TPTP Problem File: SLH0711^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Finite_Fields/0003_Finite_Fields_Preliminary_Results/prob_00547_019747__18065804_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1631 ( 521 unt; 351 typ; 0 def)
% Number of atoms : 3599 (1020 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 14868 ( 204 ~; 27 |; 120 &;12817 @)
% ( 0 <=>;1700 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 50 ( 49 usr)
% Number of type conns : 976 ( 976 >; 0 *; 0 +; 0 <<)
% Number of symbols : 304 ( 302 usr; 21 con; 0-3 aty)
% Number of variables : 2954 ( 309 ^;2612 !; 33 ?;2954 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:20:34.436
%------------------------------------------------------------------------------
% Could-be-implicit typings (49)
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id_list_b: list_b > list_b ).
thf(sy_c_Fun_Oid_001tf__b,type,
id_b: b > b ).
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thf(sy_c_Fun_Oinj__on_001tf__b_001tf__b,type,
inj_on_b_b: ( b > b ) > set_b > $o ).
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the_inv_into_a_b: set_a > ( a > b ) > b > a ).
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thf(sy_c_Ideal_Oprincipalideal_001tf__b_001tf__d,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_It__Nat__Onat_Mtf__a_J_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_It__Nat__Onat_Mtf__b_J_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_062_Itf__a_Mtf__b_J_M_Eo_J,type,
inf_inf_a_b_o: ( ( a > b ) > $o ) > ( ( a > b ) > $o ) > ( a > b ) > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_Itf__a_J_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__List__Olist_Itf__b_J_M_Eo_J,type,
inf_inf_list_b_o: ( list_b > $o ) > ( list_b > $o ) > list_b > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Nat__Onat_M_Eo_J,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_Itf__a_J_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Set__Oset_Itf__b_J_M_Eo_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_Itf__a_M_Eo_J,type,
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inf_inf_b_o: ( b > $o ) > ( b > $o ) > b > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__a_J_J,type,
inf_inf_set_nat_a: set_nat_a > set_nat_a > set_nat_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mtf__b_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
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inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__b_J,type,
inf_inf_set_b: set_b > set_b > set_b ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
append_list_a: list_list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oappend_001t__List__Olist_Itf__b_J,type,
append_list_b: list_list_b > list_list_b > list_list_b ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__b_J,type,
nil_list_b: list_list_b ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_ONil_001tf__b,type,
nil_b: list_b ).
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map_li4464339650582430090list_b: ( list_b > list_list_b ) > list_list_b > list_list_list_b ).
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map_list_b_list_b: ( list_b > list_b ) > list_list_b > list_list_b ).
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map_a_list_a: ( a > list_a ) > list_a > list_list_a ).
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map_a_b: ( a > b ) > list_a > list_b ).
thf(sy_c_List_Olist_Omap_001tf__b_001t__List__Olist_Itf__b_J,type,
map_b_list_b: ( b > list_b ) > list_b > list_list_b ).
thf(sy_c_List_Olist_Omap_001tf__b_001tf__a,type,
map_b_a: ( b > a ) > list_b > list_a ).
thf(sy_c_List_Olist_Omap_001tf__b_001tf__b,type,
map_b_b: ( b > b ) > list_b > list_b ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__b_J,type,
set_list_b2: list_list_b > set_list_b ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Olist_Oset_001tf__b,type,
set_b2: list_b > set_b ).
thf(sy_c_Multiset_Omset_001t__List__Olist_Itf__a_J,type,
mset_list_a: list_list_a > multiset_list_a ).
thf(sy_c_Multiset_Omset_001t__List__Olist_Itf__b_J,type,
mset_list_b: list_list_b > multiset_list_b ).
thf(sy_c_Multiset_Oset__mset_001t__List__Olist_Itf__a_J,type,
set_mset_list_a: multiset_list_a > set_list_a ).
thf(sy_c_Multiset_Oset__mset_001t__List__Olist_Itf__b_J,type,
set_mset_list_b: multiset_list_b > set_list_b ).
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set_mset_a: multiset_a > set_a ).
thf(sy_c_Multiset_Oset__mset_001tf__b,type,
set_mset_b: multiset_b > set_b ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__b_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_Itf__a_J_J_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__b_J_J,type,
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bound_a: a > nat > ( nat > a ) > $o ).
thf(sy_c_UnivPoly_Obound_001tf__b,type,
bound_b: b > nat > ( nat > b ) > $o ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
up_lis8464167429055313730t_unit: partia2670972154091845814t_unit > set_nat_list_a ).
thf(sy_c_UnivPoly_Oup_001t__List__Olist_Itf__b_J_001t__Product____Type__Ounit,type,
up_lis5378411187065319169t_unit: partia4026993951477142903t_unit > set_nat_list_b ).
thf(sy_c_UnivPoly_Oup_001tf__a_001tf__c,type,
up_a_c: partia8877618634411419171xt_a_c > set_nat_a ).
thf(sy_c_UnivPoly_Oup_001tf__b_001tf__d,type,
up_b_d: partia1897943568983147621xt_b_d > set_nat_b ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__b_J_J,type,
member_list_a_list_b: ( list_a > list_b ) > set_list_a_list_b > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__a_J,type,
member_list_a_a: ( list_a > a ) > set_list_a_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__a_J_Mtf__b_J,type,
member_list_a_b: ( list_a > b ) > set_list_a_b > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__b_J_Mtf__a_J,type,
member_list_b_a: ( list_b > a ) > set_list_b_a > $o ).
thf(sy_c_member_001_062_It__List__Olist_Itf__b_J_Mtf__b_J,type,
member_list_b_b: ( list_b > b ) > set_list_b_b > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__a_J_J,type,
member_nat_list_a: ( nat > list_a ) > set_nat_list_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__List__Olist_Itf__b_J_J,type,
member_nat_list_b: ( nat > list_b ) > set_nat_list_b > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__a_J,type,
member_nat_a: ( nat > a ) > set_nat_a > $o ).
thf(sy_c_member_001_062_It__Nat__Onat_Mtf__b_J,type,
member_nat_b: ( nat > b ) > set_nat_b > $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_001_062_Itf__a_Mtf__b_J,type,
member_a_b: ( a > b ) > set_a_b > $o ).
thf(sy_c_member_001_062_Itf__b_Mt__List__Olist_Itf__a_J_J,type,
member_b_list_a: ( b > list_a ) > set_b_list_a > $o ).
thf(sy_c_member_001_062_Itf__b_Mt__List__Olist_Itf__b_J_J,type,
member_b_list_b: ( b > list_b ) > set_b_list_b > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__a_J,type,
member_b_a: ( b > a ) > set_b_a > $o ).
thf(sy_c_member_001_062_Itf__b_Mtf__b_J,type,
member_b_b: ( b > b ) > set_b_b > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member5342144027231129785list_a: list_list_list_a > set_list_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_Itf__b_J_J_J,type,
member2820959508942094138list_b: list_list_list_b > set_list_list_list_b > $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__List__Olist_Itf__b_J_J,type,
member_list_list_b: list_list_b > set_list_list_b > $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__List__Olist_Itf__b_J,type,
member_list_b: list_b > set_list_b > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_R,type,
r: partia8877618634411419171xt_a_c ).
thf(sy_v_S,type,
s: partia1897943568983147621xt_b_d ).
thf(sy_v_h,type,
h: a > b ).
thf(sy_v_x____,type,
x: list_b ).
% Relevant facts (1279)
thf(fact_0_dr_Odomain__axioms,axiom,
domain_a_c @ r ).
% dr.domain_axioms
thf(fact_1_dr_Oonepideal,axiom,
principalideal_a_c @ ( partia778085601923319190xt_a_c @ r ) @ r ).
% dr.onepideal
thf(fact_2_list_Omap__id0,axiom,
( ( map_list_b_list_b @ id_list_b )
= id_list_list_b ) ).
% list.map_id0
thf(fact_3_list_Omap__id0,axiom,
( ( map_b_b @ id_b )
= id_list_b ) ).
% list.map_id0
thf(fact_4_dr_Oring__iso__restrict,axiom,
! [F: a > b,S: partia1897943568983147621xt_b_d,G: a > b] :
( ( member_a_b @ F @ ( ring_iso_a_c_b_d @ r @ S ) )
=> ( ! [R: a] :
( ( member_a @ R @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_a_b @ G @ ( ring_iso_a_c_b_d @ r @ S ) ) ) ) ).
% dr.ring_iso_restrict
thf(fact_5_map__ident,axiom,
( ( map_b_b
@ ^ [X: b] : X )
= ( ^ [Xs: list_b] : Xs ) ) ).
% map_ident
thf(fact_6_dr_Ocarrier__is__subcring,axiom,
subcring_a_c @ ( partia778085601923319190xt_a_c @ r ) @ r ).
% dr.carrier_is_subcring
thf(fact_7_dr_Oring__hom__restrict,axiom,
! [F: a > b,S: partia1897943568983147621xt_b_d,G: a > b] :
( ( member_a_b @ F @ ( ring_hom_a_c_b_d @ r @ S ) )
=> ( ! [R: a] :
( ( member_a @ R @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( F @ R )
= ( G @ R ) ) )
=> ( member_a_b @ G @ ( ring_hom_a_c_b_d @ r @ S ) ) ) ) ).
% dr.ring_hom_restrict
thf(fact_8_id__apply,axiom,
( id_list_b
= ( ^ [X: list_b] : X ) ) ).
% id_apply
thf(fact_9_id__apply,axiom,
( id_b
= ( ^ [X: b] : X ) ) ).
% id_apply
thf(fact_10_h__inj,axiom,
inj_on_a_b @ h @ ( partia778085601923319190xt_a_c @ r ) ).
% h_inj
thf(fact_11_dr_Ocgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ I @ ( cgenid547466214215511830xt_a_c @ r @ I ) ) ) ).
% dr.cgenideal_self
thf(fact_12_dr_Ocarrier__not__empty,axiom,
( ( partia778085601923319190xt_a_c @ r )
!= bot_bot_set_a ) ).
% dr.carrier_not_empty
thf(fact_13_dr_Osemiring__axioms,axiom,
semiring_a_c @ r ).
% dr.semiring_axioms
thf(fact_14_list_Omap__id,axiom,
! [T: list_list_b] :
( ( map_list_b_list_b @ id_list_b @ T )
= T ) ).
% list.map_id
thf(fact_15_list_Omap__id,axiom,
! [T: list_b] :
( ( map_b_b @ id_b @ T )
= T ) ).
% list.map_id
thf(fact_16_dr_Ocgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia778085601923319190xt_a_c @ r ) )
=> ( principalideal_a_c @ ( cgenid547466214215511830xt_a_c @ r @ I ) @ r ) ) ).
% dr.cgenideal_is_principalideal
thf(fact_17_inj__on__empty,axiom,
! [F: a > b] : ( inj_on_a_b @ F @ bot_bot_set_a ) ).
% inj_on_empty
thf(fact_18_calculation,axiom,
( ( map_a_b @ h @ ( map_b_a @ ( the_inv_into_a_b @ ( partia778085601923319190xt_a_c @ r ) @ h ) @ x ) )
= ( map_b_b
@ ^ [Y: b] : ( h @ ( the_inv_into_a_b @ ( partia778085601923319190xt_a_c @ r ) @ h @ Y ) )
@ x ) ) ).
% calculation
thf(fact_19_inj__onD,axiom,
! [F: a > b,A: set_a,X2: a,Y2: a] :
( ( inj_on_a_b @ F @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
=> ( ( member_a @ X2 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( X2 = Y2 ) ) ) ) ) ).
% inj_onD
thf(fact_20_inj__onI,axiom,
! [A: set_a,F: a > b] :
( ! [X3: a,Y3: a] :
( ( member_a @ X3 @ A )
=> ( ( member_a @ Y3 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) )
=> ( inj_on_a_b @ F @ A ) ) ).
% inj_onI
thf(fact_21_inj__on__def,axiom,
( inj_on_a_b
= ( ^ [F2: a > b,A2: set_a] :
! [X: a] :
( ( member_a @ X @ A2 )
=> ! [Y: a] :
( ( member_a @ Y @ A2 )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( X = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_22_inj__on__cong,axiom,
! [A: set_a,F: a > b,G: a > b] :
( ! [A3: a] :
( ( member_a @ A3 @ A )
=> ( ( F @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_a_b @ F @ A )
= ( inj_on_a_b @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_23_inj__on__eq__iff,axiom,
! [F: a > b,A: set_a,X2: a,Y2: a] :
( ( inj_on_a_b @ F @ A )
=> ( ( member_a @ X2 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( ( F @ X2 )
= ( F @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_24_inj__on__contraD,axiom,
! [F: a > b,A: set_a,X2: a,Y2: a] :
( ( inj_on_a_b @ F @ A )
=> ( ( X2 != Y2 )
=> ( ( member_a @ X2 @ A )
=> ( ( member_a @ Y2 @ A )
=> ( ( F @ X2 )
!= ( F @ Y2 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_25_inj__on__inverseI,axiom,
! [A: set_a,G: b > a,F: a > b] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( G @ ( F @ X3 ) )
= X3 ) )
=> ( inj_on_a_b @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_26_inj__on__id,axiom,
! [A: set_b] : ( inj_on_b_b @ id_b @ A ) ).
% inj_on_id
thf(fact_27_inj__on__id,axiom,
! [A: set_list_b] : ( inj_on_list_b_list_b @ id_list_b @ A ) ).
% inj_on_id
thf(fact_28_the__inv__into__f__eq,axiom,
! [F: a > b,A: set_a,X2: a,Y2: b] :
( ( inj_on_a_b @ F @ A )
=> ( ( ( F @ X2 )
= Y2 )
=> ( ( member_a @ X2 @ A )
=> ( ( the_inv_into_a_b @ A @ F @ Y2 )
= X2 ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_29_the__inv__into__f__f,axiom,
! [F: a > b,A: set_a,X2: a] :
( ( inj_on_a_b @ F @ A )
=> ( ( member_a @ X2 @ A )
=> ( ( the_inv_into_a_b @ A @ F @ ( F @ X2 ) )
= X2 ) ) ) ).
% the_inv_into_f_f
thf(fact_30_List_Omap_Oidentity,axiom,
( ( map_b_b
@ ^ [X: b] : X )
= id_list_b ) ).
% List.map.identity
thf(fact_31_eq__id__iff,axiom,
! [F: b > b] :
( ( ! [X: b] :
( ( F @ X )
= X ) )
= ( F = id_b ) ) ).
% eq_id_iff
thf(fact_32_eq__id__iff,axiom,
! [F: list_b > list_b] :
( ( ! [X: list_b] :
( ( F @ X )
= X ) )
= ( F = id_list_b ) ) ).
% eq_id_iff
thf(fact_33_id__def,axiom,
( id_b
= ( ^ [X: b] : X ) ) ).
% id_def
thf(fact_34_id__def,axiom,
( id_list_b
= ( ^ [X: list_b] : X ) ) ).
% id_def
thf(fact_35_list_Omap__ident,axiom,
! [T: list_b] :
( ( map_b_b
@ ^ [X: b] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_36_assms_I1_J,axiom,
member_a_b @ h @ ( ring_iso_a_c_b_d @ r @ s ) ).
% assms(1)
thf(fact_37_dr_Ocgenideal__eq__rcos,axiom,
! [I: a] :
( ( cgenid547466214215511830xt_a_c @ r @ I )
= ( r_cose3160181845575678695xt_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) @ I ) ) ).
% dr.cgenideal_eq_rcos
thf(fact_38_dr_Oideal__eq__carrier__iff,axiom,
! [A4: a] :
( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( partia778085601923319190xt_a_c @ r )
= ( cgenid547466214215511830xt_a_c @ r @ A4 ) )
= ( member_a @ A4 @ ( units_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.ideal_eq_carrier_iff
thf(fact_39_dr_Oassociated__iff__same__ideal,axiom,
! [A4: a,B: a] :
( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( associ5860276531582424204xt_a_c @ r @ A4 @ B )
= ( ( cgenid547466214215511830xt_a_c @ r @ A4 )
= ( cgenid547466214215511830xt_a_c @ r @ B ) ) ) ) ) ).
% dr.associated_iff_same_ideal
thf(fact_40_empty__iff,axiom,
! [C: nat > b] :
~ ( member_nat_b @ C @ bot_bot_set_nat_b ) ).
% empty_iff
thf(fact_41_empty__iff,axiom,
! [C: nat > a] :
~ ( member_nat_a @ C @ bot_bot_set_nat_a ) ).
% empty_iff
thf(fact_42_empty__iff,axiom,
! [C: a > b] :
~ ( member_a_b @ C @ bot_bot_set_a_b ) ).
% empty_iff
thf(fact_43_empty__iff,axiom,
! [C: set_list_a] :
~ ( member_set_list_a @ C @ bot_bo3186585308812441520list_a ) ).
% empty_iff
thf(fact_44_empty__iff,axiom,
! [C: set_b] :
~ ( member_set_b @ C @ bot_bot_set_set_b ) ).
% empty_iff
thf(fact_45_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_46_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_47_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_48_all__not__in__conv,axiom,
! [A: set_nat_b] :
( ( ! [X: nat > b] :
~ ( member_nat_b @ X @ A ) )
= ( A = bot_bot_set_nat_b ) ) ).
% all_not_in_conv
thf(fact_49_all__not__in__conv,axiom,
! [A: set_nat_a] :
( ( ! [X: nat > a] :
~ ( member_nat_a @ X @ A ) )
= ( A = bot_bot_set_nat_a ) ) ).
% all_not_in_conv
thf(fact_50_all__not__in__conv,axiom,
! [A: set_a_b] :
( ( ! [X: a > b] :
~ ( member_a_b @ X @ A ) )
= ( A = bot_bot_set_a_b ) ) ).
% all_not_in_conv
thf(fact_51_all__not__in__conv,axiom,
! [A: set_set_list_a] :
( ( ! [X: set_list_a] :
~ ( member_set_list_a @ X @ A ) )
= ( A = bot_bo3186585308812441520list_a ) ) ).
% all_not_in_conv
thf(fact_52_all__not__in__conv,axiom,
! [A: set_set_b] :
( ( ! [X: set_b] :
~ ( member_set_b @ X @ A ) )
= ( A = bot_bot_set_set_b ) ) ).
% all_not_in_conv
thf(fact_53_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_54_all__not__in__conv,axiom,
! [A: set_b] :
( ( ! [X: b] :
~ ( member_b @ X @ A ) )
= ( A = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_55_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_56_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_57_Collect__empty__eq,axiom,
! [P: set_list_a > $o] :
( ( ( collect_set_list_a @ P )
= bot_bo3186585308812441520list_a )
= ( ! [X: set_list_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_58_Collect__empty__eq,axiom,
! [P: set_b > $o] :
( ( ( collect_set_b @ P )
= bot_bot_set_set_b )
= ( ! [X: set_b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_59_Collect__empty__eq,axiom,
! [P: set_a > $o] :
( ( ( collect_set_a @ P )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_60_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_61_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_62_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_63_empty__Collect__eq,axiom,
! [P: set_list_a > $o] :
( ( bot_bo3186585308812441520list_a
= ( collect_set_list_a @ P ) )
= ( ! [X: set_list_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_64_empty__Collect__eq,axiom,
! [P: set_b > $o] :
( ( bot_bot_set_set_b
= ( collect_set_b @ P ) )
= ( ! [X: set_b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_65_empty__Collect__eq,axiom,
! [P: set_a > $o] :
( ( bot_bot_set_set_a
= ( collect_set_a @ P ) )
= ( ! [X: set_a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_66_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X: b] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_67_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X: a] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_68_dr_Ocgenideal__ideal,axiom,
! [A4: a] :
( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ideal_a_c @ ( cgenid547466214215511830xt_a_c @ r @ A4 ) @ r ) ) ).
% dr.cgenideal_ideal
thf(fact_69_dr_Osubcring__inter,axiom,
! [I2: set_a,J: set_a] :
( ( subcring_a_c @ I2 @ r )
=> ( ( subcring_a_c @ J @ r )
=> ( subcring_a_c @ ( inf_inf_set_a @ I2 @ J ) @ r ) ) ) ).
% dr.subcring_inter
thf(fact_70_dr_Oassociated__sym,axiom,
! [A4: a,B: a] :
( ( associ5860276531582424204xt_a_c @ r @ A4 @ B )
=> ( associ5860276531582424204xt_a_c @ r @ B @ A4 ) ) ).
% dr.associated_sym
thf(fact_71_dr_OUnits__closed,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_c @ r ) )
=> ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% dr.Units_closed
thf(fact_72_mem__Collect__eq,axiom,
! [A4: nat > b,P: ( nat > b ) > $o] :
( ( member_nat_b @ A4 @ ( collect_nat_b @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A4: nat > a,P: ( nat > a ) > $o] :
( ( member_nat_a @ A4 @ ( collect_nat_a @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A4: a > b,P: ( a > b ) > $o] :
( ( member_a_b @ A4 @ ( collect_a_b @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_75_mem__Collect__eq,axiom,
! [A4: set_b,P: set_b > $o] :
( ( member_set_b @ A4 @ ( collect_set_b @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_76_mem__Collect__eq,axiom,
! [A4: set_a,P: set_a > $o] :
( ( member_set_a @ A4 @ ( collect_set_a @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
! [A4: nat,P: nat > $o] :
( ( member_nat @ A4 @ ( collect_nat @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A4: b,P: b > $o] :
( ( member_b @ A4 @ ( collect_b @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A4: a,P: a > $o] :
( ( member_a @ A4 @ ( collect_a @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_80_Collect__mem__eq,axiom,
! [A: set_nat_b] :
( ( collect_nat_b
@ ^ [X: nat > b] : ( member_nat_b @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
! [A: set_nat_a] :
( ( collect_nat_a
@ ^ [X: nat > a] : ( member_nat_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A: set_a_b] :
( ( collect_a_b
@ ^ [X: a > b] : ( member_a_b @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A: set_set_b] :
( ( collect_set_b
@ ^ [X: set_b] : ( member_set_b @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A: set_b] :
( ( collect_b
@ ^ [X: b] : ( member_b @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_88_Collect__cong,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ! [X3: set_b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_b @ P )
= ( collect_set_b @ Q ) ) ) ).
% Collect_cong
thf(fact_89_Collect__cong,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_set_a @ P )
= ( collect_set_a @ Q ) ) ) ).
% Collect_cong
thf(fact_90_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_91_Collect__cong,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_b @ P )
= ( collect_b @ Q ) ) ) ).
% Collect_cong
thf(fact_92_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_93_dr_Oassoc__subst,axiom,
! [A4: a,B: a,F: a > a] :
( ( associ5860276531582424204xt_a_c @ r @ A4 @ B )
=> ( ! [A3: a,B2: a] :
( ( ( member_a @ A3 @ ( partia778085601923319190xt_a_c @ r ) )
& ( member_a @ B2 @ ( partia778085601923319190xt_a_c @ r ) )
& ( associ5860276531582424204xt_a_c @ r @ A3 @ B2 ) )
=> ( ( member_a @ ( F @ A3 ) @ ( partia778085601923319190xt_a_c @ r ) )
& ( member_a @ ( F @ B2 ) @ ( partia778085601923319190xt_a_c @ r ) )
& ( associ5860276531582424204xt_a_c @ r @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( associ5860276531582424204xt_a_c @ r @ ( F @ A4 ) @ ( F @ B ) ) ) ) ) ) ).
% dr.assoc_subst
thf(fact_94_dr_Oassociated__trans,axiom,
! [A4: a,B: a,C: a] :
( ( associ5860276531582424204xt_a_c @ r @ A4 @ B )
=> ( ( associ5860276531582424204xt_a_c @ r @ B @ C )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( associ5860276531582424204xt_a_c @ r @ A4 @ C ) ) ) ) ) ).
% dr.associated_trans
thf(fact_95_dr_Ooneideal,axiom,
ideal_a_c @ ( partia778085601923319190xt_a_c @ r ) @ r ).
% dr.oneideal
thf(fact_96_dr_OUnits__assoc,axiom,
! [A4: a,B: a] :
( ( member_a @ A4 @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_c @ r ) )
=> ( associ5860276531582424204xt_a_c @ r @ A4 @ B ) ) ) ).
% dr.Units_assoc
thf(fact_97_dr_Oi__intersect,axiom,
! [I2: set_a,J: set_a] :
( ( ideal_a_c @ I2 @ r )
=> ( ( ideal_a_c @ J @ r )
=> ( ideal_a_c @ ( inf_inf_set_a @ I2 @ J ) @ r ) ) ) ).
% dr.i_intersect
thf(fact_98_h_Ohomh,axiom,
member_a_b @ h @ ( ring_hom_a_c_b_d @ r @ s ) ).
% h.homh
thf(fact_99_dr_OUnits__cong,axiom,
! [A4: a,B: a] :
( ( member_a @ A4 @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( associ5860276531582424204xt_a_c @ r @ A4 @ B )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) ).
% dr.Units_cong
thf(fact_100_Int__iff,axiom,
! [C: nat > b,A: set_nat_b,B3: set_nat_b] :
( ( member_nat_b @ C @ ( inf_inf_set_nat_b @ A @ B3 ) )
= ( ( member_nat_b @ C @ A )
& ( member_nat_b @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_101_Int__iff,axiom,
! [C: nat > a,A: set_nat_a,B3: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A @ B3 ) )
= ( ( member_nat_a @ C @ A )
& ( member_nat_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_102_Int__iff,axiom,
! [C: a > b,A: set_a_b,B3: set_a_b] :
( ( member_a_b @ C @ ( inf_inf_set_a_b @ A @ B3 ) )
= ( ( member_a_b @ C @ A )
& ( member_a_b @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_103_Int__iff,axiom,
! [C: a,A: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B3 ) )
= ( ( member_a @ C @ A )
& ( member_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_104_Int__iff,axiom,
! [C: b,A: set_b,B3: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B3 ) )
= ( ( member_b @ C @ A )
& ( member_b @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_105_Int__iff,axiom,
! [C: list_b,A: set_list_b,B3: set_list_b] :
( ( member_list_b @ C @ ( inf_inf_set_list_b @ A @ B3 ) )
= ( ( member_list_b @ C @ A )
& ( member_list_b @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_106_Int__iff,axiom,
! [C: list_a,A: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B3 ) )
= ( ( member_list_a @ C @ A )
& ( member_list_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_107_IntI,axiom,
! [C: nat > b,A: set_nat_b,B3: set_nat_b] :
( ( member_nat_b @ C @ A )
=> ( ( member_nat_b @ C @ B3 )
=> ( member_nat_b @ C @ ( inf_inf_set_nat_b @ A @ B3 ) ) ) ) ).
% IntI
thf(fact_108_IntI,axiom,
! [C: nat > a,A: set_nat_a,B3: set_nat_a] :
( ( member_nat_a @ C @ A )
=> ( ( member_nat_a @ C @ B3 )
=> ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A @ B3 ) ) ) ) ).
% IntI
thf(fact_109_IntI,axiom,
! [C: a > b,A: set_a_b,B3: set_a_b] :
( ( member_a_b @ C @ A )
=> ( ( member_a_b @ C @ B3 )
=> ( member_a_b @ C @ ( inf_inf_set_a_b @ A @ B3 ) ) ) ) ).
% IntI
thf(fact_110_IntI,axiom,
! [C: a,A: set_a,B3: set_a] :
( ( member_a @ C @ A )
=> ( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( inf_inf_set_a @ A @ B3 ) ) ) ) ).
% IntI
thf(fact_111_IntI,axiom,
! [C: b,A: set_b,B3: set_b] :
( ( member_b @ C @ A )
=> ( ( member_b @ C @ B3 )
=> ( member_b @ C @ ( inf_inf_set_b @ A @ B3 ) ) ) ) ).
% IntI
thf(fact_112_IntI,axiom,
! [C: list_b,A: set_list_b,B3: set_list_b] :
( ( member_list_b @ C @ A )
=> ( ( member_list_b @ C @ B3 )
=> ( member_list_b @ C @ ( inf_inf_set_list_b @ A @ B3 ) ) ) ) ).
% IntI
thf(fact_113_IntI,axiom,
! [C: list_a,A: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ A )
=> ( ( member_list_a @ C @ B3 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B3 ) ) ) ) ).
% IntI
thf(fact_114_ds_Odomain__axioms,axiom,
domain_b_d @ s ).
% ds.domain_axioms
thf(fact_115_dr_Oassociated__refl,axiom,
! [A4: a] :
( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( associ5860276531582424204xt_a_c @ r @ A4 @ A4 ) ) ).
% dr.associated_refl
thf(fact_116_Int__left__commute,axiom,
! [A: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_117_Int__left__commute,axiom,
! [A: set_b,B3: set_b,C2: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B3 @ C2 ) )
= ( inf_inf_set_b @ B3 @ ( inf_inf_set_b @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_118_Int__left__commute,axiom,
! [A: set_list_b,B3: set_list_b,C2: set_list_b] :
( ( inf_inf_set_list_b @ A @ ( inf_inf_set_list_b @ B3 @ C2 ) )
= ( inf_inf_set_list_b @ B3 @ ( inf_inf_set_list_b @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_119_Int__left__commute,axiom,
! [A: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B3 @ C2 ) )
= ( inf_inf_set_list_a @ B3 @ ( inf_inf_set_list_a @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_120_Int__left__absorb,axiom,
! [A: set_a,B3: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B3 ) )
= ( inf_inf_set_a @ A @ B3 ) ) ).
% Int_left_absorb
thf(fact_121_Int__left__absorb,axiom,
! [A: set_b,B3: set_b] :
( ( inf_inf_set_b @ A @ ( inf_inf_set_b @ A @ B3 ) )
= ( inf_inf_set_b @ A @ B3 ) ) ).
% Int_left_absorb
thf(fact_122_Int__left__absorb,axiom,
! [A: set_list_b,B3: set_list_b] :
( ( inf_inf_set_list_b @ A @ ( inf_inf_set_list_b @ A @ B3 ) )
= ( inf_inf_set_list_b @ A @ B3 ) ) ).
% Int_left_absorb
thf(fact_123_Int__left__absorb,axiom,
! [A: set_list_a,B3: set_list_a] :
( ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ A @ B3 ) )
= ( inf_inf_set_list_a @ A @ B3 ) ) ).
% Int_left_absorb
thf(fact_124_Collect__conj__eq,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ( collect_set_b
@ ^ [X: set_b] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_125_Collect__conj__eq,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( collect_set_a
@ ^ [X: set_a] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_126_Collect__conj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X: nat] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_127_Collect__conj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X: a] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_128_Collect__conj__eq,axiom,
! [P: b > $o,Q: b > $o] :
( ( collect_b
@ ^ [X: b] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_129_Collect__conj__eq,axiom,
! [P: list_b > $o,Q: list_b > $o] :
( ( collect_list_b
@ ^ [X: list_b] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_list_b @ ( collect_list_b @ P ) @ ( collect_list_b @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_130_Collect__conj__eq,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( collect_list_a
@ ^ [X: list_a] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf_set_list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_131_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A2 ) ) ) ).
% Int_commute
thf(fact_132_Int__commute,axiom,
( inf_inf_set_b
= ( ^ [A2: set_b,B4: set_b] : ( inf_inf_set_b @ B4 @ A2 ) ) ) ).
% Int_commute
thf(fact_133_Int__commute,axiom,
( inf_inf_set_list_b
= ( ^ [A2: set_list_b,B4: set_list_b] : ( inf_inf_set_list_b @ B4 @ A2 ) ) ) ).
% Int_commute
thf(fact_134_Int__commute,axiom,
( inf_inf_set_list_a
= ( ^ [A2: set_list_a,B4: set_list_a] : ( inf_inf_set_list_a @ B4 @ A2 ) ) ) ).
% Int_commute
thf(fact_135_Int__Collect,axiom,
! [X2: nat > b,A: set_nat_b,P: ( nat > b ) > $o] :
( ( member_nat_b @ X2 @ ( inf_inf_set_nat_b @ A @ ( collect_nat_b @ P ) ) )
= ( ( member_nat_b @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_136_Int__Collect,axiom,
! [X2: nat > a,A: set_nat_a,P: ( nat > a ) > $o] :
( ( member_nat_a @ X2 @ ( inf_inf_set_nat_a @ A @ ( collect_nat_a @ P ) ) )
= ( ( member_nat_a @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_137_Int__Collect,axiom,
! [X2: a > b,A: set_a_b,P: ( a > b ) > $o] :
( ( member_a_b @ X2 @ ( inf_inf_set_a_b @ A @ ( collect_a_b @ P ) ) )
= ( ( member_a_b @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_138_Int__Collect,axiom,
! [X2: set_b,A: set_set_b,P: set_b > $o] :
( ( member_set_b @ X2 @ ( inf_inf_set_set_b @ A @ ( collect_set_b @ P ) ) )
= ( ( member_set_b @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_139_Int__Collect,axiom,
! [X2: set_a,A: set_set_a,P: set_a > $o] :
( ( member_set_a @ X2 @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) )
= ( ( member_set_a @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_140_Int__Collect,axiom,
! [X2: nat,A: set_nat,P: nat > $o] :
( ( member_nat @ X2 @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) )
= ( ( member_nat @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_141_Int__Collect,axiom,
! [X2: a,A: set_a,P: a > $o] :
( ( member_a @ X2 @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) )
= ( ( member_a @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_142_Int__Collect,axiom,
! [X2: b,A: set_b,P: b > $o] :
( ( member_b @ X2 @ ( inf_inf_set_b @ A @ ( collect_b @ P ) ) )
= ( ( member_b @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_143_Int__Collect,axiom,
! [X2: list_b,A: set_list_b,P: list_b > $o] :
( ( member_list_b @ X2 @ ( inf_inf_set_list_b @ A @ ( collect_list_b @ P ) ) )
= ( ( member_list_b @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_144_Int__Collect,axiom,
! [X2: list_a,A: set_list_a,P: list_a > $o] :
( ( member_list_a @ X2 @ ( inf_inf_set_list_a @ A @ ( collect_list_a @ P ) ) )
= ( ( member_list_a @ X2 @ A )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_145_Int__absorb,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_146_Int__absorb,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ A )
= A ) ).
% Int_absorb
thf(fact_147_Int__absorb,axiom,
! [A: set_list_b] :
( ( inf_inf_set_list_b @ A @ A )
= A ) ).
% Int_absorb
thf(fact_148_Int__absorb,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ A )
= A ) ).
