2 @ R )
=> ( ( subcri7763218559781929323t_unit @ J @ R )
=> ( subcri7763218559781929323t_unit @ ( inf_inf_set_list_a @ I2 @ J ) @ R ) ) ) ) ).
% ring.subcring_inter
thf(fact_175_ring_Osubcring__inter,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a,J: set_a] :
( ( ring_a_b @ R )
=> ( ( subcring_a_b @ I2 @ R )
=> ( ( subcring_a_b @ J @ R )
=> ( subcring_a_b @ ( inf_inf_set_a @ I2 @ J ) @ R ) ) ) ) ).
% ring.subcring_inter
thf(fact_176_ring_OsubcringI,axiom,
! [R: partia6043505979758434576t_unit,H: set_set_a] :
( ( ring_s5549108798478166619t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ H @ R )
=> ( ! [H1: set_a,H22: set_a] :
( ( member_set_a @ H1 @ H )
=> ( ( member_set_a @ H22 @ H )
=> ( ( mult_s7930653359683758801t_unit @ R @ H1 @ H22 )
= ( mult_s7930653359683758801t_unit @ R @ H22 @ H1 ) ) ) )
=> ( subcri4445174380595745425t_unit @ H @ R ) ) ) ) ).
% ring.subcringI
thf(fact_177_ring_OsubcringI,axiom,
! [R: partia2175431115845679010xt_a_b,H: set_a] :
( ( ring_a_b @ R )
=> ( ( subring_a_b @ H @ R )
=> ( ! [H1: a,H22: a] :
( ( member_a @ H1 @ H )
=> ( ( member_a @ H22 @ H )
=> ( ( mult_a_ring_ext_a_b @ R @ H1 @ H22 )
= ( mult_a_ring_ext_a_b @ R @ H22 @ H1 ) ) ) )
=> ( subcring_a_b @ H @ R ) ) ) ) ).
% ring.subcringI
thf(fact_178_ring_OsubcringI,axiom,
! [R: partia2670972154091845814t_unit,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ! [H1: list_a,H22: list_a] :
( ( member_list_a @ H1 @ H )
=> ( ( member_list_a @ H22 @ H )
=> ( ( mult_l7073676228092353617t_unit @ R @ H1 @ H22 )
= ( mult_l7073676228092353617t_unit @ R @ H22 @ H1 ) ) ) )
=> ( subcri7763218559781929323t_unit @ H @ R ) ) ) ) ).
% ring.subcringI
thf(fact_179_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a,P: list_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
=> ( member_set_a @ ( const_4974875009045744619t_unit @ R @ P ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_180_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_a @ ( const_term_a_b @ R @ P ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_181_domain_Oconst__term__simprules__shell_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ R @ P ) @ K ) ) ) ) ).
% domain.const_term_simprules_shell(1)
thf(fact_182_ring_Opoly__of__const__in__carrier,axiom,
! [R: partia6043505979758434576t_unit,S2: set_a] :
( ( ring_s5549108798478166619t_unit @ R )
=> ( ( member_set_a @ S2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_list_set_a @ ( poly_o9124539423244288956t_unit @ R @ S2 ) @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ).
% ring.poly_of_const_in_carrier
thf(fact_183_ring_Opoly__of__const__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,S2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ S2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( poly_of_const_a_b @ R @ S2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ).
% ring.poly_of_const_in_carrier
thf(fact_184_ring_Opoly__of__const__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,S2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ S2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( poly_o8716471131768098070t_unit @ R @ S2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ).
% ring.poly_of_const_in_carrier
thf(fact_185_ring_Opoly__of__const__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,S2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ S2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member5342144027231129785list_a @ ( poly_o1617770581224298896t_unit @ R @ S2 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ).
% ring.poly_of_const_in_carrier
thf(fact_186_domain_Ovar__closed_I1_J,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( member_list_set_a @ ( var_se2415970144172829891t_unit @ R ) @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_187_domain_Ovar__closed_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( member_list_a @ ( var_a_b @ R ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_188_domain_Ovar__closed_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( member_list_list_a @ ( var_li8453953174693405341t_unit @ R ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ).
% domain.var_closed(1)
thf(fact_189_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a,Q: list_set_a,P: list_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( ( member_list_set_a @ Q @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
=> ( ( a_minu6204024409777241716t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) @ P @ Q )
= ( a_minu6204024409777241716t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_190_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2956882679547061052t_unit,K: set_list_list_a,Q: list_list_list_a,P: list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( subrin3541368690557094692t_unit @ K @ R )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) ) )
=> ( ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ K ) @ P @ Q )
= ( a_minu4820293213911669576t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_191_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,Q: list_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_192_domain_Ouniv__poly__a__minus__consistent,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,Q: list_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q )
= ( a_minu2241224857956505934t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q ) ) ) ) ) ).
% domain.univ_poly_a_minus_consistent
thf(fact_193_x_Omonoid__cancelI,axiom,
( ! [A3: list_a,B: list_a,C: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ A3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C @ B ) )
=> ( ( member_list_a @ A3 @ ( 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 ) ) ) )
=> ( A3 = B ) ) ) ) )
=> ( ! [A3: list_a,B: list_a,C: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A3 @ C )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B @ C ) )
=> ( ( member_list_a @ A3 @ ( 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 ) ) ) )
=> ( A3 = B ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.monoid_cancelI
thf(fact_194_is__root__poly__mult__imp__is__root,axiom,
! [P: list_a,Q: list_a,X2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ X2 )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X2 ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_195_monoid__cancelI,axiom,
( ! [A3: a,B: a,C: a] :
( ( ( mult_a_ring_ext_a_b @ r @ C @ A3 )
= ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B ) ) ) ) )
=> ( ! [A3: a,B: a,C: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A3 @ C )
= ( mult_a_ring_ext_a_b @ r @ B @ C ) )
=> ( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A3 = B ) ) ) ) )
=> ( monoid5798828371819920185xt_a_b @ r ) ) ) ).
% monoid_cancelI
thf(fact_196_x_Osubalgebra__inter,axiom,
! [K: set_list_a,V: set_list_a,V2: set_list_a] :
( ( embedd1768981623711841426t_unit @ K @ V @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K @ V2 @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K @ ( inf_inf_set_list_a @ V @ V2 ) @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.subalgebra_inter
thf(fact_197_inf__bot__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_198_inf__bot__left,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X2 )
= bot_bot_set_list_a ) ).
% inf_bot_left
thf(fact_199_inf__bot__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_200_inf__bot__right,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% inf_bot_right
thf(fact_201_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_202_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X2 )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_203_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_204_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_205_map__norm__in__poly__ring__carrier,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ).
% map_norm_in_poly_ring_carrier
thf(fact_206_x_Oadd__pow__rdistr__int,axiom,
! [A: list_a,B2: list_a,K2: int] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ B2 ) )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B2 ) ) ) ) ) ).
% x.add_pow_rdistr_int
thf(fact_207_subring__inter,axiom,
! [I2: set_a,J: set_a] :
( ( subring_a_b @ I2 @ r )
=> ( ( subring_a_b @ J @ r )
=> ( subring_a_b @ ( inf_inf_set_a @ I2 @ J ) @ r ) ) ) ).
% subring_inter
thf(fact_208_subcring__inter,axiom,
! [I2: set_a,J: set_a] :
( ( subcring_a_b @ I2 @ r )
=> ( ( subcring_a_b @ J @ r )
=> ( subcring_a_b @ ( inf_inf_set_a @ I2 @ J ) @ r ) ) ) ).
% subcring_inter
thf(fact_209_inf__right__idem,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ Y )
= ( inf_inf_set_list_a @ X2 @ Y ) ) ).
% inf_right_idem
thf(fact_210_inf__right__idem,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y )
= ( inf_inf_set_a @ X2 @ Y ) ) ).
% inf_right_idem
thf(fact_211_inf_Oright__idem,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ B2 )
= ( inf_inf_set_list_a @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_212_inf_Oright__idem,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_213_inf__left__idem,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ X2 @ Y ) )
= ( inf_inf_set_list_a @ X2 @ Y ) ) ).
% inf_left_idem
thf(fact_214_inf__left__idem,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y ) )
= ( inf_inf_set_a @ X2 @ Y ) ) ).
% inf_left_idem
thf(fact_215_inf_Oleft__idem,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ A @ B2 ) )
= ( inf_inf_set_list_a @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_216_inf_Oleft__idem,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B2 ) )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_217_inf__idem,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_218_inf__idem,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_219_inf_Oidem,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_220_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_221_x_Oadd__pow__ldistr__int,axiom,
! [A: list_a,B2: list_a,K2: int] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ A ) @ B2 )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B2 ) ) ) ) ) ).
% x.add_pow_ldistr_int
thf(fact_222_x_Oadd_Oint__pow__closed,axiom,
! [X2: list_a,I: int] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.add.int_pow_closed
thf(fact_223_ring_Ois__root_Ocong,axiom,
polyno4133073214067823460ot_a_b = polyno4133073214067823460ot_a_b ).
% ring.is_root.cong
thf(fact_224_ring_Ois__root_Ocong,axiom,
polyno6951661231331188332t_unit = polyno6951661231331188332t_unit ).
% ring.is_root.cong
thf(fact_225_inf__left__commute,axiom,
! [X2: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( inf_inf_set_list_a @ Y @ ( inf_inf_set_list_a @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_226_inf__left__commute,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_227_inf_Oleft__commute,axiom,
! [B2: set_list_a,A: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ B2 @ ( inf_inf_set_list_a @ A @ C2 ) )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_228_inf_Oleft__commute,axiom,
! [B2: set_a,A: set_a,C2: set_a] :
( ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A @ C2 ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_229_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_list_a,K2: set_list_a,B2: set_list_a,A: set_list_a] :
( ( B3
= ( inf_inf_set_list_a @ K2 @ B2 ) )
=> ( ( inf_inf_set_list_a @ A @ B3 )
= ( inf_inf_set_list_a @ K2 @ ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_230_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_a,K2: set_a,B2: set_a,A: set_a] :
( ( B3
= ( inf_inf_set_a @ K2 @ B2 ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= ( inf_inf_set_a @ K2 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_231_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_list_a,K2: set_list_a,A: set_list_a,B2: set_list_a] :
( ( A2
= ( inf_inf_set_list_a @ K2 @ A ) )
=> ( ( inf_inf_set_list_a @ A2 @ B2 )
= ( inf_inf_set_list_a @ K2 @ ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_232_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K2: set_a,A: set_a,B2: set_a] :
( ( A2
= ( inf_inf_set_a @ K2 @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= ( inf_inf_set_a @ K2 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_233_inf__commute,axiom,
( inf_inf_set_list_a
= ( ^ [X: set_list_a,Y2: set_list_a] : ( inf_inf_set_list_a @ Y2 @ X ) ) ) ).
% inf_commute
thf(fact_234_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X ) ) ) ).
% inf_commute
thf(fact_235_inf_Ocommute,axiom,
( inf_inf_set_list_a
= ( ^ [A4: set_list_a,B4: set_list_a] : ( inf_inf_set_list_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_236_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_237_inf__assoc,axiom,
! [X2: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ Z )
= ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_238_inf__assoc,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Z )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_239_inf_Oassoc,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_240_inf_Oassoc,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_241_inf__sup__aci_I1_J,axiom,
( inf_inf_set_list_a
= ( ^ [X: set_list_a,Y2: set_list_a] : ( inf_inf_set_list_a @ Y2 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_242_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y2: set_a] : ( inf_inf_set_a @ Y2 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_243_inf__sup__aci_I2_J,axiom,
! [X2: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ Z )
= ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_244_inf__sup__aci_I2_J,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Z )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_245_inf__sup__aci_I3_J,axiom,
! [X2: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( inf_inf_set_list_a @ Y @ ( inf_inf_set_list_a @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_246_inf__sup__aci_I3_J,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_247_inf__sup__aci_I4_J,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ X2 @ Y ) )
= ( inf_inf_set_list_a @ X2 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_248_inf__sup__aci_I4_J,axiom,
! [X2: set_a,Y: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y ) )
= ( inf_inf_set_a @ X2 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_249_domain_Omap__norm__in__poly__ring__carrier,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a,P: list_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
=> ( member6735289990665601811_set_a @ ( map_set_a_list_set_a @ ( poly_o9124539423244288956t_unit @ R ) @ P ) @ ( partia5991483576489006735t_unit @ ( univ_p4425688966569699694t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) ) ) ) ) ) ) ).
% domain.map_norm_in_poly_ring_carrier
thf(fact_250_domain_Omap__norm__in__poly__ring__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ R ) @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ) ) ) ) ).
% domain.map_norm_in_poly_ring_carrier
thf(fact_251_domain_Omap__norm__in__poly__ring__carrier,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ ( poly_o8716471131768098070t_unit @ R ) @ P ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ) ) ) ) ).
% domain.map_norm_in_poly_ring_carrier
thf(fact_252_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia6043505979758434576t_unit,P: list_set_a,Q: list_set_a,X2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
=> ( ( member_list_set_a @ Q @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
=> ( ( polyno4890645956962836498t_unit @ R @ ( mult_l4263563202070946897t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) @ P @ Q ) @ X2 )
=> ( ( polyno4890645956962836498t_unit @ R @ P @ X2 )
| ( polyno4890645956962836498t_unit @ R @ Q @ X2 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_253_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,Q: list_list_list_a,X2: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member5342144027231129785list_a @ Q @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( polyno5142720416380192742t_unit @ R @ ( mult_l3065349954589089105t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ P @ Q ) @ X2 )
=> ( ( polyno5142720416380192742t_unit @ R @ P @ X2 )
| ( polyno5142720416380192742t_unit @ R @ Q @ X2 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_254_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a,X2: a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ R @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ P @ Q ) @ X2 )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X2 )
| ( polyno4133073214067823460ot_a_b @ R @ Q @ X2 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_255_domain_Ois__root__poly__mult__imp__is__root,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a,X2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ R @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ P @ Q ) @ X2 )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X2 )
| ( polyno6951661231331188332t_unit @ R @ Q @ X2 ) ) ) ) ) ) ).
% domain.is_root_poly_mult_imp_is_root
thf(fact_256_eval__rewrite,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ K ) @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P ) @ ( var_a_b @ r ) ) ) ) ) ).
% eval_rewrite
thf(fact_257_norm__map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_a_list_a @ F @ P ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% norm_map_in_poly_ring_carrier
thf(fact_258_x_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 ) ) ) ) ).
% x.carrier_is_subalgebra
thf(fact_259_x_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 ) ) ) ) ) ).
% x.subalgebra_in_carrier
thf(fact_260_x_Ocgenideal__prod,axiom,
! [A: list_a,B2: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( set_mu3586181839180059898t_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 ) ) @ B2 ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B2 ) ) ) ) ) ).
% x.cgenideal_prod
thf(fact_261_x_Oadd_Oint__pow__pow,axiom,
! [X2: list_a,M: int,N: int] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ M @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ N @ X2 ) )
= ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( times_times_int @ N @ M ) @ X2 ) ) ) ).
% x.add.int_pow_pow
thf(fact_262_x_Oproperfactor__prod__l,axiom,
! [A: list_a,B2: list_a,C2: list_a] :
( ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( 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 ) ) ) )
=> ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B2 ) ) ) ) ) ) ).
% x.properfactor_prod_l
thf(fact_263_x_Oproperfactor__prod__r,axiom,
! [A: list_a,B2: list_a,C2: list_a] :
( ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B2 )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( 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 ) ) ) )
=> ( proper8313688649498433056t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B2 @ C2 ) ) ) ) ) ) ).
% x.properfactor_prod_r
thf(fact_264_x_Oadd_Oint__pow__distrib,axiom,
! [X2: list_a,Y: list_a,I: int] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X2 ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ Y ) ) ) ) ) ).
% x.add.int_pow_distrib
thf(fact_265_x_Oadd_Or__cancel,axiom,
! [A: list_a,C2: list_a,B2: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ C2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B2 @ C2 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( 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 ) ) ) )
=> ( A = B2 ) ) ) ) ) ).
% x.add.r_cancel
thf(fact_266_x_Oadd_Om__lcomm,axiom,
! [X2: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Z ) ) ) ) ) ) ).
% x.add.m_lcomm
thf(fact_267_x_Oadd_Om__comm,axiom,
! [X2: list_a,Y: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X2 ) ) ) ) ).
% x.add.m_comm
thf(fact_268_x_Oadd_Om__assoc,axiom,
! [X2: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) @ Z )
= ( 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 ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.add.m_assoc
thf(fact_269_x_Oadd_Ol__cancel,axiom,
! [C2: list_a,A: list_a,B2: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ C2 @ B2 ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( 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 ) ) ) )
=> ( A = B2 ) ) ) ) ) ).
% x.add.l_cancel
thf(fact_270_x_Oeval__var,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( var_li8453953174693405341t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 )
= X2 ) ) ).
% x.eval_var
thf(fact_271_x_Or__distr,axiom,
! [X2: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X2 ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ Y ) ) ) ) ) ) ).
% x.r_distr
thf(fact_272_x_Ol__distr,axiom,
! [X2: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) @ Z )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Z ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ Z ) ) ) ) ) ) ).
% x.l_distr
thf(fact_273_x_Oadd_Oint__pow__mult__distrib,axiom,
! [X2: list_a,Y: list_a,I: int] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X2 ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X2 ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ Y ) ) ) ) ) ) ).
% x.add.int_pow_mult_distrib
thf(fact_274_x_Oeval__poly__of__const,axiom,
! [X2: list_a,Y: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 ) @ Y )
= X2 ) ) ).
% x.eval_poly_of_const
thf(fact_275_x_Oset__mult__closed,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_mu3586181839180059898t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.set_mult_closed
thf(fact_276_x_Oeval__in__carrier__2,axiom,
! [X2: list_list_a,Y: list_a] :
( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier_2
thf(fact_277_le__inf__iff,axiom,
! [X2: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ Y @ Z ) )
= ( ( ord_le8861187494160871172list_a @ X2 @ Y )
& ( ord_le8861187494160871172list_a @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_278_le__inf__iff,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X2 @ Y )
& ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_279_le__inf__iff,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_280_le__inf__iff,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X2 @ ( inf_inf_int @ Y @ Z ) )
= ( ( ord_less_eq_int @ X2 @ Y )
& ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_281_inf_Obounded__iff,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B2 @ C2 ) )
= ( ( ord_le8861187494160871172list_a @ A @ B2 )
& ( ord_le8861187494160871172list_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_282_inf_Obounded__iff,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_283_inf_Obounded__iff,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_284_inf_Obounded__iff,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B2 @ C2 ) )
= ( ( ord_less_eq_int @ A @ B2 )
& ( ord_less_eq_int @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_285_x_Oadd_Oright__cancel,axiom,
! [X2: list_a,Y: list_a,Z: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ X2 ) )
= ( Y = Z ) ) ) ) ) ).
% x.add.right_cancel
thf(fact_286_x_Oadd_Om__closed,axiom,
! [X2: list_a,Y: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.m_closed
thf(fact_287_ring_Onormalize_Ocong,axiom,
normal637505603836502915t_unit = normal637505603836502915t_unit ).
% ring.normalize.cong
thf(fact_288_ring_Onormalize_Ocong,axiom,
normalize_a_b = normalize_a_b ).
% ring.normalize.cong
thf(fact_289_ring_Oeval_Ocong,axiom,
eval_l34571156754992824t_unit = eval_l34571156754992824t_unit ).
% ring.eval.cong
thf(fact_290_ring_Oeval_Ocong,axiom,
eval_a_b = eval_a_b ).
% ring.eval.cong
thf(fact_291_subringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_292_subringE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_293_subringE_I7_J,axiom,
! [H: set_set_a,R: partia6043505979758434576t_unit,H12: set_a,H23: set_a] :
( ( subrin1511138061850335568t_unit @ H @ R )
=> ( ( member_set_a @ H12 @ H )
=> ( ( member_set_a @ H23 @ H )
=> ( member_set_a @ ( add_se3735415688806051380t_unit @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subringE(7)
thf(fact_294_inf__sup__ord_I2_J,axiom,
! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_295_inf__sup__ord_I2_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_296_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_297_inf__sup__ord_I2_J,axiom,
! [X2: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_298_inf__sup__ord_I1_J,axiom,
! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_299_inf__sup__ord_I1_J,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_300_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_301_inf__sup__ord_I1_J,axiom,
! [X2: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_302_inf__le1,axiom,
! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_303_inf__le1,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_304_inf__le1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_305_inf__le1,axiom,
! [X2: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_306_inf__le2,axiom,
! [X2: set_list_a,Y: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_307_inf__le2,axiom,
! [X2: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_308_inf__le2,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_309_inf__le2,axiom,
! [X2: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_310_le__infE,axiom,
! [X2: set_list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ A @ B2 ) )
=> ~ ( ( ord_le8861187494160871172list_a @ X2 @ A )
=> ~ ( ord_le8861187494160871172list_a @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_311_le__infE,axiom,
! [X2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B2 ) )
=> ~ ( ( ord_less_eq_set_a @ X2 @ A )
=> ~ ( ord_less_eq_set_a @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_312_le__infE,axiom,
! [X2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A )
=> ~ ( ord_less_eq_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_313_le__infE,axiom,
! [X2: int,A: int,B2: int] :
( ( ord_less_eq_int @ X2 @ ( inf_inf_int @ A @ B2 ) )
=> ~ ( ( ord_less_eq_int @ X2 @ A )
=> ~ ( ord_less_eq_int @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_314_le__infI,axiom,
! [X2: set_list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ A )
=> ( ( ord_le8861187494160871172list_a @ X2 @ B2 )
=> ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_315_le__infI,axiom,
! [X2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ B2 )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_316_le__infI,axiom,
! [X2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A )
=> ( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_317_le__infI,axiom,
! [X2: int,A: int,B2: int] :
( ( ord_less_eq_int @ X2 @ A )
=> ( ( ord_less_eq_int @ X2 @ B2 )
=> ( ord_less_eq_int @ X2 @ ( inf_inf_int @ A @ B2 ) ) ) ) ).
% le_infI
thf(fact_318_inf__mono,axiom,
! [A: set_list_a,C2: set_list_a,B2: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B2 @ D )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ ( inf_inf_set_list_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_319_inf__mono,axiom,
! [A: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_320_inf__mono,axiom,
! [A: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_321_inf__mono,axiom,
! [A: int,C2: int,B2: int,D: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ( ord_less_eq_int @ B2 @ D )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B2 ) @ ( inf_inf_int @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_322_le__infI1,axiom,
! [A: set_list_a,X2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ X2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_323_le__infI1,axiom,
! [A: set_a,X2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_324_le__infI1,axiom,
! [A: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_325_le__infI1,axiom,
! [A: int,X2: int,B2: int] :
( ( ord_less_eq_int @ A @ X2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_326_le__infI2,axiom,
! [B2: set_list_a,X2: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ X2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_327_le__infI2,axiom,
! [B2: set_a,X2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_328_le__infI2,axiom,
! [B2: nat,X2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_329_le__infI2,axiom,
! [B2: int,X2: int,A: int] :
( ( ord_less_eq_int @ B2 @ X2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_330_inf_OorderE,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( A
= ( inf_inf_set_list_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_331_inf_OorderE,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( A
= ( inf_inf_set_a @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_332_inf_OorderE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( A
= ( inf_inf_nat @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_333_inf_OorderE,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( A
= ( inf_inf_int @ A @ B2 ) ) ) ).
% inf.orderE
thf(fact_334_inf_OorderI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( A
= ( inf_inf_set_list_a @ A @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_335_inf_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% inf.orderI
thf(fact_336_inf_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( inf_inf_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% inf.orderI
thf(fact_337_inf_OorderI,axiom,
! [A: int,B2: int] :
( ( A
= ( inf_inf_int @ A @ B2 ) )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% inf.orderI
thf(fact_338_inf__unique,axiom,
! [F: set_list_a > set_list_a > set_list_a,X2: set_list_a,Y: set_list_a] :
( ! [X3: set_list_a,Y3: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_list_a,Y3: set_list_a] : ( ord_le8861187494160871172list_a @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_list_a,Y3: set_list_a,Z2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ X3 @ Z2 )
=> ( ord_le8861187494160871172list_a @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_list_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_339_inf__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_a,Y3: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Z2 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_340_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_341_inf__unique,axiom,
! [F: int > int > int,X2: int,Y: int] :
( ! [X3: int,Y3: int] : ( ord_less_eq_int @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: int,Y3: int,Z2: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ( ord_less_eq_int @ X3 @ Z2 )
=> ( ord_less_eq_int @ X3 @ ( F @ Y3 @ Z2 ) ) ) )
=> ( ( inf_inf_int @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_342_le__iff__inf,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X: set_list_a,Y2: set_list_a] :
( ( inf_inf_set_list_a @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_343_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_344_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y2: nat] :
( ( inf_inf_nat @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_345_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y2: int] :
( ( inf_inf_int @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_346_inf_Oabsorb1,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( inf_inf_set_list_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_347_inf_Oabsorb1,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( inf_inf_set_a @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_348_inf_Oabsorb1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( inf_inf_nat @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_349_inf_Oabsorb1,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( inf_inf_int @ A @ B2 )
= A ) ) ).
% inf.absorb1
thf(fact_350_inf_Oabsorb2,axiom,
! [B2: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ A )
=> ( ( inf_inf_set_list_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_351_inf_Oabsorb2,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( inf_inf_set_a @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_352_inf_Oabsorb2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( inf_inf_nat @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_353_inf_Oabsorb2,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( inf_inf_int @ A @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_354_inf__absorb1,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y )
=> ( ( inf_inf_set_list_a @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_355_inf__absorb1,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( inf_inf_set_a @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_356_inf__absorb1,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( inf_inf_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_357_inf__absorb1,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( inf_inf_int @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_358_inf__absorb2,axiom,
! [Y: set_list_a,X2: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y @ X2 )
=> ( ( inf_inf_set_list_a @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_359_inf__absorb2,axiom,
! [Y: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y @ X2 )
=> ( ( inf_inf_set_a @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_360_inf__absorb2,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_361_inf__absorb2,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( inf_inf_int @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_362_inf_OboundedE,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B2 @ C2 ) )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ~ ( ord_le8861187494160871172list_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_363_inf_OboundedE,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_364_inf_OboundedE,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A @ B2 )
=> ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_365_inf_OboundedE,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_int @ A @ B2 )
=> ~ ( ord_less_eq_int @ A @ C2 ) ) ) ).
% inf.boundedE
thf(fact_366_inf_OboundedI,axiom,
! [A: set_list_a,B2: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ord_le8861187494160871172list_a @ A @ ( inf_inf_set_list_a @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_367_inf_OboundedI,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_368_inf_OboundedI,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_369_inf_OboundedI,axiom,
! [A: int,B2: int,C2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ A @ ( inf_inf_int @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_370_inf__greatest,axiom,
! [X2: set_list_a,Y: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Z )
=> ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_371_inf__greatest,axiom,
! [X2: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
=> ( ( ord_less_eq_set_a @ X2 @ Z )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_372_inf__greatest,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_373_inf__greatest,axiom,
! [X2: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ X2 @ Z )
=> ( ord_less_eq_int @ X2 @ ( inf_inf_int @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_374_inf_Oorder__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B4: set_list_a] :
( A4
= ( inf_inf_set_list_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_375_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( A4
= ( inf_inf_set_a @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_376_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_377_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( A4
= ( inf_inf_int @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_378_inf_Ocobounded1,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_379_inf_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_380_inf_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_381_inf_Ocobounded1,axiom,
! [A: int,B2: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B2 ) @ A ) ).
% inf.cobounded1
thf(fact_382_inf_Ocobounded2,axiom,
! [A: set_list_a,B2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_383_inf_Ocobounded2,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_384_inf_Ocobounded2,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_385_inf_Ocobounded2,axiom,
! [A: int,B2: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_386_inf_Oabsorb__iff1,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B4: set_list_a] :
( ( inf_inf_set_list_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_387_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_388_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_389_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( inf_inf_int @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_390_inf_Oabsorb__iff2,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B4: set_list_a,A4: set_list_a] :
( ( inf_inf_set_list_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_391_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B4: set_a,A4: set_a] :
( ( inf_inf_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_392_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_393_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( inf_inf_int @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_394_inf_OcoboundedI1,axiom,
! [A: set_list_a,C2: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_395_inf_OcoboundedI1,axiom,
! [A: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_396_inf_OcoboundedI1,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_397_inf_OcoboundedI1,axiom,
! [A: int,C2: int,B2: int] :
( ( ord_less_eq_int @ A @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_398_inf_OcoboundedI2,axiom,
! [B2: set_list_a,C2: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B2 @ C2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_399_inf_OcoboundedI2,axiom,
! [B2: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_400_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_401_inf_OcoboundedI2,axiom,
! [B2: int,C2: int,A: int] :
( ( ord_less_eq_int @ B2 @ C2 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_402_subcringE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subcring_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_403_subcringE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subcringE(7)
thf(fact_404_subdomainE_I7_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( member_list_a @ ( add_li7652885771158616974t_unit @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_405_subdomainE_I7_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( member_a @ ( add_a_b @ R @ H12 @ H23 ) @ H ) ) ) ) ).
% subdomainE(7)
thf(fact_406_ring__iso__memE_I3_J,axiom,
! [H2: a > list_a,R: partia2175431115845679010xt_a_b,S: partia2670972154091845814t_unit,X2: a,Y: a] :
( ( member_a_list_a @ H2 @ ( ring_i4557880751517319194t_unit @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_407_ring__iso__memE_I3_J,axiom,
! [H2: a > a,R: partia2175431115845679010xt_a_b,S: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( member_a_a @ H2 @ ( ring_iso_a_b_a_b @ R @ S ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( H2 @ ( add_a_b @ R @ X2 @ Y ) )
= ( add_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_408_ring__iso__memE_I3_J,axiom,
! [H2: list_a > list_a,R: partia2670972154091845814t_unit,S: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( member_list_a_list_a @ H2 @ ( ring_i7414513579304222626t_unit @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_409_ring__iso__memE_I3_J,axiom,
! [H2: list_a > a,R: partia2670972154091845814t_unit,S: partia2175431115845679010xt_a_b,X2: list_a,Y: list_a] :
( ( member_list_a_a @ H2 @ ( ring_i7048835797181109658it_a_b @ R @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( H2 @ ( add_li7652885771158616974t_unit @ R @ X2 @ Y ) )
= ( add_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_410_ring__iso__memE_I3_J,axiom,
! [H2: list_list_a > list_a,R: partia2956882679547061052t_unit,S: partia2670972154091845814t_unit,X2: list_list_a,Y: list_list_a] :
( ( member7168557129179038582list_a @ H2 @ ( ring_i4611353245267337884t_unit @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( add_li174743652000525320t_unit @ R @ X2 @ Y ) )
= ( add_li7652885771158616974t_unit @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_411_ring__iso__memE_I3_J,axiom,
! [H2: list_list_a > a,R: partia2956882679547061052t_unit,S: partia2175431115845679010xt_a_b,X2: list_list_a,Y: list_list_a] :
( ( member_list_list_a_a @ H2 @ ( ring_i5684343068699926420it_a_b @ R @ S ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( H2 @ ( add_li174743652000525320t_unit @ R @ X2 @ Y ) )
= ( add_a_b @ S @ ( H2 @ X2 ) @ ( H2 @ Y ) ) ) ) ) ) ).
% ring_iso_memE(3)
thf(fact_412_subringE_I1_J,axiom,
! [H: set_set_a,R: partia6043505979758434576t_unit] :
( ( subrin1511138061850335568t_unit @ H @ R )
=> ( ord_le3724670747650509150_set_a @ H @ ( partia5907974310037520643t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_413_subringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subringE(1)
thf(fact_414_subringE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_415_subringE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subrin3541368690557094692t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subringE(1)
thf(fact_416_subcringE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subcringE(1)
thf(fact_417_subcringE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_418_subcringE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subcri8676831449680469861t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subcringE(1)
thf(fact_419_subdomainE_I1_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( ord_less_eq_set_a @ H @ ( partia707051561876973205xt_a_b @ R ) ) ) ).
% subdomainE(1)
thf(fact_420_subdomainE_I1_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ord_le8861187494160871172list_a @ H @ ( partia5361259788508890537t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_421_subdomainE_I1_J,axiom,
! [H: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( subdom561091866123308472t_unit @ H @ R )
=> ( ord_le8488217952732425610list_a @ H @ ( partia2464479390973590831t_unit @ R ) ) ) ).
% subdomainE(1)
thf(fact_422_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a,P: list_set_a,Q: list_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
=> ( ( member_list_set_a @ Q @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
=> ( ( const_4974875009045744619t_unit @ R @ ( add_li6048918997158153774t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) @ P @ Q ) )
= ( add_se3735415688806051380t_unit @ R @ ( const_4974875009045744619t_unit @ R @ P ) @ ( const_4974875009045744619t_unit @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_423_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a,Q: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( const_term_a_b @ R @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ R @ K ) @ P @ Q ) )
= ( add_a_b @ R @ ( const_term_a_b @ R @ P ) @ ( const_term_a_b @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_424_domain_Oconst__term__simprules__shell_I3_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a,Q: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( member_list_list_a @ Q @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( const_6738166269504826821t_unit @ R @ ( add_li174743652000525320t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ R @ ( const_6738166269504826821t_unit @ R @ P ) @ ( const_6738166269504826821t_unit @ R @ Q ) ) ) ) ) ) ) ).
% domain.const_term_simprules_shell(3)
thf(fact_425_ring_Oeval__poly__of__const,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( poly_of_const_a_b @ R @ X2 ) @ Y )
= X2 ) ) ) ).
% ring.eval_poly_of_const
thf(fact_426_ring_Oeval__poly__of__const,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( poly_o8716471131768098070t_unit @ R @ X2 ) @ Y )
= X2 ) ) ) ).
% ring.eval_poly_of_const
thf(fact_427_ring_Oeval__poly__of__const,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,Y: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ ( poly_o1617770581224298896t_unit @ R @ X2 ) @ Y )
= X2 ) ) ) ).
% ring.eval_poly_of_const
thf(fact_428_ring_Oeval__var,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( var_a_b @ R ) @ X2 )
= X2 ) ) ) ).
% ring.eval_var
thf(fact_429_ring_Oeval__var,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( var_li8453953174693405341t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% ring.eval_var
thf(fact_430_ring_Oeval__var,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ ( var_li3532061862469730199t_unit @ R ) @ X2 )
= X2 ) ) ) ).
% ring.eval_var
thf(fact_431_cring_Ocgenideal__prod,axiom,
! [R: partia3925755165846298134t_unit,A: list_set_a,B2: list_set_a] :
( ( cring_7032029500657136448t_unit @ R )
=> ( ( member_list_set_a @ A @ ( partia3893404292425143049t_unit @ R ) )
=> ( ( member_list_set_a @ B2 @ ( partia3893404292425143049t_unit @ R ) )
=> ( ( set_mu252163003341231354t_unit @ R @ ( cgenid5493546629940232227t_unit @ R @ A ) @ ( cgenid5493546629940232227t_unit @ R @ B2 ) )
= ( cgenid5493546629940232227t_unit @ R @ ( mult_l4263563202070946897t_unit @ R @ A @ B2 ) ) ) ) ) ) ).
% cring.cgenideal_prod
thf(fact_432_cring_Ocgenideal__prod,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( set_mu8047982887099575916xt_a_b @ R @ ( cgenid547466209912283029xt_a_b @ R @ A ) @ ( cgenid547466209912283029xt_a_b @ R @ B2 ) )
= ( cgenid547466209912283029xt_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ A @ B2 ) ) ) ) ) ) ).
% cring.cgenideal_prod
thf(fact_433_cring_Ocgenideal__prod,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B2: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( set_mu3586181839180059898t_unit @ R @ ( cgenid9131348535277946915t_unit @ R @ A ) @ ( cgenid9131348535277946915t_unit @ R @ B2 ) )
= ( cgenid9131348535277946915t_unit @ R @ ( mult_l7073676228092353617t_unit @ R @ A @ B2 ) ) ) ) ) ) ).
% cring.cgenideal_prod
thf(fact_434_cring_Ocgenideal__prod,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B2: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( set_mu4265036031142965114t_unit @ R @ ( cgenid24865672677839267t_unit @ R @ A ) @ ( cgenid24865672677839267t_unit @ R @ B2 ) )
= ( cgenid24865672677839267t_unit @ R @ ( mult_l4853965630390486993t_unit @ R @ A @ B2 ) ) ) ) ) ) ).
% cring.cgenideal_prod
thf(fact_435_ring_Oeval__in__carrier__2,axiom,
! [R: partia6043505979758434576t_unit,X2: list_set_a,Y: set_a] :
( ( ring_s5549108798478166619t_unit @ R )
=> ( ( member_list_set_a @ X2 @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_set_a @ ( eval_s4471923211119375966t_unit @ R @ X2 @ Y ) @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ).
% ring.eval_in_carrier_2
thf(fact_436_ring_Oeval__in__carrier__2,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_list_a,Y: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member5342144027231129785list_a @ X2 @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ( member_list_list_a @ Y @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( eval_l1088911609197519410t_unit @ R @ X2 @ Y ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% ring.eval_in_carrier_2
thf(fact_437_ring_Oeval__in__carrier__2,axiom,
! [R: partia2175431115845679010xt_a_b,X2: list_a,Y: a] :
( ( ring_a_b @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.eval_in_carrier_2
thf(fact_438_ring_Oeval__in__carrier__2,axiom,
! [R: partia2670972154091845814t_unit,X2: list_list_a,Y: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ R @ X2 @ Y ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.eval_in_carrier_2
thf(fact_439_domain_Onorm__map__in__poly__ring__carrier,axiom,
! [R: partia6043505979758434576t_unit,P: list_set_a,F: set_a > list_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_list_set_a @ ( F @ A3 ) @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) )
=> ( member6735289990665601811_set_a @ ( normal7614038334803337763t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) @ ( map_set_a_list_set_a @ F @ P ) ) @ ( partia5991483576489006735t_unit @ ( univ_p4425688966569699694t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.norm_map_in_poly_ring_carrier
thf(fact_440_domain_Onorm__map__in__poly__ring__carrier,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,F: list_list_a > list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ! [A3: list_list_a] :
( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member5342144027231129785list_a @ ( F @ A3 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) )
=> ( member6842060177613954879list_a @ ( normal5368706450718127095t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( map_li5227692475714150986list_a @ F @ P ) ) @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.norm_map_in_poly_ring_carrier
thf(fact_441_domain_Onorm__map__in__poly__ring__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,F: a > list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( map_a_list_a @ F @ P ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ) ) ).
% domain.norm_map_in_poly_ring_carrier
thf(fact_442_domain_Onorm__map__in__poly__ring__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,F: list_a > list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
=> ( member5342144027231129785list_a @ ( normal1297324897130370429t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( map_li5729356230488778442list_a @ F @ P ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ) ) ).
% domain.norm_map_in_poly_ring_carrier
thf(fact_443_domain_Oeval__rewrite,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a,P: list_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
=> ( P
= ( eval_l8695961830500448600t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) @ ( map_set_a_list_set_a @ ( poly_o9124539423244288956t_unit @ R ) @ P ) @ ( var_se2415970144172829891t_unit @ R ) ) ) ) ) ) ).
% domain.eval_rewrite
thf(fact_444_domain_Oeval__rewrite,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( P
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ R @ K ) @ ( map_a_list_a @ ( poly_of_const_a_b @ R ) @ P ) @ ( var_a_b @ R ) ) ) ) ) ) ).
% domain.eval_rewrite
thf(fact_445_domain_Oeval__rewrite,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( P
= ( eval_l1088911609197519410t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ ( map_li5729356230488778442list_a @ ( poly_o8716471131768098070t_unit @ R ) @ P ) @ ( var_li8453953174693405341t_unit @ R ) ) ) ) ) ) ).
% domain.eval_rewrite
thf(fact_446_x_Oa__lcos__m__assoc,axiom,
! [M2: set_list_a,G: list_a,H2: list_a] :
( ( ord_le8861187494160871172list_a @ M2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ G @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ H2 @ M2 ) )
= ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ G @ H2 ) @ M2 ) ) ) ) ) ).
% x.a_lcos_m_assoc
thf(fact_447_x_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 ) ) ) ) ) ) ).
% x.a_l_coset_subset_G
thf(fact_448_x_Oeval__normalize,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 ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ A )
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) ) ) ) ).
% x.eval_normalize
thf(fact_449_x_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 ) ) ) ) ).
% x.subset_Idl_subset
thf(fact_450_x_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 ) ) ) ).
% x.genideal_self
thf(fact_451_x_Oline__extension__mem__iff,axiom,
! [U: list_a,K: set_list_a,A: list_a,E: set_list_a] :
( ( member_list_a @ U @ ( embedd5150658419831591667t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K @ A @ E ) )
= ( ? [X: list_a] :
( ( member_list_a @ X @ K )
& ? [Y2: list_a] :
( ( member_list_a @ Y2 @ E )
& ( U
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X @ A ) @ Y2 ) ) ) ) ) ) ).
% x.line_extension_mem_iff
thf(fact_452_x_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 ) ) ) ) ) ) ) ).
% x.line_extension_in_carrier
thf(fact_453_const__term__simprules__shell_I3_J,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( const_term_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ) ).
% const_term_simprules_shell(3)
thf(fact_454_x_Oadd_Oint__pow__mult,axiom,
! [X2: list_a,I: int,J2: int] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( plus_plus_int @ I @ J2 ) @ X2 )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ I @ X2 ) @ ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ J2 @ X2 ) ) ) ) ).
% x.add.int_pow_mult
thf(fact_455_set__mult__closed,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_mu8047982887099575916xt_a_b @ r @ H @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_mult_closed
thf(fact_456_add_Ol__cancel,axiom,
! [C2: a,A: a,B2: a] :
( ( ( add_a_b @ r @ C2 @ A )
= ( add_a_b @ r @ C2 @ B2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B2 ) ) ) ) ) ).