% Int_absorb
thf(fact_149_Int__assoc,axiom,
! [A: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_150_Int__assoc,axiom,
! [A: set_b,B3: set_b,C2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A @ B3 ) @ C2 )
= ( inf_inf_set_b @ A @ ( inf_inf_set_b @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_151_Int__assoc,axiom,
! [A: set_list_b,B3: set_list_b,C2: set_list_b] :
( ( inf_inf_set_list_b @ ( inf_inf_set_list_b @ A @ B3 ) @ C2 )
= ( inf_inf_set_list_b @ A @ ( inf_inf_set_list_b @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_152_Int__assoc,axiom,
! [A: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ C2 )
= ( inf_inf_set_list_a @ A @ ( inf_inf_set_list_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_153_Int__def,axiom,
( inf_inf_set_nat_b
= ( ^ [A2: set_nat_b,B4: set_nat_b] :
( collect_nat_b
@ ^ [X: nat > b] :
( ( member_nat_b @ X @ A2 )
& ( member_nat_b @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_154_Int__def,axiom,
( inf_inf_set_nat_a
= ( ^ [A2: set_nat_a,B4: set_nat_a] :
( collect_nat_a
@ ^ [X: nat > a] :
( ( member_nat_a @ X @ A2 )
& ( member_nat_a @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_155_Int__def,axiom,
( inf_inf_set_a_b
= ( ^ [A2: set_a_b,B4: set_a_b] :
( collect_a_b
@ ^ [X: a > b] :
( ( member_a_b @ X @ A2 )
& ( member_a_b @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_156_Int__def,axiom,
( inf_inf_set_set_b
= ( ^ [A2: set_set_b,B4: set_set_b] :
( collect_set_b
@ ^ [X: set_b] :
( ( member_set_b @ X @ A2 )
& ( member_set_b @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_157_Int__def,axiom,
( inf_inf_set_set_a
= ( ^ [A2: set_set_a,B4: set_set_a] :
( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A2 )
& ( member_set_a @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_158_Int__def,axiom,
( inf_inf_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A2 )
& ( member_nat @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_159_Int__def,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B4: set_a] :
( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A2 )
& ( member_a @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_160_Int__def,axiom,
( inf_inf_set_b
= ( ^ [A2: set_b,B4: set_b] :
( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A2 )
& ( member_b @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_161_Int__def,axiom,
( inf_inf_set_list_b
= ( ^ [A2: set_list_b,B4: set_list_b] :
( collect_list_b
@ ^ [X: list_b] :
( ( member_list_b @ X @ A2 )
& ( member_list_b @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_162_Int__def,axiom,
( inf_inf_set_list_a
= ( ^ [A2: set_list_a,B4: set_list_a] :
( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A2 )
& ( member_list_a @ X @ B4 ) ) ) ) ) ).
% Int_def
thf(fact_163_IntD2,axiom,
! [C: nat > b,A: set_nat_b,B3: set_nat_b] :
( ( member_nat_b @ C @ ( inf_inf_set_nat_b @ A @ B3 ) )
=> ( member_nat_b @ C @ B3 ) ) ).
% IntD2
thf(fact_164_IntD2,axiom,
! [C: nat > a,A: set_nat_a,B3: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A @ B3 ) )
=> ( member_nat_a @ C @ B3 ) ) ).
% IntD2
thf(fact_165_IntD2,axiom,
! [C: a > b,A: set_a_b,B3: set_a_b] :
( ( member_a_b @ C @ ( inf_inf_set_a_b @ A @ B3 ) )
=> ( member_a_b @ C @ B3 ) ) ).
% IntD2
thf(fact_166_IntD2,axiom,
! [C: a,A: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B3 ) )
=> ( member_a @ C @ B3 ) ) ).
% IntD2
thf(fact_167_IntD2,axiom,
! [C: b,A: set_b,B3: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B3 ) )
=> ( member_b @ C @ B3 ) ) ).
% IntD2
thf(fact_168_IntD2,axiom,
! [C: list_b,A: set_list_b,B3: set_list_b] :
( ( member_list_b @ C @ ( inf_inf_set_list_b @ A @ B3 ) )
=> ( member_list_b @ C @ B3 ) ) ).
% IntD2
thf(fact_169_IntD2,axiom,
! [C: list_a,A: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B3 ) )
=> ( member_list_a @ C @ B3 ) ) ).
% IntD2
thf(fact_170_IntD1,axiom,
! [C: nat > b,A: set_nat_b,B3: set_nat_b] :
( ( member_nat_b @ C @ ( inf_inf_set_nat_b @ A @ B3 ) )
=> ( member_nat_b @ C @ A ) ) ).
% IntD1
thf(fact_171_IntD1,axiom,
! [C: nat > a,A: set_nat_a,B3: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A @ B3 ) )
=> ( member_nat_a @ C @ A ) ) ).
% IntD1
thf(fact_172_IntD1,axiom,
! [C: a > b,A: set_a_b,B3: set_a_b] :
( ( member_a_b @ C @ ( inf_inf_set_a_b @ A @ B3 ) )
=> ( member_a_b @ C @ A ) ) ).
% IntD1
thf(fact_173_IntD1,axiom,
! [C: a,A: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B3 ) )
=> ( member_a @ C @ A ) ) ).
% IntD1
thf(fact_174_IntD1,axiom,
! [C: b,A: set_b,B3: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B3 ) )
=> ( member_b @ C @ A ) ) ).
% IntD1
thf(fact_175_IntD1,axiom,
! [C: list_b,A: set_list_b,B3: set_list_b] :
( ( member_list_b @ C @ ( inf_inf_set_list_b @ A @ B3 ) )
=> ( member_list_b @ C @ A ) ) ).
% IntD1
thf(fact_176_IntD1,axiom,
! [C: list_a,A: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B3 ) )
=> ( member_list_a @ C @ A ) ) ).
% IntD1
thf(fact_177_IntE,axiom,
! [C: nat > b,A: set_nat_b,B3: set_nat_b] :
( ( member_nat_b @ C @ ( inf_inf_set_nat_b @ A @ B3 ) )
=> ~ ( ( member_nat_b @ C @ A )
=> ~ ( member_nat_b @ C @ B3 ) ) ) ).
% IntE
thf(fact_178_IntE,axiom,
! [C: nat > a,A: set_nat_a,B3: set_nat_a] :
( ( member_nat_a @ C @ ( inf_inf_set_nat_a @ A @ B3 ) )
=> ~ ( ( member_nat_a @ C @ A )
=> ~ ( member_nat_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_179_IntE,axiom,
! [C: a > b,A: set_a_b,B3: set_a_b] :
( ( member_a_b @ C @ ( inf_inf_set_a_b @ A @ B3 ) )
=> ~ ( ( member_a_b @ C @ A )
=> ~ ( member_a_b @ C @ B3 ) ) ) ).
% IntE
thf(fact_180_IntE,axiom,
! [C: a,A: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A @ B3 ) )
=> ~ ( ( member_a @ C @ A )
=> ~ ( member_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_181_IntE,axiom,
! [C: b,A: set_b,B3: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A @ B3 ) )
=> ~ ( ( member_b @ C @ A )
=> ~ ( member_b @ C @ B3 ) ) ) ).
% IntE
thf(fact_182_IntE,axiom,
! [C: list_b,A: set_list_b,B3: set_list_b] :
( ( member_list_b @ C @ ( inf_inf_set_list_b @ A @ B3 ) )
=> ~ ( ( member_list_b @ C @ A )
=> ~ ( member_list_b @ C @ B3 ) ) ) ).
% IntE
thf(fact_183_IntE,axiom,
! [C: list_a,A: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A @ B3 ) )
=> ~ ( ( member_list_a @ C @ A )
=> ~ ( member_list_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_184_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_185_bot__set__def,axiom,
( bot_bo3186585308812441520list_a
= ( collect_set_list_a @ bot_bot_set_list_a_o ) ) ).
% bot_set_def
thf(fact_186_bot__set__def,axiom,
( bot_bot_set_set_b
= ( collect_set_b @ bot_bot_set_b_o ) ) ).
% bot_set_def
thf(fact_187_bot__set__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a @ bot_bot_set_a_o ) ) ).
% bot_set_def
thf(fact_188_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_189_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_190_disjoint__iff__not__equal,axiom,
! [A: set_list_b,B3: set_list_b] :
( ( ( inf_inf_set_list_b @ A @ B3 )
= bot_bot_set_list_b )
= ( ! [X: list_b] :
( ( member_list_b @ X @ A )
=> ! [Y: list_b] :
( ( member_list_b @ Y @ B3 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_191_disjoint__iff__not__equal,axiom,
! [A: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B3 )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ! [Y: list_a] :
( ( member_list_a @ Y @ B3 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_192_disjoint__iff__not__equal,axiom,
! [A: set_set_list_a,B3: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ A @ B3 )
= bot_bo3186585308812441520list_a )
= ( ! [X: set_list_a] :
( ( member_set_list_a @ X @ A )
=> ! [Y: set_list_a] :
( ( member_set_list_a @ Y @ B3 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_193_disjoint__iff__not__equal,axiom,
! [A: set_set_b,B3: set_set_b] :
( ( ( inf_inf_set_set_b @ A @ B3 )
= bot_bot_set_set_b )
= ( ! [X: set_b] :
( ( member_set_b @ X @ A )
=> ! [Y: set_b] :
( ( member_set_b @ Y @ B3 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_194_disjoint__iff__not__equal,axiom,
! [A: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B3 )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ! [Y: set_a] :
( ( member_set_a @ Y @ B3 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_195_disjoint__iff__not__equal,axiom,
! [A: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ A @ B3 )
= bot_bot_set_b )
= ( ! [X: b] :
( ( member_b @ X @ A )
=> ! [Y: b] :
( ( member_b @ Y @ B3 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_196_disjoint__iff__not__equal,axiom,
! [A: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A @ B3 )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ! [Y: a] :
( ( member_a @ Y @ B3 )
=> ( X != Y ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_197_Int__empty__right,axiom,
! [A: set_list_b] :
( ( inf_inf_set_list_b @ A @ bot_bot_set_list_b )
= bot_bot_set_list_b ) ).
% Int_empty_right
thf(fact_198_Int__empty__right,axiom,
! [A: set_list_a] :
( ( inf_inf_set_list_a @ A @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_199_Int__empty__right,axiom,
! [A: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ A @ bot_bo3186585308812441520list_a )
= bot_bo3186585308812441520list_a ) ).
% Int_empty_right
thf(fact_200_Int__empty__right,axiom,
! [A: set_set_b] :
( ( inf_inf_set_set_b @ A @ bot_bot_set_set_b )
= bot_bot_set_set_b ) ).
% Int_empty_right
thf(fact_201_Int__empty__right,axiom,
! [A: set_set_a] :
( ( inf_inf_set_set_a @ A @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% Int_empty_right
thf(fact_202_Int__empty__right,axiom,
! [A: set_b] :
( ( inf_inf_set_b @ A @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_203_Int__empty__right,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_204_Int__empty__left,axiom,
! [B3: set_list_b] :
( ( inf_inf_set_list_b @ bot_bot_set_list_b @ B3 )
= bot_bot_set_list_b ) ).
% Int_empty_left
thf(fact_205_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_206_Int__empty__left,axiom,
! [B3: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ bot_bo3186585308812441520list_a @ B3 )
= bot_bo3186585308812441520list_a ) ).
% Int_empty_left
thf(fact_207_Int__empty__left,axiom,
! [B3: set_set_b] :
( ( inf_inf_set_set_b @ bot_bot_set_set_b @ B3 )
= bot_bot_set_set_b ) ).
% Int_empty_left
thf(fact_208_Int__empty__left,axiom,
! [B3: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ B3 )
= bot_bot_set_set_a ) ).
% Int_empty_left
thf(fact_209_Int__empty__left,axiom,
! [B3: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B3 )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_210_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_211_disjoint__iff,axiom,
! [A: set_nat_b,B3: set_nat_b] :
( ( ( inf_inf_set_nat_b @ A @ B3 )
= bot_bot_set_nat_b )
= ( ! [X: nat > b] :
( ( member_nat_b @ X @ A )
=> ~ ( member_nat_b @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_212_disjoint__iff,axiom,
! [A: set_nat_a,B3: set_nat_a] :
( ( ( inf_inf_set_nat_a @ A @ B3 )
= bot_bot_set_nat_a )
= ( ! [X: nat > a] :
( ( member_nat_a @ X @ A )
=> ~ ( member_nat_a @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_213_disjoint__iff,axiom,
! [A: set_a_b,B3: set_a_b] :
( ( ( inf_inf_set_a_b @ A @ B3 )
= bot_bot_set_a_b )
= ( ! [X: a > b] :
( ( member_a_b @ X @ A )
=> ~ ( member_a_b @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_214_disjoint__iff,axiom,
! [A: set_list_b,B3: set_list_b] :
( ( ( inf_inf_set_list_b @ A @ B3 )
= bot_bot_set_list_b )
= ( ! [X: list_b] :
( ( member_list_b @ X @ A )
=> ~ ( member_list_b @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_215_disjoint__iff,axiom,
! [A: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ A @ B3 )
= bot_bot_set_list_a )
= ( ! [X: list_a] :
( ( member_list_a @ X @ A )
=> ~ ( member_list_a @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_216_disjoint__iff,axiom,
! [A: set_set_list_a,B3: set_set_list_a] :
( ( ( inf_in4657809108759609906list_a @ A @ B3 )
= bot_bo3186585308812441520list_a )
= ( ! [X: set_list_a] :
( ( member_set_list_a @ X @ A )
=> ~ ( member_set_list_a @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_217_disjoint__iff,axiom,
! [A: set_set_b,B3: set_set_b] :
( ( ( inf_inf_set_set_b @ A @ B3 )
= bot_bot_set_set_b )
= ( ! [X: set_b] :
( ( member_set_b @ X @ A )
=> ~ ( member_set_b @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_218_disjoint__iff,axiom,
! [A: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ A @ B3 )
= bot_bot_set_set_a )
= ( ! [X: set_a] :
( ( member_set_a @ X @ A )
=> ~ ( member_set_a @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_219_disjoint__iff,axiom,
! [A: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ A @ B3 )
= bot_bot_set_b )
= ( ! [X: b] :
( ( member_b @ X @ A )
=> ~ ( member_b @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_220_disjoint__iff,axiom,
! [A: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A @ B3 )
= bot_bot_set_a )
= ( ! [X: a] :
( ( member_a @ X @ A )
=> ~ ( member_a @ X @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_221_Int__emptyI,axiom,
! [A: set_nat_b,B3: set_nat_b] :
( ! [X3: nat > b] :
( ( member_nat_b @ X3 @ A )
=> ~ ( member_nat_b @ X3 @ B3 ) )
=> ( ( inf_inf_set_nat_b @ A @ B3 )
= bot_bot_set_nat_b ) ) ).
% Int_emptyI
thf(fact_222_Int__emptyI,axiom,
! [A: set_nat_a,B3: set_nat_a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
=> ~ ( member_nat_a @ X3 @ B3 ) )
=> ( ( inf_inf_set_nat_a @ A @ B3 )
= bot_bot_set_nat_a ) ) ).
% Int_emptyI
thf(fact_223_Int__emptyI,axiom,
! [A: set_a_b,B3: set_a_b] :
( ! [X3: a > b] :
( ( member_a_b @ X3 @ A )
=> ~ ( member_a_b @ X3 @ B3 ) )
=> ( ( inf_inf_set_a_b @ A @ B3 )
= bot_bot_set_a_b ) ) ).
% Int_emptyI
thf(fact_224_Int__emptyI,axiom,
! [A: set_list_b,B3: set_list_b] :
( ! [X3: list_b] :
( ( member_list_b @ X3 @ A )
=> ~ ( member_list_b @ X3 @ B3 ) )
=> ( ( inf_inf_set_list_b @ A @ B3 )
= bot_bot_set_list_b ) ) ).
% Int_emptyI
thf(fact_225_Int__emptyI,axiom,
! [A: set_list_a,B3: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ~ ( member_list_a @ X3 @ B3 ) )
=> ( ( inf_inf_set_list_a @ A @ B3 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_226_Int__emptyI,axiom,
! [A: set_set_list_a,B3: set_set_list_a] :
( ! [X3: set_list_a] :
( ( member_set_list_a @ X3 @ A )
=> ~ ( member_set_list_a @ X3 @ B3 ) )
=> ( ( inf_in4657809108759609906list_a @ A @ B3 )
= bot_bo3186585308812441520list_a ) ) ).
% Int_emptyI
thf(fact_227_Int__emptyI,axiom,
! [A: set_set_b,B3: set_set_b] :
( ! [X3: set_b] :
( ( member_set_b @ X3 @ A )
=> ~ ( member_set_b @ X3 @ B3 ) )
=> ( ( inf_inf_set_set_b @ A @ B3 )
= bot_bot_set_set_b ) ) ).
% Int_emptyI
thf(fact_228_Int__emptyI,axiom,
! [A: set_set_a,B3: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ( member_set_a @ X3 @ B3 ) )
=> ( ( inf_inf_set_set_a @ A @ B3 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_229_Int__emptyI,axiom,
! [A: set_b,B3: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ~ ( member_b @ X3 @ B3 ) )
=> ( ( inf_inf_set_b @ A @ B3 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_230_Int__emptyI,axiom,
! [A: set_a,B3: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ~ ( member_a @ X3 @ B3 ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_231_inj__on__Int,axiom,
! [F: a > b,A: set_a,B3: set_a] :
( ( ( inj_on_a_b @ F @ A )
| ( inj_on_a_b @ F @ B3 ) )
=> ( inj_on_a_b @ F @ ( inf_inf_set_a @ A @ B3 ) ) ) ).
% inj_on_Int
thf(fact_232_ex__in__conv,axiom,
! [A: set_nat_b] :
( ( ? [X: nat > b] : ( member_nat_b @ X @ A ) )
= ( A != bot_bot_set_nat_b ) ) ).
% ex_in_conv
thf(fact_233_ex__in__conv,axiom,
! [A: set_nat_a] :
( ( ? [X: nat > a] : ( member_nat_a @ X @ A ) )
= ( A != bot_bot_set_nat_a ) ) ).
% ex_in_conv
thf(fact_234_ex__in__conv,axiom,
! [A: set_a_b] :
( ( ? [X: a > b] : ( member_a_b @ X @ A ) )
= ( A != bot_bot_set_a_b ) ) ).
% ex_in_conv
thf(fact_235_ex__in__conv,axiom,
! [A: set_set_list_a] :
( ( ? [X: set_list_a] : ( member_set_list_a @ X @ A ) )
= ( A != bot_bo3186585308812441520list_a ) ) ).
% ex_in_conv
thf(fact_236_ex__in__conv,axiom,
! [A: set_set_b] :
( ( ? [X: set_b] : ( member_set_b @ X @ A ) )
= ( A != bot_bot_set_set_b ) ) ).
% ex_in_conv
thf(fact_237_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_238_ex__in__conv,axiom,
! [A: set_b] :
( ( ? [X: b] : ( member_b @ X @ A ) )
= ( A != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_239_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_240_equals0I,axiom,
! [A: set_nat_b] :
( ! [Y3: nat > b] :
~ ( member_nat_b @ Y3 @ A )
=> ( A = bot_bot_set_nat_b ) ) ).
% equals0I
thf(fact_241_equals0I,axiom,
! [A: set_nat_a] :
( ! [Y3: nat > a] :
~ ( member_nat_a @ Y3 @ A )
=> ( A = bot_bot_set_nat_a ) ) ).
% equals0I
thf(fact_242_equals0I,axiom,
! [A: set_a_b] :
( ! [Y3: a > b] :
~ ( member_a_b @ Y3 @ A )
=> ( A = bot_bot_set_a_b ) ) ).
% equals0I
thf(fact_243_equals0I,axiom,
! [A: set_set_list_a] :
( ! [Y3: set_list_a] :
~ ( member_set_list_a @ Y3 @ A )
=> ( A = bot_bo3186585308812441520list_a ) ) ).
% equals0I
thf(fact_244_equals0I,axiom,
! [A: set_set_b] :
( ! [Y3: set_b] :
~ ( member_set_b @ Y3 @ A )
=> ( A = bot_bot_set_set_b ) ) ).
% equals0I
thf(fact_245_equals0I,axiom,
! [A: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_246_equals0I,axiom,
! [A: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A )
=> ( A = bot_bot_set_b ) ) ).
% equals0I
thf(fact_247_equals0I,axiom,
! [A: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_248_equals0D,axiom,
! [A: set_nat_b,A4: nat > b] :
( ( A = bot_bot_set_nat_b )
=> ~ ( member_nat_b @ A4 @ A ) ) ).
% equals0D
thf(fact_249_equals0D,axiom,
! [A: set_nat_a,A4: nat > a] :
( ( A = bot_bot_set_nat_a )
=> ~ ( member_nat_a @ A4 @ A ) ) ).
% equals0D
thf(fact_250_equals0D,axiom,
! [A: set_a_b,A4: a > b] :
( ( A = bot_bot_set_a_b )
=> ~ ( member_a_b @ A4 @ A ) ) ).
% equals0D
thf(fact_251_equals0D,axiom,
! [A: set_set_list_a,A4: set_list_a] :
( ( A = bot_bo3186585308812441520list_a )
=> ~ ( member_set_list_a @ A4 @ A ) ) ).
% equals0D
thf(fact_252_equals0D,axiom,
! [A: set_set_b,A4: set_b] :
( ( A = bot_bot_set_set_b )
=> ~ ( member_set_b @ A4 @ A ) ) ).
% equals0D
thf(fact_253_equals0D,axiom,
! [A: set_set_a,A4: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A4 @ A ) ) ).
% equals0D
thf(fact_254_equals0D,axiom,
! [A: set_b,A4: b] :
( ( A = bot_bot_set_b )
=> ~ ( member_b @ A4 @ A ) ) ).
% equals0D
thf(fact_255_equals0D,axiom,
! [A: set_a,A4: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A4 @ A ) ) ).
% equals0D
thf(fact_256_emptyE,axiom,
! [A4: nat > b] :
~ ( member_nat_b @ A4 @ bot_bot_set_nat_b ) ).
% emptyE
thf(fact_257_emptyE,axiom,
! [A4: nat > a] :
~ ( member_nat_a @ A4 @ bot_bot_set_nat_a ) ).
% emptyE
thf(fact_258_emptyE,axiom,
! [A4: a > b] :
~ ( member_a_b @ A4 @ bot_bot_set_a_b ) ).
% emptyE
thf(fact_259_emptyE,axiom,
! [A4: set_list_a] :
~ ( member_set_list_a @ A4 @ bot_bo3186585308812441520list_a ) ).
% emptyE
thf(fact_260_emptyE,axiom,
! [A4: set_b] :
~ ( member_set_b @ A4 @ bot_bot_set_set_b ) ).
% emptyE
thf(fact_261_emptyE,axiom,
! [A4: set_a] :
~ ( member_set_a @ A4 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_262_emptyE,axiom,
! [A4: b] :
~ ( member_b @ A4 @ bot_bot_set_b ) ).
% emptyE
thf(fact_263_emptyE,axiom,
! [A4: a] :
~ ( member_a @ A4 @ bot_bot_set_a ) ).
% emptyE
thf(fact_264_empty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X: nat] : $false ) ) ).
% empty_def
thf(fact_265_empty__def,axiom,
( bot_bo3186585308812441520list_a
= ( collect_set_list_a
@ ^ [X: set_list_a] : $false ) ) ).
% empty_def
thf(fact_266_empty__def,axiom,
( bot_bot_set_set_b
= ( collect_set_b
@ ^ [X: set_b] : $false ) ) ).
% empty_def
thf(fact_267_empty__def,axiom,
( bot_bot_set_set_a
= ( collect_set_a
@ ^ [X: set_a] : $false ) ) ).
% empty_def
thf(fact_268_empty__def,axiom,
( bot_bot_set_b
= ( collect_b
@ ^ [X: b] : $false ) ) ).
% empty_def
thf(fact_269_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X: a] : $false ) ) ).
% empty_def
thf(fact_270_ring__iso__memE_I1_J,axiom,
! [H: b > b,R2: partia1897943568983147621xt_b_d,S: partia1897943568983147621xt_b_d,X2: b] :
( ( member_b_b @ H @ ( ring_iso_b_d_b_d @ R2 @ S ) )
=> ( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( member_b @ ( H @ X2 ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_271_ring__iso__memE_I1_J,axiom,
! [H: b > a,R2: partia1897943568983147621xt_b_d,S: partia8877618634411419171xt_a_c,X2: b] :
( ( member_b_a @ H @ ( ring_iso_b_d_a_c @ R2 @ S ) )
=> ( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( member_a @ ( H @ X2 ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_272_ring__iso__memE_I1_J,axiom,
! [H: a > b,R2: partia8877618634411419171xt_a_c,S: partia1897943568983147621xt_b_d,X2: a] :
( ( member_a_b @ H @ ( ring_iso_a_c_b_d @ R2 @ S ) )
=> ( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( member_b @ ( H @ X2 ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_273_ring__iso__memE_I1_J,axiom,
! [H: a > a,R2: partia8877618634411419171xt_a_c,S: partia8877618634411419171xt_a_c,X2: a] :
( ( member_a_a @ H @ ( ring_iso_a_c_a_c @ R2 @ S ) )
=> ( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( member_a @ ( H @ X2 ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_274_ring__iso__memE_I1_J,axiom,
! [H: list_b > b,R2: partia4026993951477142903t_unit,S: partia1897943568983147621xt_b_d,X2: list_b] :
( ( member_list_b_b @ H @ ( ring_i3676047618198725658it_b_d @ R2 @ S ) )
=> ( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( member_b @ ( H @ X2 ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_275_ring__iso__memE_I1_J,axiom,
! [H: list_b > a,R2: partia4026993951477142903t_unit,S: partia8877618634411419171xt_a_c,X2: list_b] :
( ( member_list_b_a @ H @ ( ring_i6463503200171401690it_a_c @ R2 @ S ) )
=> ( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( member_a @ ( H @ X2 ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_276_ring__iso__memE_I1_J,axiom,
! [H: list_a > b,R2: partia2670972154091845814t_unit,S: partia1897943568983147621xt_b_d,X2: list_a] :
( ( member_list_a_b @ H @ ( ring_i4261380215208433627it_b_d @ R2 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_b @ ( H @ X2 ) @ ( partia8782771468121683032xt_b_d @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_277_ring__iso__memE_I1_J,axiom,
! [H: list_a > a,R2: partia2670972154091845814t_unit,S: partia8877618634411419171xt_a_c,X2: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109659it_a_c @ R2 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( member_a @ ( H @ X2 ) @ ( partia778085601923319190xt_a_c @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_278_ring__iso__memE_I1_J,axiom,
! [H: b > list_b,R2: partia1897943568983147621xt_b_d,S: partia4026993951477142903t_unit,X2: b] :
( ( member_b_list_b @ H @ ( ring_i3204478355530656794t_unit @ R2 @ S ) )
=> ( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( member_list_b @ ( H @ X2 ) @ ( partia1381092143316337258t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_279_ring__iso__memE_I1_J,axiom,
! [H: b > list_a,R2: partia1897943568983147621xt_b_d,S: partia2670972154091845814t_unit,X2: b] :
( ( member_b_list_a @ H @ ( ring_i6290234597520651355t_unit @ R2 @ S ) )
=> ( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( member_list_a @ ( H @ X2 ) @ ( partia5361259788508890537t_unit @ S ) ) ) ) ).
% ring_iso_memE(1)
thf(fact_280_dr_Oring__irreducibleE_I4_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_r999134135267193927le_a_c @ r @ R3 )
=> ~ ( member_a @ R3 @ ( units_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.ring_irreducibleE(4)
thf(fact_281_h_Ois__abelian__group__hom,axiom,
abelia4987496640480023959_c_b_d @ r @ s @ h ).
% h.is_abelian_group_hom
thf(fact_282_h_Oinj__on__domain,axiom,
( ( inj_on_a_b @ h @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( domain_b_d @ s )
=> ( domain_a_c @ r ) ) ) ).
% h.inj_on_domain
thf(fact_283_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_list_b] :
( ( inf_inf_set_list_b @ X2 @ bot_bot_set_list_b )
= bot_bot_set_list_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_284_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_285_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X2 @ bot_bo3186585308812441520list_a )
= bot_bo3186585308812441520list_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_286_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_b] :
( ( inf_inf_set_set_b @ X2 @ bot_bot_set_set_b )
= bot_bot_set_set_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_287_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_288_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_b] :
( ( inf_inf_set_b @ X2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_289_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_290_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_list_b] :
( ( inf_inf_set_list_b @ bot_bot_set_list_b @ X2 )
= bot_bot_set_list_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_291_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_292_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ bot_bo3186585308812441520list_a @ X2 )
= bot_bo3186585308812441520list_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_293_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_b] :
( ( inf_inf_set_set_b @ bot_bot_set_set_b @ X2 )
= bot_bot_set_set_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_294_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X2 )
= bot_bot_set_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_295_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X2 )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_296_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_297_inf__bot__right,axiom,
! [X2: set_list_b] :
( ( inf_inf_set_list_b @ X2 @ bot_bot_set_list_b )
= bot_bot_set_list_b ) ).
% inf_bot_right
thf(fact_298_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_299_inf__bot__right,axiom,
! [X2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ X2 @ bot_bo3186585308812441520list_a )
= bot_bo3186585308812441520list_a ) ).
% inf_bot_right
thf(fact_300_inf__bot__right,axiom,
! [X2: set_set_b] :
( ( inf_inf_set_set_b @ X2 @ bot_bot_set_set_b )
= bot_bot_set_set_b ) ).
% inf_bot_right
thf(fact_301_inf__bot__right,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ X2 @ bot_bot_set_set_a )
= bot_bot_set_set_a ) ).
% inf_bot_right
thf(fact_302_inf__bot__right,axiom,
! [X2: set_b] :
( ( inf_inf_set_b @ X2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% inf_bot_right
thf(fact_303_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_304_inf__bot__left,axiom,
! [X2: set_list_b] :
( ( inf_inf_set_list_b @ bot_bot_set_list_b @ X2 )
= bot_bot_set_list_b ) ).
% inf_bot_left
thf(fact_305_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_306_inf__bot__left,axiom,
! [X2: set_set_list_a] :
( ( inf_in4657809108759609906list_a @ bot_bo3186585308812441520list_a @ X2 )
= bot_bo3186585308812441520list_a ) ).
% inf_bot_left
thf(fact_307_inf__bot__left,axiom,
! [X2: set_set_b] :
( ( inf_inf_set_set_b @ bot_bot_set_set_b @ X2 )
= bot_bot_set_set_b ) ).
% inf_bot_left
thf(fact_308_inf__bot__left,axiom,
! [X2: set_set_a] :
( ( inf_inf_set_set_a @ bot_bot_set_set_a @ X2 )
= bot_bot_set_set_a ) ).
% inf_bot_left
thf(fact_309_inf__bot__left,axiom,
! [X2: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X2 )
= bot_bot_set_b ) ).
% inf_bot_left
thf(fact_310_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_311_h_Ohom__closed,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_b @ ( h @ X2 ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% h.hom_closed
thf(fact_312_h_Oideal__vimage,axiom,
! [I2: set_b] :
( ( ideal_b_d @ I2 @ s )
=> ( ideal_a_c
@ ( collect_a
@ ^ [R4: a] :
( ( member_a @ R4 @ ( partia778085601923319190xt_a_c @ r ) )
& ( member_b @ ( h @ R4 ) @ I2 ) ) )
@ r ) ) ).
% h.ideal_vimage
thf(fact_313_dr_Or__coset__subset__G,axiom,
! [H2: set_a,X2: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ord_less_eq_set_a @ ( r_cose3160181845575678695xt_a_c @ r @ H2 @ X2 ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% dr.r_coset_subset_G
thf(fact_314_dr_OassociatedI2,axiom,
! [U: a,A4: a,B: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( A4
= ( mult_a_ring_ext_a_c @ r @ B @ U ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( associ5860276531582424204xt_a_c @ r @ A4 @ B ) ) ) ) ).
% dr.associatedI2
thf(fact_315_ds_Ocarrier__not__empty,axiom,
( ( partia8782771468121683032xt_b_d @ s )
!= bot_bot_set_b ) ).
% ds.carrier_not_empty
thf(fact_316_ds_Oi__intersect,axiom,
! [I2: set_b,J: set_b] :
( ( ideal_b_d @ I2 @ s )
=> ( ( ideal_b_d @ J @ s )
=> ( ideal_b_d @ ( inf_inf_set_b @ I2 @ J ) @ s ) ) ) ).
% ds.i_intersect
thf(fact_317_dr_Om__lcomm,axiom,
! [X2: a,Y2: a,Z: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ X2 @ ( mult_a_ring_ext_a_c @ r @ Y2 @ Z ) )
= ( mult_a_ring_ext_a_c @ r @ Y2 @ ( mult_a_ring_ext_a_c @ r @ X2 @ Z ) ) ) ) ) ) ).
% dr.m_lcomm
thf(fact_318_dr_Om__comm,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ X2 @ Y2 )
= ( mult_a_ring_ext_a_c @ r @ Y2 @ X2 ) ) ) ) ).
% dr.m_comm
thf(fact_319_dr_Om__assoc,axiom,
! [X2: a,Y2: a,Z: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ X2 @ Y2 ) @ Z )
= ( mult_a_ring_ext_a_c @ r @ X2 @ ( mult_a_ring_ext_a_c @ r @ Y2 @ Z ) ) ) ) ) ) ).
% dr.m_assoc
thf(fact_320_ds_Ooneideal,axiom,
ideal_b_d @ ( partia8782771468121683032xt_b_d @ s ) @ s ).
% ds.oneideal
thf(fact_321_dual__order_Orefl,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_322_dual__order_Orefl,axiom,
! [A4: set_b] : ( ord_less_eq_set_b @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_323_dual__order_Orefl,axiom,
! [A4: set_list_b] : ( ord_le8932221534207217157list_b @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_324_dual__order_Orefl,axiom,
! [A4: set_list_a] : ( ord_le8861187494160871172list_a @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_325_dual__order_Orefl,axiom,
! [A4: nat] : ( ord_less_eq_nat @ A4 @ A4 ) ).
% dual_order.refl
thf(fact_326_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_327_order__refl,axiom,
! [X2: set_b] : ( ord_less_eq_set_b @ X2 @ X2 ) ).
% order_refl
thf(fact_328_order__refl,axiom,
! [X2: set_list_b] : ( ord_le8932221534207217157list_b @ X2 @ X2 ) ).
% order_refl
thf(fact_329_order__refl,axiom,
! [X2: set_list_a] : ( ord_le8861187494160871172list_a @ X2 @ X2 ) ).
% order_refl
thf(fact_330_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_331_dr_Ounit__factor,axiom,
! [A4: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_c @ r @ A4 @ B ) @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ A4 @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) ).
% dr.unit_factor
thf(fact_332_dr_Oprod__unit__r,axiom,
! [A4: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_c @ r @ A4 @ B ) @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ A4 @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) ) ).