% add.l_cancel
thf(fact_457_add_Or__cancel,axiom,
! [A: a,C2: a,B2: a] :
( ( ( add_a_b @ r @ A @ C2 )
= ( add_a_b @ r @ B2 @ C2 ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B2 ) ) ) ) ) ).
% add.r_cancel
thf(fact_458_a__assoc,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) @ Z )
= ( add_a_b @ r @ X2 @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_459_a__comm,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ Y )
= ( add_a_b @ r @ Y @ X2 ) ) ) ) ).
% a_comm
thf(fact_460_a__lcomm,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X2 @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_461_eval__poly__of__const,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_of_const_a_b @ r @ X2 ) @ Y )
= X2 ) ) ).
% eval_poly_of_const
thf(fact_462_eval__var,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( var_a_b @ r ) @ X2 )
= X2 ) ) ).
% eval_var
thf(fact_463_eval__in__carrier__2,axiom,
! [X2: list_a,Y: a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier_2
thf(fact_464_l__distr,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X2 @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_465_r__distr,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X2 @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X2 ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_466_cgenideal__prod,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_mu8047982887099575916xt_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B2 ) )
= ( cgenid547466209912283029xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) ) ) ) ) ).
% cgenideal_prod
thf(fact_467_x_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 ) ) ) ) ) ).
% x.const_term_simprules(1)
thf(fact_468_x_Onormalize__in__carrier,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.normalize_in_carrier
thf(fact_469_x_Oeval__in__carrier,axiom,
! [P: list_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( 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 ) ) ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.eval_in_carrier
thf(fact_470_local_Oadd_Oright__cancel,axiom,
! [X2: a,Y: a,Z: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X2 )
= ( add_a_b @ r @ Z @ X2 ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_471_a__closed,axiom,
! [X2: a,Y: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X2 @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_472_assms_I2_J,axiom,
ord_less_eq_set_a @ s @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(2)
thf(fact_473_ring_Osubset__Idl__subset,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a,H: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ H @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ R @ H ) @ ( genideal_a_b @ R @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_474_ring_Osubset__Idl__subset,axiom,
! [R: partia2670972154091845814t_unit,I2: set_list_a,H: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ H @ I2 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ R @ H ) @ ( genide3243992037924705879t_unit @ R @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_475_ring_Osubset__Idl__subset,axiom,
! [R: partia2956882679547061052t_unit,I2: set_list_list_a,H: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ I2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ H @ I2 )
=> ( ord_le8488217952732425610list_a @ ( genide2671672708880404049t_unit @ R @ H ) @ ( genide2671672708880404049t_unit @ R @ I2 ) ) ) ) ) ).
% ring.subset_Idl_subset
thf(fact_476_ring_Ogenideal__self,axiom,
! [R: partia2175431115845679010xt_a_b,S: set_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_477_ring_Ogenideal__self,axiom,
! [R: partia2670972154091845814t_unit,S: set_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ S @ ( genide3243992037924705879t_unit @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_478_ring_Ogenideal__self,axiom,
! [R: partia2956882679547061052t_unit,S: set_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ S @ ( partia2464479390973590831t_unit @ R ) )
=> ( ord_le8488217952732425610list_a @ S @ ( genide2671672708880404049t_unit @ R @ S ) ) ) ) ).
% ring.genideal_self
thf(fact_479_ring_Oeval__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X2: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( eval_a_b @ R @ P @ X2 ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_480_ring_Oeval__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( eval_l34571156754992824t_unit @ R @ P @ X2 ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_481_ring_Oeval__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( eval_l1088911609197519410t_unit @ R @ P @ X2 ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ).
% ring.eval_in_carrier
thf(fact_482_ring_Onormalize__in__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ R @ P ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.normalize_in_carrier
thf(fact_483_ring_Onormalize__in__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( normal637505603836502915t_unit @ R @ P ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.normalize_in_carrier
thf(fact_484_ring_Onormalize__in__carrier,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ ( normal1297324897130370429t_unit @ R @ P ) ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% ring.normalize_in_carrier
thf(fact_485_cring_Oconst__term__simprules_I1_J,axiom,
! [R: partia3925755165846298134t_unit,P: list_list_set_a] :
( ( cring_7032029500657136448t_unit @ R )
=> ( ( ord_le864617614081865828_set_a @ ( set_list_set_a2 @ P ) @ ( partia3893404292425143049t_unit @ R ) )
=> ( member_list_set_a @ ( const_6870782221686056037t_unit @ R @ P ) @ ( partia3893404292425143049t_unit @ R ) ) ) ) ).
% cring.const_term_simprules(1)
thf(fact_486_cring_Oconst__term__simprules_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( cring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ ( const_term_a_b @ R @ P ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% cring.const_term_simprules(1)
thf(fact_487_cring_Oconst__term__simprules_I1_J,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ ( const_6738166269504826821t_unit @ R @ P ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% cring.const_term_simprules(1)
thf(fact_488_cring_Oconst__term__simprules_I1_J,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ ( const_6243872422735025855t_unit @ R @ P ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% cring.const_term_simprules(1)
thf(fact_489_ring_Oeval__normalize,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,A: a] :
( ( ring_a_b @ R )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( eval_a_b @ R @ ( normalize_a_b @ R @ P ) @ A )
= ( eval_a_b @ R @ P @ A ) ) ) ) ) ).
% ring.eval_normalize
thf(fact_490_ring_Oeval__normalize,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,A: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( eval_l34571156754992824t_unit @ R @ ( normal637505603836502915t_unit @ R @ P ) @ A )
= ( eval_l34571156754992824t_unit @ R @ P @ A ) ) ) ) ) ).
% ring.eval_normalize
thf(fact_491_ring_Oeval__normalize,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,A: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ P ) @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( eval_l1088911609197519410t_unit @ R @ ( normal1297324897130370429t_unit @ R @ P ) @ A )
= ( eval_l1088911609197519410t_unit @ R @ P @ A ) ) ) ) ) ).
% ring.eval_normalize
thf(fact_492_x_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 ) ) ) ) ) ).
% x.exp_base_closed
thf(fact_493_x_Oee__sym,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ As ) ) ) ) ).
% x.ee_sym
thf(fact_494_x_Oee__trans,axiom,
! [As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Bs @ Cs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Cs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Cs ) ) ) ) ) ) ).
% x.ee_trans
thf(fact_495_x_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 ) ) ) ) ) ) ).
% x.factors_closed
thf(fact_496_x_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 ) ) ) ) ).
% x.cgenideal_eq_genideal
thf(fact_497_x_OIdl__subset__ideal_H,axiom,
! [A: list_a,B2: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( 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 @ B2 @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A @ ( genide3243992037924705879t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) ) ) ) ) ) ).
% x.Idl_subset_ideal'
thf(fact_498_x_Oa__lcos__mult__one,axiom,
! [M2: set_list_a] :
( ( ord_le8861187494160871172list_a @ M2 @ ( 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 ) ) ) @ M2 )
= M2 ) ) ).
% x.a_lcos_mult_one
thf(fact_499_normalize__in__carrier,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( normalize_a_b @ r @ P ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% normalize_in_carrier
thf(fact_500_subset__Idl__subset,axiom,
! [I2: set_a,H: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_501_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_502_eval__in__carrier,axiom,
! [P: list_a,X2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( eval_a_b @ r @ P @ X2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% eval_in_carrier
thf(fact_503_eval__normalize,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( normalize_a_b @ r @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_normalize
thf(fact_504_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_505_x_Ominus__unique,axiom,
! [Y: list_a,X2: list_a,Y4: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y4 )
= ( zero_l4142658623432671053t_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 @ Y @ ( 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 ) ) ) )
=> ( Y = Y4 ) ) ) ) ) ) ).
% x.minus_unique
thf(fact_506_x_Oadd_Or__inv__ex,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ X3 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.r_inv_ex
thf(fact_507_x_Oadd_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.one_unique
thf(fact_508_x_Oadd_Ol__inv__ex,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X3 @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.add.l_inv_ex
thf(fact_509_x_Oadd_Oinv__comm,axiom,
! [X2: list_a,Y: list_a] :
( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y )
= ( zero_l4142658623432671053t_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 @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.inv_comm
thf(fact_510_x_Oconst__term__def,axiom,
! [P: list_list_a] :
( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.const_term_def
thf(fact_511_assms_I1_J,axiom,
finite_finite_a @ s ).
% assms(1)
thf(fact_512_x_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 ) ) ) ) ).
% x.genideal_self'
thf(fact_513_x_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 ) ) ).
% x.genideal_zero
thf(fact_514_x_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 ) ) ).
% x.zeropideal
thf(fact_515_x_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 ) ) ) ).
% x.zero_closed
thf(fact_516_x_Oadd_Oint__pow__one,axiom,
! [Z: int] :
( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Z @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.add.int_pow_one
thf(fact_517_x_Or__zero,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= X2 ) ) ).
% x.r_zero
thf(fact_518_x_Ol__zero,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ X2 )
= X2 ) ) ).
% x.l_zero
thf(fact_519_x_Oadd_Or__cancel__one_H,axiom,
! [X2: list_a,A: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( X2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X2 ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one'
thf(fact_520_x_Oadd_Or__cancel__one,axiom,
! [X2: list_a,A: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ X2 )
= X2 )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.r_cancel_one
thf(fact_521_x_Oadd_Ol__cancel__one_H,axiom,
! [X2: list_a,A: list_a] :
( ( member_list_a @ X2 @ ( 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 ) ) ) )
=> ( ( X2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ A ) )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one'
thf(fact_522_x_Oadd_Ol__cancel__one,axiom,
! [X2: list_a,A: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ A )
= X2 )
= ( A
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ).
% x.add.l_cancel_one
thf(fact_523_x_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 ) ) ) ) ) ).
% x.r_null
thf(fact_524_x_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 ) ) ) ) ) ).
% x.l_null
thf(fact_525_x_Or__right__minus__eq,axiom,
! [A: list_a,B2: list_a] :
( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ B2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( A = B2 ) ) ) ) ).
% x.r_right_minus_eq
thf(fact_526_x_Oee__refl,axiom,
! [As: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ As ) ) ).
% x.ee_refl
thf(fact_527_ring_Oexp__base_Ocong,axiom,
polyno3522816881121920896t_unit = polyno3522816881121920896t_unit ).
% ring.exp_base.cong
thf(fact_528_ring_Oexp__base_Ocong,axiom,
polyno2922411391617481336se_a_b = polyno2922411391617481336se_a_b ).
% ring.exp_base.cong
thf(fact_529_subringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subringE(2)
thf(fact_530_subringE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subrin6918843898125473962t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subringE(2)
thf(fact_531_subringE_I2_J,axiom,
! [H: set_set_a,R: partia6043505979758434576t_unit] :
( ( subrin1511138061850335568t_unit @ H @ R )
=> ( member_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ H ) ) ).
% subringE(2)
thf(fact_532_subcringE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subcring_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_533_subcringE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subcri7763218559781929323t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subcringE(2)
thf(fact_534_subdomainE_I2_J,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( member_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_535_subdomainE_I2_J,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b] :
( ( subdomain_a_b @ H @ R )
=> ( member_a @ ( zero_a_b @ R ) @ H ) ) ).
% subdomainE(2)
thf(fact_536_ring_Ogenideal__zero,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( genideal_a_b @ R @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) ) ) ).
% ring.genideal_zero
thf(fact_537_ring_Ogenideal__zero,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) ) ) ).
% ring.genideal_zero
thf(fact_538_ring_Ozeropideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( principalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% ring.zeropideal
thf(fact_539_ring_Ozeropideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ).
% ring.zeropideal
thf(fact_540_subdomain_Osubintegral,axiom,
! [H: set_list_a,R: partia2670972154091845814t_unit,H12: list_a,H23: list_a] :
( ( subdom7821232466298058046t_unit @ H @ R )
=> ( ( member_list_a @ H12 @ H )
=> ( ( member_list_a @ H23 @ H )
=> ( ( ( mult_l7073676228092353617t_unit @ R @ H12 @ H23 )
= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( H12
= ( zero_l4142658623432671053t_unit @ R ) )
| ( H23
= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_541_subdomain_Osubintegral,axiom,
! [H: set_a,R: partia2175431115845679010xt_a_b,H12: a,H23: a] :
( ( subdomain_a_b @ H @ R )
=> ( ( member_a @ H12 @ H )
=> ( ( member_a @ H23 @ H )
=> ( ( ( mult_a_ring_ext_a_b @ R @ H12 @ H23 )
= ( zero_a_b @ R ) )
=> ( ( H12
= ( zero_a_b @ R ) )
| ( H23
= ( zero_a_b @ R ) ) ) ) ) ) ) ).
% subdomain.subintegral
thf(fact_542_ring_Oconst__term__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( const_term_a_b @ R @ P )
= ( eval_a_b @ R @ P @ ( zero_a_b @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_543_ring_Oconst__term__def,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( const_6738166269504826821t_unit @ R @ P )
= ( eval_l34571156754992824t_unit @ R @ P @ ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.const_term_def
thf(fact_544_domain_Oproperfactor__of__zero_I2_J,axiom,
! [R: partia6043505979758434576t_unit,B2: set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_set_a @ B2 @ ( partia5907974310037520643t_unit @ R ) )
=> ( ( proper7783390844374471584t_unit @ R @ B2 @ ( zero_s2174465271003423091t_unit @ R ) )
= ( B2
!= ( zero_s2174465271003423091t_unit @ R ) ) ) ) ) ).
% domain.properfactor_of_zero(2)
thf(fact_545_domain_Oproperfactor__of__zero_I2_J,axiom,
! [R: partia2175431115845679010xt_a_b,B2: a] :
( ( domain_a_b @ R )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( proper19828929941537682xt_a_b @ R @ B2 @ ( zero_a_b @ R ) )
= ( B2
!= ( zero_a_b @ R ) ) ) ) ) ).
% domain.properfactor_of_zero(2)
thf(fact_546_domain_Oproperfactor__of__zero_I2_J,axiom,
! [R: partia2670972154091845814t_unit,B2: list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( proper8313688649498433056t_unit @ R @ B2 @ ( zero_l4142658623432671053t_unit @ R ) )
= ( B2
!= ( zero_l4142658623432671053t_unit @ R ) ) ) ) ) ).
% domain.properfactor_of_zero(2)
thf(fact_547_domain_Oproperfactor__of__zero_I2_J,axiom,
! [R: partia2956882679547061052t_unit,B2: list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( proper6839530779442076320t_unit @ R @ B2 @ ( zero_l347298301471573063t_unit @ R ) )
= ( B2
!= ( zero_l347298301471573063t_unit @ R ) ) ) ) ) ).
% domain.properfactor_of_zero(2)
thf(fact_548_ring_Ogenideal__self_H,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_a @ I @ ( genideal_a_b @ R @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_549_ring_Ogenideal__self_H,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_a @ I @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_550_ring_Ogenideal__self_H,axiom,
! [R: partia2956882679547061052t_unit,I: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ I @ ( partia2464479390973590831t_unit @ R ) )
=> ( member_list_list_a @ I @ ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ I @ bot_bo1875519244922727510list_a ) ) ) ) ) ).
% ring.genideal_self'
thf(fact_551_principalideal_Ogenerate,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( principalideal_a_b @ I2 @ R )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ R ) )
& ( I2
= ( genideal_a_b @ R @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_552_principalideal_Ogenerate,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( princi8786919440553033881t_unit @ I2 @ R )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ R ) )
& ( I2
= ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ X3 @ bot_bot_set_list_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_553_principalideal_Ogenerate,axiom,
! [I2: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( princi2534607884127416211t_unit @ I2 @ R )
=> ? [X3: list_list_a] :
( ( member_list_list_a @ X3 @ ( partia2464479390973590831t_unit @ R ) )
& ( I2
= ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ X3 @ bot_bo1875519244922727510list_a ) ) ) ) ) ).
% principalideal.generate
thf(fact_554_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( ring_a_b @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ R @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ R @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_555_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2670972154091845814t_unit,A: list_a,B2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A @ ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_556_ring_OIdl__subset__ideal_H,axiom,
! [R: partia2956882679547061052t_unit,A: list_list_a,B2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( ord_le8488217952732425610list_a @ ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) @ ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ B2 @ bot_bo1875519244922727510list_a ) ) )
= ( member_list_list_a @ A @ ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ B2 @ bot_bo1875519244922727510list_a ) ) ) ) ) ) ) ).
% ring.Idl_subset_ideal'
thf(fact_557_cring_Ocgenideal__eq__genideal,axiom,
! [R: partia3925755165846298134t_unit,I: list_set_a] :
( ( cring_7032029500657136448t_unit @ R )
=> ( ( member_list_set_a @ I @ ( partia3893404292425143049t_unit @ R ) )
=> ( ( cgenid5493546629940232227t_unit @ R @ I )
= ( genide7749339459745170679t_unit @ R @ ( insert_list_set_a @ I @ bot_bo4397488018069675312_set_a ) ) ) ) ) ).
% cring.cgenideal_eq_genideal
thf(fact_558_cring_Ocgenideal__eq__genideal,axiom,
! [R: partia2175431115845679010xt_a_b,I: a] :
( ( cring_a_b @ R )
=> ( ( member_a @ I @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( cgenid547466209912283029xt_a_b @ R @ I )
= ( genideal_a_b @ R @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ) ).
% cring.cgenideal_eq_genideal
thf(fact_559_cring_Ocgenideal__eq__genideal,axiom,
! [R: partia2670972154091845814t_unit,I: list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( cgenid9131348535277946915t_unit @ R @ I )
= ( genide3243992037924705879t_unit @ R @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ) ).
% cring.cgenideal_eq_genideal
thf(fact_560_cring_Ocgenideal__eq__genideal,axiom,
! [R: partia2956882679547061052t_unit,I: list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( member_list_list_a @ I @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( cgenid24865672677839267t_unit @ R @ I )
= ( genide2671672708880404049t_unit @ R @ ( insert_list_list_a @ I @ bot_bo1875519244922727510list_a ) ) ) ) ) ).
% cring.cgenideal_eq_genideal
thf(fact_561_ring_Oexp__base__closed,axiom,
! [R: partia2956882679547061052t_unit,X2: list_list_a,N: nat] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ ( polyno6819740552565085946t_unit @ R @ X2 @ N ) ) @ ( partia2464479390973590831t_unit @ R ) ) ) ) ).
% ring.exp_base_closed
thf(fact_562_ring_Oexp__base__closed,axiom,
! [R: partia2670972154091845814t_unit,X2: list_a,N: nat] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ R @ X2 @ N ) ) @ ( partia5361259788508890537t_unit @ R ) ) ) ) ).
% ring.exp_base_closed
thf(fact_563_ring_Oexp__base__closed,axiom,
! [R: partia2175431115845679010xt_a_b,X2: a,N: nat] :
( ( ring_a_b @ R )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481336se_a_b @ R @ X2 @ N ) ) @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ).
% ring.exp_base_closed
thf(fact_564_p__def,axiom,
( p
= ( lagran9092808442999052491ux_a_b @ r @ s ) ) ).
% p_def
thf(fact_565_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_566_x_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 ) ) ) ) ).
% x.zeroprimeideal_domainI
thf(fact_567_x_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 ) ) ) ) ).
% x.domain_eq_zeroprimeideal
thf(fact_568_insert__disjoint_I1_J,axiom,
! [A: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ( ( inf_in7423150557312423384list_a @ ( insert_list_list_a @ A @ A2 ) @ B3 )
= bot_bo1875519244922727510list_a )
= ( ~ ( member_list_list_a @ A @ B3 )
& ( ( inf_in7423150557312423384list_a @ A2 @ B3 )
= bot_bo1875519244922727510list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_569_insert__disjoint_I1_J,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B3 )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B3 )
& ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a ) ) ) ).
% insert_disjoint(1)
thf(fact_570_insert__disjoint_I1_J,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ B3 )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A @ B3 )
& ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a ) ) ) ).
% insert_disjoint(1)
thf(fact_571_insert__disjoint_I2_J,axiom,
! [A: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ( bot_bo1875519244922727510list_a
= ( inf_in7423150557312423384list_a @ ( insert_list_list_a @ A @ A2 ) @ B3 ) )
= ( ~ ( member_list_list_a @ A @ B3 )
& ( bot_bo1875519244922727510list_a
= ( inf_in7423150557312423384list_a @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_572_insert__disjoint_I2_J,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ B3 ) )
= ( ~ ( member_a @ A @ B3 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_573_insert__disjoint_I2_J,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ B3 ) )
= ( ~ ( member_list_a @ A @ B3 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_574_disjoint__insert_I1_J,axiom,
! [B3: set_list_list_a,A: list_list_a,A2: set_list_list_a] :
( ( ( inf_in7423150557312423384list_a @ B3 @ ( insert_list_list_a @ A @ A2 ) )
= bot_bo1875519244922727510list_a )
= ( ~ ( member_list_list_a @ A @ B3 )
& ( ( inf_in7423150557312423384list_a @ B3 @ A2 )
= bot_bo1875519244922727510list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_575_disjoint__insert_I1_J,axiom,
! [B3: set_a,A: a,A2: set_a] :
( ( ( inf_inf_set_a @ B3 @ ( insert_a @ A @ A2 ) )
= bot_bot_set_a )
= ( ~ ( member_a @ A @ B3 )
& ( ( inf_inf_set_a @ B3 @ A2 )
= bot_bot_set_a ) ) ) ).