% dr.prod_unit_r
thf(fact_333_dr_Oprod__unit__l,axiom,
! [A4: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_c @ r @ A4 @ B ) @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A4 @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) ) ).
% dr.prod_unit_l
thf(fact_334_dr_Omult__cong__r,axiom,
! [B: a,B5: a,A4: a] :
( ( associ5860276531582424204xt_a_c @ r @ B @ B5 )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B5 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( associ5860276531582424204xt_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ A4 @ B ) @ ( mult_a_ring_ext_a_c @ r @ A4 @ B5 ) ) ) ) ) ) ).
% dr.mult_cong_r
thf(fact_335_dr_Omult__cong__l,axiom,
! [A4: a,A5: a,B: a] :
( ( associ5860276531582424204xt_a_c @ r @ A4 @ A5 )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ A5 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( associ5860276531582424204xt_a_c @ r @ ( mult_a_ring_ext_a_c @ r @ A4 @ B ) @ ( mult_a_ring_ext_a_c @ r @ A5 @ B ) ) ) ) ) ) ).
% dr.mult_cong_l
thf(fact_336_subset__antisym,axiom,
! [A: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_337_subset__antisym,axiom,
! [A: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_338_subset__antisym,axiom,
! [A: set_list_b,B3: set_list_b] :
( ( ord_le8932221534207217157list_b @ A @ B3 )
=> ( ( ord_le8932221534207217157list_b @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_339_subset__antisym,axiom,
! [A: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ A )
=> ( A = B3 ) ) ) ).
% subset_antisym
thf(fact_340_subsetI,axiom,
! [A: set_nat_b,B3: set_nat_b] :
( ! [X3: nat > b] :
( ( member_nat_b @ X3 @ A )
=> ( member_nat_b @ X3 @ B3 ) )
=> ( ord_le942501763763511270_nat_b @ A @ B3 ) ) ).
% subsetI
thf(fact_341_subsetI,axiom,
! [A: set_nat_a,B3: set_nat_a] :
( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
=> ( member_nat_a @ X3 @ B3 ) )
=> ( ord_le871467723717165285_nat_a @ A @ B3 ) ) ).
% subsetI
thf(fact_342_subsetI,axiom,
! [A: set_a_b,B3: set_a_b] :
( ! [X3: a > b] :
( ( member_a_b @ X3 @ A )
=> ( member_a_b @ X3 @ B3 ) )
=> ( ord_less_eq_set_a_b @ A @ B3 ) ) ).
% subsetI
thf(fact_343_subsetI,axiom,
! [A: set_a,B3: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B3 ) )
=> ( ord_less_eq_set_a @ A @ B3 ) ) ).
% subsetI
thf(fact_344_subsetI,axiom,
! [A: set_b,B3: set_b] :
( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( member_b @ X3 @ B3 ) )
=> ( ord_less_eq_set_b @ A @ B3 ) ) ).
% subsetI
thf(fact_345_subsetI,axiom,
! [A: set_list_b,B3: set_list_b] :
( ! [X3: list_b] :
( ( member_list_b @ X3 @ A )
=> ( member_list_b @ X3 @ B3 ) )
=> ( ord_le8932221534207217157list_b @ A @ B3 ) ) ).
% subsetI
thf(fact_346_subsetI,axiom,
! [A: set_list_a,B3: set_list_a] :
( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( member_list_a @ X3 @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A @ B3 ) ) ).
% subsetI
thf(fact_347_inf_Oidem,axiom,
! [A4: set_a] :
( ( inf_inf_set_a @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_348_inf_Oidem,axiom,
! [A4: set_b] :
( ( inf_inf_set_b @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_349_inf_Oidem,axiom,
! [A4: set_list_b] :
( ( inf_inf_set_list_b @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_350_inf_Oidem,axiom,
! [A4: set_list_a] :
( ( inf_inf_set_list_a @ A4 @ A4 )
= A4 ) ).
% inf.idem
thf(fact_351_inf__idem,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_352_inf__idem,axiom,
! [X2: set_b] :
( ( inf_inf_set_b @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_353_inf__idem,axiom,
! [X2: set_list_b] :
( ( inf_inf_set_list_b @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_354_inf__idem,axiom,
! [X2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_355_inf_Oleft__idem,axiom,
! [A4: set_a,B: set_a] :
( ( inf_inf_set_a @ A4 @ ( inf_inf_set_a @ A4 @ B ) )
= ( inf_inf_set_a @ A4 @ B ) ) ).
% inf.left_idem
thf(fact_356_inf_Oleft__idem,axiom,
! [A4: set_b,B: set_b] :
( ( inf_inf_set_b @ A4 @ ( inf_inf_set_b @ A4 @ B ) )
= ( inf_inf_set_b @ A4 @ B ) ) ).
% inf.left_idem
thf(fact_357_inf_Oleft__idem,axiom,
! [A4: set_list_b,B: set_list_b] :
( ( inf_inf_set_list_b @ A4 @ ( inf_inf_set_list_b @ A4 @ B ) )
= ( inf_inf_set_list_b @ A4 @ B ) ) ).
% inf.left_idem
thf(fact_358_inf_Oleft__idem,axiom,
! [A4: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ A4 @ ( inf_inf_set_list_a @ A4 @ B ) )
= ( inf_inf_set_list_a @ A4 @ B ) ) ).
% inf.left_idem
thf(fact_359_inf__left__idem,axiom,
! [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y2 ) )
= ( inf_inf_set_a @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_360_inf__left__idem,axiom,
! [X2: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X2 @ ( inf_inf_set_b @ X2 @ Y2 ) )
= ( inf_inf_set_b @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_361_inf__left__idem,axiom,
! [X2: set_list_b,Y2: set_list_b] :
( ( inf_inf_set_list_b @ X2 @ ( inf_inf_set_list_b @ X2 @ Y2 ) )
= ( inf_inf_set_list_b @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_362_inf__left__idem,axiom,
! [X2: set_list_a,Y2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ X2 @ Y2 ) )
= ( inf_inf_set_list_a @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_363_inf_Oright__idem,axiom,
! [A4: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A4 @ B ) @ B )
= ( inf_inf_set_a @ A4 @ B ) ) ).
% inf.right_idem
thf(fact_364_inf_Oright__idem,axiom,
! [A4: set_b,B: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A4 @ B ) @ B )
= ( inf_inf_set_b @ A4 @ B ) ) ).
% inf.right_idem
thf(fact_365_inf_Oright__idem,axiom,
! [A4: set_list_b,B: set_list_b] :
( ( inf_inf_set_list_b @ ( inf_inf_set_list_b @ A4 @ B ) @ B )
= ( inf_inf_set_list_b @ A4 @ B ) ) ).
% inf.right_idem
thf(fact_366_inf_Oright__idem,axiom,
! [A4: set_list_a,B: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A4 @ B ) @ B )
= ( inf_inf_set_list_a @ A4 @ B ) ) ).
% inf.right_idem
thf(fact_367_inf__right__idem,axiom,
! [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_set_a @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_368_inf__right__idem,axiom,
! [X2: set_b,Y2: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_set_b @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_369_inf__right__idem,axiom,
! [X2: set_list_b,Y2: set_list_b] :
( ( inf_inf_set_list_b @ ( inf_inf_set_list_b @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_set_list_b @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_370_inf__right__idem,axiom,
! [X2: set_list_a,Y2: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_set_list_a @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_371_dr_Ocgenideal__minimal,axiom,
! [J: set_a,A4: a] :
( ( ideal_a_c @ J @ r )
=> ( ( member_a @ A4 @ J )
=> ( ord_less_eq_set_a @ ( cgenid547466214215511830xt_a_c @ r @ A4 ) @ J ) ) ) ).
% dr.cgenideal_minimal
thf(fact_372_dr_Oring__associated__iff,axiom,
! [A4: a,B: a] :
( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( associ5860276531582424204xt_a_c @ r @ A4 @ B )
= ( ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_c @ r ) )
& ( A4
= ( mult_a_ring_ext_a_c @ r @ X @ B ) ) ) ) ) ) ) ).
% dr.ring_associated_iff
thf(fact_373_dr_OassociatedI2_H,axiom,
! [A4: a,B: a,U: a] :
( ( A4
= ( mult_a_ring_ext_a_c @ r @ B @ U ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( associ5860276531582424204xt_a_c @ r @ A4 @ B ) ) ) ) ).
% dr.associatedI2'
thf(fact_374_dr_Oring__irreducibleE_I5_J,axiom,
! [R3: a,A4: a,B: a] :
( ( member_a @ R3 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_r999134135267193927le_a_c @ r @ R3 )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_c @ r @ A4 @ B ) )
=> ( ( member_a @ A4 @ ( units_a_ring_ext_a_c @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) ) ) ) ).
% dr.ring_irreducibleE(5)
thf(fact_375_subset__empty,axiom,
! [A: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A @ bot_bo3186585308812441520list_a )
= ( A = bot_bo3186585308812441520list_a ) ) ).
% subset_empty
thf(fact_376_subset__empty,axiom,
! [A: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A @ bot_bot_set_set_b )
= ( A = bot_bot_set_set_b ) ) ).
% subset_empty
thf(fact_377_subset__empty,axiom,
! [A: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ bot_bot_set_set_a )
= ( A = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_378_subset__empty,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_379_subset__empty,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_380_subset__empty,axiom,
! [A: set_list_b] :
( ( ord_le8932221534207217157list_b @ A @ bot_bot_set_list_b )
= ( A = bot_bot_set_list_b ) ) ).
% subset_empty
thf(fact_381_subset__empty,axiom,
! [A: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ bot_bot_set_list_a )
= ( A = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_382_empty__subsetI,axiom,
! [A: set_set_list_a] : ( ord_le8877086941679407844list_a @ bot_bo3186585308812441520list_a @ A ) ).
% empty_subsetI
thf(fact_383_empty__subsetI,axiom,
! [A: set_set_b] : ( ord_le3795704787696855135_set_b @ bot_bot_set_set_b @ A ) ).
% empty_subsetI
thf(fact_384_empty__subsetI,axiom,
! [A: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A ) ).
% empty_subsetI
thf(fact_385_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_386_empty__subsetI,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% empty_subsetI
thf(fact_387_empty__subsetI,axiom,
! [A: set_list_b] : ( ord_le8932221534207217157list_b @ bot_bot_set_list_b @ A ) ).
% empty_subsetI
thf(fact_388_empty__subsetI,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A ) ).
% empty_subsetI
thf(fact_389_le__inf__iff,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) )
= ( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_390_le__inf__iff,axiom,
! [X2: set_b,Y2: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X2 @ ( inf_inf_set_b @ Y2 @ Z ) )
= ( ( ord_less_eq_set_b @ X2 @ Y2 )
& ( ord_less_eq_set_b @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_391_le__inf__iff,axiom,
! [X2: set_list_b,Y2: set_list_b,Z: set_list_b] :
( ( ord_le8932221534207217157list_b @ X2 @ ( inf_inf_set_list_b @ Y2 @ Z ) )
= ( ( ord_le8932221534207217157list_b @ X2 @ Y2 )
& ( ord_le8932221534207217157list_b @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_392_le__inf__iff,axiom,
! [X2: set_list_a,Y2: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z ) )
= ( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
& ( ord_le8861187494160871172list_a @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_393_le__inf__iff,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z ) )
= ( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_394_inf_Obounded__iff,axiom,
! [A4: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A4 @ B )
& ( ord_less_eq_set_a @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_395_inf_Obounded__iff,axiom,
! [A4: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A4 @ ( inf_inf_set_b @ B @ C ) )
= ( ( ord_less_eq_set_b @ A4 @ B )
& ( ord_less_eq_set_b @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_396_inf_Obounded__iff,axiom,
! [A4: set_list_b,B: set_list_b,C: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ ( inf_inf_set_list_b @ B @ C ) )
= ( ( ord_le8932221534207217157list_b @ A4 @ B )
& ( ord_le8932221534207217157list_b @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_397_inf_Obounded__iff,axiom,
! [A4: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ ( inf_inf_set_list_a @ B @ C ) )
= ( ( ord_le8861187494160871172list_a @ A4 @ B )
& ( ord_le8861187494160871172list_a @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_398_inf_Obounded__iff,axiom,
! [A4: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A4 @ B )
& ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_399_Int__subset__iff,axiom,
! [C2: set_a,A: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B3 ) )
= ( ( ord_less_eq_set_a @ C2 @ A )
& ( ord_less_eq_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_400_Int__subset__iff,axiom,
! [C2: set_b,A: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B3 ) )
= ( ( ord_less_eq_set_b @ C2 @ A )
& ( ord_less_eq_set_b @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_401_Int__subset__iff,axiom,
! [C2: set_list_b,A: set_list_b,B3: set_list_b] :
( ( ord_le8932221534207217157list_b @ C2 @ ( inf_inf_set_list_b @ A @ B3 ) )
= ( ( ord_le8932221534207217157list_b @ C2 @ A )
& ( ord_le8932221534207217157list_b @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_402_Int__subset__iff,axiom,
! [C2: set_list_a,A: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A @ B3 ) )
= ( ( ord_le8861187494160871172list_a @ C2 @ A )
& ( ord_le8861187494160871172list_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_403_dr_Om__closed,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_c @ r @ X2 @ Y2 ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% dr.m_closed
thf(fact_404_dr_OUnits__m__closed,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ Y2 @ ( units_a_ring_ext_a_c @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_c @ r @ X2 @ Y2 ) @ ( units_a_ring_ext_a_c @ r ) ) ) ) ).
% dr.Units_m_closed
thf(fact_405_that,axiom,
member_list_b @ x @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% that
thf(fact_406_dr_OUnits__l__cancel,axiom,
! [X2: a,Y2: a,Z: a] :
( ( member_a @ X2 @ ( units_a_ring_ext_a_c @ r ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Z @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( mult_a_ring_ext_a_c @ r @ X2 @ Y2 )
= ( mult_a_ring_ext_a_c @ r @ X2 @ Z ) )
= ( Y2 = Z ) ) ) ) ) ).
% dr.Units_l_cancel
thf(fact_407_order__antisym__conv,axiom,
! [Y2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_408_order__antisym__conv,axiom,
! [Y2: set_b,X2: set_b] :
( ( ord_less_eq_set_b @ Y2 @ X2 )
=> ( ( ord_less_eq_set_b @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_409_order__antisym__conv,axiom,
! [Y2: set_list_b,X2: set_list_b] :
( ( ord_le8932221534207217157list_b @ Y2 @ X2 )
=> ( ( ord_le8932221534207217157list_b @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_410_order__antisym__conv,axiom,
! [Y2: set_list_a,X2: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y2 @ X2 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_411_order__antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_412_linorder__le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_le_cases
thf(fact_413_ord__le__eq__subst,axiom,
! [A4: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_414_ord__le__eq__subst,axiom,
! [A4: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_415_ord__le__eq__subst,axiom,
! [A4: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_416_ord__le__eq__subst,axiom,
! [A4: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_417_ord__le__eq__subst,axiom,
! [A4: nat,B: nat,F: nat > set_b,C: set_b] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_418_ord__le__eq__subst,axiom,
! [A4: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_419_ord__le__eq__subst,axiom,
! [A4: set_a,B: set_a,F: set_a > set_b,C: set_b] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_420_ord__le__eq__subst,axiom,
! [A4: set_b,B: set_b,F: set_b > set_a,C: set_a] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_421_ord__le__eq__subst,axiom,
! [A4: set_b,B: set_b,F: set_b > set_b,C: set_b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_422_ord__le__eq__subst,axiom,
! [A4: set_list_b,B: set_list_b,F: set_list_b > nat,C: nat] :
( ( ord_le8932221534207217157list_b @ A4 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: set_list_b,Y3: set_list_b] :
( ( ord_le8932221534207217157list_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_423_ord__eq__le__subst,axiom,
! [A4: nat,F: nat > nat,B: nat,C: nat] :
( ( A4
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_424_ord__eq__le__subst,axiom,
! [A4: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( A4
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_425_ord__eq__le__subst,axiom,
! [A4: nat,F: set_b > nat,B: set_b,C: set_b] :
( ( A4
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_426_ord__eq__le__subst,axiom,
! [A4: set_a,F: nat > set_a,B: nat,C: nat] :
( ( A4
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_427_ord__eq__le__subst,axiom,
! [A4: set_b,F: nat > set_b,B: nat,C: nat] :
( ( A4
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_428_ord__eq__le__subst,axiom,
! [A4: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A4
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_429_ord__eq__le__subst,axiom,
! [A4: set_b,F: set_a > set_b,B: set_a,C: set_a] :
( ( A4
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_430_ord__eq__le__subst,axiom,
! [A4: set_a,F: set_b > set_a,B: set_b,C: set_b] :
( ( A4
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_431_ord__eq__le__subst,axiom,
! [A4: set_b,F: set_b > set_b,B: set_b,C: set_b] :
( ( A4
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_432_ord__eq__le__subst,axiom,
! [A4: nat,F: set_list_b > nat,B: set_list_b,C: set_list_b] :
( ( A4
= ( F @ B ) )
=> ( ( ord_le8932221534207217157list_b @ B @ C )
=> ( ! [X3: set_list_b,Y3: set_list_b] :
( ( ord_le8932221534207217157list_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_433_linorder__linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linorder_linear
thf(fact_434_order__eq__refl,axiom,
! [X2: set_a,Y2: set_a] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_435_order__eq__refl,axiom,
! [X2: set_b,Y2: set_b] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_b @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_436_order__eq__refl,axiom,
! [X2: set_list_b,Y2: set_list_b] :
( ( X2 = Y2 )
=> ( ord_le8932221534207217157list_b @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_437_order__eq__refl,axiom,
! [X2: set_list_a,Y2: set_list_a] :
( ( X2 = Y2 )
=> ( ord_le8861187494160871172list_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_438_order__eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_439_order__subst2,axiom,
! [A4: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_440_order__subst2,axiom,
! [A4: set_a,B: set_a,F: set_a > nat,C: nat] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_441_order__subst2,axiom,
! [A4: set_b,B: set_b,F: set_b > nat,C: nat] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_442_order__subst2,axiom,
! [A4: nat,B: nat,F: nat > set_a,C: set_a] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_443_order__subst2,axiom,
! [A4: nat,B: nat,F: nat > set_b,C: set_b] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_444_order__subst2,axiom,
! [A4: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_445_order__subst2,axiom,
! [A4: set_a,B: set_a,F: set_a > set_b,C: set_b] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_446_order__subst2,axiom,
! [A4: set_b,B: set_b,F: set_b > set_a,C: set_a] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_447_order__subst2,axiom,
! [A4: set_b,B: set_b,F: set_b > set_b,C: set_b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_448_order__subst2,axiom,
! [A4: set_list_b,B: set_list_b,F: set_list_b > nat,C: nat] :
( ( ord_le8932221534207217157list_b @ A4 @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: set_list_b,Y3: set_list_b] :
( ( ord_le8932221534207217157list_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A4 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_449_order__subst1,axiom,
! [A4: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_450_order__subst1,axiom,
! [A4: set_a,F: nat > set_a,B: nat,C: nat] :
( ( ord_less_eq_set_a @ A4 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_451_order__subst1,axiom,
! [A4: set_b,F: nat > set_b,B: nat,C: nat] :
( ( ord_less_eq_set_b @ A4 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_452_order__subst1,axiom,
! [A4: nat,F: set_a > nat,B: set_a,C: set_a] :
( ( ord_less_eq_nat @ A4 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_453_order__subst1,axiom,
! [A4: nat,F: set_b > nat,B: set_b,C: set_b] :
( ( ord_less_eq_nat @ A4 @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_454_order__subst1,axiom,
! [A4: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_455_order__subst1,axiom,
! [A4: set_a,F: set_b > set_a,B: set_b,C: set_b] :
( ( ord_less_eq_set_a @ A4 @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_456_order__subst1,axiom,
! [A4: set_b,F: set_a > set_b,B: set_a,C: set_a] :
( ( ord_less_eq_set_b @ A4 @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_457_order__subst1,axiom,
! [A4: set_b,F: set_b > set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A4 @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X3: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_458_order__subst1,axiom,
! [A4: set_list_b,F: nat > set_list_b,B: nat,C: nat] :
( ( ord_le8932221534207217157list_b @ A4 @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_le8932221534207217157list_b @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_le8932221534207217157list_b @ A4 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_459_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A6 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_460_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_b,Z2: set_b] : ( Y4 = Z2 ) )
= ( ^ [A6: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ A6 @ B6 )
& ( ord_less_eq_set_b @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_461_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_list_b,Z2: set_list_b] : ( Y4 = Z2 ) )
= ( ^ [A6: set_list_b,B6: set_list_b] :
( ( ord_le8932221534207217157list_b @ A6 @ B6 )
& ( ord_le8932221534207217157list_b @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_462_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_list_a,Z2: set_list_a] : ( Y4 = Z2 ) )
= ( ^ [A6: set_list_a,B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ A6 @ B6 )
& ( ord_le8861187494160871172list_a @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_463_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
& ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_464_Collect__mono__iff,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) )
= ( ! [X: set_b] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_465_Collect__mono__iff,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) )
= ( ! [X: set_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_466_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_467_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_468_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X: b] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_469_Collect__mono__iff,axiom,
! [P: list_b > $o,Q: list_b > $o] :
( ( ord_le8932221534207217157list_b @ ( collect_list_b @ P ) @ ( collect_list_b @ Q ) )
= ( ! [X: list_b] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_470_Collect__mono__iff,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) )
= ( ! [X: list_a] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_471_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A2: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A2 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_472_set__eq__subset,axiom,
( ( ^ [Y4: set_b,Z2: set_b] : ( Y4 = Z2 ) )
= ( ^ [A2: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ A2 @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_473_set__eq__subset,axiom,
( ( ^ [Y4: set_list_b,Z2: set_list_b] : ( Y4 = Z2 ) )
= ( ^ [A2: set_list_b,B4: set_list_b] :
( ( ord_le8932221534207217157list_b @ A2 @ B4 )
& ( ord_le8932221534207217157list_b @ B4 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_474_set__eq__subset,axiom,
( ( ^ [Y4: set_list_a,Z2: set_list_a] : ( Y4 = Z2 ) )
= ( ^ [A2: set_list_a,B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A2 ) ) ) ) ).
% set_eq_subset
thf(fact_475_antisym,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ord_less_eq_set_a @ B @ A4 )
=> ( A4 = B ) ) ) ).
% antisym
thf(fact_476_antisym,axiom,
! [A4: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( ord_less_eq_set_b @ B @ A4 )
=> ( A4 = B ) ) ) ).
% antisym
thf(fact_477_antisym,axiom,
! [A4: set_list_b,B: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ B )
=> ( ( ord_le8932221534207217157list_b @ B @ A4 )
=> ( A4 = B ) ) ) ).
% antisym
thf(fact_478_antisym,axiom,
! [A4: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ A4 )
=> ( A4 = B ) ) ) ).
% antisym
thf(fact_479_antisym,axiom,
! [A4: nat,B: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( ord_less_eq_nat @ B @ A4 )
=> ( A4 = B ) ) ) ).
% antisym
thf(fact_480_subset__trans,axiom,
! [A: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_481_subset__trans,axiom,
! [A: set_b,B3: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C2 )
=> ( ord_less_eq_set_b @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_482_subset__trans,axiom,
! [A: set_list_b,B3: set_list_b,C2: set_list_b] :
( ( ord_le8932221534207217157list_b @ A @ B3 )
=> ( ( ord_le8932221534207217157list_b @ B3 @ C2 )
=> ( ord_le8932221534207217157list_b @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_483_subset__trans,axiom,
! [A: set_list_a,B3: set_list_a,C2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B3 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ C2 )
=> ( ord_le8861187494160871172list_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_484_Collect__mono,axiom,
! [P: set_b > $o,Q: set_b > $o] :
( ! [X3: set_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3795704787696855135_set_b @ ( collect_set_b @ P ) @ ( collect_set_b @ Q ) ) ) ).
% Collect_mono
thf(fact_485_Collect__mono,axiom,
! [P: set_a > $o,Q: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q ) ) ) ).
% Collect_mono
thf(fact_486_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_487_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_488_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X3: b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_489_Collect__mono,axiom,
! [P: list_b > $o,Q: list_b > $o] :
( ! [X3: list_b] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8932221534207217157list_b @ ( collect_list_b @ P ) @ ( collect_list_b @ Q ) ) ) ).
% Collect_mono
thf(fact_490_Collect__mono,axiom,
! [P: list_a > $o,Q: list_a > $o] :
( ! [X3: list_a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8861187494160871172list_a @ ( collect_list_a @ P ) @ ( collect_list_a @ Q ) ) ) ).
% Collect_mono
thf(fact_491_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_492_subset__refl,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% subset_refl
thf(fact_493_subset__refl,axiom,
! [A: set_list_b] : ( ord_le8932221534207217157list_b @ A @ A ) ).
% subset_refl
thf(fact_494_subset__refl,axiom,
! [A: set_list_a] : ( ord_le8861187494160871172list_a @ A @ A ) ).
% subset_refl
thf(fact_495_subset__iff,axiom,
( ord_le942501763763511270_nat_b
= ( ^ [A2: set_nat_b,B4: set_nat_b] :
! [T2: nat > b] :
( ( member_nat_b @ T2 @ A2 )
=> ( member_nat_b @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_496_subset__iff,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A2: set_nat_a,B4: set_nat_a] :
! [T2: nat > a] :
( ( member_nat_a @ T2 @ A2 )
=> ( member_nat_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_497_subset__iff,axiom,
( ord_less_eq_set_a_b
= ( ^ [A2: set_a_b,B4: set_a_b] :
! [T2: a > b] :
( ( member_a_b @ T2 @ A2 )
=> ( member_a_b @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_498_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A2 )
=> ( member_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_499_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A2: set_b,B4: set_b] :
! [T2: b] :
( ( member_b @ T2 @ A2 )
=> ( member_b @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_500_subset__iff,axiom,
( ord_le8932221534207217157list_b
= ( ^ [A2: set_list_b,B4: set_list_b] :
! [T2: list_b] :
( ( member_list_b @ T2 @ A2 )
=> ( member_list_b @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_501_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A2: set_list_a,B4: set_list_a] :
! [T2: list_a] :
( ( member_list_a @ T2 @ A2 )
=> ( member_list_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_502_equalityD2,axiom,
! [A: set_a,B3: set_a] :
( ( A = B3 )
=> ( ord_less_eq_set_a @ B3 @ A ) ) ).
% equalityD2
thf(fact_503_equalityD2,axiom,
! [A: set_b,B3: set_b] :
( ( A = B3 )
=> ( ord_less_eq_set_b @ B3 @ A ) ) ).
% equalityD2
thf(fact_504_equalityD2,axiom,
! [A: set_list_b,B3: set_list_b] :
( ( A = B3 )
=> ( ord_le8932221534207217157list_b @ B3 @ A ) ) ).
% equalityD2
thf(fact_505_equalityD2,axiom,
! [A: set_list_a,B3: set_list_a] :
( ( A = B3 )
=> ( ord_le8861187494160871172list_a @ B3 @ A ) ) ).
% equalityD2
thf(fact_506_equalityD1,axiom,
! [A: set_a,B3: set_a] :
( ( A = B3 )
=> ( ord_less_eq_set_a @ A @ B3 ) ) ).
% equalityD1
thf(fact_507_equalityD1,axiom,
! [A: set_b,B3: set_b] :
( ( A = B3 )
=> ( ord_less_eq_set_b @ A @ B3 ) ) ).
% equalityD1
thf(fact_508_equalityD1,axiom,
! [A: set_list_b,B3: set_list_b] :
( ( A = B3 )
=> ( ord_le8932221534207217157list_b @ A @ B3 ) ) ).
% equalityD1
thf(fact_509_equalityD1,axiom,
! [A: set_list_a,B3: set_list_a] :
( ( A = B3 )
=> ( ord_le8861187494160871172list_a @ A @ B3 ) ) ).
% equalityD1
thf(fact_510_subset__eq,axiom,
( ord_le942501763763511270_nat_b
= ( ^ [A2: set_nat_b,B4: set_nat_b] :
! [X: nat > b] :
( ( member_nat_b @ X @ A2 )
=> ( member_nat_b @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_511_subset__eq,axiom,
( ord_le871467723717165285_nat_a
= ( ^ [A2: set_nat_a,B4: set_nat_a] :
! [X: nat > a] :
( ( member_nat_a @ X @ A2 )
=> ( member_nat_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_512_subset__eq,axiom,
( ord_less_eq_set_a_b
= ( ^ [A2: set_a_b,B4: set_a_b] :
! [X: a > b] :
( ( member_a_b @ X @ A2 )
=> ( member_a_b @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_513_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A2: set_a,B4: set_a] :
! [X: a] :
( ( member_a @ X @ A2 )
=> ( member_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_514_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A2: set_b,B4: set_b] :
! [X: b] :
( ( member_b @ X @ A2 )
=> ( member_b @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_515_subset__eq,axiom,
( ord_le8932221534207217157list_b
= ( ^ [A2: set_list_b,B4: set_list_b] :
! [X: list_b] :
( ( member_list_b @ X @ A2 )
=> ( member_list_b @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_516_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A2: set_list_a,B4: set_list_a] :
! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ( member_list_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_517_equalityE,axiom,
! [A: set_a,B3: set_a] :
( ( A = B3 )
=> ~ ( ( ord_less_eq_set_a @ A @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A ) ) ) ).
% equalityE
thf(fact_518_equalityE,axiom,
! [A: set_b,B3: set_b] :
( ( A = B3 )
=> ~ ( ( ord_less_eq_set_b @ A @ B3 )
=> ~ ( ord_less_eq_set_b @ B3 @ A ) ) ) ).
% equalityE
thf(fact_519_equalityE,axiom,
! [A: set_list_b,B3: set_list_b] :
( ( A = B3 )
=> ~ ( ( ord_le8932221534207217157list_b @ A @ B3 )
=> ~ ( ord_le8932221534207217157list_b @ B3 @ A ) ) ) ).
% equalityE
thf(fact_520_equalityE,axiom,
! [A: set_list_a,B3: set_list_a] :
( ( A = B3 )
=> ~ ( ( ord_le8861187494160871172list_a @ A @ B3 )
=> ~ ( ord_le8861187494160871172list_a @ B3 @ A ) ) ) ).
% equalityE
thf(fact_521_subsetD,axiom,
! [A: set_nat_b,B3: set_nat_b,C: nat > b] :
( ( ord_le942501763763511270_nat_b @ A @ B3 )
=> ( ( member_nat_b @ C @ A )
=> ( member_nat_b @ C @ B3 ) ) ) ).
% subsetD
thf(fact_522_subsetD,axiom,
! [A: set_nat_a,B3: set_nat_a,C: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B3 )
=> ( ( member_nat_a @ C @ A )
=> ( member_nat_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_523_subsetD,axiom,
! [A: set_a_b,B3: set_a_b,C: a > b] :
( ( ord_less_eq_set_a_b @ A @ B3 )
=> ( ( member_a_b @ C @ A )
=> ( member_a_b @ C @ B3 ) ) ) ).
% subsetD
thf(fact_524_subsetD,axiom,
! [A: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_525_subsetD,axiom,
! [A: set_b,B3: set_b,C: b] :
( ( ord_less_eq_set_b @ A @ B3 )
=> ( ( member_b @ C @ A )
=> ( member_b @ C @ B3 ) ) ) ).
% subsetD
thf(fact_526_subsetD,axiom,
! [A: set_list_b,B3: set_list_b,C: list_b] :
( ( ord_le8932221534207217157list_b @ A @ B3 )
=> ( ( member_list_b @ C @ A )
=> ( member_list_b @ C @ B3 ) ) ) ).
% subsetD
thf(fact_527_subsetD,axiom,
! [A: set_list_a,B3: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B3 )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_528_in__mono,axiom,
! [A: set_nat_b,B3: set_nat_b,X2: nat > b] :
( ( ord_le942501763763511270_nat_b @ A @ B3 )
=> ( ( member_nat_b @ X2 @ A )
=> ( member_nat_b @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_529_in__mono,axiom,
! [A: set_nat_a,B3: set_nat_a,X2: nat > a] :
( ( ord_le871467723717165285_nat_a @ A @ B3 )
=> ( ( member_nat_a @ X2 @ A )
=> ( member_nat_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_530_in__mono,axiom,
! [A: set_a_b,B3: set_a_b,X2: a > b] :
( ( ord_less_eq_set_a_b @ A @ B3 )
=> ( ( member_a_b @ X2 @ A )
=> ( member_a_b @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_531_in__mono,axiom,
! [A: set_a,B3: set_a,X2: a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_532_in__mono,axiom,
! [A: set_b,B3: set_b,X2: b] :
( ( ord_less_eq_set_b @ A @ B3 )
=> ( ( member_b @ X2 @ A )
=> ( member_b @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_533_in__mono,axiom,
! [A: set_list_b,B3: set_list_b,X2: list_b] :
( ( ord_le8932221534207217157list_b @ A @ B3 )
=> ( ( member_list_b @ X2 @ A )
=> ( member_list_b @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_534_in__mono,axiom,
! [A: set_list_a,B3: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B3 )
=> ( ( member_list_a @ X2 @ A )
=> ( member_list_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_535_dual__order_Otrans,axiom,
! [B: set_a,A4: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A4 )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_536_dual__order_Otrans,axiom,
! [B: set_b,A4: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A4 )
=> ( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_eq_set_b @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_537_dual__order_Otrans,axiom,
! [B: set_list_b,A4: set_list_b,C: set_list_b] :
( ( ord_le8932221534207217157list_b @ B @ A4 )
=> ( ( ord_le8932221534207217157list_b @ C @ B )
=> ( ord_le8932221534207217157list_b @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_538_dual__order_Otrans,axiom,
! [B: set_list_a,A4: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A4 )
=> ( ( ord_le8861187494160871172list_a @ C @ B )
=> ( ord_le8861187494160871172list_a @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_539_dual__order_Otrans,axiom,
! [B: nat,A4: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A4 )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A4 ) ) ) ).
% dual_order.trans
thf(fact_540_dual__order_Oantisym,axiom,
! [B: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B @ A4 )
=> ( ( ord_less_eq_set_a @ A4 @ B )
=> ( A4 = B ) ) ) ).
% dual_order.antisym
thf(fact_541_dual__order_Oantisym,axiom,
! [B: set_b,A4: set_b] :
( ( ord_less_eq_set_b @ B @ A4 )
=> ( ( ord_less_eq_set_b @ A4 @ B )
=> ( A4 = B ) ) ) ).
% dual_order.antisym
thf(fact_542_dual__order_Oantisym,axiom,
! [B: set_list_b,A4: set_list_b] :
( ( ord_le8932221534207217157list_b @ B @ A4 )
=> ( ( ord_le8932221534207217157list_b @ A4 @ B )
=> ( A4 = B ) ) ) ).