% disjoint_insert(1)
thf(fact_576_disjoint__insert_I1_J,axiom,
! [B3: set_list_a,A: list_a,A2: set_list_a] :
( ( ( inf_inf_set_list_a @ B3 @ ( insert_list_a @ A @ A2 ) )
= bot_bot_set_list_a )
= ( ~ ( member_list_a @ A @ B3 )
& ( ( inf_inf_set_list_a @ B3 @ A2 )
= bot_bot_set_list_a ) ) ) ).
% disjoint_insert(1)
thf(fact_577_properfactor__of__zero_I2_J,axiom,
! [B2: a] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( proper19828929941537682xt_a_b @ r @ B2 @ ( zero_a_b @ r ) )
= ( B2
!= ( zero_a_b @ r ) ) ) ) ).
% properfactor_of_zero(2)
thf(fact_578_add_Oinv__comm,axiom,
! [X2: a,Y: a] :
( ( ( add_a_b @ r @ X2 @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X2 )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_579_add_Ol__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X2 )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_580_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_581_add_Or__inv__ex,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X2 @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_582_local_Ominus__unique,axiom,
! [Y: a,X2: a,Y4: a] :
( ( ( add_a_b @ r @ Y @ X2 )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X2 @ Y4 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y4 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y4 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_583_properfactor__prod__l,axiom,
! [A: a,B2: a,C2: a] :
( ( proper19828929941537682xt_a_b @ r @ A @ B2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ C2 @ B2 ) ) ) ) ) ) ).
% properfactor_prod_l
thf(fact_584_properfactor__prod__r,axiom,
! [A: a,B2: a,C2: a] :
( ( proper19828929941537682xt_a_b @ r @ A @ B2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( proper19828929941537682xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B2 @ C2 ) ) ) ) ) ) ).
% properfactor_prod_r
thf(fact_585_local_Ointegral,axiom,
! [A: a,B2: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B2 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B2
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_586_integral__iff,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B2 )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B2
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_587_m__lcancel,axiom,
! [A: a,B2: a,C2: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B2 )
= ( mult_a_ring_ext_a_b @ r @ A @ C2 ) )
= ( B2 = C2 ) ) ) ) ) ) ).
% m_lcancel
thf(fact_588_m__rcancel,axiom,
! [A: a,B2: a,C2: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B2 @ A )
= ( mult_a_ring_ext_a_b @ r @ C2 @ A ) )
= ( B2 = C2 ) ) ) ) ) ) ).
% m_rcancel
thf(fact_589_const__term__def,axiom,
! [P: list_a] :
( ( const_term_a_b @ r @ P )
= ( eval_a_b @ r @ P @ ( zero_a_b @ r ) ) ) ).
% const_term_def
thf(fact_590_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_591_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_592_subset__antisym,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_593_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_594_subsetI,axiom,
! [A2: set_list_list_a,B3: set_list_list_a] :
( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A2 )
=> ( member_list_list_a @ X3 @ B3 ) )
=> ( ord_le8488217952732425610list_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_595_subsetI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( member_list_a @ X3 @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_596_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_597_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X: a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_598_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_599_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_600_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_601_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_602_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_603_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_604_empty__iff,axiom,
! [C2: list_list_a] :
~ ( member_list_list_a @ C2 @ bot_bo1875519244922727510list_a ) ).
% empty_iff
thf(fact_605_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_606_empty__iff,axiom,
! [C2: list_a] :
~ ( member_list_a @ C2 @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_607_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_608_Int__iff,axiom,
! [C2: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ( member_list_list_a @ C2 @ ( inf_in7423150557312423384list_a @ A2 @ B3 ) )
= ( ( member_list_list_a @ C2 @ A2 )
& ( member_list_list_a @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_609_Int__iff,axiom,
! [C2: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( ( member_list_a @ C2 @ A2 )
& ( member_list_a @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_610_Int__iff,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_611_IntI,axiom,
! [C2: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ( member_list_list_a @ C2 @ A2 )
=> ( ( member_list_list_a @ C2 @ B3 )
=> ( member_list_list_a @ C2 @ ( inf_in7423150557312423384list_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_612_IntI,axiom,
! [C2: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C2 @ A2 )
=> ( ( member_list_a @ C2 @ B3 )
=> ( member_list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_613_IntI,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B3 )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_614_Idl__subset__ideal_H,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_615_cgenideal__eq__genideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I )
= ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_616_lagrange__aux__poly,axiom,
! [S: set_a] :
( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ r @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% lagrange_aux_poly
thf(fact_617_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_618_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_619_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_620_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_621_insert__subset,axiom,
! [X2: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( insert_list_list_a @ X2 @ A2 ) @ B3 )
= ( ( member_list_list_a @ X2 @ B3 )
& ( ord_le8488217952732425610list_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_622_insert__subset,axiom,
! [X2: list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X2 @ A2 ) @ B3 )
= ( ( member_list_a @ X2 @ B3 )
& ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_623_insert__subset,axiom,
! [X2: a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X2 @ A2 ) @ B3 )
= ( ( member_a @ X2 @ B3 )
& ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_624_singletonI,axiom,
! [A: list_list_a] : ( member_list_list_a @ A @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) ) ).
% singletonI
thf(fact_625_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_626_singletonI,axiom,
! [A: list_a] : ( member_list_a @ A @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_627_Int__subset__iff,axiom,
! [C3: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( ( ord_le8861187494160871172list_a @ C3 @ A2 )
& ( ord_le8861187494160871172list_a @ C3 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_628_Int__subset__iff,axiom,
! [C3: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( ( ord_less_eq_set_a @ C3 @ A2 )
& ( ord_less_eq_set_a @ C3 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_629_Int__insert__left__if0,axiom,
! [A: list_list_a,C3: set_list_list_a,B3: set_list_list_a] :
( ~ ( member_list_list_a @ A @ C3 )
=> ( ( inf_in7423150557312423384list_a @ ( insert_list_list_a @ A @ B3 ) @ C3 )
= ( inf_in7423150557312423384list_a @ B3 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_630_Int__insert__left__if0,axiom,
! [A: list_a,C3: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ A @ C3 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B3 ) @ C3 )
= ( inf_inf_set_list_a @ B3 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_631_Int__insert__left__if0,axiom,
! [A: a,C3: set_a,B3: set_a] :
( ~ ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C3 )
= ( inf_inf_set_a @ B3 @ C3 ) ) ) ).
% Int_insert_left_if0
thf(fact_632_Int__insert__left__if1,axiom,
! [A: list_list_a,C3: set_list_list_a,B3: set_list_list_a] :
( ( member_list_list_a @ A @ C3 )
=> ( ( inf_in7423150557312423384list_a @ ( insert_list_list_a @ A @ B3 ) @ C3 )
= ( insert_list_list_a @ A @ ( inf_in7423150557312423384list_a @ B3 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_633_Int__insert__left__if1,axiom,
! [A: list_a,C3: set_list_a,B3: set_list_a] :
( ( member_list_a @ A @ C3 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B3 ) @ C3 )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ B3 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_634_Int__insert__left__if1,axiom,
! [A: a,C3: set_a,B3: set_a] :
( ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C3 )
= ( insert_a @ A @ ( inf_inf_set_a @ B3 @ C3 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_635_insert__inter__insert,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ ( insert_list_a @ A @ A2 ) @ ( insert_list_a @ A @ B3 ) )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_636_insert__inter__insert,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( insert_a @ A @ A2 ) @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_637_Int__insert__right__if0,axiom,
! [A: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ~ ( member_list_list_a @ A @ A2 )
=> ( ( inf_in7423150557312423384list_a @ A2 @ ( insert_list_list_a @ A @ B3 ) )
= ( inf_in7423150557312423384list_a @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_638_Int__insert__right__if0,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_639_Int__insert__right__if0,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_640_Int__insert__right__if1,axiom,
! [A: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ( member_list_list_a @ A @ A2 )
=> ( ( inf_in7423150557312423384list_a @ A2 @ ( insert_list_list_a @ A @ B3 ) )
= ( insert_list_list_a @ A @ ( inf_in7423150557312423384list_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_641_Int__insert__right__if1,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_642_Int__insert__right__if1,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_643_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_644_singleton__insert__inj__eq,axiom,
! [B2: list_a,A: list_a,A2: set_list_a] :
( ( ( insert_list_a @ B2 @ bot_bot_set_list_a )
= ( insert_list_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_645_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A2: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_646_singleton__insert__inj__eq_H,axiom,
! [A: list_a,A2: set_list_a,B2: list_a] :
( ( ( insert_list_a @ A @ A2 )
= ( insert_list_a @ B2 @ bot_bot_set_list_a ) )
= ( ( A = B2 )
& ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_647_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B2: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_648_disjoint__insert_I2_J,axiom,
! [A2: set_list_list_a,B2: list_list_a,B3: set_list_list_a] :
( ( bot_bo1875519244922727510list_a
= ( inf_in7423150557312423384list_a @ A2 @ ( insert_list_list_a @ B2 @ B3 ) ) )
= ( ~ ( member_list_list_a @ B2 @ A2 )
& ( bot_bo1875519244922727510list_a
= ( inf_in7423150557312423384list_a @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_649_disjoint__insert_I2_J,axiom,
! [A2: set_a,B2: a,B3: set_a] :
( ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ ( insert_a @ B2 @ B3 ) ) )
= ( ~ ( member_a @ B2 @ A2 )
& ( bot_bot_set_a
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_650_disjoint__insert_I2_J,axiom,
! [A2: set_list_a,B2: list_a,B3: set_list_a] :
( ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ B2 @ B3 ) ) )
= ( ~ ( member_list_a @ B2 @ A2 )
& ( bot_bot_set_list_a
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_651_add_Ol__cancel__one,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X2 @ A )
= X2 )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_652_add_Ol__cancel__one_H,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X2
= ( add_a_b @ r @ X2 @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_653_add_Or__cancel__one,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X2 )
= X2 )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_654_add_Or__cancel__one_H,axiom,
! [X2: a,A: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X2
= ( add_a_b @ r @ A @ X2 ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_655_l__zero,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X2 )
= X2 ) ) ).
% l_zero
thf(fact_656_r__zero,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X2 @ ( zero_a_b @ r ) )
= X2 ) ) ).
% r_zero
thf(fact_657_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_658_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_659_ring_Olagrange__basis__polynomial__aux_Ocong,axiom,
lagran9092808442999052491ux_a_b = lagran9092808442999052491ux_a_b ).
% ring.lagrange_basis_polynomial_aux.cong
thf(fact_660_primeideal_Oprimeideal,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( primei6309817859076077608t_unit @ I2 @ R ) ) ).
% primeideal.primeideal
thf(fact_661_primeideal_Oprimeideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( primeideal_a_b @ I2 @ R ) ) ).
% primeideal.primeideal
thf(fact_662_primeideal_OI__notcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_663_primeideal_OI__notcarr,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( ( partia5361259788508890537t_unit @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_664_primeideal_OI__notcarr,axiom,
! [I2: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( primei2288432046033540002t_unit @ I2 @ R )
=> ( ( partia2464479390973590831t_unit @ R )
!= I2 ) ) ).
% primeideal.I_notcarr
thf(fact_665_primeideal_Oaxioms_I2_J,axiom,
! [I2: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( primei2288432046033540002t_unit @ I2 @ R )
=> ( cring_5991999922451032090t_unit @ R ) ) ).
% primeideal.axioms(2)
thf(fact_666_primeideal_Oaxioms_I2_J,axiom,
! [I2: set_list_set_a,R: partia3925755165846298134t_unit] :
( ( primei2134727858671723208t_unit @ I2 @ R )
=> ( cring_7032029500657136448t_unit @ R ) ) ).
% primeideal.axioms(2)
thf(fact_667_primeideal_Oaxioms_I2_J,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( cring_3148771470849435808t_unit @ R ) ) ).
% primeideal.axioms(2)
thf(fact_668_primeideal_Oaxioms_I2_J,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( primeideal_a_b @ I2 @ R )
=> ( cring_a_b @ R ) ) ).
% primeideal.axioms(2)
thf(fact_669_primeideal_OI__prime,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b,A: a,B2: a] :
( ( primeideal_a_b @ I2 @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ R @ A @ B2 ) @ I2 )
=> ( ( member_a @ A @ I2 )
| ( member_a @ B2 @ I2 ) ) ) ) ) ) ).
% primeideal.I_prime
thf(fact_670_primeideal_OI__prime,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit,A: list_a,B2: list_a] :
( ( primei6309817859076077608t_unit @ I2 @ R )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ R ) )
=> ( ( member_list_a @ ( mult_l7073676228092353617t_unit @ R @ A @ B2 ) @ I2 )
=> ( ( member_list_a @ A @ I2 )
| ( member_list_a @ B2 @ I2 ) ) ) ) ) ) ).
% primeideal.I_prime
thf(fact_671_primeideal_OI__prime,axiom,
! [I2: set_list_list_a,R: partia2956882679547061052t_unit,A: list_list_a,B2: list_list_a] :
( ( primei2288432046033540002t_unit @ I2 @ R )
=> ( ( member_list_list_a @ A @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ B2 @ ( partia2464479390973590831t_unit @ R ) )
=> ( ( member_list_list_a @ ( mult_l4853965630390486993t_unit @ R @ A @ B2 ) @ I2 )
=> ( ( member_list_list_a @ A @ I2 )
| ( member_list_list_a @ B2 @ I2 ) ) ) ) ) ) ).
% primeideal.I_prime
thf(fact_672_Collect__mono__iff,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) )
= ( ! [X: list_a] :
( ( P2 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_673_Collect__mono__iff,axiom,
! [P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) )
= ( ! [X: a] :
( ( P2 @ X )
=> ( Q2 @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_674_set__eq__subset,axiom,
( ( ^ [Y5: set_list_a,Z3: set_list_a] : ( Y5 = Z3 ) )
= ( ^ [A5: set_list_a,B5: set_list_a] :
( ( ord_le8861187494160871172list_a @ A5 @ B5 )
& ( ord_le8861187494160871172list_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_675_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z3: set_a] : ( Y5 = Z3 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_676_subset__trans,axiom,
! [A2: set_list_a,B3: set_list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C3 )
=> ( ord_le8861187494160871172list_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_677_subset__trans,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_678_Collect__mono,axiom,
! [P2: list_a > $o,Q2: list_a > $o] :
( ! [X3: list_a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P2 ) @ ( collect_list_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_679_Collect__mono,axiom,
! [P2: a > $o,Q2: a > $o] :
( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_680_subset__refl,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_681_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_682_subset__iff,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A5: set_list_list_a,B5: set_list_list_a] :
! [T: list_list_a] :
( ( member_list_list_a @ T @ A5 )
=> ( member_list_list_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_683_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A5 )
=> ( member_list_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_684_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_685_equalityD2,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2 = B3 )
=> ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_686_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_687_equalityD1,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2 = B3 )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_688_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_689_subset__eq,axiom,
( ord_le8488217952732425610list_a
= ( ^ [A5: set_list_list_a,B5: set_list_list_a] :
! [X: list_list_a] :
( ( member_list_list_a @ X @ A5 )
=> ( member_list_list_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_690_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A5: set_list_a,B5: set_list_a] :
! [X: list_a] :
( ( member_list_a @ X @ A5 )
=> ( member_list_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_691_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X: a] :
( ( member_a @ X @ A5 )
=> ( member_a @ X @ B5 ) ) ) ) ).
% subset_eq
thf(fact_692_equalityE,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( A2 = B3 )
=> ~ ( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ~ ( ord_le8861187494160871172list_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_693_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_694_subsetD,axiom,
! [A2: set_list_list_a,B3: set_list_list_a,C2: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ B3 )
=> ( ( member_list_list_a @ C2 @ A2 )
=> ( member_list_list_a @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_695_subsetD,axiom,
! [A2: set_list_a,B3: set_list_a,C2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ C2 @ A2 )
=> ( member_list_a @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_696_subsetD,axiom,
! [A2: set_a,B3: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_697_in__mono,axiom,
! [A2: set_list_list_a,B3: set_list_list_a,X2: list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ B3 )
=> ( ( member_list_list_a @ X2 @ A2 )
=> ( member_list_list_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_698_in__mono,axiom,
! [A2: set_list_a,B3: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_699_in__mono,axiom,
! [A2: set_a,B3: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_700_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_701_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_702_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_703_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X: a] : ( member_a @ X @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_704_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_705_equals0I,axiom,
! [A2: set_list_list_a] :
( ! [Y3: list_list_a] :
~ ( member_list_list_a @ Y3 @ A2 )
=> ( A2 = bot_bo1875519244922727510list_a ) ) ).
% equals0I
thf(fact_706_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_707_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y3: list_a] :
~ ( member_list_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_708_equals0D,axiom,
! [A2: set_list_list_a,A: list_list_a] :
( ( A2 = bot_bo1875519244922727510list_a )
=> ~ ( member_list_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_709_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_710_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_711_emptyE,axiom,
! [A: list_list_a] :
~ ( member_list_list_a @ A @ bot_bo1875519244922727510list_a ) ).
% emptyE
thf(fact_712_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_713_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_714_domain_Olagrange__aux__poly,axiom,
! [R: partia6043505979758434576t_unit,S: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( finite_finite_set_a @ S )
=> ( ( ord_le3724670747650509150_set_a @ S @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_list_set_a @ ( lagran8431350174043022969t_unit @ R @ S ) @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_715_domain_Olagrange__aux__poly,axiom,
! [R: partia2670972154091845814t_unit,S: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( finite_finite_list_a @ S )
=> ( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( lagran3534788790333317459t_unit @ R @ S ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_716_domain_Olagrange__aux__poly,axiom,
! [R: partia2956882679547061052t_unit,S: set_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( finite1660835950917165235list_a @ S )
=> ( ( ord_le8488217952732425610list_a @ S @ ( partia2464479390973590831t_unit @ R ) )
=> ( member5342144027231129785list_a @ ( lagran8640377047181650765t_unit @ R @ S ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_717_domain_Olagrange__aux__poly,axiom,
! [R: partia2175431115845679010xt_a_b,S: set_a] :
( ( domain_a_b @ R )
=> ( ( finite_finite_a @ S )
=> ( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( lagran9092808442999052491ux_a_b @ R @ S ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ).
% domain.lagrange_aux_poly
thf(fact_718_Int__left__commute,axiom,
! [A2: set_list_a,B3: set_list_a,C3: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C3 ) )
= ( inf_inf_set_list_a @ B3 @ ( inf_inf_set_list_a @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_719_Int__left__commute,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C3 ) )
= ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A2 @ C3 ) ) ) ).
% Int_left_commute
thf(fact_720_Int__left__absorb,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_721_Int__left__absorb,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_722_Int__commute,axiom,
( inf_inf_set_list_a
= ( ^ [A5: set_list_a,B5: set_list_a] : ( inf_inf_set_list_a @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_723_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A5: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_724_Int__absorb,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_725_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_726_Int__assoc,axiom,
! [A2: set_list_a,B3: set_list_a,C3: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ C3 )
= ( inf_inf_set_list_a @ A2 @ ( inf_inf_set_list_a @ B3 @ C3 ) ) ) ).
% Int_assoc
thf(fact_727_Int__assoc,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C3 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C3 ) ) ) ).
% Int_assoc
thf(fact_728_IntD2,axiom,
! [C2: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ( member_list_list_a @ C2 @ ( inf_in7423150557312423384list_a @ A2 @ B3 ) )
=> ( member_list_list_a @ C2 @ B3 ) ) ).
% IntD2
thf(fact_729_IntD2,axiom,
! [C2: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ( member_list_a @ C2 @ B3 ) ) ).
% IntD2
thf(fact_730_IntD2,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ( member_a @ C2 @ B3 ) ) ).
% IntD2
thf(fact_731_IntD1,axiom,
! [C2: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ( member_list_list_a @ C2 @ ( inf_in7423150557312423384list_a @ A2 @ B3 ) )
=> ( member_list_list_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_732_IntD1,axiom,
! [C2: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ( member_list_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_733_IntD1,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ( member_a @ C2 @ A2 ) ) ).
% IntD1
thf(fact_734_IntE,axiom,
! [C2: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ( member_list_list_a @ C2 @ ( inf_in7423150557312423384list_a @ A2 @ B3 ) )
=> ~ ( ( member_list_list_a @ C2 @ A2 )
=> ~ ( member_list_list_a @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_735_IntE,axiom,
! [C2: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C2 @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ~ ( ( member_list_a @ C2 @ A2 )
=> ~ ( member_list_a @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_736_IntE,axiom,
! [C2: a,A2: set_a,B3: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ~ ( member_a @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_737_domain_Ozeroprimeideal,axiom,
! [R: partia6043505979758434576t_unit] :
( ( domain4236798911309298543t_unit @ R )
=> ( primei7645216761534224334t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_738_domain_Ozeroprimeideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( domain_a_b @ R )
=> ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_739_domain_Ozeroprimeideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( domain6553523120543210313t_unit @ R )
=> ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ).