% dual_order.antisym
thf(fact_543_dual__order_Oantisym,axiom,
! [B: set_list_a,A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A4 )
=> ( ( ord_le8861187494160871172list_a @ A4 @ B )
=> ( A4 = B ) ) ) ).
% dual_order.antisym
thf(fact_544_dual__order_Oantisym,axiom,
! [B: nat,A4: nat] :
( ( ord_less_eq_nat @ B @ A4 )
=> ( ( ord_less_eq_nat @ A4 @ B )
=> ( A4 = B ) ) ) ).
% dual_order.antisym
thf(fact_545_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [A6: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ A6 )
& ( ord_less_eq_set_a @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_546_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_b,Z2: set_b] : ( Y4 = Z2 ) )
= ( ^ [A6: set_b,B6: set_b] :
( ( ord_less_eq_set_b @ B6 @ A6 )
& ( ord_less_eq_set_b @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_547_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_list_b,Z2: set_list_b] : ( Y4 = Z2 ) )
= ( ^ [A6: set_list_b,B6: set_list_b] :
( ( ord_le8932221534207217157list_b @ B6 @ A6 )
& ( ord_le8932221534207217157list_b @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_548_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_list_a,Z2: set_list_a] : ( Y4 = Z2 ) )
= ( ^ [A6: set_list_a,B6: set_list_a] :
( ( ord_le8861187494160871172list_a @ B6 @ A6 )
& ( ord_le8861187494160871172list_a @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_549_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ B6 @ A6 )
& ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_550_linorder__wlog,axiom,
! [P: nat > nat > $o,A4: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A4 @ B ) ) ) ).
% linorder_wlog
thf(fact_551_order__trans,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_552_order__trans,axiom,
! [X2: set_b,Y2: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ( ord_less_eq_set_b @ Y2 @ Z )
=> ( ord_less_eq_set_b @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_553_order__trans,axiom,
! [X2: set_list_b,Y2: set_list_b,Z: set_list_b] :
( ( ord_le8932221534207217157list_b @ X2 @ Y2 )
=> ( ( ord_le8932221534207217157list_b @ Y2 @ Z )
=> ( ord_le8932221534207217157list_b @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_554_order__trans,axiom,
! [X2: set_list_a,Y2: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ( ord_le8861187494160871172list_a @ Y2 @ Z )
=> ( ord_le8861187494160871172list_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_555_order__trans,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_556_order_Otrans,axiom,
! [A4: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A4 @ C ) ) ) ).
% order.trans
thf(fact_557_order_Otrans,axiom,
! [A4: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A4 @ C ) ) ) ).
% order.trans
thf(fact_558_order_Otrans,axiom,
! [A4: set_list_b,B: set_list_b,C: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ B )
=> ( ( ord_le8932221534207217157list_b @ B @ C )
=> ( ord_le8932221534207217157list_b @ A4 @ C ) ) ) ).
% order.trans
thf(fact_559_order_Otrans,axiom,
! [A4: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A4 @ C ) ) ) ).
% order.trans
thf(fact_560_order_Otrans,axiom,
! [A4: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% order.trans
thf(fact_561_order__antisym,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_562_order__antisym,axiom,
! [X2: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ( ord_less_eq_set_b @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_563_order__antisym,axiom,
! [X2: set_list_b,Y2: set_list_b] :
( ( ord_le8932221534207217157list_b @ X2 @ Y2 )
=> ( ( ord_le8932221534207217157list_b @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_564_order__antisym,axiom,
! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ( ord_le8861187494160871172list_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_565_order__antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_566_ord__le__eq__trans,axiom,
! [A4: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_567_ord__le__eq__trans,axiom,
! [A4: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_b @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_568_ord__le__eq__trans,axiom,
! [A4: set_list_b,B: set_list_b,C: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ B )
=> ( ( B = C )
=> ( ord_le8932221534207217157list_b @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_569_ord__le__eq__trans,axiom,
! [A4: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B )
=> ( ( B = C )
=> ( ord_le8861187494160871172list_a @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_570_ord__le__eq__trans,axiom,
! [A4: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_571_ord__eq__le__trans,axiom,
! [A4: set_a,B: set_a,C: set_a] :
( ( A4 = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_572_ord__eq__le__trans,axiom,
! [A4: set_b,B: set_b,C: set_b] :
( ( A4 = B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_573_ord__eq__le__trans,axiom,
! [A4: set_list_b,B: set_list_b,C: set_list_b] :
( ( A4 = B )
=> ( ( ord_le8932221534207217157list_b @ B @ C )
=> ( ord_le8932221534207217157list_b @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_574_ord__eq__le__trans,axiom,
! [A4: set_list_a,B: set_list_a,C: set_list_a] :
( ( A4 = B )
=> ( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_575_ord__eq__le__trans,axiom,
! [A4: nat,B: nat,C: nat] :
( ( A4 = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_576_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_a,Z2: set_a] : ( Y4 = Z2 ) )
= ( ^ [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_577_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_b,Z2: set_b] : ( Y4 = Z2 ) )
= ( ^ [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
& ( ord_less_eq_set_b @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_578_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_list_b,Z2: set_list_b] : ( Y4 = Z2 ) )
= ( ^ [X: set_list_b,Y: set_list_b] :
( ( ord_le8932221534207217157list_b @ X @ Y )
& ( ord_le8932221534207217157list_b @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_579_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: set_list_a,Z2: set_list_a] : ( Y4 = Z2 ) )
= ( ^ [X: set_list_a,Y: set_list_a] :
( ( ord_le8861187494160871172list_a @ X @ Y )
& ( ord_le8861187494160871172list_a @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_580_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_581_le__cases3,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_582_nle__le,axiom,
! [A4: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A4 @ B ) )
= ( ( ord_less_eq_nat @ B @ A4 )
& ( B != A4 ) ) ) ).
% nle_le
thf(fact_583_inf__sup__ord_I2_J,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_584_inf__sup__ord_I2_J,axiom,
! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_585_inf__sup__ord_I2_J,axiom,
! [X2: set_list_b,Y2: set_list_b] : ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_586_inf__sup__ord_I2_J,axiom,
! [X2: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_587_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_588_inf__sup__ord_I1_J,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_589_inf__sup__ord_I1_J,axiom,
! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_590_inf__sup__ord_I1_J,axiom,
! [X2: set_list_b,Y2: set_list_b] : ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_591_inf__sup__ord_I1_J,axiom,
! [X2: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_592_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_593_inf__le1,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_594_inf__le1,axiom,
! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_595_inf__le1,axiom,
! [X2: set_list_b,Y2: set_list_b] : ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_596_inf__le1,axiom,
! [X2: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_597_inf__le1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_598_inf__le2,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_599_inf__le2,axiom,
! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_600_inf__le2,axiom,
! [X2: set_list_b,Y2: set_list_b] : ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_601_inf__le2,axiom,
! [X2: set_list_a,Y2: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_602_inf__le2,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_603_le__infE,axiom,
! [X2: set_a,A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A4 @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X2 @ A4 )
=> ~ ( ord_less_eq_set_a @ X2 @ B ) ) ) ).
% le_infE
thf(fact_604_le__infE,axiom,
! [X2: set_b,A4: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X2 @ ( inf_inf_set_b @ A4 @ B ) )
=> ~ ( ( ord_less_eq_set_b @ X2 @ A4 )
=> ~ ( ord_less_eq_set_b @ X2 @ B ) ) ) ).
% le_infE
thf(fact_605_le__infE,axiom,
! [X2: set_list_b,A4: set_list_b,B: set_list_b] :
( ( ord_le8932221534207217157list_b @ X2 @ ( inf_inf_set_list_b @ A4 @ B ) )
=> ~ ( ( ord_le8932221534207217157list_b @ X2 @ A4 )
=> ~ ( ord_le8932221534207217157list_b @ X2 @ B ) ) ) ).
% le_infE
thf(fact_606_le__infE,axiom,
! [X2: set_list_a,A4: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ A4 @ B ) )
=> ~ ( ( ord_le8861187494160871172list_a @ X2 @ A4 )
=> ~ ( ord_le8861187494160871172list_a @ X2 @ B ) ) ) ).
% le_infE
thf(fact_607_le__infE,axiom,
! [X2: nat,A4: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A4 @ B ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A4 )
=> ~ ( ord_less_eq_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_608_le__infI,axiom,
! [X2: set_a,A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X2 @ A4 )
=> ( ( ord_less_eq_set_a @ X2 @ B )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A4 @ B ) ) ) ) ).
% le_infI
thf(fact_609_le__infI,axiom,
! [X2: set_b,A4: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X2 @ A4 )
=> ( ( ord_less_eq_set_b @ X2 @ B )
=> ( ord_less_eq_set_b @ X2 @ ( inf_inf_set_b @ A4 @ B ) ) ) ) ).
% le_infI
thf(fact_610_le__infI,axiom,
! [X2: set_list_b,A4: set_list_b,B: set_list_b] :
( ( ord_le8932221534207217157list_b @ X2 @ A4 )
=> ( ( ord_le8932221534207217157list_b @ X2 @ B )
=> ( ord_le8932221534207217157list_b @ X2 @ ( inf_inf_set_list_b @ A4 @ B ) ) ) ) ).
% le_infI
thf(fact_611_le__infI,axiom,
! [X2: set_list_a,A4: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ A4 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ B )
=> ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ A4 @ B ) ) ) ) ).
% le_infI
thf(fact_612_le__infI,axiom,
! [X2: nat,A4: nat,B: nat] :
( ( ord_less_eq_nat @ X2 @ A4 )
=> ( ( ord_less_eq_nat @ X2 @ B )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A4 @ B ) ) ) ) ).
% le_infI
thf(fact_613_inf__mono,axiom,
! [A4: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A4 @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_614_inf__mono,axiom,
! [A4: set_b,C: set_b,B: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A4 @ C )
=> ( ( ord_less_eq_set_b @ B @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A4 @ B ) @ ( inf_inf_set_b @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_615_inf__mono,axiom,
! [A4: set_list_b,C: set_list_b,B: set_list_b,D: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ C )
=> ( ( ord_le8932221534207217157list_b @ B @ D )
=> ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A4 @ B ) @ ( inf_inf_set_list_b @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_616_inf__mono,axiom,
! [A4: set_list_a,C: set_list_a,B: set_list_a,D: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ C )
=> ( ( ord_le8861187494160871172list_a @ B @ D )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A4 @ B ) @ ( inf_inf_set_list_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_617_inf__mono,axiom,
! [A4: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_618_le__infI1,axiom,
! [A4: set_a,X2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_619_le__infI1,axiom,
! [A4: set_b,X2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A4 @ X2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A4 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_620_le__infI1,axiom,
! [A4: set_list_b,X2: set_list_b,B: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ X2 )
=> ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A4 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_621_le__infI1,axiom,
! [A4: set_list_a,X2: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ X2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A4 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_622_le__infI1,axiom,
! [A4: nat,X2: nat,B: nat] :
( ( ord_less_eq_nat @ A4 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_623_le__infI2,axiom,
! [B: set_a,X2: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_624_le__infI2,axiom,
! [B: set_b,X2: set_b,A4: set_b] :
( ( ord_less_eq_set_b @ B @ X2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A4 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_625_le__infI2,axiom,
! [B: set_list_b,X2: set_list_b,A4: set_list_b] :
( ( ord_le8932221534207217157list_b @ B @ X2 )
=> ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A4 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_626_le__infI2,axiom,
! [B: set_list_a,X2: set_list_a,A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ X2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A4 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_627_le__infI2,axiom,
! [B: nat,X2: nat,A4: nat] :
( ( ord_less_eq_nat @ B @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_628_inf_OorderE,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( A4
= ( inf_inf_set_a @ A4 @ B ) ) ) ).
% inf.orderE
thf(fact_629_inf_OorderE,axiom,
! [A4: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( A4
= ( inf_inf_set_b @ A4 @ B ) ) ) ).
% inf.orderE
thf(fact_630_inf_OorderE,axiom,
! [A4: set_list_b,B: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ B )
=> ( A4
= ( inf_inf_set_list_b @ A4 @ B ) ) ) ).
% inf.orderE
thf(fact_631_inf_OorderE,axiom,
! [A4: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B )
=> ( A4
= ( inf_inf_set_list_a @ A4 @ B ) ) ) ).
% inf.orderE
thf(fact_632_inf_OorderE,axiom,
! [A4: nat,B: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( A4
= ( inf_inf_nat @ A4 @ B ) ) ) ).
% inf.orderE
thf(fact_633_inf_OorderI,axiom,
! [A4: set_a,B: set_a] :
( ( A4
= ( inf_inf_set_a @ A4 @ B ) )
=> ( ord_less_eq_set_a @ A4 @ B ) ) ).
% inf.orderI
thf(fact_634_inf_OorderI,axiom,
! [A4: set_b,B: set_b] :
( ( A4
= ( inf_inf_set_b @ A4 @ B ) )
=> ( ord_less_eq_set_b @ A4 @ B ) ) ).
% inf.orderI
thf(fact_635_inf_OorderI,axiom,
! [A4: set_list_b,B: set_list_b] :
( ( A4
= ( inf_inf_set_list_b @ A4 @ B ) )
=> ( ord_le8932221534207217157list_b @ A4 @ B ) ) ).
% inf.orderI
thf(fact_636_inf_OorderI,axiom,
! [A4: set_list_a,B: set_list_a] :
( ( A4
= ( inf_inf_set_list_a @ A4 @ B ) )
=> ( ord_le8861187494160871172list_a @ A4 @ B ) ) ).
% inf.orderI
thf(fact_637_inf_OorderI,axiom,
! [A4: nat,B: nat] :
( ( A4
= ( inf_inf_nat @ A4 @ B ) )
=> ( ord_less_eq_nat @ A4 @ B ) ) ).
% inf.orderI
thf(fact_638_inf__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y2: 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,Z3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ( ord_less_eq_set_a @ X3 @ Z3 )
=> ( ord_less_eq_set_a @ X3 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_639_inf__unique,axiom,
! [F: set_b > set_b > set_b,X2: set_b,Y2: set_b] :
( ! [X3: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_b,Y3: set_b,Z3: set_b] :
( ( ord_less_eq_set_b @ X3 @ Y3 )
=> ( ( ord_less_eq_set_b @ X3 @ Z3 )
=> ( ord_less_eq_set_b @ X3 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_b @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_640_inf__unique,axiom,
! [F: set_list_b > set_list_b > set_list_b,X2: set_list_b,Y2: set_list_b] :
( ! [X3: set_list_b,Y3: set_list_b] : ( ord_le8932221534207217157list_b @ ( F @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: set_list_b,Y3: set_list_b] : ( ord_le8932221534207217157list_b @ ( F @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: set_list_b,Y3: set_list_b,Z3: set_list_b] :
( ( ord_le8932221534207217157list_b @ X3 @ Y3 )
=> ( ( ord_le8932221534207217157list_b @ X3 @ Z3 )
=> ( ord_le8932221534207217157list_b @ X3 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_list_b @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_641_inf__unique,axiom,
! [F: set_list_a > set_list_a > set_list_a,X2: set_list_a,Y2: 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,Z3: set_list_a] :
( ( ord_le8861187494160871172list_a @ X3 @ Y3 )
=> ( ( ord_le8861187494160871172list_a @ X3 @ Z3 )
=> ( ord_le8861187494160871172list_a @ X3 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_list_a @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_642_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: 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,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ( ord_less_eq_nat @ X3 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_643_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_644_le__iff__inf,axiom,
( ord_less_eq_set_b
= ( ^ [X: set_b,Y: set_b] :
( ( inf_inf_set_b @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_645_le__iff__inf,axiom,
( ord_le8932221534207217157list_b
= ( ^ [X: set_list_b,Y: set_list_b] :
( ( inf_inf_set_list_b @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_646_le__iff__inf,axiom,
( ord_le8861187494160871172list_a
= ( ^ [X: set_list_a,Y: set_list_a] :
( ( inf_inf_set_list_a @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_647_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ Y )
= X ) ) ) ).
% le_iff_inf
thf(fact_648_inf_Oabsorb1,axiom,
! [A4: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( inf_inf_set_a @ A4 @ B )
= A4 ) ) ).
% inf.absorb1
thf(fact_649_inf_Oabsorb1,axiom,
! [A4: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( inf_inf_set_b @ A4 @ B )
= A4 ) ) ).
% inf.absorb1
thf(fact_650_inf_Oabsorb1,axiom,
! [A4: set_list_b,B: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ B )
=> ( ( inf_inf_set_list_b @ A4 @ B )
= A4 ) ) ).
% inf.absorb1
thf(fact_651_inf_Oabsorb1,axiom,
! [A4: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B )
=> ( ( inf_inf_set_list_a @ A4 @ B )
= A4 ) ) ).
% inf.absorb1
thf(fact_652_inf_Oabsorb1,axiom,
! [A4: nat,B: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( inf_inf_nat @ A4 @ B )
= A4 ) ) ).
% inf.absorb1
thf(fact_653_inf_Oabsorb2,axiom,
! [B: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B @ A4 )
=> ( ( inf_inf_set_a @ A4 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_654_inf_Oabsorb2,axiom,
! [B: set_b,A4: set_b] :
( ( ord_less_eq_set_b @ B @ A4 )
=> ( ( inf_inf_set_b @ A4 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_655_inf_Oabsorb2,axiom,
! [B: set_list_b,A4: set_list_b] :
( ( ord_le8932221534207217157list_b @ B @ A4 )
=> ( ( inf_inf_set_list_b @ A4 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_656_inf_Oabsorb2,axiom,
! [B: set_list_a,A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ A4 )
=> ( ( inf_inf_set_list_a @ A4 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_657_inf_Oabsorb2,axiom,
! [B: nat,A4: nat] :
( ( ord_less_eq_nat @ B @ A4 )
=> ( ( inf_inf_nat @ A4 @ B )
= B ) ) ).
% inf.absorb2
thf(fact_658_inf__absorb1,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( inf_inf_set_a @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_659_inf__absorb1,axiom,
! [X2: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ( inf_inf_set_b @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_660_inf__absorb1,axiom,
! [X2: set_list_b,Y2: set_list_b] :
( ( ord_le8932221534207217157list_b @ X2 @ Y2 )
=> ( ( inf_inf_set_list_b @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_661_inf__absorb1,axiom,
! [X2: set_list_a,Y2: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ( inf_inf_set_list_a @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_662_inf__absorb1,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_663_inf__absorb2,axiom,
! [Y2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( inf_inf_set_a @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_664_inf__absorb2,axiom,
! [Y2: set_b,X2: set_b] :
( ( ord_less_eq_set_b @ Y2 @ X2 )
=> ( ( inf_inf_set_b @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_665_inf__absorb2,axiom,
! [Y2: set_list_b,X2: set_list_b] :
( ( ord_le8932221534207217157list_b @ Y2 @ X2 )
=> ( ( inf_inf_set_list_b @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_666_inf__absorb2,axiom,
! [Y2: set_list_a,X2: set_list_a] :
( ( ord_le8861187494160871172list_a @ Y2 @ X2 )
=> ( ( inf_inf_set_list_a @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_667_inf__absorb2,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_668_inf_OboundedE,axiom,
! [A4: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A4 @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A4 @ B )
=> ~ ( ord_less_eq_set_a @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_669_inf_OboundedE,axiom,
! [A4: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A4 @ ( inf_inf_set_b @ B @ C ) )
=> ~ ( ( ord_less_eq_set_b @ A4 @ B )
=> ~ ( ord_less_eq_set_b @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_670_inf_OboundedE,axiom,
! [A4: set_list_b,B: set_list_b,C: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ ( inf_inf_set_list_b @ B @ C ) )
=> ~ ( ( ord_le8932221534207217157list_b @ A4 @ B )
=> ~ ( ord_le8932221534207217157list_b @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_671_inf_OboundedE,axiom,
! [A4: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ ( inf_inf_set_list_a @ B @ C ) )
=> ~ ( ( ord_le8861187494160871172list_a @ A4 @ B )
=> ~ ( ord_le8861187494160871172list_a @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_672_inf_OboundedE,axiom,
! [A4: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A4 @ B )
=> ~ ( ord_less_eq_nat @ A4 @ C ) ) ) ).
% inf.boundedE
thf(fact_673_inf_OboundedI,axiom,
! [A4: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A4 @ B )
=> ( ( ord_less_eq_set_a @ A4 @ C )
=> ( ord_less_eq_set_a @ A4 @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_674_inf_OboundedI,axiom,
! [A4: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A4 @ B )
=> ( ( ord_less_eq_set_b @ A4 @ C )
=> ( ord_less_eq_set_b @ A4 @ ( inf_inf_set_b @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_675_inf_OboundedI,axiom,
! [A4: set_list_b,B: set_list_b,C: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ B )
=> ( ( ord_le8932221534207217157list_b @ A4 @ C )
=> ( ord_le8932221534207217157list_b @ A4 @ ( inf_inf_set_list_b @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_676_inf_OboundedI,axiom,
! [A4: set_list_a,B: set_list_a,C: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ B )
=> ( ( ord_le8861187494160871172list_a @ A4 @ C )
=> ( ord_le8861187494160871172list_a @ A4 @ ( inf_inf_set_list_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_677_inf_OboundedI,axiom,
! [A4: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A4 @ B )
=> ( ( ord_less_eq_nat @ A4 @ C )
=> ( ord_less_eq_nat @ A4 @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_678_inf__greatest,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ X2 @ Z )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_679_inf__greatest,axiom,
! [X2: set_b,Y2: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ( ord_less_eq_set_b @ X2 @ Z )
=> ( ord_less_eq_set_b @ X2 @ ( inf_inf_set_b @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_680_inf__greatest,axiom,
! [X2: set_list_b,Y2: set_list_b,Z: set_list_b] :
( ( ord_le8932221534207217157list_b @ X2 @ Y2 )
=> ( ( ord_le8932221534207217157list_b @ X2 @ Z )
=> ( ord_le8932221534207217157list_b @ X2 @ ( inf_inf_set_list_b @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_681_inf__greatest,axiom,
! [X2: set_list_a,Y2: set_list_a,Z: set_list_a] :
( ( ord_le8861187494160871172list_a @ X2 @ Y2 )
=> ( ( ord_le8861187494160871172list_a @ X2 @ Z )
=> ( ord_le8861187494160871172list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_682_inf__greatest,axiom,
! [X2: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_683_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( A6
= ( inf_inf_set_a @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_684_inf_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
( A6
= ( inf_inf_set_b @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_685_inf_Oorder__iff,axiom,
( ord_le8932221534207217157list_b
= ( ^ [A6: set_list_b,B6: set_list_b] :
( A6
= ( inf_inf_set_list_b @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_686_inf_Oorder__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( A6
= ( inf_inf_set_list_a @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_687_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] :
( A6
= ( inf_inf_nat @ A6 @ B6 ) ) ) ) ).
% inf.order_iff
thf(fact_688_inf_Ocobounded1,axiom,
! [A4: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B ) @ A4 ) ).
% inf.cobounded1
thf(fact_689_inf_Ocobounded1,axiom,
! [A4: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A4 @ B ) @ A4 ) ).
% inf.cobounded1
thf(fact_690_inf_Ocobounded1,axiom,
! [A4: set_list_b,B: set_list_b] : ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A4 @ B ) @ A4 ) ).
% inf.cobounded1
thf(fact_691_inf_Ocobounded1,axiom,
! [A4: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A4 @ B ) @ A4 ) ).
% inf.cobounded1
thf(fact_692_inf_Ocobounded1,axiom,
! [A4: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B ) @ A4 ) ).
% inf.cobounded1
thf(fact_693_inf_Ocobounded2,axiom,
! [A4: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_694_inf_Ocobounded2,axiom,
! [A4: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A4 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_695_inf_Ocobounded2,axiom,
! [A4: set_list_b,B: set_list_b] : ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A4 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_696_inf_Ocobounded2,axiom,
! [A4: set_list_a,B: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A4 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_697_inf_Ocobounded2,axiom,
! [A4: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B ) @ B ) ).
% inf.cobounded2
thf(fact_698_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B6: set_a] :
( ( inf_inf_set_a @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_699_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B6: set_b] :
( ( inf_inf_set_b @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_700_inf_Oabsorb__iff1,axiom,
( ord_le8932221534207217157list_b
= ( ^ [A6: set_list_b,B6: set_list_b] :
( ( inf_inf_set_list_b @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_701_inf_Oabsorb__iff1,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A6: set_list_a,B6: set_list_a] :
( ( inf_inf_set_list_a @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_702_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A6: nat,B6: nat] :
( ( inf_inf_nat @ A6 @ B6 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_703_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B6: set_a,A6: set_a] :
( ( inf_inf_set_a @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_704_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [B6: set_b,A6: set_b] :
( ( inf_inf_set_b @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_705_inf_Oabsorb__iff2,axiom,
( ord_le8932221534207217157list_b
= ( ^ [B6: set_list_b,A6: set_list_b] :
( ( inf_inf_set_list_b @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_706_inf_Oabsorb__iff2,axiom,
( ord_le8861187494160871172list_a
= ( ^ [B6: set_list_a,A6: set_list_a] :
( ( inf_inf_set_list_a @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_707_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B6: nat,A6: nat] :
( ( inf_inf_nat @ A6 @ B6 )
= B6 ) ) ) ).
% inf.absorb_iff2
thf(fact_708_inf_OcoboundedI1,axiom,
! [A4: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A4 @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_709_inf_OcoboundedI1,axiom,
! [A4: set_b,C: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A4 @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A4 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_710_inf_OcoboundedI1,axiom,
! [A4: set_list_b,C: set_list_b,B: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ C )
=> ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A4 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_711_inf_OcoboundedI1,axiom,
! [A4: set_list_a,C: set_list_a,B: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ C )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A4 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_712_inf_OcoboundedI1,axiom,
! [A4: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A4 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_713_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A4: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A4 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_714_inf_OcoboundedI2,axiom,
! [B: set_b,C: set_b,A4: set_b] :
( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A4 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_715_inf_OcoboundedI2,axiom,
! [B: set_list_b,C: set_list_b,A4: set_list_b] :
( ( ord_le8932221534207217157list_b @ B @ C )
=> ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A4 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_716_inf_OcoboundedI2,axiom,
! [B: set_list_a,C: set_list_a,A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B @ C )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A4 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_717_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A4: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A4 @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_718_Collect__subset,axiom,
! [A: set_nat_b,P: ( nat > b ) > $o] :
( ord_le942501763763511270_nat_b
@ ( collect_nat_b
@ ^ [X: nat > b] :
( ( member_nat_b @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_719_Collect__subset,axiom,
! [A: set_nat_a,P: ( nat > a ) > $o] :
( ord_le871467723717165285_nat_a
@ ( collect_nat_a
@ ^ [X: nat > a] :
( ( member_nat_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_720_Collect__subset,axiom,
! [A: set_a_b,P: ( a > b ) > $o] :
( ord_less_eq_set_a_b
@ ( collect_a_b
@ ^ [X: a > b] :
( ( member_a_b @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_721_Collect__subset,axiom,
! [A: set_set_b,P: set_b > $o] :
( ord_le3795704787696855135_set_b
@ ( collect_set_b
@ ^ [X: set_b] :
( ( member_set_b @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_722_Collect__subset,axiom,
! [A: set_set_a,P: set_a > $o] :
( ord_le3724670747650509150_set_a
@ ( collect_set_a
@ ^ [X: set_a] :
( ( member_set_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_723_Collect__subset,axiom,
! [A: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_724_Collect__subset,axiom,
! [A: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X: a] :
( ( member_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_725_Collect__subset,axiom,
! [A: set_b,P: b > $o] :
( ord_less_eq_set_b
@ ( collect_b
@ ^ [X: b] :
( ( member_b @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_726_Collect__subset,axiom,
! [A: set_list_b,P: list_b > $o] :
( ord_le8932221534207217157list_b
@ ( collect_list_b
@ ^ [X: list_b] :
( ( member_list_b @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_727_Collect__subset,axiom,
! [A: set_list_a,P: list_a > $o] :
( ord_le8861187494160871172list_a
@ ( collect_list_a
@ ^ [X: list_a] :
( ( member_list_a @ X @ A )
& ( P @ X ) ) )
@ A ) ).
% Collect_subset
thf(fact_728_bot_Oextremum,axiom,
! [A4: set_set_list_a] : ( ord_le8877086941679407844list_a @ bot_bo3186585308812441520list_a @ A4 ) ).
% bot.extremum
thf(fact_729_bot_Oextremum,axiom,
! [A4: set_set_b] : ( ord_le3795704787696855135_set_b @ bot_bot_set_set_b @ A4 ) ).
% bot.extremum
thf(fact_730_bot_Oextremum,axiom,
! [A4: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A4 ) ).
% bot.extremum
thf(fact_731_bot_Oextremum,axiom,
! [A4: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A4 ) ).
% bot.extremum
thf(fact_732_bot_Oextremum,axiom,
! [A4: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A4 ) ).
% bot.extremum
thf(fact_733_bot_Oextremum,axiom,
! [A4: set_list_b] : ( ord_le8932221534207217157list_b @ bot_bot_set_list_b @ A4 ) ).
% bot.extremum
thf(fact_734_bot_Oextremum,axiom,
! [A4: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A4 ) ).
% bot.extremum
thf(fact_735_bot_Oextremum,axiom,
! [A4: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A4 ) ).
% bot.extremum
thf(fact_736_bot_Oextremum__unique,axiom,
! [A4: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A4 @ bot_bo3186585308812441520list_a )
= ( A4 = bot_bo3186585308812441520list_a ) ) ).
% bot.extremum_unique
thf(fact_737_bot_Oextremum__unique,axiom,
! [A4: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A4 @ bot_bot_set_set_b )
= ( A4 = bot_bot_set_set_b ) ) ).
% bot.extremum_unique
thf(fact_738_bot_Oextremum__unique,axiom,
! [A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ bot_bot_set_set_a )
= ( A4 = bot_bot_set_set_a ) ) ).
% bot.extremum_unique
thf(fact_739_bot_Oextremum__unique,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ bot_bot_set_a )
= ( A4 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_740_bot_Oextremum__unique,axiom,
! [A4: set_b] :
( ( ord_less_eq_set_b @ A4 @ bot_bot_set_b )
= ( A4 = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_741_bot_Oextremum__unique,axiom,
! [A4: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ bot_bot_set_list_b )
= ( A4 = bot_bot_set_list_b ) ) ).
% bot.extremum_unique
thf(fact_742_bot_Oextremum__unique,axiom,
! [A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ bot_bot_set_list_a )
= ( A4 = bot_bot_set_list_a ) ) ).
% bot.extremum_unique
thf(fact_743_bot_Oextremum__unique,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ bot_bot_nat )
= ( A4 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_744_bot_Oextremum__uniqueI,axiom,
! [A4: set_set_list_a] :
( ( ord_le8877086941679407844list_a @ A4 @ bot_bo3186585308812441520list_a )
=> ( A4 = bot_bo3186585308812441520list_a ) ) ).
% bot.extremum_uniqueI
thf(fact_745_bot_Oextremum__uniqueI,axiom,
! [A4: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A4 @ bot_bot_set_set_b )
=> ( A4 = bot_bot_set_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_746_bot_Oextremum__uniqueI,axiom,
! [A4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A4 @ bot_bot_set_set_a )
=> ( A4 = bot_bot_set_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_747_bot_Oextremum__uniqueI,axiom,
! [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ bot_bot_set_a )
=> ( A4 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_748_bot_Oextremum__uniqueI,axiom,
! [A4: set_b] :
( ( ord_less_eq_set_b @ A4 @ bot_bot_set_b )
=> ( A4 = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_749_bot_Oextremum__uniqueI,axiom,
! [A4: set_list_b] :
( ( ord_le8932221534207217157list_b @ A4 @ bot_bot_set_list_b )
=> ( A4 = bot_bot_set_list_b ) ) ).
% bot.extremum_uniqueI
thf(fact_750_bot_Oextremum__uniqueI,axiom,
! [A4: set_list_a] :
( ( ord_le8861187494160871172list_a @ A4 @ bot_bot_set_list_a )
=> ( A4 = bot_bot_set_list_a ) ) ).
% bot.extremum_uniqueI
thf(fact_751_bot_Oextremum__uniqueI,axiom,
! [A4: nat] :
( ( ord_less_eq_nat @ A4 @ bot_bot_nat )
=> ( A4 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_752_Int__mono,axiom,
! [A: set_a,C2: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_753_Int__mono,axiom,
! [A: set_b,C2: set_b,B3: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A @ C2 )
=> ( ( ord_less_eq_set_b @ B3 @ D2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B3 ) @ ( inf_inf_set_b @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_754_Int__mono,axiom,
! [A: set_list_b,C2: set_list_b,B3: set_list_b,D2: set_list_b] :
( ( ord_le8932221534207217157list_b @ A @ C2 )
=> ( ( ord_le8932221534207217157list_b @ B3 @ D2 )
=> ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A @ B3 ) @ ( inf_inf_set_list_b @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_755_Int__mono,axiom,
! [A: set_list_a,C2: set_list_a,B3: set_list_a,D2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ C2 )
=> ( ( ord_le8861187494160871172list_a @ B3 @ D2 )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ ( inf_inf_set_list_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_756_Int__lower1,axiom,
! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ A ) ).
% Int_lower1
thf(fact_757_Int__lower1,axiom,
! [A: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B3 ) @ A ) ).
% Int_lower1
thf(fact_758_Int__lower1,axiom,
! [A: set_list_b,B3: set_list_b] : ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A @ B3 ) @ A ) ).
% Int_lower1
thf(fact_759_Int__lower1,axiom,
! [A: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ A ) ).
% Int_lower1
thf(fact_760_Int__lower2,axiom,
! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_761_Int__lower2,axiom,
! [A: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_762_Int__lower2,axiom,
! [A: set_list_b,B3: set_list_b] : ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_763_Int__lower2,axiom,
! [A: set_list_a,B3: set_list_a] : ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_764_Int__absorb1,axiom,
! [B3: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( ( inf_inf_set_a @ A @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_765_Int__absorb1,axiom,
! [B3: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B3 @ A )
=> ( ( inf_inf_set_b @ A @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_766_Int__absorb1,axiom,
! [B3: set_list_b,A: set_list_b] :
( ( ord_le8932221534207217157list_b @ B3 @ A )
=> ( ( inf_inf_set_list_b @ A @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_767_Int__absorb1,axiom,
! [B3: set_list_a,A: set_list_a] :
( ( ord_le8861187494160871172list_a @ B3 @ A )
=> ( ( inf_inf_set_list_a @ A @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_768_Int__absorb2,axiom,
! [A: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ( inf_inf_set_a @ A @ B3 )
= A ) ) ).