% domain.zeroprimeideal
thf(fact_740_subset__insertI2,axiom,
! [A2: set_list_a,B3: set_list_a,B2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ B2 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_741_subset__insertI2,axiom,
! [A2: set_a,B3: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_742_subset__insertI,axiom,
! [B3: set_list_a,A: list_a] : ( ord_le8861187494160871172list_a @ B3 @ ( insert_list_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_743_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_744_subset__insert,axiom,
! [X2: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ~ ( member_list_list_a @ X2 @ A2 )
=> ( ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ X2 @ B3 ) )
= ( ord_le8488217952732425610list_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_745_subset__insert,axiom,
! [X2: list_a,A2: set_list_a,B3: set_list_a] :
( ~ ( member_list_a @ X2 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ B3 ) )
= ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_746_subset__insert,axiom,
! [X2: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B3 ) )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_747_insert__mono,axiom,
! [C3: set_list_a,D2: set_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( insert_list_a @ A @ C3 ) @ ( insert_list_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_748_insert__mono,axiom,
! [C3: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_749_cring_Odomain__eq__zeroprimeideal,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( domain7810152921033798211t_unit @ R )
= ( primei2288432046033540002t_unit @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) @ R ) ) ) ).
% cring.domain_eq_zeroprimeideal
thf(fact_750_cring_Odomain__eq__zeroprimeideal,axiom,
! [R: partia3925755165846298134t_unit] :
( ( cring_7032029500657136448t_unit @ R )
=> ( ( domain5179785879681597929t_unit @ R )
= ( primei2134727858671723208t_unit @ ( insert_list_set_a @ ( zero_l6472697616691860973t_unit @ R ) @ bot_bo4397488018069675312_set_a ) @ R ) ) ) ).
% cring.domain_eq_zeroprimeideal
thf(fact_751_cring_Odomain__eq__zeroprimeideal,axiom,
! [R: partia6043505979758434576t_unit] :
( ( cring_5265999997378430022t_unit @ R )
=> ( ( domain4236798911309298543t_unit @ R )
= ( primei7645216761534224334t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R ) ) ) ).
% cring.domain_eq_zeroprimeideal
thf(fact_752_cring_Odomain__eq__zeroprimeideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( domain_a_b @ R )
= ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ) ).
% cring.domain_eq_zeroprimeideal
thf(fact_753_cring_Odomain__eq__zeroprimeideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( domain6553523120543210313t_unit @ R )
= ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ) ).
% cring.domain_eq_zeroprimeideal
thf(fact_754_cring_Ozeroprimeideal__domainI,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( primei2288432046033540002t_unit @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) @ R )
=> ( domain7810152921033798211t_unit @ R ) ) ) ).
% cring.zeroprimeideal_domainI
thf(fact_755_cring_Ozeroprimeideal__domainI,axiom,
! [R: partia3925755165846298134t_unit] :
( ( cring_7032029500657136448t_unit @ R )
=> ( ( primei2134727858671723208t_unit @ ( insert_list_set_a @ ( zero_l6472697616691860973t_unit @ R ) @ bot_bo4397488018069675312_set_a ) @ R )
=> ( domain5179785879681597929t_unit @ R ) ) ) ).
% cring.zeroprimeideal_domainI
thf(fact_756_cring_Ozeroprimeideal__domainI,axiom,
! [R: partia6043505979758434576t_unit] :
( ( cring_5265999997378430022t_unit @ R )
=> ( ( primei7645216761534224334t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R )
=> ( domain4236798911309298543t_unit @ R ) ) ) ).
% cring.zeroprimeideal_domainI
thf(fact_757_cring_Ozeroprimeideal__domainI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R )
=> ( domain_a_b @ R ) ) ) ).
% cring.zeroprimeideal_domainI
thf(fact_758_cring_Ozeroprimeideal__domainI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R )
=> ( domain6553523120543210313t_unit @ R ) ) ) ).
% cring.zeroprimeideal_domainI
thf(fact_759_singletonD,axiom,
! [B2: list_list_a,A: list_list_a] :
( ( member_list_list_a @ B2 @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_760_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_761_singletonD,axiom,
! [B2: list_a,A: list_a] :
( ( member_list_a @ B2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_762_singleton__iff,axiom,
! [B2: list_list_a,A: list_list_a] :
( ( member_list_list_a @ B2 @ ( insert_list_list_a @ A @ bot_bo1875519244922727510list_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_763_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_764_singleton__iff,axiom,
! [B2: list_a,A: list_a] :
( ( member_list_a @ B2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_765_doubleton__eq__iff,axiom,
! [A: a,B2: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_766_doubleton__eq__iff,axiom,
! [A: list_a,B2: list_a,C2: list_a,D: list_a] :
( ( ( insert_list_a @ A @ ( insert_list_a @ B2 @ bot_bot_set_list_a ) )
= ( insert_list_a @ C2 @ ( insert_list_a @ D @ bot_bot_set_list_a ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_767_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_768_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_769_singleton__inject,axiom,
! [A: a,B2: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_770_singleton__inject,axiom,
! [A: list_a,B2: list_a] :
( ( ( insert_list_a @ A @ bot_bot_set_list_a )
= ( insert_list_a @ B2 @ bot_bot_set_list_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_771_Int__Collect__mono,axiom,
! [A2: set_list_list_a,B3: set_list_list_a,P2: list_list_a > $o,Q2: list_list_a > $o] :
( ( ord_le8488217952732425610list_a @ A2 @ B3 )
=> ( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A2 )
=> ( ( P2 @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_le8488217952732425610list_a @ ( inf_in7423150557312423384list_a @ A2 @ ( collect_list_list_a @ P2 ) ) @ ( inf_in7423150557312423384list_a @ B3 @ ( collect_list_list_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_772_Int__Collect__mono,axiom,
! [A2: set_list_a,B3: set_list_a,P2: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ( ( P2 @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P2 ) ) @ ( inf_inf_set_list_a @ B3 @ ( collect_list_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_773_Int__Collect__mono,axiom,
! [A2: set_a,B3: set_a,P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( ( P2 @ X3 )
=> ( Q2 @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P2 ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_774_Int__greatest,axiom,
! [C3: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ A2 )
=> ( ( ord_le8861187494160871172list_a @ C3 @ B3 )
=> ( ord_le8861187494160871172list_a @ C3 @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_775_Int__greatest,axiom,
! [C3: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( ( ord_less_eq_set_a @ C3 @ B3 )
=> ( ord_less_eq_set_a @ C3 @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_776_Int__absorb2,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_777_Int__absorb2,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_778_Int__absorb1,axiom,
! [B3: set_list_a,A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_779_Int__absorb1,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_780_Int__lower2,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_781_Int__lower2,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_782_Int__lower1,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_783_Int__lower1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_784_Int__mono,axiom,
! [A2: set_list_a,C3: set_list_a,B3: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ ( inf_inf_set_list_a @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_785_Int__mono,axiom,
! [A2: set_a,C3: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( inf_inf_set_a @ C3 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_786_disjoint__iff__not__equal,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ! [Y2: a] :
( ( member_a @ Y2 @ B3 )
=> ( X != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_787_disjoint__iff__not__equal,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ! [Y2: list_a] :
( ( member_list_a @ Y2 @ B3 )
=> ( X != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_788_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_789_Int__empty__right,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_790_Int__empty__left,axiom,
! [B3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B3 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_791_Int__empty__left,axiom,
! [B3: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B3 )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_792_disjoint__iff,axiom,
! [A2: set_list_list_a,B3: set_list_list_a] :
( ( ( inf_in7423150557312423384list_a @ A2 @ B3 )
= bot_bo1875519244922727510list_a )
= ( ! [X: list_list_a] :
( ( member_list_list_a @ X @ A2 )
=> ~ ( member_list_list_a @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_793_disjoint__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A2 )
=> ~ ( member_a @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_794_disjoint__iff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ~ ( member_list_a @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_795_Int__emptyI,axiom,
! [A2: set_list_list_a,B3: set_list_list_a] :
( ! [X3: list_list_a] :
( ( member_list_list_a @ X3 @ A2 )
=> ~ ( member_list_list_a @ X3 @ B3 ) )
=> ( ( inf_in7423150557312423384list_a @ A2 @ B3 )
= bot_bo1875519244922727510list_a ) ) ).
% Int_emptyI
thf(fact_796_Int__emptyI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ~ ( member_a @ X3 @ B3 ) )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_797_Int__emptyI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A2 )
=> ~ ( member_list_a @ X3 @ B3 ) )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_798_Int__insert__left,axiom,
! [A: list_list_a,C3: set_list_list_a,B3: set_list_list_a] :
( ( ( member_list_list_a @ A @ C3 )
=> ( ( inf_in7423150557312423384list_a @ ( insert_list_list_a @ A @ B3 ) @ C3 )
= ( insert_list_list_a @ A @ ( inf_in7423150557312423384list_a @ B3 @ C3 ) ) ) )
& ( ~ ( member_list_list_a @ A @ C3 )
=> ( ( inf_in7423150557312423384list_a @ ( insert_list_list_a @ A @ B3 ) @ C3 )
= ( inf_in7423150557312423384list_a @ B3 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_799_Int__insert__left,axiom,
! [A: list_a,C3: set_list_a,B3: set_list_a] :
( ( ( member_list_a @ A @ C3 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B3 ) @ C3 )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ B3 @ C3 ) ) ) )
& ( ~ ( member_list_a @ A @ C3 )
=> ( ( inf_inf_set_list_a @ ( insert_list_a @ A @ B3 ) @ C3 )
= ( inf_inf_set_list_a @ B3 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_800_Int__insert__left,axiom,
! [A: a,C3: set_a,B3: set_a] :
( ( ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C3 )
= ( insert_a @ A @ ( inf_inf_set_a @ B3 @ C3 ) ) ) )
& ( ~ ( member_a @ A @ C3 )
=> ( ( inf_inf_set_a @ ( insert_a @ A @ B3 ) @ C3 )
= ( inf_inf_set_a @ B3 @ C3 ) ) ) ) ).
% Int_insert_left
thf(fact_801_Int__insert__right,axiom,
! [A: list_list_a,A2: set_list_list_a,B3: set_list_list_a] :
( ( ( member_list_list_a @ A @ A2 )
=> ( ( inf_in7423150557312423384list_a @ A2 @ ( insert_list_list_a @ A @ B3 ) )
= ( insert_list_list_a @ A @ ( inf_in7423150557312423384list_a @ A2 @ B3 ) ) ) )
& ( ~ ( member_list_list_a @ A @ A2 )
=> ( ( inf_in7423150557312423384list_a @ A2 @ ( insert_list_list_a @ A @ B3 ) )
= ( inf_in7423150557312423384list_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_802_Int__insert__right,axiom,
! [A: list_a,A2: set_list_a,B3: set_list_a] :
( ( ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( insert_list_a @ A @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) )
& ( ~ ( member_list_a @ A @ A2 )
=> ( ( inf_inf_set_list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_803_Int__insert__right,axiom,
! [A: a,A2: set_a,B3: set_a] :
( ( ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( inf_inf_set_a @ A2 @ B3 ) ) ) )
& ( ~ ( member_a @ A @ A2 )
=> ( ( inf_inf_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_804_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_805_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_806_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_807_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_808_x_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 ) ) ) ) ).
% x.maximalideal_prime
thf(fact_809_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_810_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_811_map__in__poly__ring__carrier,axiom,
! [P: list_a,F: a > list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [A3: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( F @ A3 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% map_in_poly_ring_carrier
thf(fact_812_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_813_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_814_maximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ r )
=> ( primeideal_a_b @ I2 @ r ) ) ).
% maximalideal_prime
thf(fact_815_normalize_Osimps_I1_J,axiom,
( ( normalize_a_b @ r @ nil_a )
= nil_a ) ).
% normalize.simps(1)
thf(fact_816_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_817_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_818_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_819_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_820_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_821_is__root__def,axiom,
! [P: list_a,X2: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P @ X2 )
= ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P @ X2 )
= ( zero_a_b @ r ) )
& ( P != nil_a ) ) ) ).
% is_root_def
thf(fact_822_univ__poly__zero__closed,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] : ( member_list_set_a @ nil_set_a @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_823_univ__poly__zero__closed,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] : ( member_list_a @ nil_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_824_univ__poly__zero__closed,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] : ( member_list_list_a @ nil_list_a @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ).
% univ_poly_zero_closed
thf(fact_825_maximalideal_Ois__maximalideal,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( maxima6585700282301356660t_unit @ I2 @ R )
=> ( maxima6585700282301356660t_unit @ I2 @ R ) ) ).
% maximalideal.is_maximalideal
thf(fact_826_maximalideal_Ois__maximalideal,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( maximalideal_a_b @ I2 @ R ) ) ).
% maximalideal.is_maximalideal
thf(fact_827_maximalideal_OI__notcarr,axiom,
! [I2: set_a,R: partia2175431115845679010xt_a_b] :
( ( maximalideal_a_b @ I2 @ R )
=> ( ( partia707051561876973205xt_a_b @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_828_maximalideal_OI__notcarr,axiom,
! [I2: set_list_a,R: partia2670972154091845814t_unit] :
( ( maxima6585700282301356660t_unit @ I2 @ R )
=> ( ( partia5361259788508890537t_unit @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_829_maximalideal_OI__notcarr,axiom,
! [I2: set_list_list_a,R: partia2956882679547061052t_unit] :
( ( maxima7552488817642790894t_unit @ I2 @ R )
=> ( ( partia2464479390973590831t_unit @ R )
!= I2 ) ) ).
% maximalideal.I_notcarr
thf(fact_830_univ__poly__zero,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ R @ K ) )
= nil_a ) ).
% univ_poly_zero
thf(fact_831_univ__poly__zero,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( zero_l347298301471573063t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) )
= nil_list_a ) ).
% univ_poly_zero
thf(fact_832_univ__poly__zero,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] :
( ( zero_l6472697616691860973t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) )
= nil_set_a ) ).
% univ_poly_zero
thf(fact_833_ring_Onormalize_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( normal637505603836502915t_unit @ R @ nil_list_a )
= nil_list_a ) ) ).
% ring.normalize.simps(1)
thf(fact_834_ring_Onormalize_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ nil_a )
= nil_a ) ) ).
% ring.normalize.simps(1)
thf(fact_835_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2670972154091845814t_unit] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( eval_l34571156754992824t_unit @ R @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_836_ring_Oeval_Osimps_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( ring_a_b @ R )
=> ( ( eval_a_b @ R @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ R ) ) ) ) ).
% ring.eval.simps(1)
thf(fact_837_ring_Oconst__term__not__zero,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a] :
( ( ring_a_b @ R )
=> ( ( ( const_term_a_b @ R @ P )
!= ( zero_a_b @ R ) )
=> ( P != nil_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_838_ring_Oconst__term__not__zero,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( ( const_6738166269504826821t_unit @ R @ P )
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( P != nil_list_a ) ) ) ).
% ring.const_term_not_zero
thf(fact_839_cring_Omaximalideal__prime,axiom,
! [R: partia2956882679547061052t_unit,I2: set_list_list_a] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( maxima7552488817642790894t_unit @ I2 @ R )
=> ( primei2288432046033540002t_unit @ I2 @ R ) ) ) ).
% cring.maximalideal_prime
thf(fact_840_cring_Omaximalideal__prime,axiom,
! [R: partia3925755165846298134t_unit,I2: set_list_set_a] :
( ( cring_7032029500657136448t_unit @ R )
=> ( ( maxima7437456461502928148t_unit @ I2 @ R )
=> ( primei2134727858671723208t_unit @ I2 @ R ) ) ) ).
% cring.maximalideal_prime
thf(fact_841_cring_Omaximalideal__prime,axiom,
! [R: partia2670972154091845814t_unit,I2: set_list_a] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( maxima6585700282301356660t_unit @ I2 @ R )
=> ( primei6309817859076077608t_unit @ I2 @ R ) ) ) ).
% cring.maximalideal_prime
thf(fact_842_cring_Omaximalideal__prime,axiom,
! [R: partia2175431115845679010xt_a_b,I2: set_a] :
( ( cring_a_b @ R )
=> ( ( maximalideal_a_b @ I2 @ R )
=> ( primeideal_a_b @ I2 @ R ) ) ) ).
% cring.maximalideal_prime
thf(fact_843_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_844_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_845_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_846_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_847_ring_Ois__root__def,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,X2: list_list_a] :
( ( ring_l1939023646219158831t_unit @ R )
=> ( ( polyno5142720416380192742t_unit @ R @ P @ X2 )
= ( ( member_list_list_a @ X2 @ ( partia2464479390973590831t_unit @ R ) )
& ( ( eval_l1088911609197519410t_unit @ R @ P @ X2 )
= ( zero_l347298301471573063t_unit @ R ) )
& ( P != nil_list_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_848_ring_Ois__root__def,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,X2: a] :
( ( ring_a_b @ R )
=> ( ( polyno4133073214067823460ot_a_b @ R @ P @ X2 )
= ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ R ) )
& ( ( eval_a_b @ R @ P @ X2 )
= ( zero_a_b @ R ) )
& ( P != nil_a ) ) ) ) ).
% ring.is_root_def
thf(fact_849_ring_Ois__root__def,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,X2: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( polyno6951661231331188332t_unit @ R @ P @ X2 )
= ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R ) )
& ( ( eval_l34571156754992824t_unit @ R @ P @ X2 )
= ( zero_l4142658623432671053t_unit @ R ) )
& ( P != nil_list_a ) ) ) ) ).
% ring.is_root_def
thf(fact_850_field_Ozeromaximalideal,axiom,
! [R: partia6043505979758434576t_unit] :
( ( field_6045675692312731021t_unit @ R )
=> ( maxima2253313296322093082t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_851_field_Ozeromaximalideal,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( field_a_b @ R )
=> ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_852_field_Ozeromaximalideal,axiom,
! [R: partia2670972154091845814t_unit] :
( ( field_6388047844668329575t_unit @ R )
=> ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R ) ) ).
% field.zeromaximalideal
thf(fact_853_cring_Ozeromaximalideal__fieldI,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( maxima7552488817642790894t_unit @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) @ R )
=> ( field_1861437471013600865t_unit @ R ) ) ) ).
% cring.zeromaximalideal_fieldI
thf(fact_854_cring_Ozeromaximalideal__fieldI,axiom,
! [R: partia3925755165846298134t_unit] :
( ( cring_7032029500657136448t_unit @ R )
=> ( ( maxima7437456461502928148t_unit @ ( insert_list_set_a @ ( zero_l6472697616691860973t_unit @ R ) @ bot_bo4397488018069675312_set_a ) @ R )
=> ( field_3588249815925144839t_unit @ R ) ) ) ).
% cring.zeromaximalideal_fieldI
thf(fact_855_cring_Ozeromaximalideal__fieldI,axiom,
! [R: partia6043505979758434576t_unit] :
( ( cring_5265999997378430022t_unit @ R )
=> ( ( maxima2253313296322093082t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R )
=> ( field_6045675692312731021t_unit @ R ) ) ) ).
% cring.zeromaximalideal_fieldI
thf(fact_856_cring_Ozeromaximalideal__fieldI,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R )
=> ( field_a_b @ R ) ) ) ).
% cring.zeromaximalideal_fieldI
thf(fact_857_cring_Ozeromaximalideal__fieldI,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R )
=> ( field_6388047844668329575t_unit @ R ) ) ) ).
% cring.zeromaximalideal_fieldI
thf(fact_858_cring_Ozeromaximalideal__eq__field,axiom,
! [R: partia2956882679547061052t_unit] :
( ( cring_5991999922451032090t_unit @ R )
=> ( ( maxima7552488817642790894t_unit @ ( insert_list_list_a @ ( zero_l347298301471573063t_unit @ R ) @ bot_bo1875519244922727510list_a ) @ R )
= ( field_1861437471013600865t_unit @ R ) ) ) ).
% cring.zeromaximalideal_eq_field
thf(fact_859_cring_Ozeromaximalideal__eq__field,axiom,
! [R: partia3925755165846298134t_unit] :
( ( cring_7032029500657136448t_unit @ R )
=> ( ( maxima7437456461502928148t_unit @ ( insert_list_set_a @ ( zero_l6472697616691860973t_unit @ R ) @ bot_bo4397488018069675312_set_a ) @ R )
= ( field_3588249815925144839t_unit @ R ) ) ) ).
% cring.zeromaximalideal_eq_field
thf(fact_860_cring_Ozeromaximalideal__eq__field,axiom,
! [R: partia6043505979758434576t_unit] :
( ( cring_5265999997378430022t_unit @ R )
=> ( ( maxima2253313296322093082t_unit @ ( insert_set_a @ ( zero_s2174465271003423091t_unit @ R ) @ bot_bot_set_set_a ) @ R )
= ( field_6045675692312731021t_unit @ R ) ) ) ).