% Int_absorb2
thf(fact_769_Int__absorb2,axiom,
! [A: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A @ B3 )
=> ( ( inf_inf_set_b @ A @ B3 )
= A ) ) ).
% Int_absorb2
thf(fact_770_Int__absorb2,axiom,
! [A: set_list_b,B3: set_list_b] :
( ( ord_le8932221534207217157list_b @ A @ B3 )
=> ( ( inf_inf_set_list_b @ A @ B3 )
= A ) ) ).
% Int_absorb2
thf(fact_771_Int__absorb2,axiom,
! [A: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ A @ B3 )
=> ( ( inf_inf_set_list_a @ A @ B3 )
= A ) ) ).
% Int_absorb2
thf(fact_772_Int__greatest,axiom,
! [C2: set_a,A: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_773_Int__greatest,axiom,
! [C2: set_b,A: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ C2 @ A )
=> ( ( ord_less_eq_set_b @ C2 @ B3 )
=> ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_774_Int__greatest,axiom,
! [C2: set_list_b,A: set_list_b,B3: set_list_b] :
( ( ord_le8932221534207217157list_b @ C2 @ A )
=> ( ( ord_le8932221534207217157list_b @ C2 @ B3 )
=> ( ord_le8932221534207217157list_b @ C2 @ ( inf_inf_set_list_b @ A @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_775_Int__greatest,axiom,
! [C2: set_list_a,A: set_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C2 @ A )
=> ( ( ord_le8861187494160871172list_a @ C2 @ B3 )
=> ( ord_le8861187494160871172list_a @ C2 @ ( inf_inf_set_list_a @ A @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_776_Int__Collect__mono,axiom,
! [A: set_nat_b,B3: set_nat_b,P: ( nat > b ) > $o,Q: ( nat > b ) > $o] :
( ( ord_le942501763763511270_nat_b @ A @ B3 )
=> ( ! [X3: nat > b] :
( ( member_nat_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le942501763763511270_nat_b @ ( inf_inf_set_nat_b @ A @ ( collect_nat_b @ P ) ) @ ( inf_inf_set_nat_b @ B3 @ ( collect_nat_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_777_Int__Collect__mono,axiom,
! [A: set_nat_a,B3: set_nat_a,P: ( nat > a ) > $o,Q: ( nat > a ) > $o] :
( ( ord_le871467723717165285_nat_a @ A @ B3 )
=> ( ! [X3: nat > a] :
( ( member_nat_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le871467723717165285_nat_a @ ( inf_inf_set_nat_a @ A @ ( collect_nat_a @ P ) ) @ ( inf_inf_set_nat_a @ B3 @ ( collect_nat_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_778_Int__Collect__mono,axiom,
! [A: set_a_b,B3: set_a_b,P: ( a > b ) > $o,Q: ( a > b ) > $o] :
( ( ord_less_eq_set_a_b @ A @ B3 )
=> ( ! [X3: a > b] :
( ( member_a_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a_b @ ( inf_inf_set_a_b @ A @ ( collect_a_b @ P ) ) @ ( inf_inf_set_a_b @ B3 @ ( collect_a_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_779_Int__Collect__mono,axiom,
! [A: set_set_b,B3: set_set_b,P: set_b > $o,Q: set_b > $o] :
( ( ord_le3795704787696855135_set_b @ A @ B3 )
=> ( ! [X3: set_b] :
( ( member_set_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3795704787696855135_set_b @ ( inf_inf_set_set_b @ A @ ( collect_set_b @ P ) ) @ ( inf_inf_set_set_b @ B3 @ ( collect_set_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_780_Int__Collect__mono,axiom,
! [A: set_set_a,B3: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A @ B3 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B3 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_781_Int__Collect__mono,axiom,
! [A: set_nat,B3: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B3 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_782_Int__Collect__mono,axiom,
! [A: set_a,B3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A @ B3 )
=> ( ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_783_Int__Collect__mono,axiom,
! [A: set_b,B3: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A @ B3 )
=> ( ! [X3: b] :
( ( member_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B3 @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_784_Int__Collect__mono,axiom,
! [A: set_list_b,B3: set_list_b,P: list_b > $o,Q: list_b > $o] :
( ( ord_le8932221534207217157list_b @ A @ B3 )
=> ( ! [X3: list_b] :
( ( member_list_b @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8932221534207217157list_b @ ( inf_inf_set_list_b @ A @ ( collect_list_b @ P ) ) @ ( inf_inf_set_list_b @ B3 @ ( collect_list_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_785_Int__Collect__mono,axiom,
! [A: set_list_a,B3: set_list_a,P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A @ B3 )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ A )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B3 @ ( collect_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_786_inj__on__subset,axiom,
! [F: a > b,A: set_a,B3: set_a] :
( ( inj_on_a_b @ F @ A )
=> ( ( ord_less_eq_set_a @ B3 @ A )
=> ( inj_on_a_b @ F @ B3 ) ) ) ).
% inj_on_subset
thf(fact_787_subset__inj__on,axiom,
! [F: a > b,B3: set_a,A: set_a] :
( ( inj_on_a_b @ F @ B3 )
=> ( ( ord_less_eq_set_a @ A @ B3 )
=> ( inj_on_a_b @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_788_inf__set__def,axiom,
( inf_inf_set_nat_b
= ( ^ [A2: set_nat_b,B4: set_nat_b] :
( collect_nat_b
@ ( inf_inf_nat_b_o
@ ^ [X: nat > b] : ( member_nat_b @ X @ A2 )
@ ^ [X: nat > b] : ( member_nat_b @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_789_inf__set__def,axiom,
( inf_inf_set_nat_a
= ( ^ [A2: set_nat_a,B4: set_nat_a] :
( collect_nat_a
@ ( inf_inf_nat_a_o
@ ^ [X: nat > a] : ( member_nat_a @ X @ A2 )
@ ^ [X: nat > a] : ( member_nat_a @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_790_inf__set__def,axiom,
( inf_inf_set_a_b
= ( ^ [A2: set_a_b,B4: set_a_b] :
( collect_a_b
@ ( inf_inf_a_b_o
@ ^ [X: a > b] : ( member_a_b @ X @ A2 )
@ ^ [X: a > b] : ( member_a_b @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_791_inf__set__def,axiom,
( inf_inf_set_set_b
= ( ^ [A2: set_set_b,B4: set_set_b] :
( collect_set_b
@ ( inf_inf_set_b_o
@ ^ [X: set_b] : ( member_set_b @ X @ A2 )
@ ^ [X: set_b] : ( member_set_b @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_792_inf__set__def,axiom,
( inf_inf_set_set_a
= ( ^ [A2: set_set_a,B4: set_set_a] :
( collect_set_a
@ ( inf_inf_set_a_o
@ ^ [X: set_a] : ( member_set_a @ X @ A2 )
@ ^ [X: set_a] : ( member_set_a @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_793_inf__set__def,axiom,
( inf_inf_set_nat
= ( ^ [A2: set_nat,B4: set_nat] :
( collect_nat
@ ( inf_inf_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A2 )
@ ^ [X: nat] : ( member_nat @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_794_inf__set__def,axiom,
( inf_inf_set_a
= ( ^ [A2: set_a,B4: set_a] :
( collect_a
@ ( inf_inf_a_o
@ ^ [X: a] : ( member_a @ X @ A2 )
@ ^ [X: a] : ( member_a @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_795_inf__set__def,axiom,
( inf_inf_set_b
= ( ^ [A2: set_b,B4: set_b] :
( collect_b
@ ( inf_inf_b_o
@ ^ [X: b] : ( member_b @ X @ A2 )
@ ^ [X: b] : ( member_b @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_796_inf__set__def,axiom,
( inf_inf_set_list_b
= ( ^ [A2: set_list_b,B4: set_list_b] :
( collect_list_b
@ ( inf_inf_list_b_o
@ ^ [X: list_b] : ( member_list_b @ X @ A2 )
@ ^ [X: list_b] : ( member_list_b @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_797_inf__set__def,axiom,
( inf_inf_set_list_a
= ( ^ [A2: set_list_a,B4: set_list_a] :
( collect_list_a
@ ( inf_inf_list_a_o
@ ^ [X: list_a] : ( member_list_a @ X @ A2 )
@ ^ [X: list_a] : ( member_list_a @ X @ B4 ) ) ) ) ) ).
% inf_set_def
thf(fact_798_ring__iso__memE_I2_J,axiom,
! [H: b > b,R2: partia1897943568983147621xt_b_d,S: partia1897943568983147621xt_b_d,X2: b,Y2: b] :
( ( member_b_b @ H @ ( ring_iso_b_d_b_d @ R2 @ S ) )
=> ( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( H @ ( mult_b_ring_ext_b_d @ R2 @ X2 @ Y2 ) )
= ( mult_b_ring_ext_b_d @ S @ ( H @ X2 ) @ ( H @ Y2 ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_799_ring__iso__memE_I2_J,axiom,
! [H: b > a,R2: partia1897943568983147621xt_b_d,S: partia8877618634411419171xt_a_c,X2: b,Y2: b] :
( ( member_b_a @ H @ ( ring_iso_b_d_a_c @ R2 @ S ) )
=> ( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( H @ ( mult_b_ring_ext_b_d @ R2 @ X2 @ Y2 ) )
= ( mult_a_ring_ext_a_c @ S @ ( H @ X2 ) @ ( H @ Y2 ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_800_ring__iso__memE_I2_J,axiom,
! [H: a > a,R2: partia8877618634411419171xt_a_c,S: partia8877618634411419171xt_a_c,X2: a,Y2: a] :
( ( member_a_a @ H @ ( ring_iso_a_c_a_c @ R2 @ S ) )
=> ( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( ( H @ ( mult_a_ring_ext_a_c @ R2 @ X2 @ Y2 ) )
= ( mult_a_ring_ext_a_c @ S @ ( H @ X2 ) @ ( H @ Y2 ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_801_ring__iso__memE_I2_J,axiom,
! [H: a > b,R2: partia8877618634411419171xt_a_c,S: partia1897943568983147621xt_b_d,X2: a,Y2: a] :
( ( member_a_b @ H @ ( ring_iso_a_c_b_d @ R2 @ S ) )
=> ( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( ( H @ ( mult_a_ring_ext_a_c @ R2 @ X2 @ Y2 ) )
= ( mult_b_ring_ext_b_d @ S @ ( H @ X2 ) @ ( H @ Y2 ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_802_ring__iso__memE_I2_J,axiom,
! [H: list_b > b,R2: partia4026993951477142903t_unit,S: partia1897943568983147621xt_b_d,X2: list_b,Y2: list_b] :
( ( member_list_b_b @ H @ ( ring_i3676047618198725658it_b_d @ R2 @ S ) )
=> ( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( ( H @ ( mult_l1800058939208302097t_unit @ R2 @ X2 @ Y2 ) )
= ( mult_b_ring_ext_b_d @ S @ ( H @ X2 ) @ ( H @ Y2 ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_803_ring__iso__memE_I2_J,axiom,
! [H: list_b > a,R2: partia4026993951477142903t_unit,S: partia8877618634411419171xt_a_c,X2: list_b,Y2: list_b] :
( ( member_list_b_a @ H @ ( ring_i6463503200171401690it_a_c @ R2 @ S ) )
=> ( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( ( H @ ( mult_l1800058939208302097t_unit @ R2 @ X2 @ Y2 ) )
= ( mult_a_ring_ext_a_c @ S @ ( H @ X2 ) @ ( H @ Y2 ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_804_ring__iso__memE_I2_J,axiom,
! [H: list_a > b,R2: partia2670972154091845814t_unit,S: partia1897943568983147621xt_b_d,X2: list_a,Y2: list_a] :
( ( member_list_a_b @ H @ ( ring_i4261380215208433627it_b_d @ R2 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R2 @ X2 @ Y2 ) )
= ( mult_b_ring_ext_b_d @ S @ ( H @ X2 ) @ ( H @ Y2 ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_805_ring__iso__memE_I2_J,axiom,
! [H: list_a > a,R2: partia2670972154091845814t_unit,S: partia8877618634411419171xt_a_c,X2: list_a,Y2: list_a] :
( ( member_list_a_a @ H @ ( ring_i7048835797181109659it_a_c @ R2 @ S ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( H @ ( mult_l7073676228092353617t_unit @ R2 @ X2 @ Y2 ) )
= ( mult_a_ring_ext_a_c @ S @ ( H @ X2 ) @ ( H @ Y2 ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_806_ring__iso__memE_I2_J,axiom,
! [H: b > list_b,R2: partia1897943568983147621xt_b_d,S: partia4026993951477142903t_unit,X2: b,Y2: b] :
( ( member_b_list_b @ H @ ( ring_i3204478355530656794t_unit @ R2 @ S ) )
=> ( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( H @ ( mult_b_ring_ext_b_d @ R2 @ X2 @ Y2 ) )
= ( mult_l1800058939208302097t_unit @ S @ ( H @ X2 ) @ ( H @ Y2 ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_807_ring__iso__memE_I2_J,axiom,
! [H: b > list_a,R2: partia1897943568983147621xt_b_d,S: partia2670972154091845814t_unit,X2: b,Y2: b] :
( ( member_b_list_a @ H @ ( ring_i6290234597520651355t_unit @ R2 @ S ) )
=> ( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( H @ ( mult_b_ring_ext_b_d @ R2 @ X2 @ Y2 ) )
= ( mult_l7073676228092353617t_unit @ S @ ( H @ X2 ) @ ( H @ Y2 ) ) ) ) ) ) ).
% ring_iso_memE(2)
thf(fact_808_inf__sup__aci_I4_J,axiom,
! [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y2 ) )
= ( inf_inf_set_a @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_809_inf__sup__aci_I4_J,axiom,
! [X2: set_b,Y2: set_b] :
( ( inf_inf_set_b @ X2 @ ( inf_inf_set_b @ X2 @ Y2 ) )
= ( inf_inf_set_b @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_810_inf__sup__aci_I4_J,axiom,
! [X2: set_list_b,Y2: set_list_b] :
( ( inf_inf_set_list_b @ X2 @ ( inf_inf_set_list_b @ X2 @ Y2 ) )
= ( inf_inf_set_list_b @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_811_inf__sup__aci_I4_J,axiom,
! [X2: set_list_a,Y2: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ X2 @ Y2 ) )
= ( inf_inf_set_list_a @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_812_inf__sup__aci_I3_J,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) )
= ( inf_inf_set_a @ Y2 @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_813_inf__sup__aci_I3_J,axiom,
! [X2: set_b,Y2: set_b,Z: set_b] :
( ( inf_inf_set_b @ X2 @ ( inf_inf_set_b @ Y2 @ Z ) )
= ( inf_inf_set_b @ Y2 @ ( inf_inf_set_b @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_814_inf__sup__aci_I3_J,axiom,
! [X2: set_list_b,Y2: set_list_b,Z: set_list_b] :
( ( inf_inf_set_list_b @ X2 @ ( inf_inf_set_list_b @ Y2 @ Z ) )
= ( inf_inf_set_list_b @ Y2 @ ( inf_inf_set_list_b @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_815_inf__sup__aci_I3_J,axiom,
! [X2: set_list_a,Y2: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z ) )
= ( inf_inf_set_list_a @ Y2 @ ( inf_inf_set_list_a @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_816_inf__sup__aci_I2_J,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_817_inf__sup__aci_I2_J,axiom,
! [X2: set_b,Y2: set_b,Z: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_b @ X2 @ ( inf_inf_set_b @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_818_inf__sup__aci_I2_J,axiom,
! [X2: set_list_b,Y2: set_list_b,Z: set_list_b] :
( ( inf_inf_set_list_b @ ( inf_inf_set_list_b @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_list_b @ X2 @ ( inf_inf_set_list_b @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_819_inf__sup__aci_I2_J,axiom,
! [X2: set_list_a,Y2: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_820_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y: set_a] : ( inf_inf_set_a @ Y @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_821_inf__sup__aci_I1_J,axiom,
( inf_inf_set_b
= ( ^ [X: set_b,Y: set_b] : ( inf_inf_set_b @ Y @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_822_inf__sup__aci_I1_J,axiom,
( inf_inf_set_list_b
= ( ^ [X: set_list_b,Y: set_list_b] : ( inf_inf_set_list_b @ Y @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_823_inf__sup__aci_I1_J,axiom,
( inf_inf_set_list_a
= ( ^ [X: set_list_a,Y: set_list_a] : ( inf_inf_set_list_a @ Y @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_824_inf_Oassoc,axiom,
! [A4: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A4 @ B ) @ C )
= ( inf_inf_set_a @ A4 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_825_inf_Oassoc,axiom,
! [A4: set_b,B: set_b,C: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ A4 @ B ) @ C )
= ( inf_inf_set_b @ A4 @ ( inf_inf_set_b @ B @ C ) ) ) ).
% inf.assoc
thf(fact_826_inf_Oassoc,axiom,
! [A4: set_list_b,B: set_list_b,C: set_list_b] :
( ( inf_inf_set_list_b @ ( inf_inf_set_list_b @ A4 @ B ) @ C )
= ( inf_inf_set_list_b @ A4 @ ( inf_inf_set_list_b @ B @ C ) ) ) ).
% inf.assoc
thf(fact_827_inf_Oassoc,axiom,
! [A4: set_list_a,B: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ A4 @ B ) @ C )
= ( inf_inf_set_list_a @ A4 @ ( inf_inf_set_list_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_828_inf__assoc,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_829_inf__assoc,axiom,
! [X2: set_b,Y2: set_b,Z: set_b] :
( ( inf_inf_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_b @ X2 @ ( inf_inf_set_b @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_830_inf__assoc,axiom,
! [X2: set_list_b,Y2: set_list_b,Z: set_list_b] :
( ( inf_inf_set_list_b @ ( inf_inf_set_list_b @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_list_b @ X2 @ ( inf_inf_set_list_b @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_831_inf__assoc,axiom,
! [X2: set_list_a,Y2: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ ( inf_inf_set_list_a @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_832_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B6: set_a] : ( inf_inf_set_a @ B6 @ A6 ) ) ) ).
% inf.commute
thf(fact_833_inf_Ocommute,axiom,
( inf_inf_set_b
= ( ^ [A6: set_b,B6: set_b] : ( inf_inf_set_b @ B6 @ A6 ) ) ) ).
% inf.commute
thf(fact_834_inf_Ocommute,axiom,
( inf_inf_set_list_b
= ( ^ [A6: set_list_b,B6: set_list_b] : ( inf_inf_set_list_b @ B6 @ A6 ) ) ) ).
% inf.commute
thf(fact_835_inf_Ocommute,axiom,
( inf_inf_set_list_a
= ( ^ [A6: set_list_a,B6: set_list_a] : ( inf_inf_set_list_a @ B6 @ A6 ) ) ) ).
% inf.commute
thf(fact_836_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X: set_a,Y: set_a] : ( inf_inf_set_a @ Y @ X ) ) ) ).
% inf_commute
thf(fact_837_inf__commute,axiom,
( inf_inf_set_b
= ( ^ [X: set_b,Y: set_b] : ( inf_inf_set_b @ Y @ X ) ) ) ).
% inf_commute
thf(fact_838_inf__commute,axiom,
( inf_inf_set_list_b
= ( ^ [X: set_list_b,Y: set_list_b] : ( inf_inf_set_list_b @ Y @ X ) ) ) ).
% inf_commute
thf(fact_839_inf__commute,axiom,
( inf_inf_set_list_a
= ( ^ [X: set_list_a,Y: set_list_a] : ( inf_inf_set_list_a @ Y @ X ) ) ) ).
% inf_commute
thf(fact_840_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_a,K: set_a,A4: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ K @ A4 ) )
=> ( ( inf_inf_set_a @ A @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A4 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_841_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_b,K: set_b,A4: set_b,B: set_b] :
( ( A
= ( inf_inf_set_b @ K @ A4 ) )
=> ( ( inf_inf_set_b @ A @ B )
= ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A4 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_842_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_list_b,K: set_list_b,A4: set_list_b,B: set_list_b] :
( ( A
= ( inf_inf_set_list_b @ K @ A4 ) )
=> ( ( inf_inf_set_list_b @ A @ B )
= ( inf_inf_set_list_b @ K @ ( inf_inf_set_list_b @ A4 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_843_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_list_a,K: set_list_a,A4: set_list_a,B: set_list_a] :
( ( A
= ( inf_inf_set_list_a @ K @ A4 ) )
=> ( ( inf_inf_set_list_a @ A @ B )
= ( inf_inf_set_list_a @ K @ ( inf_inf_set_list_a @ A4 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_844_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_a,K: set_a,B: set_a,A4: set_a] :
( ( B3
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A4 @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A4 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_845_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_b,K: set_b,B: set_b,A4: set_b] :
( ( B3
= ( inf_inf_set_b @ K @ B ) )
=> ( ( inf_inf_set_b @ A4 @ B3 )
= ( inf_inf_set_b @ K @ ( inf_inf_set_b @ A4 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_846_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_list_b,K: set_list_b,B: set_list_b,A4: set_list_b] :
( ( B3
= ( inf_inf_set_list_b @ K @ B ) )
=> ( ( inf_inf_set_list_b @ A4 @ B3 )
= ( inf_inf_set_list_b @ K @ ( inf_inf_set_list_b @ A4 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_847_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_list_a,K: set_list_a,B: set_list_a,A4: set_list_a] :
( ( B3
= ( inf_inf_set_list_a @ K @ B ) )
=> ( ( inf_inf_set_list_a @ A4 @ B3 )
= ( inf_inf_set_list_a @ K @ ( inf_inf_set_list_a @ A4 @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_848_inf_Oleft__commute,axiom,
! [B: set_a,A4: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A4 @ C ) )
= ( inf_inf_set_a @ A4 @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_849_inf_Oleft__commute,axiom,
! [B: set_b,A4: set_b,C: set_b] :
( ( inf_inf_set_b @ B @ ( inf_inf_set_b @ A4 @ C ) )
= ( inf_inf_set_b @ A4 @ ( inf_inf_set_b @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_850_inf_Oleft__commute,axiom,
! [B: set_list_b,A4: set_list_b,C: set_list_b] :
( ( inf_inf_set_list_b @ B @ ( inf_inf_set_list_b @ A4 @ C ) )
= ( inf_inf_set_list_b @ A4 @ ( inf_inf_set_list_b @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_851_inf_Oleft__commute,axiom,
! [B: set_list_a,A4: set_list_a,C: set_list_a] :
( ( inf_inf_set_list_a @ B @ ( inf_inf_set_list_a @ A4 @ C ) )
= ( inf_inf_set_list_a @ A4 @ ( inf_inf_set_list_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_852_inf__left__commute,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) )
= ( inf_inf_set_a @ Y2 @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_853_inf__left__commute,axiom,
! [X2: set_b,Y2: set_b,Z: set_b] :
( ( inf_inf_set_b @ X2 @ ( inf_inf_set_b @ Y2 @ Z ) )
= ( inf_inf_set_b @ Y2 @ ( inf_inf_set_b @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_854_inf__left__commute,axiom,
! [X2: set_list_b,Y2: set_list_b,Z: set_list_b] :
( ( inf_inf_set_list_b @ X2 @ ( inf_inf_set_list_b @ Y2 @ Z ) )
= ( inf_inf_set_list_b @ Y2 @ ( inf_inf_set_list_b @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_855_inf__left__commute,axiom,
! [X2: set_list_a,Y2: set_list_a,Z: set_list_a] :
( ( inf_inf_set_list_a @ X2 @ ( inf_inf_set_list_a @ Y2 @ Z ) )
= ( inf_inf_set_list_a @ Y2 @ ( inf_inf_set_list_a @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_856_ds_Oonepideal,axiom,
principalideal_b_d @ ( partia8782771468121683032xt_b_d @ s ) @ s ).
% ds.onepideal
thf(fact_857__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Acarrier_A_Ipoly__ring_AR_J_A_092_060Longrightarrow_062_Amap_A_Ithe__inv__into_A_Icarrier_AR_J_Ah_J_A_Imap_Ah_Ax_J_A_061_Ax_092_060close_062,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( map_b_a @ ( the_inv_into_a_b @ ( partia778085601923319190xt_a_c @ r ) @ h ) @ ( map_a_b @ h @ X2 ) )
= X2 ) ) ).
% \<open>\<And>x. x \<in> carrier (poly_ring R) \<Longrightarrow> map (the_inv_into (carrier R) h) (map h x) = x\<close>
thf(fact_858_dr_Omonoid__cancelI,axiom,
( ! [A3: a,B2: a,C3: a] :
( ( ( mult_a_ring_ext_a_c @ r @ C3 @ A3 )
= ( mult_a_ring_ext_a_c @ r @ C3 @ B2 ) )
=> ( ( member_a @ A3 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C3 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: a,B2: a,C3: a] :
( ( ( mult_a_ring_ext_a_c @ r @ A3 @ C3 )
= ( mult_a_ring_ext_a_c @ r @ B2 @ C3 ) )
=> ( ( member_a @ A3 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C3 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid5798828376123148986xt_a_c @ r ) ) ) ).
% dr.monoid_cancelI
thf(fact_859_dr_Oideal__is__subalgebra,axiom,
! [K2: set_a,I2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ideal_a_c @ I2 @ r )
=> ( embedd9027525575939734155ra_a_c @ K2 @ I2 @ r ) ) ) ).
% dr.ideal_is_subalgebra
thf(fact_860_domain_Oring__irreducibleE_I5_J,axiom,
! [R2: partia2956882679547061052t_unit,R3: list_list_a,A4: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R2 )
=> ( ( member_list_list_a @ R3 @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( ring_r360171070648044744t_unit @ R2 @ R3 )
=> ( ( member_list_list_a @ A4 @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( R3
= ( mult_l4853965630390486993t_unit @ R2 @ A4 @ B ) )
=> ( ( member_list_list_a @ A4 @ ( units_4903515905731149798t_unit @ R2 ) )
| ( member_list_list_a @ B @ ( units_4903515905731149798t_unit @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_861_domain_Oring__irreducibleE_I5_J,axiom,
! [R2: partia8877618634411419171xt_a_c,R3: a,A4: a,B: a] :
( ( domain_a_c @ R2 )
=> ( ( member_a @ R3 @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( ( ring_r999134135267193927le_a_c @ R2 @ R3 )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( ( R3
= ( mult_a_ring_ext_a_c @ R2 @ A4 @ B ) )
=> ( ( member_a @ A4 @ ( units_a_ring_ext_a_c @ R2 ) )
| ( member_a @ B @ ( units_a_ring_ext_a_c @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_862_domain_Oring__irreducibleE_I5_J,axiom,
! [R2: partia1897943568983147621xt_b_d,R3: b,A4: b,B: b] :
( ( domain_b_d @ R2 )
=> ( ( member_b @ R3 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( ring_r7435050590149293703le_b_d @ R2 @ R3 )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( R3
= ( mult_b_ring_ext_b_d @ R2 @ A4 @ B ) )
=> ( ( member_b @ A4 @ ( units_b_ring_ext_b_d @ R2 ) )
| ( member_b @ B @ ( units_b_ring_ext_b_d @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_863_domain_Oring__irreducibleE_I5_J,axiom,
! [R2: partia2670972154091845814t_unit,R3: list_a,A4: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( ring_r932985474545269838t_unit @ R2 @ R3 )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ R2 @ A4 @ B ) )
=> ( ( member_list_a @ A4 @ ( units_2932844235741507942t_unit @ R2 ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_864_domain_Oring__irreducibleE_I5_J,axiom,
! [R2: partia4026993951477142903t_unit,R3: list_b,A4: list_b,B: list_b] :
( ( domain3467766878553215752t_unit @ R2 )
=> ( ( member_list_b @ R3 @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( ( ring_r7070601269410051085t_unit @ R2 @ R3 )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( ( R3
= ( mult_l1800058939208302097t_unit @ R2 @ A4 @ B ) )
=> ( ( member_list_b @ A4 @ ( units_6882598983712232230t_unit @ R2 ) )
| ( member_list_b @ B @ ( units_6882598983712232230t_unit @ R2 ) ) ) ) ) ) ) ) ) ).
% domain.ring_irreducibleE(5)
thf(fact_865_domain_Oring__associated__iff,axiom,
! [R2: partia2956882679547061052t_unit,A4: list_list_a,B: list_list_a] :
( ( domain7810152921033798211t_unit @ R2 )
=> ( ( member_list_list_a @ A4 @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( member_list_list_a @ B @ ( partia2464479390973590831t_unit @ R2 ) )
=> ( ( associ5603075271488036121t_unit @ R2 @ A4 @ B )
= ( ? [X: list_list_a] :
( ( member_list_list_a @ X @ ( units_4903515905731149798t_unit @ R2 ) )
& ( A4
= ( mult_l4853965630390486993t_unit @ R2 @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_866_domain_Oring__associated__iff,axiom,
! [R2: partia4026993951477142903t_unit,A4: list_b,B: list_b] :
( ( domain3467766878553215752t_unit @ R2 )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ R2 ) )
=> ( ( associ3133968390036396889t_unit @ R2 @ A4 @ B )
= ( ? [X: list_b] :
( ( member_list_b @ X @ ( units_6882598983712232230t_unit @ R2 ) )
& ( A4
= ( mult_l1800058939208302097t_unit @ R2 @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_867_domain_Oring__associated__iff,axiom,
! [R2: partia2670972154091845814t_unit,A4: list_a,B: list_a] :
( ( domain6553523120543210313t_unit @ R2 )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ R2 ) )
=> ( ( associ8407585678920448409t_unit @ R2 @ A4 @ B )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ R2 ) )
& ( A4
= ( mult_l7073676228092353617t_unit @ R2 @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_868_domain_Oring__associated__iff,axiom,
! [R2: partia1897943568983147621xt_b_d,A4: b,B: b] :
( ( domain_b_d @ R2 )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ R2 ) )
=> ( ( associ9192668908051667533xt_b_d @ R2 @ A4 @ B )
= ( ? [X: b] :
( ( member_b @ X @ ( units_b_ring_ext_b_d @ R2 ) )
& ( A4
= ( mult_b_ring_ext_b_d @ R2 @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_869_domain_Oring__associated__iff,axiom,
! [R2: partia8877618634411419171xt_a_c,A4: a,B: a] :
( ( domain_a_c @ R2 )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ R2 ) )
=> ( ( associ5860276531582424204xt_a_c @ R2 @ A4 @ B )
= ( ? [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_c @ R2 ) )
& ( A4
= ( mult_a_ring_ext_a_c @ R2 @ X @ B ) ) ) ) ) ) ) ) ).
% domain.ring_associated_iff
thf(fact_870_dr_Ogenideal__ideal,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ideal_a_c @ ( genideal_a_c @ r @ S ) @ r ) ) ).
% dr.genideal_ideal
thf(fact_871_dr_OIdl__subset__ideal,axiom,
! [I2: set_a,H2: set_a] :
( ( ideal_a_c @ I2 @ r )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_c @ r @ H2 ) @ I2 )
= ( ord_less_eq_set_a @ H2 @ I2 ) ) ) ) ).
% dr.Idl_subset_ideal
thf(fact_872_h__img,axiom,
( ( image_a_b @ h @ ( partia778085601923319190xt_a_c @ r ) )
= ( partia8782771468121683032xt_b_d @ s ) ) ).
% h_img
thf(fact_873_h_Ozero__closed,axiom,
member_b @ ( h @ ( zero_a_c @ r ) ) @ ( partia8782771468121683032xt_b_d @ s ) ).
% h.zero_closed
thf(fact_874_ds_OUnits__assoc,axiom,
! [A4: b,B: b] :
( ( member_b @ A4 @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ B @ ( units_b_ring_ext_b_d @ s ) )
=> ( associ9192668908051667533xt_b_d @ s @ A4 @ B ) ) ) ).
% ds.Units_assoc
thf(fact_875_ds_Oassociated__sym,axiom,
! [A4: b,B: b] :
( ( associ9192668908051667533xt_b_d @ s @ A4 @ B )
=> ( associ9192668908051667533xt_b_d @ s @ B @ A4 ) ) ).
% ds.associated_sym
thf(fact_876_ds_OassociatedI2,axiom,
! [U: b,A4: b,B: b] :
( ( member_b @ U @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( A4
= ( mult_b_ring_ext_b_d @ s @ B @ U ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( associ9192668908051667533xt_b_d @ s @ A4 @ B ) ) ) ) ).
% ds.associatedI2
thf(fact_877_ds_OassociatedI2_H,axiom,
! [A4: b,B: b,U: b] :
( ( A4
= ( mult_b_ring_ext_b_d @ s @ B @ U ) )
=> ( ( member_b @ U @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( associ9192668908051667533xt_b_d @ s @ A4 @ B ) ) ) ) ).
% ds.associatedI2'
thf(fact_878_ds_Oring__associated__iff,axiom,
! [A4: b,B: b] :
( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( associ9192668908051667533xt_b_d @ s @ A4 @ B )
= ( ? [X: b] :
( ( member_b @ X @ ( units_b_ring_ext_b_d @ s ) )
& ( A4
= ( mult_b_ring_ext_b_d @ s @ X @ B ) ) ) ) ) ) ) ).
% ds.ring_associated_iff
thf(fact_879_ds_Oring__irreducibleE_I5_J,axiom,
! [R3: b,A4: b,B: b] :
( ( member_b @ R3 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_r7435050590149293703le_b_d @ s @ R3 )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( R3
= ( mult_b_ring_ext_b_d @ s @ A4 @ B ) )
=> ( ( member_b @ A4 @ ( units_b_ring_ext_b_d @ s ) )
| ( member_b @ B @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) ) ) ) ).
% ds.ring_irreducibleE(5)
thf(fact_880_ds_OUnits__cong,axiom,
! [A4: b,B: b] :
( ( member_b @ A4 @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( associ9192668908051667533xt_b_d @ s @ A4 @ B )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ B @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) ).
% ds.Units_cong
thf(fact_881_ds_Omult__cong__r,axiom,
! [B: b,B5: b,A4: b] :
( ( associ9192668908051667533xt_b_d @ s @ B @ B5 )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B5 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( associ9192668908051667533xt_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ A4 @ B ) @ ( mult_b_ring_ext_b_d @ s @ A4 @ B5 ) ) ) ) ) ) ).
% ds.mult_cong_r
thf(fact_882_ds_Omult__cong__l,axiom,
! [A4: b,A5: b,B: b] :
( ( associ9192668908051667533xt_b_d @ s @ A4 @ A5 )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ A5 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( associ9192668908051667533xt_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ A4 @ B ) @ ( mult_b_ring_ext_b_d @ s @ A5 @ B ) ) ) ) ) ) ).
% ds.mult_cong_l
thf(fact_883_ds_Ounit__factor,axiom,
! [A4: b,B: b] :
( ( member_b @ ( mult_b_ring_ext_b_d @ s @ A4 @ B ) @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ A4 @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) ).