% cring.zeromaximalideal_eq_field
thf(fact_861_cring_Ozeromaximalideal__eq__field,axiom,
! [R: partia2175431115845679010xt_a_b] :
( ( cring_a_b @ R )
=> ( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ R ) @ bot_bot_set_a ) @ R )
= ( field_a_b @ R ) ) ) ).
% cring.zeromaximalideal_eq_field
thf(fact_862_cring_Ozeromaximalideal__eq__field,axiom,
! [R: partia2670972154091845814t_unit] :
( ( cring_3148771470849435808t_unit @ R )
=> ( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ R ) @ bot_bot_set_list_a ) @ R )
= ( field_6388047844668329575t_unit @ R ) ) ) ).
% cring.zeromaximalideal_eq_field
thf(fact_863_domain_Omap__in__poly__ring__carrier,axiom,
! [R: partia6043505979758434576t_unit,P: list_set_a,F: set_a > list_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) )
=> ( ! [A3: set_a] :
( ( member_set_a @ A3 @ ( partia5907974310037520643t_unit @ R ) )
=> ( member_list_set_a @ ( F @ A3 ) @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) )
=> ( ! [A3: set_a] :
( ( A3
!= ( zero_s2174465271003423091t_unit @ R ) )
=> ( ( F @ A3 )
!= nil_set_a ) )
=> ( member6735289990665601811_set_a @ ( map_set_a_list_set_a @ F @ P ) @ ( partia5991483576489006735t_unit @ ( univ_p4425688966569699694t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ ( partia5907974310037520643t_unit @ R ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_864_domain_Omap__in__poly__ring__carrier,axiom,
! [R: partia2956882679547061052t_unit,P: list_list_list_a,F: list_list_a > list_list_list_a] :
( ( domain7810152921033798211t_unit @ R )
=> ( ( member5342144027231129785list_a @ P @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) )
=> ( ! [A3: list_list_a] :
( ( member_list_list_a @ A3 @ ( partia2464479390973590831t_unit @ R ) )
=> ( member5342144027231129785list_a @ ( F @ A3 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) )
=> ( ! [A3: list_list_a] :
( ( A3
!= ( zero_l347298301471573063t_unit @ R ) )
=> ( ( F @ A3 )
!= nil_list_list_a ) )
=> ( member6842060177613954879list_a @ ( map_li5227692475714150986list_a @ F @ P ) @ ( partia2412307164297199803t_unit @ ( univ_p3766428211910075458t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ R @ ( partia2464479390973590831t_unit @ R ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_865_domain_Omap__in__poly__ring__carrier,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,F: a > list_a] :
( ( domain_a_b @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ R ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) )
=> ( ! [A3: a] :
( ( A3
!= ( zero_a_b @ R ) )
=> ( ( F @ A3 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ ( partia707051561876973205xt_a_b @ R ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_866_domain_Omap__in__poly__ring__carrier,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,F: list_a > list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ R ) )
=> ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) )
=> ( ! [A3: list_a] :
( ( A3
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( F @ A3 )
!= nil_list_a ) )
=> ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ F @ P ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ ( partia5361259788508890537t_unit @ R ) ) ) ) ) ) ) ) ) ) ).
% domain.map_in_poly_ring_carrier
thf(fact_867_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_868_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_869_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_870_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_871_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_872_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_873_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_874_x_Onormalize_Osimps_I1_J,axiom,
( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
= nil_list_a ) ).
% x.normalize.simps(1)
thf(fact_875_x_Oeval_Osimps_I1_J,axiom,
( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a )
= ( ^ [Uu: list_a] : ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.eval.simps(1)
thf(fact_876_x_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 ) ) ).
% x.const_term_not_zero
thf(fact_877_x_Ois__root__def,axiom,
! [P: list_list_a,X2: list_a] :
( ( polyno6951661231331188332t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X2 )
= ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
& ( P != nil_list_a ) ) ) ).
% x.is_root_def
thf(fact_878_x_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 ) ) ) ) ).
% x.zeromaximalideal_eq_field
thf(fact_879_x_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 ) ) ) ) ).
% x.zeromaximalideal_fieldI
thf(fact_880_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_881_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_882_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_883_domain_Ouniv__poly__not__field,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ R @ K ) ) ) ) ).
% domain.univ_poly_not_field
thf(fact_884_domain_Ouniv__poly__not__field,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ~ ( field_1861437471013600865t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_not_field
thf(fact_885_domain_Ouniv__poly__not__field,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ~ ( field_3588249815925144839t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) ) ) ).
% domain.univ_poly_not_field
thf(fact_886_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_887_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_888_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_889_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_890_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_891_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_892_ring_Oring__primeI,axiom,
! [R: partia2175431115845679010xt_a_b,P: a] :
( ( ring_a_b @ R )
=> ( ( P
!= ( zero_a_b @ R ) )
=> ( ( prime_a_ring_ext_a_b @ R @ P )
=> ( ring_ring_prime_a_b @ R @ P ) ) ) ) ).
% ring.ring_primeI
thf(fact_893_ring_Oring__primeI,axiom,
! [R: partia2670972154091845814t_unit,P: list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( P
!= ( zero_l4142658623432671053t_unit @ R ) )
=> ( ( prime_2011924034616061926t_unit @ R @ P )
=> ( ring_r6430282645014804837t_unit @ R @ P ) ) ) ) ).
% ring.ring_primeI
thf(fact_894_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_895_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_896_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_897_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_898_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_899_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_900_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_901_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_902_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_903_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_904_x_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 ) ) ) ).
% x.ring_primeI
thf(fact_905_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_906_set__empty,axiom,
! [Xs: list_list_a] :
( ( ( set_list_a2 @ Xs )
= bot_bot_set_list_a )
= ( Xs = nil_list_a ) ) ).
% set_empty
thf(fact_907_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_908_set__empty2,axiom,
! [Xs: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% set_empty2
thf(fact_909_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_910_x_Ofactors__mult,axiom,
! [Fa: list_list_a,A: list_a,Fb: list_list_a,B2: 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 @ B2 )
=> ( ( 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 @ B2 ) ) ) ) ) ) ).
% x.factors_mult
thf(fact_911_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_912_x_Onormalize__idem,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ Q ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ Q ) ) ) ).
% x.normalize_idem
thf(fact_913_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_914_Diff__cancel,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ A2 )
= bot_bot_set_list_a ) ).
% Diff_cancel
thf(fact_915_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_916_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_917_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_918_Diff__empty,axiom,
! [A2: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Diff_empty
thf(fact_919_map__eq__conv,axiom,
! [F: list_a > list_a,Xs: list_list_a,G: list_a > list_a] :
( ( ( map_list_a_list_a @ F @ Xs )
= ( map_list_a_list_a @ G @ Xs ) )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_920_map__eq__conv,axiom,
! [F: a > list_a,Xs: list_a,G: a > list_a] :
( ( ( map_a_list_a @ F @ Xs )
= ( map_a_list_a @ G @ Xs ) )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_921_map__eq__conv,axiom,
! [F: a > a,Xs: list_a,G: a > a] :
( ( ( map_a_a @ F @ Xs )
= ( map_a_a @ G @ Xs ) )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_922_map__is__Nil__conv,axiom,
! [F: list_a > a,Xs: list_list_a] :
( ( ( map_list_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_list_a ) ) ).
% map_is_Nil_conv
thf(fact_923_map__is__Nil__conv,axiom,
! [F: a > list_a,Xs: list_a] :
( ( ( map_a_list_a @ F @ Xs )
= nil_list_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_924_map__is__Nil__conv,axiom,
! [F: list_a > list_a,Xs: list_list_a] :
( ( ( map_list_a_list_a @ F @ Xs )
= nil_list_a )
= ( Xs = nil_list_a ) ) ).
% map_is_Nil_conv
thf(fact_925_map__is__Nil__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( ( map_a_a @ F @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% map_is_Nil_conv
thf(fact_926_Nil__is__map__conv,axiom,
! [F: list_a > a,Xs: list_list_a] :
( ( nil_a
= ( map_list_a_a @ F @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% Nil_is_map_conv
thf(fact_927_Nil__is__map__conv,axiom,
! [F: a > list_a,Xs: list_a] :
( ( nil_list_a
= ( map_a_list_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_928_Nil__is__map__conv,axiom,
! [F: list_a > list_a,Xs: list_list_a] :
( ( nil_list_a
= ( map_list_a_list_a @ F @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% Nil_is_map_conv
thf(fact_929_Nil__is__map__conv,axiom,
! [F: a > a,Xs: list_a] :
( ( nil_a
= ( map_a_a @ F @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_map_conv
thf(fact_930_list_Omap__disc__iff,axiom,
! [F: list_a > a,A: list_list_a] :
( ( ( map_list_a_a @ F @ A )
= nil_a )
= ( A = nil_list_a ) ) ).
% list.map_disc_iff
thf(fact_931_list_Omap__disc__iff,axiom,
! [F: a > list_a,A: list_a] :
( ( ( map_a_list_a @ F @ A )
= nil_list_a )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_932_list_Omap__disc__iff,axiom,
! [F: list_a > list_a,A: list_list_a] :
( ( ( map_list_a_list_a @ F @ A )
= nil_list_a )
= ( A = nil_list_a ) ) ).
% list.map_disc_iff
thf(fact_933_list_Omap__disc__iff,axiom,
! [F: a > a,A: list_a] :
( ( ( map_a_a @ F @ A )
= nil_a )
= ( A = nil_a ) ) ).
% list.map_disc_iff
thf(fact_934_map__append,axiom,
! [F: list_a > a,Xs: list_list_a,Ys: list_list_a] :
( ( map_list_a_a @ F @ ( append_list_a @ Xs @ Ys ) )
= ( append_a @ ( map_list_a_a @ F @ Xs ) @ ( map_list_a_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_935_map__append,axiom,
! [F: a > list_a,Xs: list_a,Ys: list_a] :
( ( map_a_list_a @ F @ ( append_a @ Xs @ Ys ) )
= ( append_list_a @ ( map_a_list_a @ F @ Xs ) @ ( map_a_list_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_936_map__append,axiom,
! [F: list_a > list_a,Xs: list_list_a,Ys: list_list_a] :
( ( map_list_a_list_a @ F @ ( append_list_a @ Xs @ Ys ) )
= ( append_list_a @ ( map_list_a_list_a @ F @ Xs ) @ ( map_list_a_list_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_937_map__append,axiom,
! [F: a > a,Xs: list_a,Ys: list_a] :
( ( map_a_a @ F @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( map_a_a @ F @ Xs ) @ ( map_a_a @ F @ Ys ) ) ) ).
% map_append
thf(fact_938_Diff__eq__empty__iff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( minus_646659088055828811list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_939_Diff__eq__empty__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( minus_minus_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_940_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_941_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_942_Diff__disjoint,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B3 @ A2 ) )
= bot_bot_set_a ) ).
% Diff_disjoint
thf(fact_943_Diff__disjoint,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ ( minus_646659088055828811list_a @ B3 @ A2 ) )
= bot_bot_set_list_a ) ).
% Diff_disjoint
thf(fact_944_map__eq__append__conv,axiom,
! [F: list_a > a,Xs: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( map_list_a_a @ F @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ? [Us: list_list_a,Vs: list_list_a] :
( ( Xs
= ( append_list_a @ Us @ Vs ) )
& ( Ys
= ( map_list_a_a @ F @ Us ) )
& ( Zs
= ( map_list_a_a @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_945_map__eq__append__conv,axiom,
! [F: a > list_a,Xs: list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( map_a_list_a @ F @ Xs )
= ( append_list_a @ Ys @ Zs ) )
= ( ? [Us: list_a,Vs: list_a] :
( ( Xs
= ( append_a @ Us @ Vs ) )
& ( Ys
= ( map_a_list_a @ F @ Us ) )
& ( Zs
= ( map_a_list_a @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_946_map__eq__append__conv,axiom,
! [F: list_a > list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ( ( map_list_a_list_a @ F @ Xs )
= ( append_list_a @ Ys @ Zs ) )
= ( ? [Us: list_list_a,Vs: list_list_a] :
( ( Xs
= ( append_list_a @ Us @ Vs ) )
& ( Ys
= ( map_list_a_list_a @ F @ Us ) )
& ( Zs
= ( map_list_a_list_a @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_947_map__eq__append__conv,axiom,
! [F: a > a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( map_a_a @ F @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ? [Us: list_a,Vs: list_a] :
( ( Xs
= ( append_a @ Us @ Vs ) )
& ( Ys
= ( map_a_a @ F @ Us ) )
& ( Zs
= ( map_a_a @ F @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_948_append__eq__map__conv,axiom,
! [Ys: list_a,Zs: list_a,F: list_a > a,Xs: list_list_a] :
( ( ( append_a @ Ys @ Zs )
= ( map_list_a_a @ F @ Xs ) )
= ( ? [Us: list_list_a,Vs: list_list_a] :
( ( Xs
= ( append_list_a @ Us @ Vs ) )
& ( Ys
= ( map_list_a_a @ F @ Us ) )
& ( Zs
= ( map_list_a_a @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_949_append__eq__map__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,F: a > list_a,Xs: list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( map_a_list_a @ F @ Xs ) )
= ( ? [Us: list_a,Vs: list_a] :
( ( Xs
= ( append_a @ Us @ Vs ) )
& ( Ys
= ( map_a_list_a @ F @ Us ) )
& ( Zs
= ( map_a_list_a @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_950_append__eq__map__conv,axiom,
! [Ys: list_list_a,Zs: list_list_a,F: list_a > list_a,Xs: list_list_a] :
( ( ( append_list_a @ Ys @ Zs )
= ( map_list_a_list_a @ F @ Xs ) )
= ( ? [Us: list_list_a,Vs: list_list_a] :
( ( Xs
= ( append_list_a @ Us @ Vs ) )
& ( Ys
= ( map_list_a_list_a @ F @ Us ) )
& ( Zs
= ( map_list_a_list_a @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_951_append__eq__map__conv,axiom,
! [Ys: list_a,Zs: list_a,F: a > a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( map_a_a @ F @ Xs ) )
= ( ? [Us: list_a,Vs: list_a] :
( ( Xs
= ( append_a @ Us @ Vs ) )
& ( Ys
= ( map_a_a @ F @ Us ) )
& ( Zs
= ( map_a_a @ F @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_952_Int__Diff,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C3 )
= ( inf_inf_set_a @ A2 @ ( minus_minus_set_a @ B3 @ C3 ) ) ) ).
% Int_Diff
thf(fact_953_Int__Diff,axiom,
! [A2: set_list_a,B3: set_list_a,C3: set_list_a] :
( ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ C3 )
= ( inf_inf_set_list_a @ A2 @ ( minus_646659088055828811list_a @ B3 @ C3 ) ) ) ).
% Int_Diff
thf(fact_954_Diff__Int2,axiom,
! [A2: set_a,C3: set_a,B3: set_a] :
( ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C3 ) @ ( inf_inf_set_a @ B3 @ C3 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C3 ) @ B3 ) ) ).
% Diff_Int2
thf(fact_955_Diff__Int2,axiom,
! [A2: set_list_a,C3: set_list_a,B3: set_list_a] :
( ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A2 @ C3 ) @ ( inf_inf_set_list_a @ B3 @ C3 ) )
= ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A2 @ C3 ) @ B3 ) ) ).
% Diff_Int2
thf(fact_956_Diff__Diff__Int,axiom,
! [A2: set_a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_957_Diff__Diff__Int,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
= ( inf_inf_set_list_a @ A2 @ B3 ) ) ).
% Diff_Diff_Int
thf(fact_958_Diff__Int__distrib,axiom,
! [C3: set_a,A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ C3 @ ( minus_minus_set_a @ A2 @ B3 ) )
= ( minus_minus_set_a @ ( inf_inf_set_a @ C3 @ A2 ) @ ( inf_inf_set_a @ C3 @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_959_Diff__Int__distrib,axiom,
! [C3: set_list_a,A2: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ C3 @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
= ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ C3 @ A2 ) @ ( inf_inf_set_list_a @ C3 @ B3 ) ) ) ).
% Diff_Int_distrib
thf(fact_960_Diff__Int__distrib2,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( inf_inf_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ C3 )
= ( minus_minus_set_a @ ( inf_inf_set_a @ A2 @ C3 ) @ ( inf_inf_set_a @ B3 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_961_Diff__Int__distrib2,axiom,
! [A2: set_list_a,B3: set_list_a,C3: set_list_a] :
( ( inf_inf_set_list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) @ C3 )
= ( minus_646659088055828811list_a @ ( inf_inf_set_list_a @ A2 @ C3 ) @ ( inf_inf_set_list_a @ B3 @ C3 ) ) ) ).
% Diff_Int_distrib2
thf(fact_962_Diff__mono,axiom,
! [A2: set_list_a,C3: set_list_a,D2: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ C3 )
=> ( ( ord_le8861187494160871172list_a @ D2 @ B3 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) @ ( minus_646659088055828811list_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_963_Diff__mono,axiom,
! [A2: set_a,C3: set_a,D2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ D2 @ B3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_964_Diff__subset,axiom,
! [A2: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_965_Diff__subset,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_966_double__diff,axiom,
! [A2: set_list_a,B3: set_list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C3 )
=> ( ( minus_646659088055828811list_a @ B3 @ ( minus_646659088055828811list_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_967_double__diff,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_968_diff__shunt__var,axiom,
! [X2: set_list_a,Y: set_list_a] :
( ( ( minus_646659088055828811list_a @ X2 @ Y )
= bot_bot_set_list_a )
= ( ord_le8861187494160871172list_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_969_diff__shunt__var,axiom,
! [X2: set_a,Y: set_a] :
( ( ( minus_minus_set_a @ X2 @ Y )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X2 @ Y ) ) ).
% diff_shunt_var
thf(fact_970_subset__Diff__insert,axiom,
! [A2: set_list_list_a,B3: set_list_list_a,X2: list_list_a,C3: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( minus_5335179877275218001list_a @ B3 @ ( insert_list_list_a @ X2 @ C3 ) ) )
= ( ( ord_le8488217952732425610list_a @ A2 @ ( minus_5335179877275218001list_a @ B3 @ C3 ) )
& ~ ( member_list_list_a @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_971_subset__Diff__insert,axiom,
! [A2: set_list_a,B3: set_list_a,X2: list_a,C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B3 @ ( insert_list_a @ X2 @ C3 ) ) )
= ( ( ord_le8861187494160871172list_a @ A2 @ ( minus_646659088055828811list_a @ B3 @ C3 ) )
& ~ ( member_list_a @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_972_subset__Diff__insert,axiom,
! [A2: set_a,B3: set_a,X2: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ ( insert_a @ X2 @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ C3 ) )
& ~ ( member_a @ X2 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_973_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_974_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_975_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_976_Diff__insert2,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_977_Diff__insert2,axiom,
! [A2: set_list_a,A: list_a,B3: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_978_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_979_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_980_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_981_Diff__insert,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_982_Diff__insert,axiom,
! [A2: set_list_a,A: list_a,B3: set_list_a] :
( ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ A @ B3 ) )
= ( minus_646659088055828811list_a @ ( minus_646659088055828811list_a @ A2 @ B3 ) @ ( insert_list_a @ A @ bot_bot_set_list_a ) ) ) ).
% Diff_insert
thf(fact_983_Int__Diff__disjoint,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ A2 @ B3 ) )
= bot_bot_set_a ) ).
% Int_Diff_disjoint
thf(fact_984_Int__Diff__disjoint,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A2 @ B3 ) @ ( minus_646659088055828811list_a @ A2 @ B3 ) )
= bot_bot_set_list_a ) ).
% Int_Diff_disjoint
thf(fact_985_Diff__triv,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
=> ( ( minus_minus_set_a @ A2 @ B3 )
= A2 ) ) ).
% Diff_triv
thf(fact_986_Diff__triv,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a )
=> ( ( minus_646659088055828811list_a @ A2 @ B3 )
= A2 ) ) ).
% Diff_triv
thf(fact_987_ring_Onormalize__idem,axiom,
! [R: partia2670972154091845814t_unit,P: list_list_a,Q: list_list_a] :
( ( ring_l6212528067271185461t_unit @ R )
=> ( ( normal637505603836502915t_unit @ R @ ( append_list_a @ ( normal637505603836502915t_unit @ R @ P ) @ Q ) )
= ( normal637505603836502915t_unit @ R @ ( append_list_a @ P @ Q ) ) ) ) ).
% ring.normalize_idem
thf(fact_988_ring_Onormalize__idem,axiom,
! [R: partia2175431115845679010xt_a_b,P: list_a,Q: list_a] :
( ( ring_a_b @ R )
=> ( ( normalize_a_b @ R @ ( append_a @ ( normalize_a_b @ R @ P ) @ Q ) )
= ( normalize_a_b @ R @ ( append_a @ P @ Q ) ) ) ) ).