% ds.unit_factor
thf(fact_884_ds_Oprod__unit__r,axiom,
! [A4: b,B: b] :
( ( member_b @ ( mult_b_ring_ext_b_d @ s @ A4 @ B ) @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ B @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ A4 @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) ) ).
% ds.prod_unit_r
thf(fact_885_ds_Oprod__unit__l,axiom,
! [A4: b,B: b] :
( ( member_b @ ( mult_b_ring_ext_b_d @ s @ A4 @ B ) @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A4 @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ B @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) ) ).
% ds.prod_unit_l
thf(fact_886_ds_Oring__irreducibleE_I4_J,axiom,
! [R3: b] :
( ( member_b @ R3 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_r7435050590149293703le_b_d @ s @ R3 )
=> ~ ( member_b @ R3 @ ( units_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.ring_irreducibleE(4)
thf(fact_887_ds_Oassociated__trans,axiom,
! [A4: b,B: b,C: b] :
( ( associ9192668908051667533xt_b_d @ s @ A4 @ B )
=> ( ( associ9192668908051667533xt_b_d @ s @ B @ C )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( associ9192668908051667533xt_b_d @ s @ A4 @ C ) ) ) ) ) ).
% ds.associated_trans
thf(fact_888_ds_Oassoc__subst,axiom,
! [A4: b,B: b,F: b > b] :
( ( associ9192668908051667533xt_b_d @ s @ A4 @ B )
=> ( ! [A3: b,B2: b] :
( ( ( member_b @ A3 @ ( partia8782771468121683032xt_b_d @ s ) )
& ( member_b @ B2 @ ( partia8782771468121683032xt_b_d @ s ) )
& ( associ9192668908051667533xt_b_d @ s @ A3 @ B2 ) )
=> ( ( member_b @ ( F @ A3 ) @ ( partia8782771468121683032xt_b_d @ s ) )
& ( member_b @ ( F @ B2 ) @ ( partia8782771468121683032xt_b_d @ s ) )
& ( associ9192668908051667533xt_b_d @ s @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( associ9192668908051667533xt_b_d @ s @ ( F @ A4 ) @ ( F @ B ) ) ) ) ) ) ).
% ds.assoc_subst
thf(fact_889_ds_OUnits__closed,axiom,
! [X2: b] :
( ( member_b @ X2 @ ( units_b_ring_ext_b_d @ s ) )
=> ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% ds.Units_closed
thf(fact_890_ds_Om__lcomm,axiom,
! [X2: b,Y2: b,Z: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ X2 @ ( mult_b_ring_ext_b_d @ s @ Y2 @ Z ) )
= ( mult_b_ring_ext_b_d @ s @ Y2 @ ( mult_b_ring_ext_b_d @ s @ X2 @ Z ) ) ) ) ) ) ).
% ds.m_lcomm
thf(fact_891_ds_Om__comm,axiom,
! [X2: b,Y2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ X2 @ Y2 )
= ( mult_b_ring_ext_b_d @ s @ Y2 @ X2 ) ) ) ) ).
% ds.m_comm
thf(fact_892_ds_Om__assoc,axiom,
! [X2: b,Y2: b,Z: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ ( mult_b_ring_ext_b_d @ s @ X2 @ Y2 ) @ Z )
= ( mult_b_ring_ext_b_d @ s @ X2 @ ( mult_b_ring_ext_b_d @ s @ Y2 @ Z ) ) ) ) ) ) ).
% ds.m_assoc
thf(fact_893_pdr_OUnits__assoc,axiom,
! [A4: list_a,B: list_a] :
( ( member_list_a @ A4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) ) ) ).
% pdr.Units_assoc
thf(fact_894_pdr_Oassociated__sym,axiom,
! [A4: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ A4 ) ) ).
% pdr.associated_sym
thf(fact_895_pdr_Odomain__axioms,axiom,
domain6553523120543210313t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.domain_axioms
thf(fact_896_pds_OUnits__assoc,axiom,
! [A4: list_b,B: list_b] :
( ( member_list_b @ A4 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) ) ) ).
% pds.Units_assoc
thf(fact_897_pds_Oassociated__sym,axiom,
! [A4: list_b,B: list_b] :
( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B )
=> ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ A4 ) ) ).
% pds.associated_sym
thf(fact_898_pds_Odomain__axioms,axiom,
domain3467766878553215752t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.domain_axioms
thf(fact_899_dr_Osubalgebra__inter,axiom,
! [K2: set_a,V: set_a,V2: set_a] :
( ( embedd9027525575939734155ra_a_c @ K2 @ V @ r )
=> ( ( embedd9027525575939734155ra_a_c @ K2 @ V2 @ r )
=> ( embedd9027525575939734155ra_a_c @ K2 @ ( inf_inf_set_a @ V @ V2 ) @ r ) ) ) ).
% dr.subalgebra_inter
thf(fact_900_pdr_Oring__associated__iff,axiom,
! [A4: list_a,B: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B )
= ( ? [X: list_a] :
( ( member_list_a @ X @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( A4
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X @ B ) ) ) ) ) ) ) ).
% pdr.ring_associated_iff
thf(fact_901_pdr_OassociatedI2_H,axiom,
! [A4: list_a,B: list_a,U: list_a] :
( ( A4
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ U ) )
=> ( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) ) ) ) ).
% pdr.associatedI2'
thf(fact_902_pdr_OassociatedI2,axiom,
! [U: list_a,A4: list_a,B: list_a] :
( ( member_list_a @ U @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( A4
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ U ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) ) ) ) ).
% pdr.associatedI2
thf(fact_903_pdr_Oring__irreducibleE_I5_J,axiom,
! [R3: list_a,A4: list_a,B: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ R3 )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) )
=> ( ( member_list_a @ A4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
| ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ) ) ).
% pdr.ring_irreducibleE(5)
thf(fact_904_pdr_OUnits__cong,axiom,
! [A4: list_a,B: list_a] :
( ( member_list_a @ A4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.Units_cong
thf(fact_905_pdr_Omult__cong__l,axiom,
! [A4: list_a,A5: list_a,B: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ A5 )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A5 @ B ) ) ) ) ) ) ).
% pdr.mult_cong_l
thf(fact_906_pdr_Omult__cong__r,axiom,
! [B: list_a,B5: list_a,A4: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ B5 )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B5 ) ) ) ) ) ) ).
% pdr.mult_cong_r
thf(fact_907_pdr_Oprod__unit__l,axiom,
! [A4: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.prod_unit_l
thf(fact_908_pdr_Oprod__unit__r,axiom,
! [A4: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ A4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.prod_unit_r
thf(fact_909_pdr_Ounit__factor,axiom,
! [A4: list_a,B: list_a] :
( ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ A4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.unit_factor
thf(fact_910_pdr_Oring__irreducibleE_I4_J,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ R3 )
=> ~ ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.ring_irreducibleE(4)
thf(fact_911_pdr_Oassoc__subst,axiom,
! [A4: list_a,B: list_a,F: list_a > list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B )
=> ( ! [A3: list_a,B2: list_a] :
( ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A3 @ B2 ) )
=> ( ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( member_list_a @ ( F @ B2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( F @ A4 ) @ ( F @ B ) ) ) ) ) ) ).
% pdr.assoc_subst
thf(fact_912_pdr_Oassociated__trans,axiom,
! [A4: list_a,B: list_a,C: list_a] :
( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ C )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ C ) ) ) ) ) ).
% pdr.associated_trans
thf(fact_913_pdr_OUnits__closed,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.Units_closed
thf(fact_914_pdr_Om__assoc,axiom,
! [X2: list_a,Y2: list_a,Z: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Y2 ) @ Z )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y2 @ Z ) ) ) ) ) ) ).
% pdr.m_assoc
thf(fact_915_pdr_Om__comm,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Y2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y2 @ X2 ) ) ) ) ).
% pdr.m_comm
thf(fact_916_pdr_Om__lcomm,axiom,
! [X2: list_a,Y2: list_a,Z: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y2 @ Z ) )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Z ) ) ) ) ) ) ).
% pdr.m_lcomm
thf(fact_917_pdr_Ocarrier__not__empty,axiom,
( ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= bot_bot_set_list_a ) ).
% pdr.carrier_not_empty
thf(fact_918_pds_OassociatedI2,axiom,
! [U: list_b,A4: list_b,B: list_b] :
( ( member_list_b @ U @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( A4
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ U ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) ) ) ) ).
% pds.associatedI2
thf(fact_919_pds_OassociatedI2_H,axiom,
! [A4: list_b,B: list_b,U: list_b] :
( ( A4
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ U ) )
=> ( ( member_list_b @ U @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) ) ) ) ).
% pds.associatedI2'
thf(fact_920_pds_Oring__associated__iff,axiom,
! [A4: list_b,B: list_b] :
( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B )
= ( ? [X: list_b] :
( ( member_list_b @ X @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( A4
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X @ B ) ) ) ) ) ) ) ).
% pds.ring_associated_iff
thf(fact_921_pds_Oring__irreducibleE_I5_J,axiom,
! [R3: list_b,A4: list_b,B: list_b] :
( ( member_list_b @ R3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ R3 )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( R3
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) )
=> ( ( member_list_b @ A4 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
| ( member_list_b @ B @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ) ) ).
% pds.ring_irreducibleE(5)
thf(fact_922_pds_OUnits__cong,axiom,
! [A4: list_b,B: list_b] :
( ( member_list_b @ A4 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ B @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.Units_cong
thf(fact_923_pds_Omult__cong__l,axiom,
! [A4: list_b,A5: list_b,B: list_b] :
( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ A5 )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A5 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A5 @ B ) ) ) ) ) ) ).
% pds.mult_cong_l
thf(fact_924_pds_Omult__cong__r,axiom,
! [B: list_b,B5: list_b,A4: list_b] :
( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ B5 )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B5 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B5 ) ) ) ) ) ) ).
% pds.mult_cong_r
thf(fact_925_pds_Oprod__unit__l,axiom,
! [A4: list_b,B: list_b] :
( ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A4 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ B @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.prod_unit_l
thf(fact_926_pds_Oprod__unit__r,axiom,
! [A4: list_b,B: list_b] :
( ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ A4 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.prod_unit_r
thf(fact_927_pds_Ounit__factor,axiom,
! [A4: list_b,B: list_b] :
( ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ A4 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.unit_factor
thf(fact_928_pds_Oring__irreducibleE_I4_J,axiom,
! [R3: list_b] :
( ( member_list_b @ R3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ R3 )
=> ~ ( member_list_b @ R3 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.ring_irreducibleE(4)
thf(fact_929_pds_Oassoc__subst,axiom,
! [A4: list_b,B: list_b,F: list_b > list_b] :
( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B )
=> ( ! [A3: list_b,B2: list_b] :
( ( ( member_list_b @ A3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( member_list_b @ B2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A3 @ B2 ) )
=> ( ( member_list_b @ ( F @ A3 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( member_list_b @ ( F @ B2 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( F @ A3 ) @ ( F @ B2 ) ) ) )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( F @ A4 ) @ ( F @ B ) ) ) ) ) ) ).
% pds.assoc_subst
thf(fact_930_pds_Oassociated__trans,axiom,
! [A4: list_b,B: list_b,C: list_b] :
( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B )
=> ( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ C )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ C ) ) ) ) ) ).
% pds.associated_trans
thf(fact_931_pds_OUnits__closed,axiom,
! [X2: list_b] :
( ( member_list_b @ X2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.Units_closed
thf(fact_932_pds_Om__assoc,axiom,
! [X2: list_b,Y2: list_b,Z: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Y2 ) @ Z )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y2 @ Z ) ) ) ) ) ) ).
% pds.m_assoc
thf(fact_933_pds_Om__comm,axiom,
! [X2: list_b,Y2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Y2 )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y2 @ X2 ) ) ) ) ).
% pds.m_comm
thf(fact_934_pds_Om__lcomm,axiom,
! [X2: list_b,Y2: list_b,Z: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y2 @ Z ) )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y2 @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Z ) ) ) ) ) ) ).
% pds.m_lcomm
thf(fact_935_pds_Ocarrier__not__empty,axiom,
( ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= bot_bot_set_list_b ) ).
% pds.carrier_not_empty
thf(fact_936_dr_Ointegral,axiom,
! [A4: a,B: a] :
( ( ( mult_a_ring_ext_a_c @ r @ A4 @ B )
= ( zero_a_c @ r ) )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( A4
= ( zero_a_c @ r ) )
| ( B
= ( zero_a_c @ r ) ) ) ) ) ) ).
% dr.integral
thf(fact_937_dr_Ointegral__iff,axiom,
! [A4: a,B: a] :
( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( mult_a_ring_ext_a_c @ r @ A4 @ B )
= ( zero_a_c @ r ) )
= ( ( A4
= ( zero_a_c @ r ) )
| ( B
= ( zero_a_c @ r ) ) ) ) ) ) ).
% dr.integral_iff
thf(fact_938_dr_Om__lcancel,axiom,
! [A4: a,B: a,C: a] :
( ( A4
!= ( zero_a_c @ r ) )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( mult_a_ring_ext_a_c @ r @ A4 @ B )
= ( mult_a_ring_ext_a_c @ r @ A4 @ C ) )
= ( B = C ) ) ) ) ) ) ).
% dr.m_lcancel
thf(fact_939_dr_Om__rcancel,axiom,
! [A4: a,B: a,C: a] :
( ( A4
!= ( zero_a_c @ r ) )
=> ( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ C @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( mult_a_ring_ext_a_c @ r @ B @ A4 )
= ( mult_a_ring_ext_a_c @ r @ C @ A4 ) )
= ( B = C ) ) ) ) ) ) ).
% dr.m_rcancel
thf(fact_940_image__eqI,axiom,
! [B: a > b,F: b > a > b,X2: b,A: set_b] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A )
=> ( member_a_b @ B @ ( image_b_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_941_image__eqI,axiom,
! [B: b,F: b > b,X2: b,A: set_b] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A )
=> ( member_b @ B @ ( image_b_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_942_image__eqI,axiom,
! [B: a,F: b > a,X2: b,A: set_b] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A )
=> ( member_a @ B @ ( image_b_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_943_image__eqI,axiom,
! [B: nat > b,F: a > nat > b,X2: a,A: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_nat_b @ B @ ( image_a_nat_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_944_image__eqI,axiom,
! [B: nat > a,F: a > nat > a,X2: a,A: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_nat_a @ B @ ( image_a_nat_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_945_image__eqI,axiom,
! [B: a > b,F: a > a > b,X2: a,A: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a_b @ B @ ( image_a_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_946_image__eqI,axiom,
! [B: b,F: a > b,X2: a,A: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_b @ B @ ( image_a_b @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_947_image__eqI,axiom,
! [B: a,F: a > a,X2: a,A: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A )
=> ( member_a @ B @ ( image_a_a @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_948_dr_Ogenideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_c @ r @ S ) ) ) ).
% dr.genideal_self
thf(fact_949_dr_Osubset__Idl__subset,axiom,
! [I2: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_c @ r @ H2 ) @ ( genideal_a_c @ r @ I2 ) ) ) ) ).
% dr.subset_Idl_subset
thf(fact_950_dr_Ocarrier__is__subalgebra,axiom,
! [K2: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( embedd9027525575939734155ra_a_c @ K2 @ ( partia778085601923319190xt_a_c @ r ) @ r ) ) ).
% dr.carrier_is_subalgebra
thf(fact_951_dr_Osubalgebra__in__carrier,axiom,
! [K2: set_a,V: set_a] :
( ( embedd9027525575939734155ra_a_c @ K2 @ V @ r )
=> ( ord_less_eq_set_a @ V @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% dr.subalgebra_in_carrier
thf(fact_952_dr_Oring__irreducibleE_I1_J,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_r999134135267193927le_a_c @ r @ R3 )
=> ( R3
!= ( zero_a_c @ r ) ) ) ) ).
% dr.ring_irreducibleE(1)
thf(fact_953_dr_Ogenideal__minimal,axiom,
! [I2: set_a,S: set_a] :
( ( ideal_a_c @ I2 @ r )
=> ( ( ord_less_eq_set_a @ S @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_c @ r @ S ) @ I2 ) ) ) ).
% dr.genideal_minimal
thf(fact_954_ds_OUnits__m__closed,axiom,
! [X2: b,Y2: b] :
( ( member_b @ X2 @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ Y2 @ ( units_b_ring_ext_b_d @ s ) )
=> ( member_b @ ( mult_b_ring_ext_b_d @ s @ X2 @ Y2 ) @ ( units_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.Units_m_closed
thf(fact_955_ds_OUnits__l__cancel,axiom,
! [X2: b,Y2: b,Z: b] :
( ( member_b @ X2 @ ( units_b_ring_ext_b_d @ s ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Z @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( mult_b_ring_ext_b_d @ s @ X2 @ Y2 )
= ( mult_b_ring_ext_b_d @ s @ X2 @ Z ) )
= ( Y2 = Z ) ) ) ) ) ).
% ds.Units_l_cancel
thf(fact_956_ds_Oassociated__refl,axiom,
! [A4: b] :
( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( associ9192668908051667533xt_b_d @ s @ A4 @ A4 ) ) ).
% ds.associated_refl
thf(fact_957_ds_Om__closed,axiom,
! [X2: b,Y2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ ( mult_b_ring_ext_b_d @ s @ X2 @ Y2 ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% ds.m_closed
thf(fact_958__092_060open_062_092_060And_062y_Ax_O_A_092_060lbrakk_062x_A_092_060in_062_Acarrier_A_Ipoly__ring_AR_J_059_Ay_A_092_060in_062_Acarrier_A_Ipoly__ring_AR_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Amap_Ah_A_Ix_A_092_060otimes_062_092_060_094bsub_062poly__ring_AR_092_060_094esub_062_Ay_J_A_061_Amap_Ah_Ax_A_092_060otimes_062_092_060_094bsub_062poly__ring_AS_092_060_094esub_062_Amap_Ah_Ay_092_060close_062,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( map_a_b @ h @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Y2 ) )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( map_a_b @ h @ X2 ) @ ( map_a_b @ h @ Y2 ) ) ) ) ) ).
% \<open>\<And>y x. \<lbrakk>x \<in> carrier (poly_ring R); y \<in> carrier (poly_ring R)\<rbrakk> \<Longrightarrow> map h (x \<otimes>\<^bsub>poly_ring R\<^esub> y) = map h x \<otimes>\<^bsub>poly_ring S\<^esub> map h y\<close>
thf(fact_959_t__1,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_b @ ( map_a_b @ h @ X2 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% t_1
thf(fact_960_pdr_OUnits__m__closed,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Y2 ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.Units_m_closed
thf(fact_961_pds_OUnits__m__closed,axiom,
! [X2: list_b,Y2: list_b] :
( ( member_list_b @ X2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Y2 ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.Units_m_closed
thf(fact_962_dr_Ozero__closed,axiom,
member_a @ ( zero_a_c @ r ) @ ( partia778085601923319190xt_a_c @ r ) ).
% dr.zero_closed
thf(fact_963__092_060open_062_092_060And_062x_O_Ax_A_092_060in_062_Acarrier_A_Ipoly__ring_AS_J_A_092_060Longrightarrow_062_Amap_A_Ithe__inv__into_A_Icarrier_AR_J_Ah_J_Ax_A_092_060in_062_Acarrier_A_Ipoly__ring_AR_J_092_060close_062,axiom,
! [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_a @ ( map_b_a @ ( the_inv_into_a_b @ ( partia778085601923319190xt_a_c @ r ) @ h ) @ X2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% \<open>\<And>x. x \<in> carrier (poly_ring S) \<Longrightarrow> map (the_inv_into (carrier R) h) x \<in> carrier (poly_ring R)\<close>
thf(fact_964_pdr_OUnits__l__cancel,axiom,
! [X2: list_a,Y2: list_a,Z: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Z @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Y2 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Z ) )
= ( Y2 = Z ) ) ) ) ) ).
% pdr.Units_l_cancel
thf(fact_965_pdr_Oassociated__refl,axiom,
! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ A4 ) ) ).
% pdr.associated_refl
thf(fact_966_pdr_Om__closed,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Y2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.m_closed
thf(fact_967_pds_OUnits__l__cancel,axiom,
! [X2: list_b,Y2: list_b,Z: list_b] :
( ( member_list_b @ X2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Z @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Y2 )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Z ) )
= ( Y2 = Z ) ) ) ) ) ).
% pds.Units_l_cancel
thf(fact_968_pds_Oassociated__refl,axiom,
! [A4: list_b] :
( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ A4 ) ) ).
% pds.associated_refl
thf(fact_969_pds_Om__closed,axiom,
! [X2: list_b,Y2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Y2 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.m_closed
thf(fact_970_dr_Ol__null,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ ( zero_a_c @ r ) @ X2 )
= ( zero_a_c @ r ) ) ) ).
% dr.l_null
thf(fact_971_dr_Or__null,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( mult_a_ring_ext_a_c @ r @ X2 @ ( zero_a_c @ r ) )
= ( zero_a_c @ r ) ) ) ).
% dr.r_null
thf(fact_972_h_Ohom__mult,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( h @ ( mult_a_ring_ext_a_c @ r @ X2 @ Y2 ) )
= ( mult_b_ring_ext_b_d @ s @ ( h @ X2 ) @ ( h @ Y2 ) ) ) ) ) ).
% h.hom_mult
thf(fact_973_pds_Oonepideal,axiom,
princi5701163198563039320t_unit @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.onepideal
thf(fact_974_pdr_Oonepideal,axiom,
princi8786919440553033881t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.onepideal
thf(fact_975_dr_Oring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_ring_prime_a_c @ r @ P2 )
=> ( P2
!= ( zero_a_c @ r ) ) ) ) ).
% dr.ring_primeE(1)
thf(fact_976_h_Oimg__is__subalgebra,axiom,
! [K2: set_a,V: set_a] :
( ( ord_less_eq_set_a @ K2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( embedd9027525575939734155ra_a_c @ K2 @ V @ r )
=> ( embedd6240069993967058123ra_b_d @ ( image_a_b @ h @ K2 ) @ ( image_a_b @ h @ V ) @ s ) ) ) ).
% h.img_is_subalgebra
thf(fact_977_pds_Omonoid__cancelI,axiom,
( ! [A3: list_b,B2: list_b,C3: list_b] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C3 @ A3 )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C3 @ B2 ) )
=> ( ( member_list_b @ A3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: list_b,B2: list_b,C3: list_b] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A3 @ C3 )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B2 @ C3 ) )
=> ( ( member_list_b @ A3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid8253019609946410375t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.monoid_cancelI
thf(fact_978_pdr_Omonoid__cancelI,axiom,
( ! [A3: list_a,B2: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C3 @ A3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C3 @ B2 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: list_a,B2: list_a,C3: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A3 @ C3 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B2 @ C3 ) )
=> ( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid4303264861975686087t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.monoid_cancelI
thf(fact_979__092_060open_062map_Ah_A_092_060in_062_Acarrier_A_Ipoly__ring_AR_J_A_092_060rightarrow_062_Acarrier_A_Ipoly__ring_AS_J_092_060close_062,axiom,
( member_list_a_list_b @ ( map_a_b @ h )
@ ( pi_list_a_list_b @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
@ ^ [Uu: list_a] : ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% \<open>map h \<in> carrier (poly_ring R) \<rightarrow> carrier (poly_ring S)\<close>
thf(fact_980_ds_Ointegral,axiom,
! [A4: b,B: b] :
( ( ( mult_b_ring_ext_b_d @ s @ A4 @ B )
= ( zero_b_d @ s ) )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( A4
= ( zero_b_d @ s ) )
| ( B
= ( zero_b_d @ s ) ) ) ) ) ) ).
% ds.integral
thf(fact_981_ds_Ointegral__iff,axiom,
! [A4: b,B: b] :
( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( mult_b_ring_ext_b_d @ s @ A4 @ B )
= ( zero_b_d @ s ) )
= ( ( A4
= ( zero_b_d @ s ) )
| ( B
= ( zero_b_d @ s ) ) ) ) ) ) ).
% ds.integral_iff
thf(fact_982_ds_Om__lcancel,axiom,
! [A4: b,B: b,C: b] :
( ( A4
!= ( zero_b_d @ s ) )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( mult_b_ring_ext_b_d @ s @ A4 @ B )
= ( mult_b_ring_ext_b_d @ s @ A4 @ C ) )
= ( B = C ) ) ) ) ) ) ).
% ds.m_lcancel
thf(fact_983_ds_Om__rcancel,axiom,
! [A4: b,B: b,C: b] :
( ( A4
!= ( zero_b_d @ s ) )
=> ( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( mult_b_ring_ext_b_d @ s @ B @ A4 )
= ( mult_b_ring_ext_b_d @ s @ C @ A4 ) )
= ( B = C ) ) ) ) ) ) ).
% ds.m_rcancel
thf(fact_984_ds_Ocarrier__is__subalgebra,axiom,
! [K2: set_b] :
( ( ord_less_eq_set_b @ K2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( embedd6240069993967058123ra_b_d @ K2 @ ( partia8782771468121683032xt_b_d @ s ) @ s ) ) ).
% ds.carrier_is_subalgebra
thf(fact_985_ds_Osubalgebra__in__carrier,axiom,
! [K2: set_b,V: set_b] :
( ( embedd6240069993967058123ra_b_d @ K2 @ V @ s )
=> ( ord_less_eq_set_b @ V @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% ds.subalgebra_in_carrier
thf(fact_986_ds_Oring__irreducibleE_I1_J,axiom,
! [R3: b] :
( ( member_b @ R3 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_r7435050590149293703le_b_d @ s @ R3 )
=> ( R3
!= ( zero_b_d @ s ) ) ) ) ).
% ds.ring_irreducibleE(1)
thf(fact_987_ds_Osubalgebra__inter,axiom,
! [K2: set_b,V: set_b,V2: set_b] :
( ( embedd6240069993967058123ra_b_d @ K2 @ V @ s )
=> ( ( embedd6240069993967058123ra_b_d @ K2 @ V2 @ s )
=> ( embedd6240069993967058123ra_b_d @ K2 @ ( inf_inf_set_b @ V @ V2 ) @ s ) ) ) ).
% ds.subalgebra_inter
thf(fact_988_ds_Oideal__is__subalgebra,axiom,
! [K2: set_b,I2: set_b] :
( ( ord_less_eq_set_b @ K2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ideal_b_d @ I2 @ s )
=> ( embedd6240069993967058123ra_b_d @ K2 @ I2 @ s ) ) ) ).
% ds.ideal_is_subalgebra
thf(fact_989_pdr_Om__rcancel,axiom,
! [A4: list_a,B: list_a,C: list_a] :
( ( A4
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B @ A4 )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ C @ A4 ) )
= ( B = C ) ) ) ) ) ) ).
% pdr.m_rcancel
thf(fact_990_pdr_Om__lcancel,axiom,
! [A4: list_a,B: list_a,C: list_a] :
( ( A4
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B )
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ C ) )
= ( B = C ) ) ) ) ) ) ).
% pdr.m_lcancel
thf(fact_991_pdr_Ointegral__iff,axiom,
! [A4: list_a,B: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( ( A4
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
| ( B
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.integral_iff
thf(fact_992_pdr_Ointegral,axiom,
! [A4: list_a,B: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( A4
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
| ( B
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% pdr.integral
thf(fact_993_pds_Om__rcancel,axiom,
! [A4: list_b,B: list_b,C: list_b] :
( ( A4
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B @ A4 )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ C @ A4 ) )
= ( B = C ) ) ) ) ) ) ).
% pds.m_rcancel
thf(fact_994_pds_Om__lcancel,axiom,
! [A4: list_b,B: list_b,C: list_b] :
( ( A4
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ C @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B )
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ C ) )
= ( B = C ) ) ) ) ) ) ).
% pds.m_lcancel
thf(fact_995_pds_Ointegral__iff,axiom,
! [A4: list_b,B: list_b] :
( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( ( A4
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
| ( B
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.integral_iff
thf(fact_996_pds_Ointegral,axiom,
! [A4: list_b,B: list_b] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( A4
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
| ( B
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% pds.integral
thf(fact_997_pdr_Oring__irreducibleE_I1_J,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ R3 )
=> ( R3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.ring_irreducibleE(1)
thf(fact_998_pds_Oring__irreducibleE_I1_J,axiom,
! [R3: list_b] :
( ( member_list_b @ R3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ R3 )
=> ( R3
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.ring_irreducibleE(1)
thf(fact_999_h__non__zero__iff,axiom,
! [X2: a] :
( ( X2
!= ( zero_a_c @ r ) )
=> ( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( h @ X2 )
!= ( zero_b_d @ s ) ) ) ) ).
% h_non_zero_iff
thf(fact_1000_ds_Ozero__closed,axiom,
member_b @ ( zero_b_d @ s ) @ ( partia8782771468121683032xt_b_d @ s ) ).
% ds.zero_closed
thf(fact_1001_ds_Ol__null,axiom,
! [X2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ ( zero_b_d @ s ) @ X2 )
= ( zero_b_d @ s ) ) ) ).
% ds.l_null
thf(fact_1002_ds_Or__null,axiom,
! [X2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( mult_b_ring_ext_b_d @ s @ X2 @ ( zero_b_d @ s ) )
= ( zero_b_d @ s ) ) ) ).
% ds.r_null
thf(fact_1003_pdr_Ozero__closed,axiom,
member_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% pdr.zero_closed
thf(fact_1004_pds_Ozero__closed,axiom,
member_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% pds.zero_closed
thf(fact_1005_pdr_Or__null,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.r_null
thf(fact_1006_pdr_Ol__null,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ X2 )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.l_null
thf(fact_1007_pds_Or__null,axiom,
! [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.r_null
thf(fact_1008_pds_Ol__null,axiom,
! [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ X2 )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.l_null
thf(fact_1009_h_Ohom__zero,axiom,
( ( h @ ( zero_a_c @ r ) )
= ( zero_b_d @ s ) ) ).
% h.hom_zero
thf(fact_1010_ds_Oring__primeE_I1_J,axiom,
! [P2: b] :
( ( member_b @ P2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_ring_prime_b_d @ s @ P2 )
=> ( P2
!= ( zero_b_d @ s ) ) ) ) ).
% ds.ring_primeE(1)
thf(fact_1011_dr_Oring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_c @ r ) )
=> ( ( prime_a_ring_ext_a_c @ r @ P2 )
=> ( ring_ring_prime_a_c @ r @ P2 ) ) ) ).
% dr.ring_primeI
thf(fact_1012_dr_Oring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ring_ring_prime_a_c @ r @ P2 )
=> ( prime_a_ring_ext_a_c @ r @ P2 ) ) ) ).
% dr.ring_primeE(3)
thf(fact_1013_ds_Oassociated__polynomials__imp__same__is__root,axiom,
! [P2: list_b,Q2: list_b,X2: b] :
( ( member_list_b @ P2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Q2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 @ Q2 )
=> ( ( polyno1345617632095147429ot_b_d @ s @ P2 @ X2 )
= ( polyno1345617632095147429ot_b_d @ s @ Q2 @ X2 ) ) ) ) ) ).
% ds.associated_polynomials_imp_same_is_root
thf(fact_1014_dr_Oassociated__polynomials__imp__same__is__root,axiom,
! [P2: list_a,Q2: list_a,X2: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 @ Q2 )
=> ( ( polyno4133073214067823461ot_a_c @ r @ P2 @ X2 )
= ( polyno4133073214067823461ot_a_c @ r @ Q2 @ X2 ) ) ) ) ) ).
% dr.associated_polynomials_imp_same_is_root
thf(fact_1015_dr_Ozero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_c @ r @ ( zero_a_c @ r ) ).
% dr.zero_is_prime(1)
thf(fact_1016_pdr_Oring__primeE_I1_J,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 )
=> ( P2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.ring_primeE(1)
thf(fact_1017_pds_Oring__primeE_I1_J,axiom,
! [P2: list_b] :
( ( member_list_b @ P2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 )
=> ( P2
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.ring_primeE(1)
thf(fact_1018_dr_Ois__root__poly__mult__imp__is__root,axiom,
! [P2: list_a,Q2: list_a,X2: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( polyno4133073214067823461ot_a_c @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 @ Q2 ) @ X2 )
=> ( ( polyno4133073214067823461ot_a_c @ r @ P2 @ X2 )
| ( polyno4133073214067823461ot_a_c @ r @ Q2 @ X2 ) ) ) ) ) ).
% dr.is_root_poly_mult_imp_is_root
thf(fact_1019_ds_Ois__root__poly__mult__imp__is__root,axiom,
! [P2: list_b,Q2: list_b,X2: b] :
( ( member_list_b @ P2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Q2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( polyno1345617632095147429ot_b_d @ s @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 @ Q2 ) @ X2 )
=> ( ( polyno1345617632095147429ot_b_d @ s @ P2 @ X2 )
| ( polyno1345617632095147429ot_b_d @ s @ Q2 @ X2 ) ) ) ) ) ).
% ds.is_root_poly_mult_imp_is_root
thf(fact_1020_pds_Oring__primeI,axiom,
! [P2: list_b] :
( ( P2
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( prime_5961678782586786214t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 )
=> ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 ) ) ) ).
% pds.ring_primeI
thf(fact_1021_pds_Ozero__is__prime_I1_J,axiom,
prime_5961678782586786214t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% pds.zero_is_prime(1)
thf(fact_1022_pdr_Oring__primeI,axiom,
! [P2: list_a] :
( ( P2
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( prime_2011924034616061926t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 ) ) ) ).
% pdr.ring_primeI
thf(fact_1023_pdr_Ozero__is__prime_I1_J,axiom,
prime_2011924034616061926t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% pdr.zero_is_prime(1)
thf(fact_1024_pds_Oring__primeE_I3_J,axiom,
! [P2: list_b] :
( ( member_list_b @ P2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 )
=> ( prime_5961678782586786214t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 ) ) ) ).
% pds.ring_primeE(3)
thf(fact_1025_ds_Ozero__is__prime_I1_J,axiom,
prime_b_ring_ext_b_d @ s @ ( zero_b_d @ s ) ).
% ds.zero_is_prime(1)
thf(fact_1026_ds_Oring__primeE_I3_J,axiom,
! [P2: b] :
( ( member_b @ P2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ring_ring_prime_b_d @ s @ P2 )
=> ( prime_b_ring_ext_b_d @ s @ P2 ) ) ) ).
% ds.ring_primeE(3)
thf(fact_1027_ds_Oring__primeI,axiom,
! [P2: b] :
( ( P2
!= ( zero_b_d @ s ) )
=> ( ( prime_b_ring_ext_b_d @ s @ P2 )
=> ( ring_ring_prime_b_d @ s @ P2 ) ) ) ).
% ds.ring_primeI
thf(fact_1028_pdr_Oring__primeE_I3_J,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 )
=> ( prime_2011924034616061926t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 ) ) ) ).
% pdr.ring_primeE(3)
thf(fact_1029_set__x,axiom,
ord_less_eq_set_b @ ( set_b2 @ x ) @ ( image_a_b @ h @ ( partia778085601923319190xt_a_c @ r ) ) ).
% set_x
thf(fact_1030_pdr_Oassociated__polynomials__imp__same__is__root,axiom,
! [P2: list_list_a,Q2: list_list_a,X2: list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ P2 @ Q2 )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 @ X2 )
= ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Q2 @ X2 ) ) ) ) ) ).
% pdr.associated_polynomials_imp_same_is_root
thf(fact_1031_pdr_Ois__root__poly__mult__imp__is__root,axiom,
! [P2: list_list_a,Q2: list_list_a,X2: list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( mult_l4853965630390486993t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ P2 @ Q2 ) @ X2 )
=> ( ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 @ X2 )
| ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Q2 @ X2 ) ) ) ) ) ).
% pdr.is_root_poly_mult_imp_is_root
thf(fact_1032_pds_Oassociated__polynomials__imp__same__is__root,axiom,
! [P2: list_list_b,Q2: list_list_b,X2: list_b] :
( ( member_list_list_b @ P2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( member_list_list_b @ Q2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( associ7557723228729174617t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ P2 @ Q2 )
=> ( ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 @ X2 )
= ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Q2 @ X2 ) ) ) ) ) ).
% pds.associated_polynomials_imp_same_is_root
thf(fact_1033_pds_Ois__root__poly__mult__imp__is__root,axiom,
! [P2: list_list_b,Q2: list_list_b,X2: list_b] :
( ( member_list_list_b @ P2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( member_list_list_b @ Q2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( mult_l6808613587631625489t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ P2 @ Q2 ) @ X2 )
=> ( ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 @ X2 )
| ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Q2 @ X2 ) ) ) ) ) ).
% pds.is_root_poly_mult_imp_is_root
thf(fact_1034_ds_Oexp__base__closed,axiom,
! [X2: b,N: nat] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ord_less_eq_set_b @ ( set_b2 @ ( polyno134955809644805305se_b_d @ s @ X2 @ N ) ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% ds.exp_base_closed
thf(fact_1035_pds_Oa__lcos__mult__one,axiom,
! [M: set_list_b] :
( ( ord_le8932221534207217157list_b @ M @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( a_l_co3923087131696239825t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ M )
= M ) ) ).
% pds.a_lcos_mult_one
thf(fact_1036_pdr_Oa__lcos__mult__one,axiom,
! [M: set_list_a] :
( ( ord_le8861187494160871172list_a @ M @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ M )
= M ) ) ).
% pdr.a_lcos_mult_one
thf(fact_1037_pds_Oup__smult__closed,axiom,
! [A4: list_b,P2: nat > list_b] :
( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_nat_list_b @ P2 @ ( up_lis5378411187065319169t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_nat_list_b
@ ^ [I3: nat] : ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ ( P2 @ I3 ) )
@ ( up_lis5378411187065319169t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.up_smult_closed
thf(fact_1038_pdr_Oa__l__coset__subset__G,axiom,
! [H2: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( a_l_co7008843373686234386t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ H2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.a_l_coset_subset_G
thf(fact_1039_pds_Oa__l__coset__subset__G,axiom,
! [H2: set_list_b,X2: list_b] :
( ( ord_le8932221534207217157list_b @ H2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ord_le8932221534207217157list_b @ ( a_l_co3923087131696239825t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ H2 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.a_l_coset_subset_G
thf(fact_1040_pds_Obound__upD,axiom,
! [F: nat > list_b] :
( ( member_nat_list_b @ F @ ( up_lis5378411187065319169t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ? [N2: nat] : ( bound_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ N2 @ F ) ) ).
% pds.bound_upD
thf(fact_1041_dr_Oexp__base__closed,axiom,
! [X2: a,N: nat] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( polyno2922411391617481337se_a_c @ r @ X2 @ N ) ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% dr.exp_base_closed
thf(fact_1042_pds_Oexp__base__closed,axiom,
! [X2: list_b,N: nat] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ord_le8932221534207217157list_b @ ( set_list_b2 @ ( polyno437060639131926335t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ N ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.exp_base_closed
thf(fact_1043_pdr_Oexp__base__closed,axiom,
! [X2: list_a,N: nat] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( polyno3522816881121920896t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ N ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.exp_base_closed
thf(fact_1044_dr_Oa__l__coset__subset__G,axiom,
! [H2: set_a,X2: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ord_less_eq_set_a @ ( a_l_coset_a_c @ r @ X2 @ H2 ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% dr.a_l_coset_subset_G
thf(fact_1045_ds_Oa__l__coset__subset__G,axiom,
! [H2: set_b,X2: b] :
( ( ord_less_eq_set_b @ H2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ord_less_eq_set_b @ ( a_l_coset_b_d @ s @ X2 @ H2 ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% ds.a_l_coset_subset_G
thf(fact_1046_dr_Oup__smult__closed,axiom,
! [A4: a,P2: nat > a] :
( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_nat_a @ P2 @ ( up_a_c @ r ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( mult_a_ring_ext_a_c @ r @ A4 @ ( P2 @ I3 ) )
@ ( up_a_c @ r ) ) ) ) ).
% dr.up_smult_closed
thf(fact_1047_ds_Oup__smult__closed,axiom,
! [A4: b,P2: nat > b] :
( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_nat_b @ P2 @ ( up_b_d @ s ) )
=> ( member_nat_b
@ ^ [I3: nat] : ( mult_b_ring_ext_b_d @ s @ A4 @ ( P2 @ I3 ) )
@ ( up_b_d @ s ) ) ) ) ).
% ds.up_smult_closed
thf(fact_1048_dr_Oa__lcos__mult__one,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( a_l_coset_a_c @ r @ ( zero_a_c @ r ) @ M )
= M ) ) ).
% dr.a_lcos_mult_one
thf(fact_1049_ds_Oa__lcos__mult__one,axiom,
! [M: set_b] :
( ( ord_less_eq_set_b @ M @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( a_l_coset_b_d @ s @ ( zero_b_d @ s ) @ M )
= M ) ) ).
% ds.a_lcos_mult_one
thf(fact_1050_pdr_Oup__smult__closed,axiom,
! [A4: list_a,P2: nat > list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [I3: nat] : ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ ( P2 @ I3 ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.up_smult_closed
thf(fact_1051_pdr_Obound__upD,axiom,
! [F: nat > list_a] :
( ( member_nat_list_a @ F @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ? [N2: nat] : ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ N2 @ F ) ) ).
% pdr.bound_upD
thf(fact_1052_dr_Obound__upD,axiom,
! [F: nat > a] :
( ( member_nat_a @ F @ ( up_a_c @ r ) )
=> ? [N2: nat] : ( bound_a @ ( zero_a_c @ r ) @ N2 @ F ) ) ).
% dr.bound_upD
thf(fact_1053_ds_Obound__upD,axiom,
! [F: nat > b] :
( ( member_nat_b @ F @ ( up_b_d @ s ) )
=> ? [N2: nat] : ( bound_b @ ( zero_b_d @ s ) @ N2 @ F ) ) ).
% ds.bound_upD
thf(fact_1054_pds_Oee__sym,axiom,
! [As: list_list_b,Bs: list_list_b] :
( ( essent4653736668955345094t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ As @ Bs )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ As ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Bs ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( essent4653736668955345094t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Bs @ As ) ) ) ) ).
% pds.ee_sym
thf(fact_1055_pds_Oee__trans,axiom,
! [As: list_list_b,Bs: list_list_b,Cs: list_list_b] :
( ( essent4653736668955345094t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ As @ Bs )
=> ( ( essent4653736668955345094t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Bs @ Cs )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ As ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Bs ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Cs ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( essent4653736668955345094t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ As @ Cs ) ) ) ) ) ) ).
% pds.ee_trans
thf(fact_1056_pds_Oee__refl,axiom,
! [As: list_list_b] :
( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ As ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( essent4653736668955345094t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ As @ As ) ) ).
% pds.ee_refl
thf(fact_1057_pdr_Oee__trans,axiom,
! [As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ As @ Bs )
=> ( ( essent703981920984620806t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Bs @ Cs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Cs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ As @ Cs ) ) ) ) ) ) ).
% pdr.ee_trans
thf(fact_1058_pdr_Oee__sym,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( essent703981920984620806t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ As @ Bs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Bs @ As ) ) ) ) ).
% pdr.ee_sym
thf(fact_1059_pds_Oassociated__polynomials__imp__same__roots,axiom,
! [P2: list_list_b,Q2: list_list_b] :
( ( member_list_list_b @ P2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( member_list_list_b @ Q2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( associ7557723228729174617t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ P2 @ Q2 )
=> ( ( polyno4772666585000257442t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 )
= ( polyno4772666585000257442t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Q2 ) ) ) ) ) ).
% pds.associated_polynomials_imp_same_roots
thf(fact_1060_pdr_Oassociated__polynomials__imp__same__roots,axiom,
! [P2: list_list_a,Q2: list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( member_list_list_a @ Q2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( associ5603075271488036121t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ P2 @ Q2 )
=> ( ( polyno7858422826990252003t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 )
= ( polyno7858422826990252003t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Q2 ) ) ) ) ) ).
% pdr.associated_polynomials_imp_same_roots
thf(fact_1061_pdr_Oee__refl,axiom,
! [As: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( essent703981920984620806t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ As @ As ) ) ).
% pdr.ee_refl
thf(fact_1062_pds_Oup__minus__closed,axiom,
! [P2: nat > list_b,Q2: nat > list_b] :
( ( member_nat_list_b @ P2 @ ( up_lis5378411187065319169t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_nat_list_b @ Q2 @ ( up_lis5378411187065319169t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_nat_list_b
@ ^ [I3: nat] : ( a_minu898264511480707987t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_lis5378411187065319169t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.up_minus_closed
thf(fact_1063_pdr_Oup__minus__closed,axiom,
! [P2: nat > list_a,Q2: nat > list_a] :
( ( member_nat_list_a @ P2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_nat_list_a @ Q2 @ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_nat_list_a
@ ^ [I3: nat] : ( a_minu3984020753470702548t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_lis8464167429055313730t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.up_minus_closed
thf(fact_1064_pds_Ocarrier__is__subalgebra,axiom,
! [K2: set_list_b] :
( ( ord_le8932221534207217157list_b @ K2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( embedd7906597418576622673t_unit @ K2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.carrier_is_subalgebra
thf(fact_1065_pds_Osubalgebra__in__carrier,axiom,
! [K2: set_list_b,V: set_list_b] :
( ( embedd7906597418576622673t_unit @ K2 @ V @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ord_le8932221534207217157list_b @ V @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.subalgebra_in_carrier
thf(fact_1066_pds_Osubalgebra__inter,axiom,
! [K2: set_list_b,V: set_list_b,V2: set_list_b] :
( ( embedd7906597418576622673t_unit @ K2 @ V @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( embedd7906597418576622673t_unit @ K2 @ V2 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( embedd7906597418576622673t_unit @ K2 @ ( inf_inf_set_list_b @ V @ V2 ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.subalgebra_inter
thf(fact_1067_dr_Oassociated__polynomials__imp__same__roots,axiom,
! [P2: list_a,Q2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Q2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 @ Q2 )
=> ( ( polynomial_roots_a_c @ r @ P2 )
= ( polynomial_roots_a_c @ r @ Q2 ) ) ) ) ) ).
% dr.associated_polynomials_imp_same_roots
thf(fact_1068_ds_Oassociated__polynomials__imp__same__roots,axiom,
! [P2: list_b,Q2: list_b] :
( ( member_list_b @ P2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Q2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 @ Q2 )
=> ( ( polynomial_roots_b_d @ s @ P2 )
= ( polynomial_roots_b_d @ s @ Q2 ) ) ) ) ) ).
% ds.associated_polynomials_imp_same_roots
thf(fact_1069_pdr_Ominus__closed,axiom,
! [X2: list_a,Y2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Y2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.minus_closed
thf(fact_1070_pds_Ominus__closed,axiom,
! [X2: list_b,Y2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ ( a_minu898264511480707987t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Y2 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.minus_closed
thf(fact_1071_pdr_Or__right__minus__eq,axiom,
! [A4: list_a,B: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( A4 = B ) ) ) ) ).
% pdr.r_right_minus_eq
thf(fact_1072_pds_Or__right__minus__eq,axiom,
! [A4: list_b,B: list_b] :
( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( a_minu898264511480707987t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( A4 = B ) ) ) ) ).
% pds.r_right_minus_eq
thf(fact_1073_pdr_Osubalgebra__in__carrier,axiom,
! [K2: set_list_a,V: set_list_a] :
( ( embedd1768981623711841426t_unit @ K2 @ V @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ord_le8861187494160871172list_a @ V @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.subalgebra_in_carrier
thf(fact_1074_pdr_Ocarrier__is__subalgebra,axiom,
! [K2: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( embedd1768981623711841426t_unit @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.carrier_is_subalgebra
thf(fact_1075_pds_Omap__in__poly__ring__carrier,axiom,
! [P2: list_list_b,F: list_b > list_list_b] :
( ( member_list_list_b @ P2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ! [A3: list_b] :
( ( member_list_b @ A3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_list_b @ ( F @ A3 ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) )
=> ( ! [A3: list_b] :
( ( A3
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( F @ A3 )
!= nil_list_b ) )
=> ( member2820959508942094138list_b @ ( map_li4464339650582430090list_b @ F @ P2 ) @ ( partia7846567554643003126t_unit @ ( univ_p6451808943148412807t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ) ) ) ).
% pds.map_in_poly_ring_carrier
thf(fact_1076_pdr_Omap__in__poly__ring__carrier,axiom,
! [P2: list_list_a,F: list_a > list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) )
=> ( ! [A3: list_a] :
( ( A3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( F @ A3 )
!= nil_list_a ) )
=> ( member5342144027231129785list_a @ ( map_li5729356230488778442list_a @ F @ P2 ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ) ) ) ).
% pdr.map_in_poly_ring_carrier
thf(fact_1077_pdr_Osubalgebra__inter,axiom,
! [K2: set_list_a,V: set_list_a,V2: set_list_a] :
( ( embedd1768981623711841426t_unit @ K2 @ V @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( embedd1768981623711841426t_unit @ K2 @ V2 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K2 @ ( inf_inf_set_list_a @ V @ V2 ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.subalgebra_inter
thf(fact_1078_pds_Onorm__map__in__poly__ring__carrier,axiom,
! [P2: list_list_b,F: list_b > list_list_b] :
( ( member_list_list_b @ P2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ! [A3: list_b] :
( ( member_list_b @ A3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_list_b @ ( F @ A3 ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) )
=> ( member2820959508942094138list_b @ ( normal5498541872298712508t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ ( map_li4464339650582430090list_b @ F @ P2 ) ) @ ( partia7846567554643003126t_unit @ ( univ_p6451808943148412807t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ) ) ).
% pds.norm_map_in_poly_ring_carrier
thf(fact_1079_pdr_Onorm__map__in__poly__ring__carrier,axiom,
! [P2: list_list_a,F: list_a > list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_list_a @ ( F @ A3 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) )
=> ( member5342144027231129785list_a @ ( normal1297324897130370429t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ ( map_li5729356230488778442list_a @ F @ P2 ) ) @ ( partia5038748322285217333t_unit @ ( univ_p2250591967980070728t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ) ) ).
% pdr.norm_map_in_poly_ring_carrier
thf(fact_1080_ds_Omap__in__poly__ring__carrier,axiom,
! [P2: list_b,F: b > list_b] :
( ( member_list_b @ P2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_list_b @ ( F @ A3 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) )
=> ( ! [A3: b] :
( ( A3
!= ( zero_b_d @ s ) )
=> ( ( F @ A3 )
!= nil_b ) )
=> ( member_list_list_b @ ( map_b_list_b @ F @ P2 ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ) ).
% ds.map_in_poly_ring_carrier
thf(fact_1081_dr_Omap__in__poly__ring__carrier,axiom,
! [P2: list_a,F: a > list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) )
=> ( ! [A3: a] :
( ( A3
!= ( zero_a_c @ r ) )
=> ( ( F @ A3 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ) ).
% dr.map_in_poly_ring_carrier
thf(fact_1082_pds_Ofinite__number__of__roots,axiom,
! [P2: list_list_b] :
( ( member_list_list_b @ P2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( finite_finite_list_b @ ( collect_list_b @ ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 ) ) ) ) ).
% pds.finite_number_of_roots
thf(fact_1083_pdr_Ofinite__number__of__roots,axiom,
! [P2: list_list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( finite_finite_list_a @ ( collect_list_a @ ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 ) ) ) ) ).
% pdr.finite_number_of_roots
thf(fact_1084_pds_Ofactors__closed,axiom,
! [Fs: list_list_b,A4: list_b] :
( ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fs @ A4 )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Fs ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.factors_closed
thf(fact_1085_pdr_Ofactors__closed,axiom,
! [Fs: list_list_a,A4: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fs @ A4 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.factors_closed
thf(fact_1086_pdr_Onormalize_Osimps_I1_J,axiom,
( ( normal637505603836502915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ nil_list_a )
= nil_list_a ) ).
% pdr.normalize.simps(1)
thf(fact_1087_pds_Onormalize_Osimps_I1_J,axiom,
( ( normal6775121398701284162t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ nil_list_b )
= nil_list_b ) ).
% pds.normalize.simps(1)
thf(fact_1088_pdr_Onormalize__in__carrier,axiom,
! [P2: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ P2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ ( normal637505603836502915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.normalize_in_carrier
thf(fact_1089_pds_Onormalize__in__carrier,axiom,
! [P2: list_list_b] :
( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ P2 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ord_le8932221534207217157list_b @ ( set_list_b2 @ ( normal6775121398701284162t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.normalize_in_carrier
thf(fact_1090_dr_Onorm__map__in__poly__ring__carrier,axiom,
! [P2: list_a,F: a > list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) )
=> ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( map_a_list_a @ F @ P2 ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ) ).
% dr.norm_map_in_poly_ring_carrier
thf(fact_1091_ds_Onorm__map__in__poly__ring__carrier,axiom,
! [P2: list_b,F: b > list_b] :
( ( member_list_b @ P2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ! [A3: b] :
( ( member_b @ A3 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_list_b @ ( F @ A3 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) )
=> ( member_list_list_b @ ( normal6775121398701284162t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( map_b_list_b @ F @ P2 ) ) @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ) ).
% ds.norm_map_in_poly_ring_carrier
thf(fact_1092_pdr_Ofinite__ring__finite__units,axiom,
( ( finite_finite_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( finite_finite_list_a @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.finite_ring_finite_units
thf(fact_1093_pds_Ofinite__ring__finite__units,axiom,
( ( finite_finite_list_b @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( finite_finite_list_b @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.finite_ring_finite_units
thf(fact_1094_pds_Ofactors__mult,axiom,
! [Fa: list_list_b,A4: list_b,Fb: list_list_b,B: list_b] :
( ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fa @ A4 )
=> ( ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fb @ B )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Fa ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Fb ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( append_list_b @ Fa @ Fb ) @ ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) ) ) ) ) ) ).
% pds.factors_mult
thf(fact_1095_pdr_Ofactors__mult,axiom,
! [Fa: list_list_a,A4: list_a,Fb: list_list_a,B: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fa @ A4 )
=> ( ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fb @ B )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fa ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fb ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( append_list_a @ Fa @ Fb ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) ) ) ) ) ) ).
% pdr.factors_mult
thf(fact_1096_pds_Owfactors__factors,axiom,
! [As: list_list_b,A4: list_b] :
( ( wfacto7783783145806120978t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ As @ A4 )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ As ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ? [A7: list_b] :
( ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ As @ A7 )
& ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A7 @ A4 ) ) ) ) ).
% pds.wfactors_factors
thf(fact_1097_pdr_Owfactors__factors,axiom,
! [As: list_list_a,A4: list_a] :
( ( wfacto3834028397835396690t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ As @ A4 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ? [A7: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ As @ A7 )
& ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A7 @ A4 ) ) ) ) ).
% pdr.wfactors_factors
thf(fact_1098_pdr_Onormalize__idem,axiom,
! [P2: list_list_a,Q2: list_list_a] :
( ( normal637505603836502915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( append_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 ) @ Q2 ) )
= ( normal637505603836502915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( append_list_a @ P2 @ Q2 ) ) ) ).
% pdr.normalize_idem
thf(fact_1099_pds_Onormalize__idem,axiom,
! [P2: list_list_b,Q2: list_list_b] :
( ( normal6775121398701284162t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( append_list_b @ ( normal6775121398701284162t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 ) @ Q2 ) )
= ( normal6775121398701284162t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( append_list_b @ P2 @ Q2 ) ) ) ).
% pds.normalize_idem
thf(fact_1100_dr_Ofinite__number__of__roots,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( finite_finite_a @ ( collect_a @ ( polyno4133073214067823461ot_a_c @ r @ P2 ) ) ) ) ).
% dr.finite_number_of_roots
thf(fact_1101_ds_Ofinite__number__of__roots,axiom,
! [P2: list_b] :
( ( member_list_b @ P2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( finite_finite_b @ ( collect_b @ ( polyno1345617632095147429ot_b_d @ s @ P2 ) ) ) ) ).
% ds.finite_number_of_roots
thf(fact_1102_pdr_Ounit__wfactors,axiom,
! [A4: list_a] :
( ( member_list_a @ A4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( wfacto3834028397835396690t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ nil_list_a @ A4 ) ) ).
% pdr.unit_wfactors
thf(fact_1103_pds_Ounit__wfactors,axiom,
! [A4: list_b] :
( ( member_list_b @ A4 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( wfacto7783783145806120978t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ nil_list_b @ A4 ) ) ).
% pds.unit_wfactors
thf(fact_1104_pdr_Owfactors__cong__r,axiom,
! [Fs: list_list_a,A4: list_a,A5: list_a] :
( ( wfacto3834028397835396690t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fs @ A4 )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ A5 )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ A5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( wfacto3834028397835396690t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fs @ A5 ) ) ) ) ) ) ).
% pdr.wfactors_cong_r
thf(fact_1105_pds_Owfactors__cong__r,axiom,
! [Fs: list_list_b,A4: list_b,A5: list_b] :
( ( wfacto7783783145806120978t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fs @ A4 )
=> ( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ A5 )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ A5 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Fs ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( wfacto7783783145806120978t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fs @ A5 ) ) ) ) ) ) ).
% pds.wfactors_cong_r
thf(fact_1106_pdr_Ofactors__wfactors,axiom,
! [As: list_list_a,A4: list_a] :
( ( factor7181967632740204193t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ As @ A4 )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( wfacto3834028397835396690t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ As @ A4 ) ) ) ).
% pdr.factors_wfactors
thf(fact_1107_pds_Ofactors__wfactors,axiom,
! [As: list_list_b,A4: list_b] :
( ( factor1908350343856152673t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ As @ A4 )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ As ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( wfacto7783783145806120978t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ As @ A4 ) ) ) ).
% pds.factors_wfactors
thf(fact_1108_dr_Ofinite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia778085601923319190xt_a_c @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_c @ r ) ) ) ).
% dr.finite_ring_finite_units
thf(fact_1109_ds_Ofinite__ring__finite__units,axiom,
( ( finite_finite_b @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( finite_finite_b @ ( units_b_ring_ext_b_d @ s ) ) ) ).
% ds.finite_ring_finite_units
thf(fact_1110_dr_Onoetherian__ringI,axiom,
( ! [I4: set_a] :
( ( ideal_a_c @ I4 @ r )
=> ? [A8: set_a] :
( ( ord_less_eq_set_a @ A8 @ ( partia778085601923319190xt_a_c @ r ) )
& ( finite_finite_a @ A8 )
& ( I4
= ( genideal_a_c @ r @ A8 ) ) ) )
=> ( ring_n3639167112692572310ng_a_c @ r ) ) ).
% dr.noetherian_ringI
thf(fact_1111_ds_Oup__minus__closed,axiom,
! [P2: nat > b,Q2: nat > b] :
( ( member_nat_b @ P2 @ ( up_b_d @ s ) )
=> ( ( member_nat_b @ Q2 @ ( up_b_d @ s ) )
=> ( member_nat_b
@ ^ [I3: nat] : ( a_minus_b_d @ s @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_b_d @ s ) ) ) ) ).
% ds.up_minus_closed
thf(fact_1112_dr_Oup__minus__closed,axiom,
! [P2: nat > a,Q2: nat > a] :
( ( member_nat_a @ P2 @ ( up_a_c @ r ) )
=> ( ( member_nat_a @ Q2 @ ( up_a_c @ r ) )
=> ( member_nat_a
@ ^ [I3: nat] : ( a_minus_a_c @ r @ ( P2 @ I3 ) @ ( Q2 @ I3 ) )
@ ( up_a_c @ r ) ) ) ) ).
% dr.up_minus_closed
thf(fact_1113_pds_Oroots__mem__iff__is__root,axiom,
! [P2: list_list_b,X2: list_b] :
( ( member_list_list_b @ P2 @ ( partia8406058877693316080t_unit @ ( univ_p4867482214140432013t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) )
=> ( ( member_list_b @ X2 @ ( set_mset_list_b @ ( polyno4772666585000257442t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 ) ) )
= ( polyno3865904989341193771t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 @ X2 ) ) ) ).
% pds.roots_mem_iff_is_root
thf(fact_1114_dr_Ominus__closed,axiom,
! [X2: a,Y2: a] :
( ( member_a @ X2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ Y2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ ( a_minus_a_c @ r @ X2 @ Y2 ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% dr.minus_closed
thf(fact_1115_ds_Ominus__closed,axiom,
! [X2: b,Y2: b] :
( ( member_b @ X2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ Y2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ ( a_minus_b_d @ s @ X2 @ Y2 ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% ds.minus_closed
thf(fact_1116_dr_Or__right__minus__eq,axiom,
! [A4: a,B: a] :
( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ( a_minus_a_c @ r @ A4 @ B )
= ( zero_a_c @ r ) )
= ( A4 = B ) ) ) ) ).
% dr.r_right_minus_eq
thf(fact_1117_ds_Or__right__minus__eq,axiom,
! [A4: b,B: b] :
( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( a_minus_b_d @ s @ A4 @ B )
= ( zero_b_d @ s ) )
= ( A4 = B ) ) ) ) ).
% ds.r_right_minus_eq
thf(fact_1118_pdr_Oroots__mem__iff__is__root,axiom,
! [P2: list_list_a,X2: list_a] :
( ( member_list_list_a @ P2 @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) )
=> ( ( member_list_a @ X2 @ ( set_mset_list_a @ ( polyno7858422826990252003t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 ) ) )
= ( polyno6951661231331188332t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 @ X2 ) ) ) ).
% pdr.roots_mem_iff_is_root
thf(fact_1119_pds_Operm__wfactorsD,axiom,
! [As: list_list_b,Bs: list_list_b,A4: list_b,B: list_b] :
( ( ( mset_list_b @ As )
= ( mset_list_b @ Bs ) )
=> ( ( wfacto7783783145806120978t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ As @ A4 )
=> ( ( wfacto7783783145806120978t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Bs @ B )
=> ( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ As ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B ) ) ) ) ) ) ) ).
% pds.perm_wfactorsD
thf(fact_1120_pdr_Operm__wfactorsD,axiom,
! [As: list_list_a,Bs: list_list_a,A4: list_a,B: list_a] :
( ( ( mset_list_a @ As )
= ( mset_list_a @ Bs ) )
=> ( ( wfacto3834028397835396690t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ As @ A4 )
=> ( ( wfacto3834028397835396690t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Bs @ B )
=> ( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B ) ) ) ) ) ) ) ).
% pdr.perm_wfactorsD
thf(fact_1121_ds_Ogenideal__self,axiom,
! [S: set_b] :
( ( ord_less_eq_set_b @ S @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ord_less_eq_set_b @ S @ ( genideal_b_d @ s @ S ) ) ) ).
% ds.genideal_self
thf(fact_1122_ds_Osubset__Idl__subset,axiom,
! [I2: set_b,H2: set_b] :
( ( ord_less_eq_set_b @ I2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ord_less_eq_set_b @ H2 @ I2 )
=> ( ord_less_eq_set_b @ ( genideal_b_d @ s @ H2 ) @ ( genideal_b_d @ s @ I2 ) ) ) ) ).
% ds.subset_Idl_subset
thf(fact_1123_ds_Ogenideal__minimal,axiom,
! [I2: set_b,S: set_b] :
( ( ideal_b_d @ I2 @ s )
=> ( ( ord_less_eq_set_b @ S @ I2 )
=> ( ord_less_eq_set_b @ ( genideal_b_d @ s @ S ) @ I2 ) ) ) ).
% ds.genideal_minimal
thf(fact_1124_pdr_Ooneideal,axiom,
ideal_8896367198367571637t_unit @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.oneideal
thf(fact_1125_pds_Ooneideal,axiom,
ideal_5810610956377577076t_unit @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.oneideal
thf(fact_1126_pdr_Ogenideal__minimal,axiom,
! [I2: set_list_a,S: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I2 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ S @ I2 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ S ) @ I2 ) ) ) ).
% pdr.genideal_minimal
thf(fact_1127_pds_Ogenideal__minimal,axiom,
! [I2: set_list_b,S: set_list_b] :
( ( ideal_5810610956377577076t_unit @ I2 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( ord_le8932221534207217157list_b @ S @ I2 )
=> ( ord_le8932221534207217157list_b @ ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ S ) @ I2 ) ) ) ).
% pds.genideal_minimal
thf(fact_1128_ds_Ogenideal__ideal,axiom,
! [S: set_b] :
( ( ord_less_eq_set_b @ S @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ideal_b_d @ ( genideal_b_d @ s @ S ) @ s ) ) ).
% ds.genideal_ideal
thf(fact_1129_ds_OIdl__subset__ideal,axiom,
! [I2: set_b,H2: set_b] :
( ( ideal_b_d @ I2 @ s )
=> ( ( ord_less_eq_set_b @ H2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ord_less_eq_set_b @ ( genideal_b_d @ s @ H2 ) @ I2 )
= ( ord_less_eq_set_b @ H2 @ I2 ) ) ) ) ).
% ds.Idl_subset_ideal
thf(fact_1130_pdr_Oi__intersect,axiom,
! [I2: set_list_a,J: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I2 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( ideal_8896367198367571637t_unit @ J @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ideal_8896367198367571637t_unit @ ( inf_inf_set_list_a @ I2 @ J ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.i_intersect
thf(fact_1131_pds_Oi__intersect,axiom,
! [I2: set_list_b,J: set_list_b] :
( ( ideal_5810610956377577076t_unit @ I2 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( ideal_5810610956377577076t_unit @ J @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ideal_5810610956377577076t_unit @ ( inf_inf_set_list_b @ I2 @ J ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.i_intersect
thf(fact_1132_pdr_Ogenideal__ideal,axiom,
! [S: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ideal_8896367198367571637t_unit @ ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ S ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.genideal_ideal
thf(fact_1133_pdr_OIdl__subset__ideal,axiom,
! [I2: set_list_a,H2: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I2 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H2 ) @ I2 )
= ( ord_le8861187494160871172list_a @ H2 @ I2 ) ) ) ) ).
% pdr.Idl_subset_ideal
thf(fact_1134_pdr_Osubset__Idl__subset,axiom,
! [I2: set_list_a,H2: set_list_a] :
( ( ord_le8861187494160871172list_a @ I2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ H2 @ I2 )
=> ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ H2 ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ I2 ) ) ) ) ).
% pdr.subset_Idl_subset
thf(fact_1135_pdr_Ogenideal__self,axiom,
! [S: set_list_a] :
( ( ord_le8861187494160871172list_a @ S @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ S @ ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ S ) ) ) ).
% pdr.genideal_self
thf(fact_1136_pds_Ogenideal__ideal,axiom,
! [S: set_list_b] :
( ( ord_le8932221534207217157list_b @ S @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ideal_5810610956377577076t_unit @ ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ S ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.genideal_ideal
thf(fact_1137_pds_OIdl__subset__ideal,axiom,
! [I2: set_list_b,H2: set_list_b] :
( ( ideal_5810610956377577076t_unit @ I2 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( ord_le8932221534207217157list_b @ H2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H2 ) @ I2 )
= ( ord_le8932221534207217157list_b @ H2 @ I2 ) ) ) ) ).
% pds.Idl_subset_ideal
thf(fact_1138_pds_Osubset__Idl__subset,axiom,
! [I2: set_list_b,H2: set_list_b] :
( ( ord_le8932221534207217157list_b @ I2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ H2 @ I2 )
=> ( ord_le8932221534207217157list_b @ ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ H2 ) @ ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ I2 ) ) ) ) ).
% pds.subset_Idl_subset
thf(fact_1139_pds_Ogenideal__self,axiom,
! [S: set_list_b] :
( ( ord_le8932221534207217157list_b @ S @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ord_le8932221534207217157list_b @ S @ ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ S ) ) ) ).
% pds.genideal_self
thf(fact_1140_ds_Onoetherian__ringI,axiom,
( ! [I4: set_b] :
( ( ideal_b_d @ I4 @ s )
=> ? [A8: set_b] :
( ( ord_less_eq_set_b @ A8 @ ( partia8782771468121683032xt_b_d @ s ) )
& ( finite_finite_b @ A8 )
& ( I4
= ( genideal_b_d @ s @ A8 ) ) ) )
=> ( ring_n851711530719896278ng_b_d @ s ) ) ).
% ds.noetherian_ringI
thf(fact_1141_pdr_Onoetherian__ringI,axiom,
( ! [I4: set_list_a] :
( ( ideal_8896367198367571637t_unit @ I4 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ? [A8: set_list_a] :
( ( ord_le8861187494160871172list_a @ A8 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( finite_finite_list_a @ A8 )
& ( I4
= ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A8 ) ) ) )
=> ( ring_n5188127996776581661t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.noetherian_ringI
thf(fact_1142_pds_Onoetherian__ringI,axiom,
( ! [I4: set_list_b] :
( ( ideal_5810610956377577076t_unit @ I4 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ? [A8: set_list_b] :
( ( ord_le8932221534207217157list_b @ A8 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( finite_finite_list_b @ A8 )
& ( I4
= ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A8 ) ) ) )
=> ( ring_n2102371754786587100t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.noetherian_ringI
thf(fact_1143_pdr_Oideal__is__subalgebra,axiom,
! [K2: set_list_a,I2: set_list_a] :
( ( ord_le8861187494160871172list_a @ K2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ideal_8896367198367571637t_unit @ I2 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( embedd1768981623711841426t_unit @ K2 @ I2 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.ideal_is_subalgebra
thf(fact_1144_pds_Oideal__is__subalgebra,axiom,
! [K2: set_list_b,I2: set_list_b] :
( ( ord_le8932221534207217157list_b @ K2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ideal_5810610956377577076t_unit @ I2 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( embedd7906597418576622673t_unit @ K2 @ I2 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.ideal_is_subalgebra
thf(fact_1145_dr_Oroots__mem__iff__is__root,axiom,
! [P2: list_a,X2: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_a @ X2 @ ( set_mset_a @ ( polynomial_roots_a_c @ r @ P2 ) ) )
= ( polyno4133073214067823461ot_a_c @ r @ P2 @ X2 ) ) ) ).