% ring.normalize_idem
thf(fact_989_subset__insert__iff,axiom,
! [A2: set_list_list_a,X2: list_list_a,B3: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ A2 @ ( insert_list_list_a @ X2 @ B3 ) )
= ( ( ( member_list_list_a @ X2 @ A2 )
=> ( ord_le8488217952732425610list_a @ ( minus_5335179877275218001list_a @ A2 @ ( insert_list_list_a @ X2 @ bot_bo1875519244922727510list_a ) ) @ B3 ) )
& ( ~ ( member_list_list_a @ X2 @ A2 )
=> ( ord_le8488217952732425610list_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_990_subset__insert__iff,axiom,
! [A2: set_list_a,X2: list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ B3 ) )
= ( ( ( member_list_a @ X2 @ A2 )
=> ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) @ B3 ) )
& ( ~ ( member_list_a @ X2 @ A2 )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_991_subset__insert__iff,axiom,
! [A2: set_a,X2: a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B3 ) )
= ( ( ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_992_Diff__single__insert,axiom,
! [A2: set_list_a,X2: list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( minus_646659088055828811list_a @ A2 @ ( insert_list_a @ X2 @ bot_bot_set_list_a ) ) @ B3 )
=> ( ord_le8861187494160871172list_a @ A2 @ ( insert_list_a @ X2 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_993_Diff__single__insert,axiom,
! [A2: set_a,X2: a,B3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X2 @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_994_domain_OpirreducibleE_I1_J,axiom,
! [R: partia6043505979758434576t_unit,K: set_set_a,P: list_set_a] :
( ( domain4236798911309298543t_unit @ R )
=> ( ( subrin1511138061850335568t_unit @ K @ R )
=> ( ( member_list_set_a @ P @ ( partia3893404292425143049t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) ) )
=> ( ( ring_r8677422918745460462t_unit @ ( univ_p6720748963476187508t_unit @ R @ K ) @ P )
=> ( P != nil_set_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_995_domain_OpirreducibleE_I1_J,axiom,
! [R: partia2670972154091845814t_unit,K: set_list_a,P: list_list_a] :
( ( domain6553523120543210313t_unit @ R )
=> ( ( subrin6918843898125473962t_unit @ K @ R )
=> ( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) ) )
=> ( ( ring_r360171070648044744t_unit @ ( univ_p7953238456130426574t_unit @ R @ K ) @ P )
=> ( P != nil_list_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_996_domain_OpirreducibleE_I1_J,axiom,
! [R: partia2175431115845679010xt_a_b,K: set_a,P: list_a] :
( ( domain_a_b @ R )
=> ( ( subring_a_b @ K @ R )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ R @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ R @ K ) @ P )
=> ( P != nil_a ) ) ) ) ) ).
% domain.pirreducibleE(1)
thf(fact_997_subset__code_I1_J,axiom,
! [Xs: list_list_list_a,B3: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ ( set_list_list_a2 @ Xs ) @ B3 )
= ( ! [X: list_list_a] :
( ( member_list_list_a @ X @ ( set_list_list_a2 @ Xs ) )
=> ( member_list_list_a @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_998_subset__code_I1_J,axiom,
! [Xs: list_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B3 )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_999_subset__code_I1_J,axiom,
! [Xs: list_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B3 )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( member_a @ X @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_1000_ex__map__conv,axiom,
! [Ys: list_list_a,F: a > list_a] :
( ( ? [Xs2: list_a] :
( Ys
= ( map_a_list_a @ F @ Xs2 ) ) )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
=> ? [Y2: a] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1001_ex__map__conv,axiom,
! [Ys: list_list_a,F: list_a > list_a] :
( ( ? [Xs2: list_list_a] :
( Ys
= ( map_list_a_list_a @ F @ Xs2 ) ) )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
=> ? [Y2: list_a] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1002_ex__map__conv,axiom,
! [Ys: list_a,F: a > a] :
( ( ? [Xs2: list_a] :
( Ys
= ( map_a_a @ F @ Xs2 ) ) )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Ys ) )
=> ? [Y2: a] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_1003_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_1004_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_1005_x_Oeval__append__aux,axiom,
! [P: list_list_a,B2: 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 @ B2 @ ( 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 ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ ( cons_list_a @ B2 @ nil_list_a ) ) @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ A ) @ B2 ) ) ) ) ) ).
% x.eval_append_aux
thf(fact_1006_x_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 ) ) ) ).
% x.const_term_eq_last
thf(fact_1007_x_Onormalize_Ocases,axiom,
! [X2: list_list_a] :
( ( X2 != nil_list_a )
=> ~ ! [V3: list_a,Va: list_list_a] :
( X2
!= ( cons_list_a @ V3 @ Va ) ) ) ).
% x.normalize.cases
thf(fact_1008_local_Onormalize__idem,axiom,
! [P: list_a,Q: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( normalize_a_b @ r @ P ) @ Q ) )
= ( normalize_a_b @ r @ ( append_a @ P @ Q ) ) ) ).
% local.normalize_idem
thf(fact_1009_x_Opoly__of__const__def,axiom,
( ( poly_o8716471131768098070t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) )
= ( ^ [K3: list_a] : ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ K3 @ nil_list_a ) ) ) ) ).
% x.poly_of_const_def
thf(fact_1010_x_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 )
=> ~ ! [P3: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( P
!= ( append_list_a @ P3 @ ( cons_list_a @ A @ nil_list_a ) ) ) ) ) ) ) ).
% x.const_term_explicit
thf(fact_1011_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_1012_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_1013_x_Ofactors__mult__single,axiom,
! [A: list_a,Fb: list_list_a,B2: 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 @ B2 )
=> ( ( 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 @ B2 ) ) ) ) ) ).
% x.factors_mult_single
thf(fact_1014_x_Opoly__add__append__zero,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_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 ) ) @ ( append_list_a @ Q @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ).
% x.poly_add_append_zero
thf(fact_1015_normalize_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ~ ! [V3: a,Va: list_a] :
( X2
!= ( cons_a @ V3 @ Va ) ) ) ).
% normalize.cases
thf(fact_1016_poly__of__const__def,axiom,
( ( poly_of_const_a_b @ r )
= ( ^ [K3: a] : ( normalize_a_b @ r @ ( cons_a @ K3 @ nil_a ) ) ) ) ).
% poly_of_const_def
thf(fact_1017_x_Opoly__add__in__carrier,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( 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 ) ) @ P1 @ P22 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.poly_add_in_carrier
thf(fact_1018_x_Opoly__add__comm,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 @ P1 ) ) ) ) ).
% x.poly_add_comm
thf(fact_1019_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 )
=> ~ ! [P3: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( P
!= ( append_a @ P3 @ ( cons_a @ A @ nil_a ) ) ) ) ) ) ) ).
% const_term_explicit
thf(fact_1020_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_1021_x_Opoly__add__normalize__aux,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 ) @ P22 ) ) ) ) ).
% x.poly_add_normalize_aux
thf(fact_1022_x_Opoly__add__normalize_I2_J,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 ) ) ) ) ) ).
% x.poly_add_normalize(2)
thf(fact_1023_x_Opoly__add__normalize_I3_J,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 ) ) ) ) ) ).
% x.poly_add_normalize(3)
thf(fact_1024_x_Oeval__poly__add,axiom,
! [P: list_list_a,Q: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( 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 ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A ) ) ) ) ) ) ).
% x.eval_poly_add
thf(fact_1025_eval__append__aux,axiom,
! [P: list_a,B2: a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ ( cons_a @ B2 @ nil_a ) ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ A ) @ B2 ) ) ) ) ) ).
% eval_append_aux
thf(fact_1026_x_Opoly__add__zero_H_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 ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.poly_add_zero'(1)
thf(fact_1027_x_Opoly__add__zero_H_I2_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) ) ).
% x.poly_add_zero'(2)
thf(fact_1028_x_Oconst__term__simprules_I3_J,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ).
% x.const_term_simprules(3)
thf(fact_1029_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_1030_x_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 ) ) ).
% x.FactRing_zeroideal(2)
thf(fact_1031_x_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 ) ) ).
% x.FactRing_zeroideal(1)
thf(fact_1032_x_Oeval__poly__add__aux,axiom,
! [P: list_list_a,Q: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( size_s349497388124573686list_a @ P )
= ( size_s349497388124573686list_a @ Q ) )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A ) ) ) ) ) ) ) ).
% x.eval_poly_add_aux
thf(fact_1033_x_Opoly__mult__append__zero,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_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 ) ) @ Q )
= ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ ( cons_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ nil_list_a ) ) ) ) ) ) ).
% x.poly_mult_append_zero
thf(fact_1034_factors__mult,axiom,
! [Fa: list_a,A: a,Fb: list_a,B2: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fa @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B2 )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fa ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fb ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( append_a @ Fa @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) ) ) ) ) ) ).
% factors_mult
thf(fact_1035_x_Ocombine__append__zero,axiom,
! [Us2: list_list_a,Ks: 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 @ Ks @ ( 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 ) ) @ Ks @ Us2 ) ) ) ).
% x.combine_append_zero
thf(fact_1036_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_1037_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_1038_poly__add__comm,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ P22 @ P1 ) ) ) ) ).
% poly_add_comm
thf(fact_1039_poly__add__in__carrier,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_add_a_b @ r @ P1 @ P22 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_add_in_carrier
thf(fact_1040_poly__add__normalize__aux,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P22 ) ) ) ) ).
% poly_add_normalize_aux
thf(fact_1041_poly__add__normalize_I2_J,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ P1 @ ( normalize_a_b @ r @ P22 ) ) ) ) ) ).
% poly_add_normalize(2)
thf(fact_1042_poly__add__normalize_I3_J,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P1 @ P22 )
= ( poly_add_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ ( normalize_a_b @ r @ P22 ) ) ) ) ) ).
% poly_add_normalize(3)
thf(fact_1043_x_Onormalize__length__le,axiom,
! [P: list_list_a] : ( ord_less_eq_nat @ ( size_s349497388124573686list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) ) @ ( size_s349497388124573686list_a @ P ) ) ).
% x.normalize_length_le
thf(fact_1044_x_Opoly__mult_Osimps_I1_J,axiom,
! [P22: list_list_a] :
( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P22 )
= nil_list_a ) ).
% x.poly_mult.simps(1)
thf(fact_1045_factors__closed,axiom,
! [Fs: list_a,A: a] :
( ( factor5638265376665762323xt_a_b @ r @ Fs @ A )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Fs ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% factors_closed
thf(fact_1046_x_Oee__length,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ As @ Bs )
=> ( ( size_s349497388124573686list_a @ As )
= ( size_s349497388124573686list_a @ Bs ) ) ) ).
% x.ee_length
thf(fact_1047_factors__mult__single,axiom,
! [A: a,Fb: list_a,B2: a] :
( ( irredu6211895646901577903xt_a_b @ r @ A )
=> ( ( factor5638265376665762323xt_a_b @ r @ Fb @ B2 )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor5638265376665762323xt_a_b @ r @ ( cons_a @ A @ Fb ) @ ( mult_a_ring_ext_a_b @ r @ A @ B2 ) ) ) ) ) ).
% factors_mult_single
thf(fact_1048_eval__poly__add,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_add
thf(fact_1049_x_Ocombine_Osimps_I3_J,axiom,
! [Ks: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ nil_list_a )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% x.combine.simps(3)
thf(fact_1050_x_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 ) ) ) ) ).
% x.combine.simps(2)
thf(fact_1051_poly__add__zero_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ P @ nil_a )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(1)
thf(fact_1052_poly__add__zero_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ nil_a @ P )
= ( normalize_a_b @ r @ P ) ) ) ).
% poly_add_zero'(2)
thf(fact_1053_const__term__simprules_I3_J,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) )
= ( add_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(3)
thf(fact_1054_x_Opoly__mult__in__carrier,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.poly_mult_in_carrier
thf(fact_1055_x_Ocombine_Osimps_I1_J,axiom,
! [K2: list_a,Ks: list_list_a,U: list_a,Us2: list_list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( cons_list_a @ K2 @ Ks ) @ ( cons_list_a @ U @ Us2 ) )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ U ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us2 ) ) ) ).
% x.combine.simps(1)
thf(fact_1056_x_Ocombine__eq__eval,axiom,
! [Ks: list_list_a,X2: list_a] :
( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( size_s349497388124573686list_a @ Ks ) ) )
= ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ X2 ) ) ).
% x.combine_eq_eval
thf(fact_1057_x_Ocombine__r__distr,axiom,
! [Ks: list_list_a,Us2: list_list_a,K2: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ ( 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 @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks @ Us2 ) )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_list_a_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K2 ) @ Ks ) @ Us2 ) ) ) ) ) ).
% x.combine_r_distr
thf(fact_1058_x_Opoly__mult__zero_I2_J,axiom,
! [P: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ nil_list_a )
= nil_list_a ) ) ).
% x.poly_mult_zero(2)
thf(fact_1059_x_Opoly__mult__zero_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 ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ nil_list_a @ P )
= nil_list_a ) ) ).
% x.poly_mult_zero(1)
thf(fact_1060_x_Opoly__mult__l__distr_H,axiom,
! [P1: list_list_a,P22: list_list_a,P32: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P32 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 ) @ P32 )
= ( poly_a7601779127272115787t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P32 ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P22 @ P32 ) ) ) ) ) ) ).
% x.poly_mult_l_distr'
thf(fact_1061_x_Opoly__mult__normalize,axiom,
! [P1: list_list_a,P22: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P1 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P22 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 @ P22 )
= ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P1 ) @ P22 ) ) ) ) ).
% x.poly_mult_normalize
thf(fact_1062_x_Ocombine_Oelims,axiom,
! [X2: list_list_a,Xa: list_list_a,Y: list_a] :
( ( ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Xa )
= Y )
=> ( ! [K4: list_a,Ks2: list_list_a] :
( ( X2
= ( cons_list_a @ K4 @ Ks2 ) )
=> ! [U2: list_a,Us3: list_list_a] :
( ( Xa
= ( cons_list_a @ U2 @ Us3 ) )
=> ( Y
!= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K4 @ U2 ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks2 @ Us3 ) ) ) ) )
=> ( ( ( X2 = nil_list_a )
=> ( Y
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ~ ( ( Xa = nil_list_a )
=> ( Y
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% x.combine.elims
thf(fact_1063_poly__add__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_add_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ ( append_a @ Q @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) )
= ( normalize_a_b @ r @ ( append_a @ ( poly_add_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_add_append_zero
thf(fact_1064_x_Oeval__poly__mult,axiom,
! [P: list_list_a,Q: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( 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 ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A ) ) ) ) ) ) ).
% x.eval_poly_mult
thf(fact_1065_x_Oconst__term__simprules_I2_J,axiom,
! [P: list_list_a,Q: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( poly_m7087347720095500472t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ Q ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P ) @ ( const_6738166269504826821t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q ) ) ) ) ) ).
% x.const_term_simprules(2)
thf(fact_1066_x_Ocombine__append,axiom,
! [Ks: list_list_a,Us2: list_list_a,Ks3: list_list_a,Vs2: list_list_a] :
( ( ( size_s349497388124573686list_a @ Ks )
= ( size_s349497388124573686list_a @ Us2 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ ( 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 @ Ks3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Vs2 ) @ ( 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 ) ) @ Ks @ Us2 ) @ ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Ks3 @ Vs2 ) )
= ( embedd2435972518007585703t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ Ks @ Ks3 ) @ ( append_list_a @ Us2 @ Vs2 ) ) ) ) ) ) ) ) ).
% x.combine_append
thf(fact_1067_x_Ocombine__in__carrier,axiom,
! [Ks: list_list_a,Us2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ks ) @ ( 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 ) ) @ Ks @ Us2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.combine_in_carrier
thf(fact_1068_x_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 ) ) ) ) ).
% x.poly_add_monom
thf(fact_1069_x_Odegree__oneE,axiom,
! [P: list_list_a,K: set_list_a] :
( ( member_list_list_a @ P @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_s349497388124573686list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A3: list_a] :
( ( member_list_a @ A3 @ K )
=> ( ( A3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ! [B: list_a] :
( ( member_list_a @ B @ K )
=> ( P
!= ( cons_list_a @ A3 @ ( cons_list_a @ B @ nil_list_a ) ) ) ) ) ) ) ) ).
% x.degree_oneE
thf(fact_1070_poly__mult_Osimps_I1_J,axiom,
! [P22: list_a] :
( ( poly_mult_a_b @ r @ nil_a @ P22 )
= nil_a ) ).
% poly_mult.simps(1)
thf(fact_1071_combine_Osimps_I2_J,axiom,
! [Us2: list_a] :
( ( embedded_combine_a_b @ r @ nil_a @ Us2 )
= ( zero_a_b @ r ) ) ).
% combine.simps(2)
thf(fact_1072_combine_Osimps_I3_J,axiom,
! [Ks: list_a] :
( ( embedded_combine_a_b @ r @ Ks @ nil_a )
= ( zero_a_b @ r ) ) ).
% combine.simps(3)
thf(fact_1073_normalize__length__le,axiom,
! [P: list_a] : ( ord_less_eq_nat @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) @ ( size_size_list_a @ P ) ) ).
% normalize_length_le
thf(fact_1074_combine__eq__eval,axiom,
! [Ks: list_a,X2: a] :
( ( embedded_combine_a_b @ r @ Ks @ ( polyno2922411391617481336se_a_b @ r @ X2 @ ( size_size_list_a @ Ks ) ) )
= ( eval_a_b @ r @ Ks @ X2 ) ) ).
% combine_eq_eval
thf(fact_1075_poly__mult__in__carrier,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( poly_mult_a_b @ r @ P1 @ P22 ) ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% poly_mult_in_carrier
thf(fact_1076_poly__mult__comm,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ P22 @ P1 ) ) ) ) ).
% poly_mult_comm
thf(fact_1077_combine_Osimps_I1_J,axiom,
! [K2: a,Ks: list_a,U: a,Us2: list_a] :
( ( embedded_combine_a_b @ r @ ( cons_a @ K2 @ Ks ) @ ( cons_a @ U @ Us2 ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K2 @ U ) @ ( embedded_combine_a_b @ r @ Ks @ Us2 ) ) ) ).
% combine.simps(1)
thf(fact_1078_combine__r__distr,axiom,
! [Ks: list_a,Us2: list_a,K2: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ K2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ K2 @ ( embedded_combine_a_b @ r @ Ks @ Us2 ) )
= ( embedded_combine_a_b @ r @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ K2 ) @ Ks ) @ Us2 ) ) ) ) ) ).
% combine_r_distr
thf(fact_1079_poly__mult__zero_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ nil_a )
= nil_a ) ) ).
% poly_mult_zero(2)
thf(fact_1080_poly__mult__zero_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ nil_a @ P )
= nil_a ) ) ).
% poly_mult_zero(1)
thf(fact_1081_poly__mult__normalize,axiom,
! [P1: list_a,P22: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ P22 )
= ( poly_mult_a_b @ r @ ( normalize_a_b @ r @ P1 ) @ P22 ) ) ) ) ).
% poly_mult_normalize
thf(fact_1082_poly__mult__l__distr_H,axiom,
! [P1: list_a,P22: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_add_a_b @ r @ P1 @ P22 ) @ P32 )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P32 ) @ ( poly_mult_a_b @ r @ P22 @ P32 ) ) ) ) ) ) ).
% poly_mult_l_distr'
thf(fact_1083_poly__mult__r__distr_H,axiom,
! [P1: list_a,P22: list_a,P32: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P1 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P22 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ P32 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P1 @ ( poly_add_a_b @ r @ P22 @ P32 ) )
= ( poly_add_a_b @ r @ ( poly_mult_a_b @ r @ P1 @ P22 ) @ ( poly_mult_a_b @ r @ P1 @ P32 ) ) ) ) ) ) ).
% poly_mult_r_distr'
thf(fact_1084_eval__poly__mult,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) @ A )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_poly_mult
thf(fact_1085_poly__mult__semiassoc,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( cons_a @ A @ nil_a ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_semiassoc
thf(fact_1086_combine__append,axiom,
! [Ks: list_a,Us2: list_a,Ks3: list_a,Vs2: list_a] :
( ( ( size_size_list_a @ Ks )
= ( size_size_list_a @ Us2 ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Ks3 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Vs2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( embedded_combine_a_b @ r @ Ks @ Us2 ) @ ( embedded_combine_a_b @ r @ Ks3 @ Vs2 ) )
= ( embedded_combine_a_b @ r @ ( append_a @ Ks @ Ks3 ) @ ( append_a @ Us2 @ Vs2 ) ) ) ) ) ) ) ) ).
% combine_append
thf(fact_1087_combine_Oelims,axiom,
! [X2: list_a,Xa: list_a,Y: a] :
( ( ( embedded_combine_a_b @ r @ X2 @ Xa )
= Y )
=> ( ! [K4: a,Ks2: list_a] :
( ( X2
= ( cons_a @ K4 @ Ks2 ) )
=> ! [U2: a,Us3: list_a] :
( ( Xa
= ( cons_a @ U2 @ Us3 ) )
=> ( Y
!= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ K4 @ U2 ) @ ( embedded_combine_a_b @ r @ Ks2 @ Us3 ) ) ) ) )
=> ( ( ( X2 = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) )
=> ~ ( ( Xa = nil_a )
=> ( Y
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% combine.elims
thf(fact_1088_const__term__simprules_I2_J,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( const_term_a_b @ r @ ( poly_mult_a_b @ r @ P @ Q ) )
= ( mult_a_ring_ext_a_b @ r @ ( const_term_a_b @ r @ P ) @ ( const_term_a_b @ r @ Q ) ) ) ) ) ).
% const_term_simprules(2)
thf(fact_1089_eval__poly__add__aux,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( size_size_list_a @ P )
= ( size_size_list_a @ Q ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( poly_add_a_b @ r @ P @ Q ) @ A )
= ( add_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ) ).
% eval_poly_add_aux
thf(fact_1090_poly__add__degree__le,axiom,
! [X2: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X2 ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% poly_add_degree_le
thf(fact_1091_poly__mult__degree__le,axiom,
! [X2: list_a,Y: list_a,N: nat,M: nat] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X2 ) @ one_one_nat ) @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ M )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ N @ M ) ) ) ) ) ) ).
% poly_mult_degree_le
thf(fact_1092_poly__mult__degree__le__1,axiom,
! [X2: list_a,Y: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) ) @ one_one_nat ) @ ( plus_plus_nat @ ( minus_minus_nat @ ( size_size_list_a @ X2 ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) ) ) ) ) ).
% poly_mult_degree_le_1
thf(fact_1093_degree__oneE,axiom,
! [P: list_a,K: set_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ~ ! [A3: a] :
( ( member_a @ A3 @ K )
=> ( ( A3
!= ( zero_a_b @ r ) )
=> ! [B: a] :
( ( member_a @ B @ K )
=> ( P
!= ( cons_a @ A3 @ ( cons_a @ B @ nil_a ) ) ) ) ) ) ) ) ).
% degree_oneE
thf(fact_1094_combine__append__zero,axiom,
! [Us2: list_a,Ks: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( append_a @ Ks @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Us2 )
= ( embedded_combine_a_b @ r @ Ks @ Us2 ) ) ) ).
% combine_append_zero
thf(fact_1095_poly__sub__degree__le,axiom,
! [X2: list_a,N: nat,Y: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ X2 ) @ one_one_nat ) @ N )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Y ) @ one_one_nat ) @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) ) @ one_one_nat ) @ N ) ) ) ) ) ).
% poly_sub_degree_le
thf(fact_1096_poly__mult__append__zero,axiom,
! [P: list_a,Q: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) @ Q )
= ( normalize_a_b @ r @ ( append_a @ ( poly_mult_a_b @ r @ P @ Q ) @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ) ).
% poly_mult_append_zero
thf(fact_1097_poly__mult__var_H_I1_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( var_a_b @ r ) @ P )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(1)
thf(fact_1098_poly__mult__var_H_I2_J,axiom,
! [P: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( var_a_b @ r ) )
= ( normalize_a_b @ r @ ( append_a @ P @ ( cons_a @ ( zero_a_b @ r ) @ nil_a ) ) ) ) ) ).
% poly_mult_var'(2)
thf(fact_1099_combine__in__carrier,axiom,
! [Ks: list_a,Us2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ks ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( embedded_combine_a_b @ r @ Ks @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% combine_in_carrier
thf(fact_1100_x_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 ) ) ) ) ) ).
% x.monom_in_carrier
thf(fact_1101_pirreducible__imp__not__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
!= one_one_nat )
=> ~ ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ).
% pirreducible_imp_not_splitted
thf(fact_1102_degree__one__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= one_one_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_one_imp_splitted
thf(fact_1103_poly__add__monom,axiom,
! [P: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( poly_add_a_b @ r @ ( monom_a_b @ r @ A @ ( size_size_list_a @ P ) ) @ P )
= ( cons_a @ A @ P ) ) ) ) ).
% poly_add_monom
thf(fact_1104_subalgebra__inter,axiom,
! [K: set_a,V: set_a,V2: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ( embedd9027525575939734154ra_a_b @ K @ V2 @ r )
=> ( embedd9027525575939734154ra_a_b @ K @ ( inf_inf_set_a @ V @ V2 ) @ r ) ) ) ).
% subalgebra_inter
thf(fact_1105_line__extension__in__carrier,axiom,
! [K: set_a,A: a,E: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ E @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% line_extension_in_carrier
thf(fact_1106_carrier__is__subalgebra,axiom,
! [K: set_a] :
( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( embedd9027525575939734154ra_a_b @ K @ ( partia707051561876973205xt_a_b @ r ) @ r ) ) ).
% carrier_is_subalgebra
thf(fact_1107_subalgebra__in__carrier,axiom,
! [K: set_a,V: set_a] :
( ( embedd9027525575939734154ra_a_b @ K @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subalgebra_in_carrier
thf(fact_1108_line__extension__mem__iff,axiom,
! [U: a,K: set_a,A: a,E: set_a] :
( ( member_a @ U @ ( embedd971793762689825387on_a_b @ r @ K @ A @ E ) )
= ( ? [X: a] :
( ( member_a @ X @ K )
& ? [Y2: a] :
( ( member_a @ Y2 @ E )
& ( U
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ A ) @ Y2 ) ) ) ) ) ) ).
% line_extension_mem_iff
thf(fact_1109_poly__mult__monom__assoc,axiom,
! [P: list_a,Q: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P ) @ Q )
= ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ ( poly_mult_a_b @ r @ P @ Q ) ) ) ) ) ) ).
% poly_mult_monom_assoc
thf(fact_1110_x_Oadd_Oint__pow__1,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( add_po2638046716968164713it_int @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ one_one_int @ X2 )
= X2 ) ) ).
% x.add.int_pow_1
thf(fact_1111_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_1112_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_1113_degree__zero__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% degree_zero_imp_splitted
thf(fact_1114_degree__zero__imp__not__is__root,axiom,
! [P: list_a,X2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat )
= zero_zero_nat )
=> ~ ( polyno4133073214067823460ot_a_b @ r @ P @ X2 ) ) ) ).
% degree_zero_imp_not_is_root
thf(fact_1115_x_Onat__pow__pow,axiom,
! [X2: list_a,N: nat,M: nat] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N ) @ M )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( times_times_nat @ N @ M ) ) ) ) ).
% x.nat_pow_pow
thf(fact_1116_x_Onat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.nat_pow_zero
thf(fact_1117_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_1118_x_Opow__mult__distrib,axiom,
! [X2: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X2 ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N ) ) ) ) ) ) ).
% x.pow_mult_distrib
thf(fact_1119_x_Onat__pow__distrib,axiom,
! [X2: list_a,Y: list_a,N: nat] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y ) @ N )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ N ) ) ) ) ) ).
% x.nat_pow_distrib
thf(fact_1120_x_Onat__pow__comm,axiom,
! [X2: list_a,N: nat,M: 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 @ M ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ M ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N ) ) ) ) ).
% x.nat_pow_comm
thf(fact_1121_x_Ogroup__commutes__pow,axiom,
! [X2: list_a,Y: list_a,N: nat] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ X2 ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Y @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N ) @ Y )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Y @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ N ) ) ) ) ) ) ).
% x.group_commutes_pow
thf(fact_1122_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_1123_var__pow__closed,axiom,
! [K: set_a,N: nat] :
( ( subring_a_b @ K @ r )
=> ( member_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ ( var_a_b @ r ) @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ).
% var_pow_closed
thf(fact_1124_x_Onat__pow__mult,axiom,
! [X2: list_a,N: nat,M: 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 @ M ) )
= ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% x.nat_pow_mult
thf(fact_1125_polynomial__pow__degree,axiom,
! [P: list_a,N: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ).
% polynomial_pow_degree
thf(fact_1126_subring__polynomial__pow__degree,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 ) ) )
=> ( ( minus_minus_nat @ ( size_size_list_a @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ K ) @ P @ N ) ) @ one_one_nat )
= ( times_times_nat @ N @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) ) ) ) ) ).
% subring_polynomial_pow_degree
thf(fact_1127_x_Oeval__monom,axiom,
! [B2: list_a,A: list_a,N: nat] :
( ( member_list_a @ B2 @ ( 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 ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( monom_7446464087056152608t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B2 @ N ) @ A )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ B2 @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ N ) ) ) ) ) ).
% x.eval_monom
thf(fact_1128_x_Oeval__append,axiom,
! [P: list_list_a,Q: list_list_a,A: list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Q ) @ ( 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 ) ) ) )
=> ( ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( append_list_a @ P @ Q ) @ A )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ A ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ A @ ( size_s349497388124573686list_a @ Q ) ) ) @ ( eval_l34571156754992824t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q @ A ) ) ) ) ) ) ).
% x.eval_append
thf(fact_1129_x_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 ) ) ) ) ) ).
% x.nat_pow_closed
thf(fact_1130_x_Onat__pow__eone,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ X2 @ one_one_nat )
= X2 ) ) ).
% x.nat_pow_eone
thf(fact_1131_splitted__imp__trivial__factors,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( P != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ P )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q @ P )
=> ( ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat )
= one_one_nat ) ) ) ) ) ) ) ).
% splitted_imp_trivial_factors
thf(fact_1132_trivial__factors__imp__splitted,axiom,
! [P: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [Q3: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ Q3 )
=> ( ( polyno5814909790663948098es_a_b @ r @ Q3 @ P )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ Q3 ) @ one_one_nat ) @ one_one_nat ) ) ) )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ).
% trivial_factors_imp_splitted
thf(fact_1133_euclidean__function,axiom,
! [A: a,B2: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q3: a,R4: a] :
( ( member_a @ Q3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B2 @ Q3 ) @ R4 ) )
& ( ( R4
= ( zero_a_b @ r ) )
| ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ) ) ).
% euclidean_function
thf(fact_1134_nat__pow__pow,axiom,
! [X2: a,N: nat,M: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ M )
= ( pow_a_1026414303147256608_b_nat @ r @ X2 @ ( times_times_nat @ N @ M ) ) ) ) ).
% nat_pow_pow
thf(fact_1135_nat__pow__zero,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( zero_a_b @ r ) @ N )
= ( zero_a_b @ r ) ) ) ).
% nat_pow_zero
thf(fact_1136_group__commutes__pow,axiom,
! [X2: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X2 ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) ) ) ) ) ) ).
% group_commutes_pow
thf(fact_1137_nat__pow__comm,axiom,
! [X2: a,N: nat,M: 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 @ M ) )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ M ) @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) ) ) ) ).
% nat_pow_comm
thf(fact_1138_nat__pow__distrib,axiom,
! [X2: a,Y: a,N: nat] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ).
% nat_pow_distrib
thf(fact_1139_pow__mult__distrib,axiom,
! [X2: a,Y: a,N: nat] :
( ( ( mult_a_ring_ext_a_b @ r @ X2 @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X2 ) )
=> ( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ ( mult_a_ring_ext_a_b @ r @ X2 @ Y ) @ N )
= ( mult_a_ring_ext_a_b @ r @ ( pow_a_1026414303147256608_b_nat @ r @ X2 @ N ) @ ( pow_a_1026414303147256608_b_nat @ r @ Y @ N ) ) ) ) ) ) ).
% pow_mult_distrib
thf(fact_1140_zero__pdivides,axiom,
! [P: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P )
= ( P = nil_a ) ) ).
% zero_pdivides
thf(fact_1141_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_1142_nat__pow__mult,axiom,
! [X2: a,N: nat,M: 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 @ M ) )
= ( pow_a_1026414303147256608_b_nat @ r @ X2 @ ( plus_plus_nat @ N @ M ) ) ) ) ).
% nat_pow_mult
thf(fact_1143_eval__monom,axiom,
! [B2: a,A: a,N: nat] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( monom_a_b @ r @ B2 @ N ) @ A )
= ( mult_a_ring_ext_a_b @ r @ B2 @ ( pow_a_1026414303147256608_b_nat @ r @ A @ N ) ) ) ) ) ).
% eval_monom
thf(fact_1144_pdivides__zero,axiom,
! [K: set_a,P: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_1145_pdivides__imp__root__sharing,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( eval_a_b @ r @ P @ A )
= ( zero_a_b @ r ) )
=> ( ( eval_a_b @ r @ Q @ A )
= ( zero_a_b @ r ) ) ) ) ) ) ).
% pdivides_imp_root_sharing
thf(fact_1146_polynomial__pow__division,axiom,
! [P: list_a,N: nat,M: nat] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( polyno5814909790663948098es_a_b @ r @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ N ) @ ( pow_li1142815632869257134it_nat @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P @ M ) ) ) ) ).
% polynomial_pow_division
thf(fact_1147_pdivides__imp__splitted,axiom,
! [P: list_a,Q: list_a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( polyno8329700637149614481ed_a_b @ r @ P ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_1148_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_1149_eval__append,axiom,
! [P: list_a,Q: list_a,A: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ Q ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ P @ Q ) @ A )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ P @ A ) @ ( pow_a_1026414303147256608_b_nat @ r @ A @ ( size_size_list_a @ Q ) ) ) @ ( eval_a_b @ r @ Q @ A ) ) ) ) ) ) ).
% eval_append
thf(fact_1150_pdivides__imp__degree__le,axiom,
! [K: set_a,P: list_a,Q: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno5814909790663948098es_a_b @ r @ P @ Q )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_size_list_a @ Q ) @ one_one_nat ) ) ) ) ) ) ) ).
% pdivides_imp_degree_le
thf(fact_1151_nat__pow__eone,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( pow_a_1026414303147256608_b_nat @ r @ X2 @ one_one_nat )
= X2 ) ) ).
% nat_pow_eone
thf(fact_1152_euclidean__domainI,axiom,
! [Phi: a > nat] :
( ! [A3: a,B: a] :
( ( member_a @ A3 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ B @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ? [Q4: a,R5: a] :
( ( member_a @ Q4 @ ( partia707051561876973205xt_a_b @ r ) )
& ( member_a @ R5 @ ( partia707051561876973205xt_a_b @ r ) )
& ( A3
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ B @ Q4 ) @ R5 ) )
& ( ( R5
= ( zero_a_b @ r ) )
| ( ord_less_nat @ ( Phi @ R5 ) @ ( Phi @ B ) ) ) ) ) )
=> ( ring_e8745995371659049232in_a_b @ r @ Phi ) ) ).
% euclidean_domainI
thf(fact_1153_x_Oorder__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_3240872107759947550t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
= ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ).
% x.order_gt_0_iff_finite
thf(fact_1154_order__gt__0__iff__finite,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( order_a_ring_ext_a_b @ r ) )
= ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% order_gt_0_iff_finite
thf(fact_1155_x_OboundD__carrier,axiom,
! [N: nat,F: nat > list_a,M: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_list_a @ ( F @ M ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% x.boundD_carrier
thf(fact_1156_alg__mult__gt__zero__iff__is__root,axiom,
! [P: list_a,X2: a] :
( ( member_list_a @ P @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( polyno4422430861927485590lt_a_b @ r @ P @ X2 ) )
= ( polyno4133073214067823460ot_a_b @ r @ P @ X2 ) ) ) ).
% alg_mult_gt_zero_iff_is_root
thf(fact_1157_boundD__carrier,axiom,
! [N: nat,F: nat > a,M: nat] :
( ( bound_a @ ( zero_a_b @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M )
=> ( member_a @ ( F @ M ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% boundD_carrier
thf(fact_1158_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1159_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1160_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( K2 = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1161_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J2: nat,K2: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K2 ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J2 ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_1162_nat__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1163_nat__mult__eq__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1164_nat__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1165_nat__diff__add__eq2,axiom,
! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1166_nat__diff__add__eq1,axiom,
! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1167_nat__le__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1168_nat__le__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1169_nat__eq__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1170_nat__eq__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1171_nat__less__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J2 @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J2 ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1172_nat__less__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1173_mult__le__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1174_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_1175_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1176_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1177_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1178_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times_nat @ M @ K2 )
= ( times_times_nat @ N @ K2 ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1179_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K2 @ M )
= ( times_times_nat @ K2 @ N ) )
= ( ( M = N )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1180_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1181_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1182_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1183_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1184_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1185_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1186_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1187_mult__less__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1188_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1189_Nat_Odiff__diff__right,axiom,
! [K2: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J2 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1190_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1191_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1192_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1193_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1194_le__trans,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_1195_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1196_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1197_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1198_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P2 @ X3 )
& ! [Y6: nat] :
( ( P2 @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1199_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1200_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1201_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times_int @ K2 @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1202_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1203_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1204_zmult__zless__mono2,axiom,
! [I: int,J2: int,K2: int] :
( ( ord_less_int @ I @ J2 )
=> ( ( ord_less_int @ zero_zero_int @ K2 )
=> ( ord_less_int @ ( times_times_int @ K2 @ I ) @ ( times_times_int @ K2 @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1205_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1206_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1207_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1208_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1209_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1210_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1211_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1212_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1213_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1214_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J2: nat] :
( ! [I3: nat,J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1215_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1216_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1217_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1218_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1219_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1220_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1221_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_1222_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1223_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1224_trans__le__add2,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1225_trans__le__add1,axiom,
! [I: nat,J2: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J2 @ M ) ) ) ).
% trans_le_add1
thf(fact_1226_add__le__mono1,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1227_add__le__mono,axiom,
! [I: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1228_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1229_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1230_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1231_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1232_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1233_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1234_mult__le__mono2,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1235_mult__le__mono1,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1236_mult__le__mono,axiom,
! [I: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1237_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1238_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1239_diff__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1240_diff__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1241_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1242_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1243_add__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times_nat @ K2 @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K2 @ M ) @ ( times_times_nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1244_add__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K2 )
= ( plus_plus_nat @ ( times_times_nat @ M @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_1245_int__le__induct,axiom,
! [I: int,K2: int,P2: int > $o] :
( ( ord_less_eq_int @ I @ K2 )
=> ( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_le_induct
thf(fact_1246_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1247_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1248_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1249_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1250_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K4: nat] :
( ( ord_less_eq_nat @ K4 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K4 )
=> ~ ( P2 @ I4 ) )
& ( P2 @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1251_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1252_diff__less__mono,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1253_mult__less__mono1,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1254_mult__less__mono2,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1255_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1256_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I )
= K2 )
= ( J2
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1257_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1258_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J2 ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J2 @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1259_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J2 @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1260_le__diff__conv,axiom,
! [J2: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1261_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1262_int__ge__induct,axiom,
! [K2: int,I: int,P2: int > $o] :
( ( ord_less_eq_int @ K2 @ I )
=> ( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1263_int__induct,axiom,
! [P2: int > $o,K2: int,I: int] :
( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K2 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P2 @ I ) ) ) ) ).
% int_induct
thf(fact_1264_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_1265_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_1266_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_1267_less__diff__conv2,axiom,
! [K2: nat,J2: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I )
= ( ord_less_nat @ J2 @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1268_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N2: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1269_poly__mult__monom_H,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( monom_a_b @ r @ A @ N ) @ P )
= ( normalize_a_b @ r @ ( append_a @ ( map_a_a @ ( mult_a_ring_ext_a_b @ r @ A ) @ P ) @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ) ) ) ).
% poly_mult_monom'
thf(fact_1270_normalize__replicate__zero,axiom,
! [N: nat,P: list_a] :
( ( normalize_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) )
= ( normalize_a_b @ r @ P ) ) ).
% normalize_replicate_zero
thf(fact_1271_local_Omonom__def,axiom,
! [A: a,N: nat] :
( ( monom_a_b @ r @ A @ N )
= ( cons_a @ A @ ( replicate_a @ N @ ( zero_a_b @ r ) ) ) ) ).
% local.monom_def
thf(fact_1272_normalize__trick,axiom,
! [P: list_a] :
( P
= ( append_a @ ( replicate_a @ ( minus_minus_nat @ ( size_size_list_a @ P ) @ ( size_size_list_a @ ( normalize_a_b @ r @ P ) ) ) @ ( zero_a_b @ r ) ) @ ( normalize_a_b @ r @ P ) ) ) ).
% normalize_trick
thf(fact_1273_poly__mult__replicate__zero_I2_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ P @ ( replicate_a @ N @ ( zero_a_b @ r ) ) )
= nil_a ) ) ).
% poly_mult_replicate_zero(2)
thf(fact_1274_poly__mult__replicate__zero_I1_J,axiom,
! [P: list_a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( poly_mult_a_b @ r @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P )
= nil_a ) ) ).
% poly_mult_replicate_zero(1)
thf(fact_1275_eval__replicate,axiom,
! [P: list_a,A: a,N: nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ P ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( eval_a_b @ r @ ( append_a @ ( replicate_a @ N @ ( zero_a_b @ r ) ) @ P ) @ A )
= ( eval_a_b @ r @ P @ A ) ) ) ) ).
% eval_replicate
thf(fact_1276_combine__replicate,axiom,
! [Us2: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Us2 ) @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( embedded_combine_a_b @ r @ ( replicate_a @ ( size_size_list_a @ Us2 ) @ ( zero_a_b @ r ) ) @ Us2 )
= ( zero_a_b @ r ) ) ) ).
% combine_replicate
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
cring_3148771470849435808t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ).
%------------------------------------------------------------------------------