% dr.roots_mem_iff_is_root
thf(fact_1146_ds_Oroots__mem__iff__is__root,axiom,
! [P2: list_b,X2: b] :
( ( member_list_b @ P2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_b @ X2 @ ( set_mset_b @ ( polynomial_roots_b_d @ s @ P2 ) ) )
= ( polyno1345617632095147429ot_b_d @ s @ P2 @ X2 ) ) ) ).
% ds.roots_mem_iff_is_root
thf(fact_1147_pdr_Operm__closed,axiom,
! [As: list_list_a,Bs: list_list_a] :
( ( ( mset_list_a @ As )
= ( mset_list_a @ Bs ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ As ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Bs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.perm_closed
thf(fact_1148_pds_Operm__closed,axiom,
! [As: list_list_b,Bs: list_list_b] :
( ( ( mset_list_b @ As )
= ( mset_list_b @ Bs ) )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ As ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Bs ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.perm_closed
thf(fact_1149_pdr_Owfactors__perm__cong__l,axiom,
! [Fs: list_list_a,A4: list_a,Fs2: list_list_a] :
( ( wfacto3834028397835396690t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fs @ A4 )
=> ( ( ( mset_list_a @ Fs )
= ( mset_list_a @ Fs2 ) )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Fs ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( wfacto3834028397835396690t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Fs2 @ A4 ) ) ) ) ).
% pdr.wfactors_perm_cong_l
thf(fact_1150_pds_Owfactors__perm__cong__l,axiom,
! [Fs: list_list_b,A4: list_b,Fs2: list_list_b] :
( ( wfacto7783783145806120978t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fs @ A4 )
=> ( ( ( mset_list_b @ Fs )
= ( mset_list_b @ Fs2 ) )
=> ( ( ord_le8932221534207217157list_b @ ( set_list_b2 @ Fs ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( wfacto7783783145806120978t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Fs2 @ A4 ) ) ) ) ).
% pds.wfactors_perm_cong_l
thf(fact_1151_ds_Omonoid__cancelI,axiom,
( ! [A3: b,B2: b,C3: b] :
( ( ( mult_b_ring_ext_b_d @ s @ C3 @ A3 )
= ( mult_b_ring_ext_b_d @ s @ C3 @ B2 ) )
=> ( ( member_b @ A3 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C3 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: b,B2: b,C3: b] :
( ( ( mult_b_ring_ext_b_d @ s @ A3 @ C3 )
= ( mult_b_ring_ext_b_d @ s @ B2 @ C3 ) )
=> ( ( member_b @ A3 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ C3 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid9131220752592392315xt_b_d @ s ) ) ) ).
% ds.monoid_cancelI
thf(fact_1152_pds_OboundD__carrier,axiom,
! [N: nat,F: nat > list_b,M2: nat] :
( ( bound_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M2 )
=> ( member_list_b @ ( F @ M2 ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.boundD_carrier
thf(fact_1153_dr_OboundD__carrier,axiom,
! [N: nat,F: nat > a,M2: nat] :
( ( bound_a @ ( zero_a_c @ r ) @ N @ F )
=> ( ( ord_less_nat @ N @ M2 )
=> ( member_a @ ( F @ M2 ) @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% dr.boundD_carrier
thf(fact_1154_ds_OboundD__carrier,axiom,
! [N: nat,F: nat > b,M2: nat] :
( ( bound_b @ ( zero_b_d @ s ) @ N @ F )
=> ( ( ord_less_nat @ N @ M2 )
=> ( member_b @ ( F @ M2 ) @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% ds.boundD_carrier
thf(fact_1155_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_1156_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1157_pdr_OboundD__carrier,axiom,
! [N: nat,F: nat > list_a,M2: nat] :
( ( bound_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ N @ F )
=> ( ( ord_less_nat @ N @ M2 )
=> ( member_list_a @ ( F @ M2 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.boundD_carrier
thf(fact_1158_pds_OIdl__subset__ideal_H,axiom,
! [A4: list_b,B: list_b] :
( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ord_le8932221534207217157list_b @ ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( insert_list_b @ A4 @ bot_bot_set_list_b ) ) @ ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( insert_list_b @ B @ bot_bot_set_list_b ) ) )
= ( member_list_b @ A4 @ ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( insert_list_b @ B @ bot_bot_set_list_b ) ) ) ) ) ) ).
% pds.Idl_subset_ideal'
thf(fact_1159_pdr_OIdl__subset__ideal_H,axiom,
! [A4: list_a,B: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ord_le8861187494160871172list_a @ ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( insert_list_a @ A4 @ bot_bot_set_list_a ) ) @ ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) )
= ( member_list_a @ A4 @ ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( insert_list_a @ B @ bot_bot_set_list_a ) ) ) ) ) ) ).
% pdr.Idl_subset_ideal'
thf(fact_1160_pds_Ozeropideal,axiom,
princi5701163198563039320t_unit @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.zeropideal
thf(fact_1161_pdr_Ozeroideal,axiom,
ideal_8896367198367571637t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.zeroideal
thf(fact_1162_pds_Ozeroideal,axiom,
ideal_5810610956377577076t_unit @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.zeroideal
thf(fact_1163_pdr_Ogenideal__self_H,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ I @ ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% pdr.genideal_self'
thf(fact_1164_pds_Ogenideal__self_H,axiom,
! [I: list_b] :
( ( member_list_b @ I @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ I @ ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( insert_list_b @ I @ bot_bot_set_list_b ) ) ) ) ).
% pds.genideal_self'
thf(fact_1165_pdr_Ogenideal__zero,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) ) ).
% pdr.genideal_zero
thf(fact_1166_pds_Ogenideal__zero,axiom,
( ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) )
= ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) ) ).
% pds.genideal_zero
thf(fact_1167_pdr_Ozeropideal,axiom,
princi8786919440553033881t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.zeropideal
thf(fact_1168_pds_Ozeroprimeideal__domainI,axiom,
( ( primei3224061617086083047t_unit @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( domain3467766878553215752t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.zeroprimeideal_domainI
thf(fact_1169_pds_Odomain__eq__zeroprimeideal,axiom,
( ( domain3467766878553215752t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( primei3224061617086083047t_unit @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.domain_eq_zeroprimeideal
thf(fact_1170_dr_Ozeroideal,axiom,
ideal_a_c @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) @ r ).
% dr.zeroideal
thf(fact_1171_ds_Ozeroideal,axiom,
ideal_b_d @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) @ s ).
% ds.zeroideal
thf(fact_1172_dr_Ogenideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia778085601923319190xt_a_c @ r ) )
=> ( member_a @ I @ ( genideal_a_c @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% dr.genideal_self'
thf(fact_1173_ds_Ogenideal__self_H,axiom,
! [I: b] :
( ( member_b @ I @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ I @ ( genideal_b_d @ s @ ( insert_b @ I @ bot_bot_set_b ) ) ) ) ).
% ds.genideal_self'
thf(fact_1174_dr_Ogenideal__zero,axiom,
( ( genideal_a_c @ r @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) ) ).
% dr.genideal_zero
thf(fact_1175_ds_Ogenideal__zero,axiom,
( ( genideal_b_d @ s @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) )
= ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) ) ).
% ds.genideal_zero
thf(fact_1176_dr_Ozeropideal,axiom,
principalideal_a_c @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) @ r ).
% dr.zeropideal
thf(fact_1177_ds_Ozeropideal,axiom,
principalideal_b_d @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) @ s ).
% ds.zeropideal
thf(fact_1178_dr_OIdl__subset__ideal_H,axiom,
! [A4: a,B: a] :
( ( member_a @ A4 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_c @ r @ ( insert_a @ A4 @ bot_bot_set_a ) ) @ ( genideal_a_c @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A4 @ ( genideal_a_c @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% dr.Idl_subset_ideal'
thf(fact_1179_ds_OIdl__subset__ideal_H,axiom,
! [A4: b,B: b] :
( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ord_less_eq_set_b @ ( genideal_b_d @ s @ ( insert_b @ A4 @ bot_bot_set_b ) ) @ ( genideal_b_d @ s @ ( insert_b @ B @ bot_bot_set_b ) ) )
= ( member_b @ A4 @ ( genideal_b_d @ s @ ( insert_b @ B @ bot_bot_set_b ) ) ) ) ) ) ).
% ds.Idl_subset_ideal'
thf(fact_1180_dr_Ocgenideal__eq__genideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( cgenid547466214215511830xt_a_c @ r @ I )
= ( genideal_a_c @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% dr.cgenideal_eq_genideal
thf(fact_1181_pds_Ozeroprimeideal,axiom,
primei3224061617086083047t_unit @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ).
% pds.zeroprimeideal
thf(fact_1182_pds_Omaximalideal__prime,axiom,
! [I2: set_list_b] :
( ( maxima3499944040311362099t_unit @ I2 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( primei3224061617086083047t_unit @ I2 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.maximalideal_prime
thf(fact_1183_pdr_Ozeroprimeideal__domainI,axiom,
( ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.zeroprimeideal_domainI
thf(fact_1184_pdr_Ozeroprimeideal,axiom,
primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ).
% pdr.zeroprimeideal
thf(fact_1185_pdr_Odomain__eq__zeroprimeideal,axiom,
( ( domain6553523120543210313t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( primei6309817859076077608t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.domain_eq_zeroprimeideal
thf(fact_1186_pdr_Omaximalideal__prime,axiom,
! [I2: set_list_a] :
( ( maxima6585700282301356660t_unit @ I2 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( primei6309817859076077608t_unit @ I2 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.maximalideal_prime
thf(fact_1187_h_Onon__trivial__field__hom__imp__inj,axiom,
( ( field_a_c @ r )
=> ( ( ( image_a_b @ h @ ( partia778085601923319190xt_a_c @ r ) )
!= ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) )
=> ( inj_on_a_b @ h @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% h.non_trivial_field_hom_imp_inj
thf(fact_1188_dr_Otrivialideals__fieldI,axiom,
( ( ( partia778085601923319190xt_a_c @ r )
!= ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) )
=> ( ( ( collect_set_a
@ ^ [I5: set_a] : ( ideal_a_c @ I5 @ r ) )
= ( insert_set_a @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia778085601923319190xt_a_c @ r ) @ bot_bot_set_set_a ) ) )
=> ( field_a_c @ r ) ) ) ).
% dr.trivialideals_fieldI
thf(fact_1189_dr_Otrivialideals__eq__field,axiom,
( ( ( partia778085601923319190xt_a_c @ r )
!= ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) )
=> ( ( ( collect_set_a
@ ^ [I5: set_a] : ( ideal_a_c @ I5 @ r ) )
= ( insert_set_a @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) @ ( insert_set_a @ ( partia778085601923319190xt_a_c @ r ) @ bot_bot_set_set_a ) ) )
= ( field_a_c @ r ) ) ) ).
% dr.trivialideals_eq_field
thf(fact_1190_dr_Ozeromaximalideal__fieldI,axiom,
( ( maximalideal_a_c @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) @ r )
=> ( field_a_c @ r ) ) ).
% dr.zeromaximalideal_fieldI
thf(fact_1191_dr_Ozeromaximalideal__eq__field,axiom,
( ( maximalideal_a_c @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) @ r )
= ( field_a_c @ r ) ) ).
% dr.zeromaximalideal_eq_field
thf(fact_1192_ds_Otrivialideals__eq__field,axiom,
( ( ( partia8782771468121683032xt_b_d @ s )
!= ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) )
=> ( ( ( collect_set_b
@ ^ [I5: set_b] : ( ideal_b_d @ I5 @ s ) )
= ( insert_set_b @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) @ ( insert_set_b @ ( partia8782771468121683032xt_b_d @ s ) @ bot_bot_set_set_b ) ) )
= ( field_b_d @ s ) ) ) ).
% ds.trivialideals_eq_field
thf(fact_1193_ds_Otrivialideals__fieldI,axiom,
( ( ( partia8782771468121683032xt_b_d @ s )
!= ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) )
=> ( ( ( collect_set_b
@ ^ [I5: set_b] : ( ideal_b_d @ I5 @ s ) )
= ( insert_set_b @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) @ ( insert_set_b @ ( partia8782771468121683032xt_b_d @ s ) @ bot_bot_set_set_b ) ) )
=> ( field_b_d @ s ) ) ) ).
% ds.trivialideals_fieldI
thf(fact_1194_pdr_Ozeromaximalideal__eq__field,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.zeromaximalideal_eq_field
thf(fact_1195_pdr_Ozeromaximalideal__fieldI,axiom,
( ( maxima6585700282301356660t_unit @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.zeromaximalideal_fieldI
thf(fact_1196_pds_Ozeromaximalideal__fieldI,axiom,
( ( maxima3499944040311362099t_unit @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( field_3302291602678335014t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.zeromaximalideal_fieldI
thf(fact_1197_pds_Ozeromaximalideal__eq__field,axiom,
( ( maxima3499944040311362099t_unit @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( field_3302291602678335014t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.zeromaximalideal_eq_field
thf(fact_1198_pdr_Otrivialideals__fieldI,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( ( ( collect_set_list_a
@ ^ [I5: set_list_a] : ( ideal_8896367198367571637t_unit @ I5 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( insert_set_list_a @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) @ ( insert_set_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bo3186585308812441520list_a ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.trivialideals_fieldI
thf(fact_1199_pdr_Otrivialideals__eq__field,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( ( ( collect_set_list_a
@ ^ [I5: set_list_a] : ( ideal_8896367198367571637t_unit @ I5 @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( insert_set_list_a @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) @ ( insert_set_list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bo3186585308812441520list_a ) ) )
= ( field_6388047844668329575t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.trivialideals_eq_field
thf(fact_1200_pds_Otrivialideals__fieldI,axiom,
( ( ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) )
=> ( ( ( collect_set_list_b
@ ^ [I5: set_list_b] : ( ideal_5810610956377577076t_unit @ I5 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( insert_set_list_b @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) @ ( insert_set_list_b @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bo665400790523405873list_b ) ) )
=> ( field_3302291602678335014t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.trivialideals_fieldI
thf(fact_1201_pds_Otrivialideals__eq__field,axiom,
( ( ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) )
=> ( ( ( collect_set_list_b
@ ^ [I5: set_list_b] : ( ideal_5810610956377577076t_unit @ I5 @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= ( insert_set_list_b @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) @ ( insert_set_list_b @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bo665400790523405873list_b ) ) )
= ( field_3302291602678335014t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.trivialideals_eq_field
thf(fact_1202_ds_Ozeromaximalideal__eq__field,axiom,
( ( maximalideal_b_d @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) @ s )
= ( field_b_d @ s ) ) ).
% ds.zeromaximalideal_eq_field
thf(fact_1203_ds_Ozeromaximalideal__fieldI,axiom,
( ( maximalideal_b_d @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) @ s )
=> ( field_b_d @ s ) ) ).
% ds.zeromaximalideal_fieldI
thf(fact_1204_pds_Oring__irreducibleI,axiom,
! [R3: list_b] :
( ( member_list_b @ R3 @ ( minus_717693128102174796list_b @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) ) )
=> ( ~ ( member_list_b @ R3 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ! [A3: list_b,B2: list_b] :
( ( member_list_b @ A3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( R3
= ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A3 @ B2 ) )
=> ( ( member_list_b @ A3 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
| ( member_list_b @ B2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) )
=> ( ring_r7070601269410051085t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ R3 ) ) ) ) ).
% pds.ring_irreducibleI
thf(fact_1205_pds_Ocring__fieldI,axiom,
( ( ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( minus_717693128102174796list_b @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) ) )
=> ( field_3302291602678335014t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.cring_fieldI
thf(fact_1206_pdr_Oring__irreducibleI,axiom,
! [R3: list_a] :
( ( member_list_a @ R3 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ~ ( member_list_a @ R3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ! [A3: list_a,B2: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( R3
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A3 @ B2 ) )
=> ( ( member_list_a @ A3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
| ( member_list_a @ B2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ R3 ) ) ) ) ).
% pdr.ring_irreducibleI
thf(fact_1207_pds_Oprimeideal__iff__prime,axiom,
! [P2: list_b] :
( ( member_list_b @ P2 @ ( minus_717693128102174796list_b @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) ) )
=> ( ( primei3224061617086083047t_unit @ ( cgenid3857731246393895395t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( ring_r3344526403024810276t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ P2 ) ) ) ).
% pds.primeideal_iff_prime
thf(fact_1208_ds_Ocgenideal__self,axiom,
! [I: b] :
( ( member_b @ I @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( member_b @ I @ ( cgenid3879858590684755159xt_b_d @ s @ I ) ) ) ).
% ds.cgenideal_self
thf(fact_1209_dr_Omaximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_c @ I2 @ r )
=> ( primeideal_a_c @ I2 @ r ) ) ).
% dr.maximalideal_prime
thf(fact_1210_ds_Omaximalideal__prime,axiom,
! [I2: set_b] :
( ( maximalideal_b_d @ I2 @ s )
=> ( primeideal_b_d @ I2 @ s ) ) ).
% ds.maximalideal_prime
thf(fact_1211_ds_Oideal__eq__carrier__iff,axiom,
! [A4: b] :
( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( ( partia8782771468121683032xt_b_d @ s )
= ( cgenid3879858590684755159xt_b_d @ s @ A4 ) )
= ( member_b @ A4 @ ( units_b_ring_ext_b_d @ s ) ) ) ) ).
% ds.ideal_eq_carrier_iff
thf(fact_1212_ds_Oassociated__iff__same__ideal,axiom,
! [A4: b,B: b] :
( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( associ9192668908051667533xt_b_d @ s @ A4 @ B )
= ( ( cgenid3879858590684755159xt_b_d @ s @ A4 )
= ( cgenid3879858590684755159xt_b_d @ s @ B ) ) ) ) ) ).
% ds.associated_iff_same_ideal
thf(fact_1213_ds_Ocgenideal__ideal,axiom,
! [A4: b] :
( ( member_b @ A4 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ideal_b_d @ ( cgenid3879858590684755159xt_b_d @ s @ A4 ) @ s ) ) ).
% ds.cgenideal_ideal
thf(fact_1214_ds_Ocgenideal__minimal,axiom,
! [J: set_b,A4: b] :
( ( ideal_b_d @ J @ s )
=> ( ( member_b @ A4 @ J )
=> ( ord_less_eq_set_b @ ( cgenid3879858590684755159xt_b_d @ s @ A4 ) @ J ) ) ) ).
% ds.cgenideal_minimal
thf(fact_1215_ds_Ocgenideal__is__principalideal,axiom,
! [I: b] :
( ( member_b @ I @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( principalideal_b_d @ ( cgenid3879858590684755159xt_b_d @ s @ I ) @ s ) ) ).
% ds.cgenideal_is_principalideal
thf(fact_1216_pdr_Ocgenideal__self,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( member_list_a @ I @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ I ) ) ) ).
% pdr.cgenideal_self
thf(fact_1217_dr_Ozeroprimeideal,axiom,
primeideal_a_c @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) @ r ).
% dr.zeroprimeideal
thf(fact_1218_ds_Ozeroprimeideal,axiom,
primeideal_b_d @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) @ s ).
% ds.zeroprimeideal
thf(fact_1219_pdr_Oideal__eq__carrier__iff,axiom,
! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 ) )
= ( member_list_a @ A4 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.ideal_eq_carrier_iff
thf(fact_1220_pdr_Oassociated__iff__same__ideal,axiom,
! [A4: list_a,B: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( associ8407585678920448409t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 @ B )
= ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 )
= ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ B ) ) ) ) ) ).
% pdr.associated_iff_same_ideal
thf(fact_1221_pdr_Ocgenideal__ideal,axiom,
! [A4: list_a] :
( ( member_list_a @ A4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ideal_8896367198367571637t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.cgenideal_ideal
thf(fact_1222_pds_Ocgenideal__self,axiom,
! [I: list_b] :
( ( member_list_b @ I @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( member_list_b @ I @ ( cgenid3857731246393895395t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ I ) ) ) ).
% pds.cgenideal_self
thf(fact_1223_pdr_Ocgenideal__minimal,axiom,
! [J: set_list_a,A4: list_a] :
( ( ideal_8896367198367571637t_unit @ J @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
=> ( ( member_list_a @ A4 @ J )
=> ( ord_le8861187494160871172list_a @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A4 ) @ J ) ) ) ).
% pdr.cgenideal_minimal
thf(fact_1224_ds_Ocgenideal__eq__genideal,axiom,
! [I: b] :
( ( member_b @ I @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( cgenid3879858590684755159xt_b_d @ s @ I )
= ( genideal_b_d @ s @ ( insert_b @ I @ bot_bot_set_b ) ) ) ) ).
% ds.cgenideal_eq_genideal
thf(fact_1225_ds_Ozeroprimeideal__domainI,axiom,
( ( primeideal_b_d @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) @ s )
=> ( domain_b_d @ s ) ) ).
% ds.zeroprimeideal_domainI
thf(fact_1226_ds_Odomain__eq__zeroprimeideal,axiom,
( ( domain_b_d @ s )
= ( primeideal_b_d @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) @ s ) ) ).
% ds.domain_eq_zeroprimeideal
thf(fact_1227_dr_Ozeroprimeideal__domainI,axiom,
( ( primeideal_a_c @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) @ r )
=> ( domain_a_c @ r ) ) ).
% dr.zeroprimeideal_domainI
thf(fact_1228_dr_Odomain__eq__zeroprimeideal,axiom,
( ( domain_a_c @ r )
= ( primeideal_a_c @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) @ r ) ) ).
% dr.domain_eq_zeroprimeideal
thf(fact_1229_pdr_Ocgenideal__is__principalideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( princi8786919440553033881t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ I ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.cgenideal_is_principalideal
thf(fact_1230_pds_Oideal__eq__carrier__iff,axiom,
! [A4: list_b] :
( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( cgenid3857731246393895395t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 ) )
= ( member_list_b @ A4 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.ideal_eq_carrier_iff
thf(fact_1231_pds_Oassociated__iff__same__ideal,axiom,
! [A4: list_b,B: list_b] :
( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ B @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( associ3133968390036396889t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 @ B )
= ( ( cgenid3857731246393895395t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 )
= ( cgenid3857731246393895395t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ B ) ) ) ) ) ).
% pds.associated_iff_same_ideal
thf(fact_1232_pds_Ocgenideal__ideal,axiom,
! [A4: list_b] :
( ( member_list_b @ A4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ideal_5810610956377577076t_unit @ ( cgenid3857731246393895395t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.cgenideal_ideal
thf(fact_1233_pds_Ocgenideal__minimal,axiom,
! [J: set_list_b,A4: list_b] :
( ( ideal_5810610956377577076t_unit @ J @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
=> ( ( member_list_b @ A4 @ J )
=> ( ord_le8932221534207217157list_b @ ( cgenid3857731246393895395t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A4 ) @ J ) ) ) ).
% pds.cgenideal_minimal
thf(fact_1234_ds_Oprimeideal__iff__prime,axiom,
! [P2: b] :
( ( member_b @ P2 @ ( minus_minus_set_b @ ( partia8782771468121683032xt_b_d @ s ) @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) ) )
=> ( ( primeideal_b_d @ ( cgenid3879858590684755159xt_b_d @ s @ P2 ) @ s )
= ( ring_ring_prime_b_d @ s @ P2 ) ) ) ).
% ds.primeideal_iff_prime
thf(fact_1235_pds_Ocgenideal__is__principalideal,axiom,
! [I: list_b] :
( ( member_list_b @ I @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( princi5701163198563039320t_unit @ ( cgenid3857731246393895395t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ I ) @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.cgenideal_is_principalideal
thf(fact_1236_pdr_Ocgenideal__eq__genideal,axiom,
! [I: list_a] :
( ( member_list_a @ I @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ I )
= ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( insert_list_a @ I @ bot_bot_set_list_a ) ) ) ) ).
% pdr.cgenideal_eq_genideal
thf(fact_1237_dr_Ocring__fieldI,axiom,
( ( ( units_a_ring_ext_a_c @ r )
= ( minus_minus_set_a @ ( partia778085601923319190xt_a_c @ r ) @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_c @ r ) ) ).
% dr.cring_fieldI
thf(fact_1238_ds_Ocring__fieldI,axiom,
( ( ( units_b_ring_ext_b_d @ s )
= ( minus_minus_set_b @ ( partia8782771468121683032xt_b_d @ s ) @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) ) )
=> ( field_b_d @ s ) ) ).
% ds.cring_fieldI
thf(fact_1239_dr_Oprimeideal__iff__prime,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia778085601923319190xt_a_c @ r ) @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_c @ ( cgenid547466214215511830xt_a_c @ r @ P2 ) @ r )
= ( ring_ring_prime_a_c @ r @ P2 ) ) ) ).
% dr.primeideal_iff_prime
thf(fact_1240_pds_Ocgenideal__eq__genideal,axiom,
! [I: list_b] :
( ( member_list_b @ I @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( cgenid3857731246393895395t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ I )
= ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( insert_list_b @ I @ bot_bot_set_list_b ) ) ) ) ).
% pds.cgenideal_eq_genideal
thf(fact_1241_dr_Oring__irreducibleI,axiom,
! [R3: a] :
( ( member_a @ R3 @ ( minus_minus_set_a @ ( partia778085601923319190xt_a_c @ r ) @ ( insert_a @ ( zero_a_c @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R3 @ ( units_a_ring_ext_a_c @ r ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( member_a @ B2 @ ( partia778085601923319190xt_a_c @ r ) )
=> ( ( R3
= ( mult_a_ring_ext_a_c @ r @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_c @ r ) )
| ( member_a @ B2 @ ( units_a_ring_ext_a_c @ r ) ) ) ) ) )
=> ( ring_r999134135267193927le_a_c @ r @ R3 ) ) ) ) ).
% dr.ring_irreducibleI
thf(fact_1242_ds_Oring__irreducibleI,axiom,
! [R3: b] :
( ( member_b @ R3 @ ( minus_minus_set_b @ ( partia8782771468121683032xt_b_d @ s ) @ ( insert_b @ ( zero_b_d @ s ) @ bot_bot_set_b ) ) )
=> ( ~ ( member_b @ R3 @ ( units_b_ring_ext_b_d @ s ) )
=> ( ! [A3: b,B2: b] :
( ( member_b @ A3 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( member_b @ B2 @ ( partia8782771468121683032xt_b_d @ s ) )
=> ( ( R3
= ( mult_b_ring_ext_b_d @ s @ A3 @ B2 ) )
=> ( ( member_b @ A3 @ ( units_b_ring_ext_b_d @ s ) )
| ( member_b @ B2 @ ( units_b_ring_ext_b_d @ s ) ) ) ) ) )
=> ( ring_r7435050590149293703le_b_d @ s @ R3 ) ) ) ) ).
% ds.ring_irreducibleI
thf(fact_1243_pdr_Ocring__fieldI,axiom,
( ( ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.cring_fieldI
thf(fact_1244_pdr_Oprimeideal__iff__prime,axiom,
! [P2: list_a] :
( ( member_list_a @ P2 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( ( primei6309817859076077608t_unit @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 ) @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( ring_r6430282645014804837t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ P2 ) ) ) ).
% pdr.primeideal_iff_prime
thf(fact_1245_pds_Ofield__intro2,axiom,
( ( ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ! [X3: list_b] :
( ( member_list_b @ X3 @ ( minus_717693128102174796list_b @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) ) )
=> ( member_list_b @ X3 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) )
=> ( field_3302291602678335014t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.field_intro2
thf(fact_1246_pdr_Ofield__intro2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( minus_646659088055828811list_a @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) ) )
=> ( member_list_a @ X3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.field_intro2
thf(fact_1247_pdr_Ozero__not__one,axiom,
( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.zero_not_one
thf(fact_1248_pds_Ozero__not__one,axiom,
( ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.zero_not_one
thf(fact_1249_pdr_Oone__unique,axiom,
! [U: list_a] :
( ( member_list_a @ U @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ! [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.one_unique
thf(fact_1250_pdr_Oinv__unique,axiom,
! [Y2: list_a,X2: list_a,Y5: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y2 @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Y5 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y5 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( Y2 = Y5 ) ) ) ) ) ) ).
% pdr.inv_unique
thf(fact_1251_pds_Oone__unique,axiom,
! [U: list_b] :
( ( member_list_b @ U @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ! [X3: list_b] :
( ( member_list_b @ X3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.one_unique
thf(fact_1252_pds_Oinv__unique,axiom,
! [Y2: list_b,X2: list_b,Y5: list_b] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y2 @ X2 )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Y5 )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y5 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( Y2 = Y5 ) ) ) ) ) ) ).
% pds.inv_unique
thf(fact_1253_pdr_OUnits__inv__comm,axiom,
! [X2: list_a,Y2: list_a] :
( ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ Y2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( member_list_a @ Y2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ Y2 @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ) ).
% pdr.Units_inv_comm
thf(fact_1254_pds_OUnits__inv__comm,axiom,
! [X2: list_b,Y2: list_b] :
( ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ Y2 )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ X2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( member_list_b @ Y2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ Y2 @ X2 )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ) ).
% pds.Units_inv_comm
thf(fact_1255_pdr_OUnits__l__inv__ex,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X3 @ X2 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.Units_l_inv_ex
thf(fact_1256_pdr_OUnits__r__inv__ex,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ? [X3: list_a] :
( ( member_list_a @ X3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ X3 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) ).
% pdr.Units_r_inv_ex
thf(fact_1257_pds_OUnits__r__inv__ex,axiom,
! [X2: list_b] :
( ( member_list_b @ X2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ? [X3: list_b] :
( ( member_list_b @ X3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ X3 )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.Units_r_inv_ex
thf(fact_1258_pds_OUnits__l__inv__ex,axiom,
! [X2: list_b] :
( ( member_list_b @ X2 @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ? [X3: list_b] :
( ( member_list_b @ X3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X3 @ X2 )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) ).
% pds.Units_l_inv_ex
thf(fact_1259_pdr_Ocarrier__one__not__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.carrier_one_not_zero
thf(fact_1260_pdr_Ocarrier__one__zero,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) )
= ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.carrier_one_zero
thf(fact_1261_pdr_Oone__zeroD,axiom,
( ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) ) ) ).
% pdr.one_zeroD
thf(fact_1262_pdr_Oone__zeroI,axiom,
( ( ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( insert_list_a @ ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) )
=> ( ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.one_zeroI
thf(fact_1263_pds_Ocarrier__one__not__zero,axiom,
( ( ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) )
= ( ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.carrier_one_not_zero
thf(fact_1264_pds_Ocarrier__one__zero,axiom,
( ( ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) )
= ( ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.carrier_one_zero
thf(fact_1265_pds_Oone__zeroD,axiom,
( ( ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) ) ) ).
% pds.one_zeroD
thf(fact_1266_pds_Oone__zeroI,axiom,
( ( ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( insert_list_b @ ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) )
=> ( ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.one_zeroI
thf(fact_1267_pdr_Ocring__fieldI2,axiom,
( ( ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) )
!= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ! [A3: list_a] :
( ( member_list_a @ A3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( A3
!= ( zero_l4142658623432671053t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
& ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ A3 @ X4 )
= ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ) )
=> ( field_6388047844668329575t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ) ).
% pdr.cring_fieldI2
thf(fact_1268_pds_Ocring__fieldI2,axiom,
( ( ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) )
!= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ! [A3: list_b] :
( ( member_list_b @ A3 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( A3
!= ( zero_l1056902381442676492t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ? [X4: list_b] :
( ( member_list_b @ X4 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
& ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ A3 @ X4 )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ) )
=> ( field_3302291602678335014t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ) ).
% pds.cring_fieldI2
thf(fact_1269_pdr_Ogenideal__one,axiom,
( ( genide3243992037924705879t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( insert_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ bot_bot_set_list_a ) )
= ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ) ).
% pdr.genideal_one
thf(fact_1270_pds_Ogenideal__one,axiom,
( ( genide158235795934711318t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ ( insert_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ bot_bot_set_list_b ) )
= ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% pds.genideal_one
thf(fact_1271__092_060open_062map_Ah_A_092_060one_062_092_060_094bsub_062poly__ring_AR_092_060_094esub_062_A_061_A_092_060one_062_092_060_094bsub_062poly__ring_AS_092_060_094esub_062_092_060close_062,axiom,
( ( map_a_b @ h @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ) ).
% \<open>map h \<one>\<^bsub>poly_ring R\<^esub> = \<one>\<^bsub>poly_ring S\<^esub>\<close>
thf(fact_1272_pdr_Oone__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% pdr.one_closed
thf(fact_1273_pds_Oone__closed,axiom,
member_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% pds.one_closed
thf(fact_1274_pdr_OUnits__one__closed,axiom,
member_list_a @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ ( units_2932844235741507942t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) ).
% pdr.Units_one_closed
thf(fact_1275_pds_OUnits__one__closed,axiom,
member_list_b @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) @ ( units_6882598983712232230t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) ).
% pds.Units_one_closed
thf(fact_1276_pdr_Or__one,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ X2 @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
= X2 ) ) ).
% pdr.r_one
thf(fact_1277_pdr_Ol__one,axiom,
! [X2: list_a] :
( ( member_list_a @ X2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) )
=> ( ( mult_l7073676228092353617t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) @ ( one_li8328186300101108157t_unit @ ( univ_poly_a_c @ r @ ( partia778085601923319190xt_a_c @ r ) ) ) @ X2 )
= X2 ) ) ).
% pdr.l_one
thf(fact_1278_pds_Or__one,axiom,
! [X2: list_b] :
( ( member_list_b @ X2 @ ( partia1381092143316337258t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
=> ( ( mult_l1800058939208302097t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) @ X2 @ ( one_li3054569011217056637t_unit @ ( univ_poly_b_d @ s @ ( partia8782771468121683032xt_b_d @ s ) ) ) )
= X2 ) ) ).
% pds.r_one
% Conjectures (1)
thf(conj_0,conjecture,
( ( map_b_b
@ ^ [Y: b] : ( h @ ( the_inv_into_a_b @ ( partia778085601923319190xt_a_c @ r ) @ h @ Y ) )
@ x )
= ( map_b_b @ id_b @ x ) ) ).
%------------------------------------------------------------------------------